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abstract: 'The recoil correction of order $(Z\alpha)^6(m/M)m$ to the hydrogen energy levels is recalculated and a discrepancy existing in the literature on this correction for the $1S$ energy level, is resolved. An analytic expression for the correction to the $S$-levels with arbitrary principal quantum number is obtained.'
address:
- |
Department of Physics, Pennsylvania State University, University Park, PA 16802, USA[^1]\
and Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188350, Russia[^2]
- 'Department of Physics, Pennsylvania State University, University Park, PA 16802, USA'
author:
- 'Michael I. Eides [^3]'
- 'Howard Grotch[^4]'
date: 'November, 1996'
title: 'Recoil Corrections of Order $(Z\alpha)^6(m/M)m$ to the Hydrogen Energy Levels Revisited'
---
Introduction
============
The calculation of the recoil corrections of order $(Z\alpha)^6(m/M)m$ to the hydrogen energy levels has a long history Refs.[@gy; @eg; @dgo; @dge; @kmy; @fkmy]. After initial disagreements consensus was achieved in Ref.[@pg], where one and the same result was obtained in two apparently different frameworks. The first, more traditional approach, used earlier in Refs.[@eg; @dgo; @dge], starts with an effective Dirac equation in the external field. Corrections to the Dirac energy levels are calculated with the help of a systematic diagrammatic procedure. The other logically independent calculational framework, also used in Ref.[@pg], starts with an exact expression for all recoil corrections of the first order in the mass ratio of the light and heavy particles $m/M$. This remarkable expression, which is exact in $Z\alpha$, was first discovered by M. A. Braun [@braun], and rederived later in different ways in a number of papers Refs.[@shab; @yelkh; @pg].
The agreement on the $(Z\alpha)^6(m/M)m$ contribution achieved in [@pg] seemed to put an end to all problems connected with this correction. However, it was claimed in a recent work [@elkh], that the result of [@pg] is in error. The discrepancy between the results of Refs.[@pg; @elkh] is confusing since the calculation in [@elkh] is performed in the same framework as the one employed in [@pg], namely it is based on a particularly nice form of the Braun formula obtained by the author earlier [@yelkh],
$$\label{braun}
\Delta E_{rec}=-\frac{1}{M}Re\int\frac{d\omega}{2\pi i}
<{n}|({\bf p}-{\bf\hat D}(\omega))G(E+\omega)({\bf
p}-{\bf\hat D}(\omega))|n>,$$
where summation over all intermediate states is understood, $G(E+\omega)$ is the Coulomb-Green function in the Coulomb gauge, which in the momentum space has the form
$${\bf\hat D}(\omega,k)=-4\pi Z\alpha(\mbox{\boldmath$\alpha$}-\frac{{\bf
k}(\mbox{\boldmath$\alpha k$})}{{\bf k}^2})\frac{1}{\omega^2-{\bf k}^2+i0}
\equiv -4\pi Z\alpha\frac{\mbox{\boldmath$\alpha_k$}}{\omega^2-{\bf k}^2+i0},$$
and
$$\alpha_i=\gamma^0\gamma^i.$$
Note that ${\bf\hat D}(\omega,k)$ is nothing more than the transverse photon propagator with the source at the proton position, and integration over the exchanged photon momentum $\bf k$ is implicit in the expression above. Below we will explicitly perform multiplication in the matrix element in Eq.(\[braun\]). Respective contributions to the energy levels will be called Coulomb (corresponds to $\bf pp$), magnetic (corresponds to $\bf
p\hat D$ and $\bf\hat Dp$), and seagull (corresponds to $\bf\hat D \hat D$).
It is the aim of this paper to resolve the above noted discrepancy on the recoil correction of order $(Z\alpha)^6(m/M)m$ to the $1S$ energy level, and also to obtain this correction for the $S$-levels with arbitrary principal quantum number (it was earlier calculated only for $n=1,2$ [@pg]).
Two Approaches to the Braun Formula
===================================
Calculation of the recoil contribution of order $(Z\alpha)^6$ generated by the Braun formula was performed in [@pg] in a most straightforward way since separation of the high- and low-frequency contributions was made in the framework of the $\epsilon$-method developed by one of the authors earlier [@pach]. Hence, not only were contributions of order $(Z\alpha)^6(m/M)m$ obtained in Ref.[@pg], but also linear in $m/M$ parts of recoil corrections of orders $(Z\alpha)^4$ and $(Z\alpha)^5$ (ref.[@salp]) were reproduced for the $1S$-state. Note that the Braun formula, despite its obvious advantages, in its present form sums only contributions linear in the mass ratio. Hence, old methods are more adequate for obtaining the proper mass dependence of the contributions of orders $(Z\alpha)^4$ and $(Z\alpha)^5$, which were worked out in Ref.[@gy]. Calculations in Ref.[@pg] turned out to be rather lengthy and tedious just because all corrections of previous orders in $Z\alpha$ were reproduced.
The most significant feature of the recoil corrections of order $(Z\alpha)^6$, which made the whole approach of Ref.[@elkh] possible, is connected with the absence of [*logarithmic*]{} recoil corrections of this order, as was proved in [@fkmy]. Unlike [@pg], the calculations in [@elkh] are organized in such a way that one explicitly makes approximations inadequate for calculation of the contributions of the previous orders in $Z\alpha$, significantly simplifying calculation of the correction of order $(Z\alpha)^6$. Due to absence of the logarithmic contributions of order $(Z\alpha)^6$, infrared divergences connected with the crude approximations unadequate for calculation of the contributions of the previous orders would be powerlike and can be safely thrown away. Next, absence of logarithmic corrections of order $(Z\alpha)^6$ means that it is not necessary to worry too much about matching the low- and high-frequency (long- and short-distance in terms of Ref.[@elkh]) contributions, since each region will produce only nonlogarithmic contributions and correction terms would be suppressed as powers of the separation parameter. We would like to emphasize once more that this approach would be doomed if the logarithmic divergences were present, since in such a case one could not hope to calculate an additive constant to the log, since the exact value of the integration cutoff would not be known.
We are going to perform below calculation of the recoil contribution of order $(Z\alpha)^6$ in the framework of Ref.[@elkh], and to discover the source of discrepancy between the results of Ref.[@pg] and Ref.[@elkh]. In order to really implement such program we need to have a regular method to qualify all terms which will be thrown away. To this end we will use a slight generalization of the ordinary approach to calculation of the leading order contribution to the Lamb shift.
It may be proved that all corrections of order $(Z\alpha)^6(m/M)m$ are generated by the exchange of photons with momenta larger than $m(Z\alpha)^2$, so we will consider below only this integration region. In the spirit of the common approach to the Lamb shift calculations we will split the integration region over the exchanged photon momenta (and when necessary over frequencies) with the help of an auxiliary parameter $\sigma$ which satisfies the conditions
$$mZ\alpha\ll\sigma\ll m,$$
and we will call the photons with momenta smaller than $\sigma$ low-frequency (or long-distance) photons, and the photons with momenta larger than $\sigma$ will be called high-frequency (or short-distance) photons. Considering low-frequency photons we may expand over the ratio $k/m$ since for such photons $k/m\leq\sigma/m\ll 1$[^5]. On the other hand, for the high-frequency photons $mZ\alpha/k\leq mZ\alpha/\sigma\ll1$, and we may expand over this parameter. Note that for momenta of order $\sigma$ both expansions are valid simultaneously, and, hence, we may match the expansions and get rid of the auxiliary parameter $\sigma$. However, the problem under consideration is in a sense even simpler than calculation of the leading order contribution to the Lamb shift, and due to absence of the logarithmic contributions of order $(Z\alpha)^6(m/M)m$, precise matching of the high- and low-frequency contributions is unnecessary. Below we will consider calculation only of the low-frequency ($mZ\alpha<k<\sigma$) contribution to the energy shift, since for the high frequency contribution the results of Ref.[@pg] and Ref.[@elkh] nicely coincide.
Main Recoil Contribution
========================
With the help of the Braun formula one may easily obtain an expression for the leading recoil correction which is linear in the mass ratio and which includes all terms of order $(Z\alpha)^4$ and lower (see Ref.[@shab]). To this end we rewrite the Coulomb contribution in Eq.(\[braun\]) in the form
$$\Delta E_{Coul}=
\frac{1}{2M}<{n}|{\bf p}^2|n>-\frac{1}{M}<{n}|{\bf p}\Lambda^-{\bf p}]|n>$$
$$\equiv \Delta E_{c1}+\Delta E_{c2}.$$
We also extract the nonretarded Breit part from the magnetic contribution in Eq.(\[braun\] )
$$\Delta E_{magn}=\Delta E_{Br}+\Delta E_{magn,r},$$
where
$$\Delta E_{Br}=
-\frac{1}{2M}
<{n}|{\bf p}{\bf\hat D}(0,k)
+{\bf\hat D}(0,k){\bf p}|n>,$$
and
$$\Delta E_{magn,r}=
-\frac{1}{M}\int\frac{d\omega}{2\pi i}
<{n}|[V,{\bf p}]G(E+\omega){\bf\hat D}(\omega,k)$$
$$-{\bf\hat D}(\omega,k)G(E+\omega)[V,{\bf p}]|n>
\frac{1}{\omega+i0},$$
where $V$ is the Coulomb potential ($V=-Z\alpha/r$).
Now it is not difficult to check with the help of the virial relations (see, e.g., Ref.[@ee]), that the sum of the main part of the Coulomb term and of the Breit contribution acquires a very nice form
$$\label{gys}
\Delta E_{c1}+\Delta E_{Br}=\frac{m^2-E^2}{2M},$$
where $E$ is the value of the energy given by the Dirac equation. As we will see below, all other recoil contributions to the energy level start at least with the term of order $(Z\alpha)^5$, and, hence, the formula above correctly describes all contributions of order $(Z\alpha)^4$ and lower. However, this formula describes only contributions linear in the mass ratio. A more precise expression which takes into account corrections of higher order in $m/M$, was obtained in Ref.[@gy].
It is easy to see that the expression in Eq.(\[gys\]) also contains the correction of order $(Z\alpha)^6$, which for the $nS$-states has the form
$$\label{gy}
\Delta
E_{GY}=(\frac{1}{8}+\frac{3}{8n}-\frac{1}{n^2}+\frac{1}{2n^3})
\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m.$$
This contribution was originally obtained in Ref.[@gy].
The remaining part of the Coulomb contribution has the form
$$\Delta E_{c2}=-\frac{1}{M}<n|{\bf p}\Lambda_-{\bf p}|n>.$$
Let us check that this term leads to corrections of higher order than $(Z\alpha)^6$ when the intermediate momenta are of the atomic scale. We want to exploit the large (of order $2m$) value of the energy gap between positive and negative states in comparison with the typical energy splittings (of order $m(Z\alpha)^2$) in the positive energy spectrum. First, let us note that
$$<n|[{\bf p},V]\Lambda_-[{\bf p},V]|n>=<n|[{\bf p},H-E]\Lambda_-[{\bf
p},H-E]|n>$$
$$= -<n|{\bf p}\sum_-|m><m|(E_n-E_m)^2{\bf p}|n>.$$
However, $(E_n-E_m)^2>4m^2(1-c\alpha^2)$, and, hence,
$$|<n|[{\bf p},V]\Lambda_-[{\bf p},V]|n>|=|<n|{\bf
p}\sum_-|m><m|(E_n-E_m)^2{\bf p}|n>|$$
$$\geq |<n|{\bf p}\Lambda_-{\bf p}|n>|4m^2(1-c\alpha^2).$$
Then
$$|<n|{\bf p}\Lambda_-{\bf p}|n>|\leq\frac{1}{4m^2(1-c\alpha^2)}
|<n|[{\bf p},V]\Lambda_-[{\bf p},V]|n>|.$$
We know that at the atomic scale the Coulomb potential is of order $(Z\alpha)^2$, the momentum operators are of order $Z\alpha$, and, hence, we explicitly have the factor $(Z\alpha)^6$. Note that this approach would not work if we had a projector on the positive energy states. In such a case the energy differences would be of order $(Z\alpha)^2$ themselves and we would not get any suppression, since the factors $(Z\alpha)^2$ would cancel in the numerator and denominator.
Returning to our case, it is easy to realize that the projector on the negative energy states leads to additional suppression in the nonrelativistic limit, and, hence, the term under consideration does not produce any contribution of order $(Z\alpha)^6$ at the atomic scale.
There is complete agreement between the results of Ref.[@pg] and Ref.[@elkh] for the corrections discussed in this section.
Seagull Contribution
====================
Following Ref.[@elkh] let us again start with the Braun expression Eq.(\[braun\]) for the seagull contribution and perform the integration by closing the contour each time around one of the transverse photon poles
$$\label{seagull}
\Delta E_{s}=-\frac{1}{M}\int\frac{d\omega}{2\pi i}
<{n}|{\bf\hat D}(\omega,k)G(E+\omega){\bf\hat D}(\omega,k)|n>.$$
Substituting the pole representation for the Coulomb Green function we obtain in accordance with Ref.[@elkh]
$$\label{seagullpos}
\Delta E_{s}=
\frac{(Z\alpha)^2}{2M}
<{n}|\frac{4\pi\mbox{\boldmath
$\alpha_{k'}$}}{k'}[
\sum_+\frac{|m><m|}{(E-k'-E_m)(E-k-E_m)}(1+\frac{E_m-E}{k'+k})$$
$$-\sum_-\frac{|m><m|}{(E+k'-E_m)(E+k-E_m)}(1-\frac{E_m-E}{k'+k})]
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k}|n>.$$
Let us consider positive- and negative-energy parts of this expression separately.
We may expand the positive energy part in $(E-E_m)/k$ and $(E-E_m)/k$, taking into account that in the low-frequency integration region $mZ\alpha<k<\sigma$. In the first order of this expansion we get
$$\Delta E^+_{s}=\frac{(Z\alpha)^2}{2M}
<{n}|\frac{4\pi\mbox{\boldmath $\alpha_{k'}$}}{k'^2}
\Lambda_+\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^2}|n>.$$
Calculation of this contribution will be considered below. Let us turn to the negative-energy contribution. Energy differences are large for the negative energy contribution ($|E-E_m|\approx 2m(1-c\alpha^2)$), so we expand the negative energy term in $k/(E-E_m)$
$$\sum_-\frac{|m><m|}{(E+k'-E_m)(E+k-E_m)}(1-\frac{E_m-E}{k'+k})$$
$$=\sum_-\frac{|m><m|}{E-E_m}[\frac{1}{k+k'}+\frac{(k+k')^2}{2(E-E_m)^3}].$$
In accordance with Ref.[@elkh] the terms linear in $k/2m$ cancel, and the negative energy contribution acquires the form
$$\Delta E^-_{s}
=-\frac{(Z\alpha)^2}{4mM(1+c\alpha^2)}
<{n}|\frac{4\pi\mbox{\boldmath $\alpha_{k'}$}}{k'}
\Lambda_-[\frac{1}{k+k'}+\frac{(k+k')^2}
{2[2m(1+c\alpha^2)]^3}]
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k}|n>.$$
It may be shown (compare below consideration of the negative energy contribution in the case of the one transverse exchange) that the first term produces the contribution of order $(Z\alpha)^5$ while the second is of order $(Z\alpha)^7$. Only terms linear in $k$, $k'$ are capable of producing contributions of order $(Z\alpha)^6$, but these terms cancel each other, as we have just seen.
Let us now return to the positive energy contribution. The idea of Ref.[@elkh] is to consider matrix elements and to calculate them in the nonrelativistic approximation, which produces the leading low-frequency contribution. All matrix elements under consideration have common structure. In general they are the products of matrix elements of $\gamma$-matrices in the momentum space. Each such matrix element in the nonrelativistic limit may easily be reduced to an explicit function of momenta and $\sigma$-matrices, then transformed into coordinate space and calculated between Coulomb-Schrodinger wave functions.
We have performed an explicit calculation along these lines and obtained in complete accord with Ref.[@elkh]
$$\label{seagullop}
\Delta E^+_{s}=\frac{(Z\alpha)^2}{4m^2M}<n|2{\bf
p}\frac{1}{r^2}{\bf p}+\frac{1}{r^4} -\frac{3{\bf l}^2+2\mbox{\boldmath
$\sigma l$}}{2r^4}|n>.$$
This expression is singular at the origin. This singularity produces linear and logarithmic ultraviolet divergences in momentum space as well as a constant contribution, and hence the contribution under consideration cannot be calculated unambiguously in the general case. It is necessary to realize at this stage that the initial expression for the seagull contribution in Eq.(\[seagull\]) was defined unambiguously. Even separation of the integration region with the help of the auxiliary parameter $\sigma$ could not lead to an ultraviolet divergence in the low-frequency region since all momentum integrations are cut off from above by $\sigma$ and should generate not power divergent but power suppressed terms. It is clear that the apparent divergence is connected with our inaccurate calculation of the singularity at large momenta or small distances. Hence, we have to return to the initial momentum space expression for the positive energy seagull contribution and perform all calculations directly in the momentum space. The result of such a calculation may be later interpreted as an unambiguous prescription for the proper regularization of the coordinate space operators for the $S$-states.
Note, that for the non-$S$ states, wave functions vanish at the origin, the operators above are well defined on such wave functions, and lead to unambiguous results. Of course, any regularization at small distances will not influence the value of the non-$S$ matrix elements of the operator in Eq.(\[seagullop\]), and will not influence the agreement between the $P$-level energy shift calculated in Ref.[@elkh], and the same shift obtained earlier in another framework in Ref.[@gkmy].
Accurate Calculation with Momentum Space Cutoff
-----------------------------------------------
Direct calculation of the positive energy seagull contribution Eq.(\[seagullpos\]) in momentum space leads to the following expression for the $S$-state contribution
$$\Delta E^+_{s}=
=\frac{(Z\alpha)^2}{m^2M}\int
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}(2\pi)^3\delta({\bf p'-p-k-k'})\frac{8\pi^2}{k'^2k^2}$$
$$\psi(p')[-{\bf p'p} +\frac{\bf (k'k)(p'k') (pk)}{k'^2k^2}-\frac{\bf
k'k}{2}]\psi(p)\equiv \Delta E_{s1}+\Delta E_{s2}+\Delta E_{1/r^4}.$$
The first two terms in the integrand do not rise too rapidly with $k$ and $k'$, and we may unambiguously calculate them using the Fourier transforms discussed above. For the first term we have
$$\Delta E_{s1}=
-\frac{(Z\alpha)^2}{m^2M}\int d^3r
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}e^{i\bf r(-p'+p+k+k')})\frac{8\pi^2}{k'^2k^2}$$
$$\psi(p'){\bf p'p}\psi(p)
=-\frac{(Z\alpha)^2}{2m^2M}\int d^3r
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}e^{i\bf r(-p'+p)})
\frac{1}{r^2}
\psi(p'){\bf p'p}\psi(p).$$
The remaining integration over $p'$ and $p$ simply returns us to the coordinate space wave functions, and we may rewrite the expression above in the operator notation[^6]
$$\label{s1}
\Delta E_{s1}=\frac{(Z\alpha)^2}{2m^2M}<n|{\bf p}\frac{1}{r^2}{\bf p}|n>.$$
This contribution exactly reproduces the nonsingular operator obtained in the previous chapter.
Next we calculate the second contribution in the same manner as above
$$\Delta E_{s2}
=\frac{(Z\alpha)^2}{m^2M}\int d^3r\int
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}
e^{i\bf r(-p'+p+k+k')}
\psi(p')\frac{8\pi^2\bf (k'k)(p'k') (pk)}{k'^4k^4}\psi(p)$$
$$=\frac{(Z\alpha)^2}{2m^2M}\int d^3r\int
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}
e^{i\bf r(-p'+p)})
\psi(p')\frac{p'_jp_m}{4r^2}(\delta_{ij}-\frac{r_ir_j}{r^2})
(\delta_{im}-\frac{r_ir_m}{r^2})\psi(p)$$ $$=\frac{(Z\alpha)^2}{2m^2M}\int d^3r\int
\frac{d^3p'}{(2\pi)^3}\frac{d^3p}{(2\pi)^3}
e^{i\bf r(-p'+p)})
\psi(p')\frac{1}{4r^2}({\bf p'p}-\frac{\bf(p'r)(pr)}{r^2})\psi(p).$$
Now we use the formula
$${\bf (rp')(rp)=-[r\times p'][r\times p]+r^2(p'p)},$$
and omit the terms with the vector product since we are considering only $S$-states now. Then we obtain
$$\Delta E_{s2}=0.$$
Next we have to calculate the third contribution, which corresponds to the $1/r^4$ term in the naive result above in Eq.(\[seagullop\]). This time we cannot use Fourier transformations over exchanged momenta for calculation of this integral, since this leads to a singular expression in coordinate space. So we first perform the safe Fourier transformations over the wave function momenta, and then directly evaluate the exchanged momenta integrals, taking into account that they are cut from above by $\sigma\ll m$,
$$\Delta E^{1/r^4}_{s}
=-\frac{(Z\alpha)^2}{m^2M}
\int \frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}\frac{4\pi^2({\bf k'k})}{k'^2k^2}
<n({\bf r})|e^{i{\bf (k+k')r}}|n({\bf r})>.$$
In order to preserve the transparency of the presentation we will perform the calculation only for $n=1$ here. The general case of arbitrary principal quantum number will be considered at the end of the paper. We substitute explicit expressions for the $1S$-wave functions in the formula above, and do the coordinate-space integral
$$\Delta E^{1/r^4}_{s}=
-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
\int\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}\frac{4\pi^2({\bf k'k})}{k'^2k^2}\int d^3r
e^{i{\bf (k+k')r}}e^{-2\gamma r}$$
$$=-\frac{64\pi^3(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
\int \frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}\frac{{\bf k'k}}{k'^2k^2[({\bf
k+k'})^2+(2\gamma)^2]^2},$$
where $\gamma=mZ\alpha$.
Symmetrical integrals over the exchanged momenta are cut from above by the parameter $\sigma$. However, first integration, say over $\bf k'$, is convergent at high momenta and the cutoff may be safely ignored
$$\label{s3}
\Delta E^{1/r^4}_{s}=-\frac{16\pi(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
\int \frac{d^3k}{(2\pi)^3k^2}\int_0^\infty
dk'\int_{-1}^1dx\frac{k'kx}{[k^2+k'^2+2kk'x+(2\gamma)^2]^2}$$
$$=-\frac{8\pi^2(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
\int
\frac{d^3k}{(2\pi)^3k^2}[\frac{arctan\frac{k}{2\gamma}}{k}-\frac{1}{2\gamma}]
=-\frac{4(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
\int_0^\sigma dk[\frac{arctan\frac{k}{2\gamma}}{k}-\frac{1}{2\gamma}]$$ $$=-\frac{(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
[2{\pi}\ln\frac{\sigma}{2\gamma} -2\frac{\sigma}{\gamma}].$$
Nonlogarithmic term of order $(Z\alpha)^5$ in this expression is additionally suppressed by the small ratio $\sigma/m$, and may be safely ignored. Thus, we see that the properly regularized operator $1/r^4$ in the seagull diagram does not generate a constant contribution. The logarithmic divergence above should cancel with the respective contribution of the one-transverse (magnetic) diagram.
Magnetic Contribution
=====================
This time we start with the Braun expression for the one transverse photon in Eq.(\[braun\])
$$\Delta E_{magn}=
\frac{1}{M}Re\int\frac{d\omega}{2\pi i}
<{n}|{\bf p}G(E+\omega){\bf\hat D}(\omega,k)
+{\bf\hat D}(\omega,k)G(E+\omega){\bf p}|n>$$
and first calculate the contour integral[^7]
$$\label{tr}
\Delta E_{magn}=
-\frac{Z\alpha}{2M}<{n}|{\bf p}[\sum_+\frac{|m><m|}{k+E_m-E}
-\sum_-\frac{|m><m|}{E-E_m+k}]\frac{4\pi\mbox{\boldmath
$\alpha_k$}}{k}|{n}>+h.c..$$
As we are again calculating the low-frequency corrections to the Breit potential let us expand the positive energy term in $(E_m-E)/k$
$$\Delta E^+_{magn}$$
$$=-\frac{Z\alpha}{2M}<{n}|{\bf p}\sum_+|m><m|
[\frac{1}{k}-\frac{E_m-E}{k^2}+\frac{(E_m-E)^2}{k^3}
+\ldots]\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k}|{n}>+h.c..$$
The first term in this expansion may be written in the form
$$\label{breittr}
\Delta E^+_{magn1}=-\frac{Z\alpha}{2M}<{n}|{\bf p}\Lambda_+
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^2}|{n}>+h.c.$$
$$=-\frac{Z\alpha}{2M}<{n}|{\bf p}\frac{4\pi
\mbox{\boldmath $\alpha_k$}}{k^2}|{n}>
+\frac{Z\alpha}{2M}<{n}|{\bf p}\Lambda_-
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^2}|{n}>+h.c.$$ $$=\Delta E_{Br}+\Delta E^+_{magn1-},$$
and it is now evident that the first (Breit) term here coincides with that part of transverse exchange which cancels with the respective term in the Coulomb contribution.
Remaining positive-energy contributions are given by the expression
$$\Delta E^+_{magnr}=-\frac{Z\alpha}{2M}<{n}|{\bf p}\sum_+|m><m|
[-\frac{E_m-E}{k^2}+\frac{(E_m-E)^2}{k^3}
+\ldots]\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k}|{n}>+h.c.$$
$$\equiv \Delta E^+_{magn2} +\Delta E^+_{magn3}+\ldots.$$
Positive Energy Contribution
----------------------------
In accordance with Ref.[@elkh] one may check that the term $\Delta
E^+_{magn2}$ does not lead to the contributions of order $(Z\alpha)^6$. We have
$$\Delta E^+_{magn2}=\frac{Z\alpha}{2M}<{n}|{\bf p}\sum_+|m><m|
(E_m-E)\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^3}|{n}>+h.c.$$
$$=-\frac{(Z\alpha)^2}{2M}<{n}|\frac{4\pi\mbox{\boldmath $k'$}}{k'^2}
\Lambda_+\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^3}|{n}>+h.c..$$
The simplest way to estimate this matrix element is to make a Fourier transformation. Then we need an infrared divergent Fourier transform of $1/k^3$. All momentum integrals in the low-frequency region are cut off from below by $m(Z\alpha)^2$, and it is easy to check that the leading term in the infrared divergent Fourier transform generates a logarithmic divergent contribution of order $(Z\alpha)^5$ in accordance with Ref.[@elkh]. The next terms vanish with the infrared cutoff and cannot produce contributions of order $(Z\alpha)^6$.
Let us turn now to the term $\Delta E^+_{magn3}$. Naive calculation in the coordinate space in accordance with the result in Ref.[@elkh] leads to the result
$$\label{magn3}
\Delta E^+_{magn3}=-\frac{Z\alpha}{2M}<{n}|{\bf p}\sum_+(E_m-E)^2|m><m|
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^4}|{n}>+h.c.$$
$$=-\frac{(Z\alpha)^2}{4m^2M}
<{n}|2{\bf p}\frac{1}{r^2}{\bf p}-\frac{7\bf l^2}{2r^4}
-\frac{\mbox{\boldmath $\sigma l$}}{r^4}|{n}>.$$
This expression contains only operators which are nonsingular at the origin for $S$-states. Hence, they are well defined, and there is no need for a careful momentum space consideration in this case.
Negative Energy Contribution
----------------------------
There are two negative-energy contributions connected with the magnetic term, one in Eq.(\[tr\]), and the other in Eq.(\[breittr\]).
Let us consider first
$$\Delta E^-_{magn}=\frac{Z\alpha}{2M}<{n}|{\bf p}
\sum_-\frac{|m><m|}{E-E_m+k}\frac{4\pi\mbox{\boldmath
$\alpha_k$}}{k}|{n}>+h.c..$$
We have checked, in accordance with Ref.[@elkh], that this term leads at most to contributions of order $(Z\alpha)^6$, and, hence, is of no interest.
We still have to calculate one more negative energy contribution, contained in Eq.(\[breittr\])
$$\label{negencontr}
\Delta E^+_{magn1-}
=\frac{Z\alpha}{2M}<{n}|{\bf p}\Lambda_-
\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^2}|{n}>+h.c.$$
$$=\frac{(Z\alpha)^2}{8m^2M}
<{n}|\frac{4\pi\mbox{\boldmath $\alpha_{k'}$}}{k'^2}\frac{4\pi{\bf
k}\mbox{\boldmath $(\alpha k)$}}{k^2}|{n}>+h.c..$$
Naive calculation with the help of the Fourier transformation leads, in accordance with Ref.[@elkh], to the expression
$$\label{naivemagn}
\Delta E^+_{magn1-}
=\frac{(Z\alpha)^2}{4m^2M}
<{n}|\frac{4\pi\delta({\bf r})}{r}-\frac{1}{r^4}|n>.$$
However, this expression, as in the case of the seagull contribution, contains singular operators at the origin, and does not have unambiguous meaning for the $S$-states. A more careful calculation, which explicitly takes into account a momentum space cutoff $\sigma$, is needed.
First we transform the negative energy contribution in Eq.(\[negencontr\]) to the form
$$\Delta E^+_{magn1-}
=-\frac{Z\alpha}{4mM}
<{n}|[{\bf p},V]\Lambda_-\frac{4\pi\mbox{\boldmath $\alpha_k$}}{k^2}|{n}>
+h.c.$$
Next we substitute the negative energy projection operator in the nonrelativistic approximation $\Lambda_-({\bf p})\approx{1}/{2}
-({\mbox{\boldmath $\alpha p$}+\beta m})/{2m}$ and use the trivial identity
$$[p,V]\Lambda_-=\Lambda_-[p,V]-[\Lambda_-,[p,V]]=\Lambda_-[p,V]+
[\frac{\mbox{\boldmath $\alpha p$}}{2m},[p,V]].$$
Note that the first term on the right hand side vanishes applied to the ket-vector, and the negative energy contribution reduces in the nonrelativistic approximation to
$$\Delta E^+_{magn1-}
=-\frac{Z\alpha}{2m^2M}\int\frac{d^3k}{(2\pi)^3}
<{n}|{\mbox{\boldmath $p_k$}}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}|{n}>.$$
Then we use
$$<n(r)|{\bf p_k}=-i\gamma<n(r)|\frac{\bf r_k}{r},$$
$$[{\bf p},V]=-i(Z\alpha)\frac{\bf r}{r^3},$$
and obtain
$$\Delta E^+_{magn1-}
=\frac{(Z\alpha)^2}{2m^2M}\gamma\int\frac{d^3k}{(2\pi)^3}\int d^3r
\psi(r)^2\frac{{\mbox{\boldmath ($r_kr$)}}}{r^4}\frac{4\pi e^{i{\bf
kr}}}{k^2}.$$
As in the case of the singular seagull contribution we will perform the calculation for $n=1$ first, postponing consideration of the general case to the next chapter. We substitute explicit expressions for the wave functions and obtain
$$\label{magnsing}
\Delta E^+_{magn1-}
=\frac{2\pi(Z\alpha)^2}{2m^2M}\gamma|\psi(0)|^2
\int\frac{d^3k}{(2\pi)^3}\frac{4\pi }{k^2}
\int_{-1}^1dx (1-x^2)\int_0^\infty dre^{-2\gamma r}e^{ikrx}$$
$$=\frac{4\pi(Z\alpha)^2}{2m^2M}\gamma^2|\psi(0)|^2
\int\frac{d^3k}{(2\pi)^3}\frac{4\pi }{k^2}
[\frac{(4\gamma^2+k^2)\arctan\frac{k}{2\gamma}}{\gamma k^3}-\frac{2}{k^2}]$$ $$=\frac{4\pi(Z\alpha)^2}{2m^2M}\frac{(4\pi)^2}{(2\pi)^3}\gamma^2|\psi(0)|^2
\int_0^\sigma dk
[\frac{(4\gamma^2+k^2)\arctan\frac{k}{2\gamma}}{\gamma k^3}-\frac{2}{k^2}]$$ $$=\frac{\pi(Z\alpha)^2}{m^2M}\gamma|\psi(0)|^2
[2\ln\frac{\sigma}{2\gamma} -1].$$
Again, as in the case of the seagull contribution, this term may be understood as a proper regularization of the naive singular in the coordinate space operator from Eq.(\[naivemagn\]).
Calculations for Arbitrary Principal Quantum Number
===================================================
The total low-frequency contribution for the $1S$-state is given by the sum of the results in Eq.(\[gy\]), Eq.(\[s1\]), Eq.(\[s3\]), Eq.(\[magn3\]) and Eq.(\[magnsing\])
$$\label{totlow1s}
\Delta E_{low-freq}(1S)=
-{(Z\alpha)^6}\frac{m}{M}m,$$
and coincides with the result obtained earlier for the low-frequency contribution in Ref.[@pg]. We see that the seagull and magnetic contributions partially cancel each other. This reflects cancellation of the $1/r^4$ terms in the language of Ref.[@elkh]. However, the contribution $(-1)$ survives. This contribution is connected with the $\delta$-function term in Ref.[@elkh], and the error in Ref.[@elkh] is due to an improper regularization of this contribution. Note that from the point of view of the coordinate representation after the Fourier transformation is done the proper regularization is highly nontrivial. One could never obtain this contribution with a naive [ *ad hoc*]{} regularization in coordinate space.
The result in Eq.(\[totlow1s\]) is valid only for the $1S$-state. We are going to generalize it to an arbitrary principal quantum number.
Seagull Contribution for Arbitrary $nS$-Level
---------------------------------------------
The general expression for the wave function of an $nS$-level has the form
$$\psi_n(r)=(\frac{\gamma^3}{\pi n^3})^\frac{1}{2}e^{-\frac{\gamma r}{n}}
[1-\frac{n-1}{n}\gamma r+\ldots].$$
Let us introduce $\beta\equiv \gamma/n$. Almost all calculations above for $n=1$ immediately turn into calculations for arbitrary $n$ after substitution $\gamma\rightarrow \beta$ [@ey]. The wave function has the form
$$\psi_n(r)=(\frac{\beta^3}{\pi})^\frac{1}{2}e^{-{\beta r}}
[1-(n-1)\beta r+\ldots]\equiv \psi_n(0)e^{-{\beta r}}
[1-(n-1)\beta r+\ldots].$$
Quadratic and higher order terms in $r$ in the postexponential factor in the wave function do not produce any contribution to the energy level connected with the singular operator in the naive expression in Eq.(\[seagullop\]), and we will ignore them below. The only difference between the general case and the case of $n=1$ is connected with the linear term in the postexponential factor. Let us find out how it changes the result for the seagull contribution. First, let us write down the singular seagull contribution induced by the purely exponential part of the wave function for arbitrary $n$ in the form
$$\Delta E^{1/r^4}_{s}=
-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
\int\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}\frac{4\pi^2({\bf k'k})}{k'^2k^2}\int d^3r
e^{i{\bf (k+k')r}}e^{-2\beta r}$$
$$=-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2\epsilon^{1/r^4}_{s},$$
where
$$\epsilon^{1/r^4}_{s}=2{\pi}\beta\ln\frac{\sigma}{2\beta}$$
The linear terms in the wave functions lead to an additional contribution
$$\Delta E^{1/r^4}_{s,corr}=
-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
\int\frac{d^3k'}{(2\pi)^3}
\frac{d^3k}{(2\pi)^3}\frac{4\pi^2({\bf k'k})}{k'^2k^2}\int d^3r
e^{i{\bf (k+k')r}}e^{-2\beta r}[-2(n-1)\beta r]$$
$$=-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2(n-1)\beta
\frac{\partial}{\partial\beta}\epsilon^{1/r^4}_{s}
=-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2(n-1)[2\pi\beta\ln\frac{\sigma}{2\beta}
-2\pi\beta],$$
and the total seagull contribution to the energy shift is equal to
$$\label{s3n}
\Delta E^{1/r^4}_{s,tot}=\Delta E^{1/r^4}_{s}
+\Delta E^{1/r^4}_{s,corr}=-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[\epsilon^{1/r^4}_{s}+(n-1)\beta
\frac{\partial}{\partial\beta}\epsilon^{1/r^4}_{s}]$$
$$=-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2[2\pi\gamma\ln\frac{\sigma}{2\beta}
-2\pi(n-1)\beta].$$
Magnetic Contribution for Arbitrary $nS$-Level
----------------------------------------------
As in the case of the seagull contribution the only difference of the general case from the case of $n=1$ is connected with the linear term in the postexponential factor in the wave function. The purely exponential part of the wave function leads to the following singular magnetic contribution for arbitrary $n$
$$\Delta E^+_{magn1-}
=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\beta\ln\frac{\sigma}{2\beta} -\pi\beta]
\equiv\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2\epsilon^+_{magn1-}.$$
The new contribution induced by the linear term in the wave function has the form
$$\Delta E^+_{magn1-,cor}
=-\frac{Z\alpha}{2m^2M}\int\frac{d^3k}{(2\pi)^3}
[-(n-1)\beta]\{<{n}|r{\mbox{\boldmath $p_k$}}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}|{n}>$$
$$+<{n}|{\mbox{\boldmath $p_k$}}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}r|{n}>\}.$$
Next we write
$$r{\bf p_k}={\bf p_k}r-[{\bf p_k},r],$$
and using the commutation relation
$$[{\bf p_k},r]=-i\frac{\bf r_k}{r},$$
obtain
$$\Delta E^+_{magn1-,cor}
=-\frac{Z\alpha}{2m^2M}\int\frac{d^3k}{(2\pi)^3}
[-(n-1)\beta]\{<{n}|i\frac{\bf r_k}{r}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}|{n}>$$
$$+<{n}|{\mbox{\boldmath $p_k$}}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}2r|{n}>\}.$$ $$=-\frac{Z\alpha}{2m^2M}\int\frac{d^3k}{(2\pi)^3}
(n-1)\{<{n}|(-i\beta\frac{\bf r_k}{r})[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}|{n}>$$ $$-(n-1)\beta<{n}|{\mbox{\boldmath $p_k$}}[{\bf
p},V]\frac{4\pi e^{i{\bf kr}}}{k^2}2r|{n}>\}$$ $$=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
(n-1)[\epsilon^+_{magn1-}+\beta^2\frac{\partial}{\partial\beta}
(\frac{\epsilon^+_{magn1-}}{\beta})]$$ $$=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\beta(n-1)\ln\frac{\sigma}{2\beta}
-(n-1)\pi\beta-(n-1)2\pi\beta]$$ $$=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\beta(n-1)\ln\frac{\sigma}{2\beta}
-3(n-1)\pi\beta].$$
Then the total singular magnetic contribution is equal to
$$\label{magnsingn}
\Delta E^+_{magn1-,tot}=\Delta E^+_{magn1-}+\Delta E^+_{magn1-,cor}
=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\beta\ln\frac{\sigma}{2\beta} -\pi\beta$$
$$+2\pi\beta(n-1)\ln\frac{\sigma}{2\beta} -3(n-1)\pi\beta]
=\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\gamma\ln\frac{\sigma}{2\beta} -\pi\beta-3(n-1)\pi\beta].$$
Total Recoil Correction
========================
The total low-frequency contribution of order $(Z\alpha)^6(m/M)m$ for arbitrary $nS$-state is given by the sum of the terms in Eq.(\[gy\]), Eq.(\[s1\]), Eq.(\[s3n\]), Eq.(\[magn3\]) and Eq.(\[magnsingn\])
$$\label{totlow}
\Delta E_{low-freq}=
(\frac{1}{8}+\frac{3}{8n}-\frac{1}{n^2}+\frac{1}{2n^3})
\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m
+\frac{(Z\alpha)^2}{2m^2M}<n|{\bf p}\frac{1}{r^2}{\bf p}|n>$$
$$-\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2[2\pi\gamma\ln\frac{\sigma}{2\beta}
-2\pi(n-1)\beta-2\sigma]
-\frac{(Z\alpha)^2}{4m^2M}
<{n}|2{\bf p}\frac{1}{r^2}{\bf p}|{n}>$$ $$+\frac{(Z\alpha)^2}{m^2M}|\psi(0)|^2
[2\pi\gamma\ln\frac{\sigma}{2\beta} -\pi\beta-3(n-1)\pi\beta]$$ $$=(\frac{1}{8}+\frac{3}{8n}-\frac{1}{n^2}+\frac{1}{2n^3})
\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m-\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m.$$
Note that the last term connected with the naive singular operators in the coordinate space turned out to be state-independent.
To obtain the total recoil correction of order $(Z\alpha)^6(m/M)m$ it is also necessary to calculate the high-frequency (or short-distance) contribution to the energy shift. The simplest way is to use again the Braun formula Eq.(\[braun\]), but this time in the Feynman gauge. This calculation is quite straightforward if one again uses the auxiliary parameter $\sigma$ introduced above in order to qualify would be infrared divergences. Such a calculation was performed explicitly in ref.[@elkh] and led to the result
$$\label{tothigh}
\Delta E_{high-freq}=(4\ln2-\frac{5}{2})\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m,$$
in complete agreement with Ref.[@pg].
Then total correction of order $(Z\alpha)^6(m/M)m$ to the energy levels is given by the sum of the results in Eq.(\[totlow\]) and Eq.(\[tothigh\])
$$\label{tot}
\Delta E_{tot}=(\frac{1}{8}+\frac{3}{8n}-\frac{1}{n^2}+\frac{1}{2n^3})
\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m
+(4\ln2-\frac{7}{2})\frac{(Z\alpha)^6}{n^3}\frac{m}{M}m.$$
For $n=1,2$ this result nicely coincides with the one obtained in Ref.[@pg].
In conclusion let us emphasize that discrepancies between the different results for the correction of order $(Z\alpha)^6(m/M)$ to the energy levels of the hydrogenlike ions are resolved and the correction of this order is now firmly established.
M. E. is deeply grateful for the kind hospitality of the Physics Department at Penn State University, where this work was performed. The authors appreciate the support of this work by the National Science Foundation under grant number PHY-9421408.
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[^1]: Temporary address.
[^2]: Permanent address.
[^3]: E-mail address: eides@phys.psu.edu, eides@lnpi.spb.su
[^4]: E-mail address: h1g@psuvm.psu.edu
[^5]: Note that the apparent linear divergences in this region of the form $\sigma/m$ are really parametrically small.
[^6]: One has to take into account that the apparent sign of the expression below changes, since the momenta in the exponent have opposite signs.
[^7]: Note that the overall minus sign is connected with the respective sign in the definition of the transverse propagator.
|
---
abstract: '**Symmetry-broken three-dimensional topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl Fermion quantum transport. Here using high-resolution angle-resolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd$_3$As$_2$, which has been predicted to be the parent material, from which many unusual topological phases can be derived. We observe a highly linear bulk band crossing to form a three-dimensional dispersive Dirac cone projected at the Brillouin zone center by studying the (001)-cleaved surface. Remarkably, an unusually in-plane high Fermi velocity up to 1.5 $\times$ 10$^{6}$ ms$^{-1}$ is observed in our samples, where the mobility is known up to 40,000 cm$^2$V$^{-1}$s$^{-1}$ suggesting that Cd$_3$As$_2$ can be a promising candidate as an anisotropic-hypercone (3D) high spin-orbit analog of graphene. Our experimental identification of the Dirac-like bulk topological semimetal phase in Cd$_2$As$_2$ opens the door for exploring higher dimensional spin- orbit Dirac physics in a real material.**'
author:
- 'Madhab Neupane\*'
- 'Su-Yang Xu\*'
- 'Raman Sankar\*'
- Nasser Alidoust
- Guang Bian
- Chang Liu
- Ilya Belopolski
- 'Tay-Rong Chang'
- 'Horng-Tay Jeng'
- Hsin Lin
- Arun Bansil
- Fangcheng Chou
- 'M. Zahid Hasan'
title: 'Discovery of a three-dimensional topological Dirac semimetal phase in high-mobility Cd$_3$As$_2$'
---
Two-dimensional (2D) Dirac electron systems exhibiting many exotic quantum phenomena constitute one of the most active topics in condensed matter physics [@Graphene; @Weyl; @RMP; @Zhang_RMP; @David_nature; @Xia; @Chen_Science; @Hasan2; @HsiehSci; @Dirac_3D; @Bismuth; @Dirac_semi; @3D_Dirac; @Dai; @Neupane; @Volovik; @Fang; @Ashvin; @Balent]. The notable examples are graphene and the surface states of topological insulators (TI). Three-dimensional (3D) Dirac fermion metals, sometimes noted as the topological bulk Dirac semimetal (BDS) phases, are also of great interest if the material possesses 3D isotropic or anisotropic relativistic dispersion in the presence of strong spin-orbit coupling. It has been theoretically predicted that a topological (spin-orbit) 3D spin-orbit Dirac semimetal can be viewed as a composite of two sets of Weyl fermions where broken time-reversal or space inversion symmetry can lead to a surface Fermi-arc semimetal phase or a topological insulator [@Dai]. In the absence of spin-orbit coupling, topological phases cannot be derived from a 3D Dirac semimetal. Thus the parent BDS phase with strong spin-orbit coupling is of great interest. Despite their predicted existence [@3D_Dirac; @Dirac_semi; @Dai], experimental studies on the massless BDS phase have been lacking since it has been difficult to realize this phase in real materials, especially in stoichiometric single crystalline non-metastable systems with high mobility. It has also been noted that the BDS state can be achieved at the critical point of a topological phase transition [@Suyang; @Ando] between a normal insulator and a topological insulator which requires fine-tuning of the chemical doping/alloying composition thus by effectively varying the spin-orbit coupling strength. This approach also introduces chemical disorder into the system. In stoichiometric bulk materials, the known 3D Dirac fermions in bismuth are in fact of massive variety since there clearly exists a band gap in the bulk Dirac spectrum [@Bismuth]. On the other hand, the bulk Dirac fermions in the Bi$_{1-x}$Sb$_x$ system coexist with additional Fermi surfaces [@David_nature]. Therefore, to this date, identification of a gapless BDS phase in stoichiometric materials remains experimentally elusive.
In this article, we present the experimental identification of a gapless Dirac-like 3D topological (spin-orbit) semimetal phase in stoichiometric single crystalline system of Cd$_3$As$_2$, which is protected by the $C_4$ crystalline (crystal structure) symmetry and spin-orbit coupling as predicted in theory [@Dai]. Using high-resolution angle-resolved photoemission spectroscopy (ARPES), we show that Cd$_3$As$_2$ features a bulk band Dirac-like cone locating at the center of the (001) surface projected Brillouin zone (BZ). Remarkably, we observe that the band velocity of the bulk Dirac spectrum is as high as $\sim$ 10 $\textrm{\AA}{\cdot}$eV, which along with its massless character favorably contributes to its natural high mobility ($\sim$ $10^5$ cm$^2$V$^{-1}$s$^{-1}$ [@Mobi; @Mobi2]). We further compare and contrast the observed crystalline-symmetry-protected BDS phase in Cd$_3$As$_2$ with those of in the Bi-based 3D-TI systems such as in BiTl(S$_{1-\delta}$Se$_{\delta}$)$_2$ and (Bi$_{1-\delta}$In$_{\delta}$)$_2$Se$_3$ systems. Our experimental identification and band-structure measurements of the Dirac-like bulk semimetal phase and its clear contrast with Bi$_2$Se$_3$ and 2D graphene discovered previously, opens the door for exploring higher dimensional spin-orbit Dirac physics in a stoichiometric material. These new directions are uniquely enabled by our observation of strongly spin-orbit coupled 3D massless Dirac semimetal phase protected by the $C_4$ symmetry, which is not possible in the 2D Dirac fermions in graphene and the surfaces of topological insulators, or weak spin-orbit 3D Dirac fermions in other materials.
**Results**
**Crystalline symmetry protected topological Dirac phase**
The crystal structure of Cd$_3$As$_2$ has a tetragonal unit cell with $a= 12.67$ $\AA$ and $c= 25.48$ $\AA$ for $Z= 32$ with symmetry of space group $I4_1$cd (see Figs. 1a and b). In this structure, arsenic ions are approximately cubic close-packed and Cd ions are tetrahedrally coordinated, which can be described in parallel to a fluorite structure of systematic Cd/As vacancies. There are four layers per unit and the missing Cd-As$_4$ tetrahedra are arranged without the central symmetry as shown with the (001) projection view in Fig.1b, with the two vacant sites being at diagonally opposite corners of a cube face [@crys_str]. The corresponding Brillouin zone (BZ) is shown in Fig. 1d, where the center of the BZ is the $\Gamma$ point, the centers of the top and bottom square surfaces are the $Z$ points, and other high symmetry points are also noted. Cd$_3$As$_2$ has attracted attention in electrical transport due to its high mobility of $10^5$ cm$^2$V$^{-1}$s$^{-1}$ reported in previous studies [@Mobi; @Mobi2]. The carrier density and mobility of our Cd$_3$As$_2$ samples (shown in Fig. 1 and 2) are characterized to be of $5.2\times10^{18}$ cm$^{-3}$ and $42850$ cm$^2$V$^{-1}$s$^{-1}$, respectively, at temperature of 130 K, consistent with previous reports [@Mobi; @Mobi2], which provide an evidence for the high quality of our single crystalline samples. In band theoretical calculations, Cd$_3$As$_2$ is also of interest since it features an inverted band structure [@Inverted]. More interestingly, a very recent theoretical prediction [@Dai] which motivated this work, has shown that the spin-orbit interaction in Cd$_3$As$_2$ cannot open up a full energy gap between the inverted bulk conduction and valence bands due to the protection of an additional crystallographic symmetry [@Dirac_3D] (in the case of Cd$_3$As$_2$ it is the $C_4$ rotational symmetry along the $k_z$ direction [@Dai]), which is in contrast to other band-inverted systems such as HgTe [@RMP]. This theory predicts [@Dai] that the $C_4$ rotational symmetry protects two bulk (3D) Dirac band touching points at two special $\mathbf{k}$ points along the $\Gamma-Z$ momentum space cut-direction, as shown by the red crossings in Fig. 1d. Therefore, Cd$_3$As$_2$ serves a candidate for a spacegroup or crystal structure symmetry protected $C_4$ bulk Dirac semimetal (BDS) phase.
**Observation of bulk Dirac cone**
In order to experimentally identify such a BDS phase, we systematically study the electronic structure of Cd$_3$As$_2$ on the cleaved (001) surface. Fig. 1c shows momentum-integrated ARPES spectral intensity over a wide energy window. Sharp ARPES intensity peaks at binding energies of $E_\textrm{B} \simeq 11$ eV and $41$ eV that correspond to the cadmium $4d$ and the arsenic $3d$ core levels are observed, confirming the chemical composition of our samples. We study the overall electronic structure of the valence band. Fig. 1e shows the second derivative image of an ARPES dispersion map in a 3 eV binding energy window, where the dispersion of several valence bands are identified. Moreover, a low-lying small feature that crosses the Fermi level is observed. In order to resolve it, high-resolution ARPES dispersion measurements are performed in the close vicinity of the Fermi level as shown in Fig. 1f. Remarkably, a linearly dispersive upper Dirac cone is observed at the surface BZ center $\bar{\Gamma}$ point, whose Dirac node is found to locate at a binding energy of $E_{\textrm{B}}\simeq0.2$ eV. At the Fermi level, only the upper Dirac band but no other electronic states are observed. On the other hand, the linearly dispersive lower Dirac cone is found to coexist with another parabolic bulk valence band, which can be seen from Fig. 1e. From the observed steep Dirac dispersion (Fig. 1f), we obtain a surprisingly high Fermi velocity of about 9.8 eV$\cdot$$\AA$ ($\simeq1.5\times10^{6}$ ms$^{-1}$). This is more than 10-fold larger than the theoretical prediction of 0.15 eV$\cdot$ $\AA$ at the corresponding location of the chemical potential [@Dai]. Compared to the much-studied 2D Dirac systems, the Fermi velocity of the 3D Dirac fermions in Cd$_3$As$_2$ is thus about 3 times higher than that of in the topological surface states (TSS) of Bi$_2$Se$_3$ [@Xia], $1.5$ times higher than in graphene [@Eli] and 30 times higher than that in the topological Kondo insulator phase in SmB$_6$ [@SmB6; @SmB6_Hasan]. The observed large Fermi velocity of the 3D Dirac band provides clues to understand Cd$_3$As$_2$’s unusually high mobility reported in previous transport experiments [@Mobi; @Mobi2]. Therefore one can expect to observe unusual magneto-electrical and quantum Hall transport properties under high magnetic field. It is well-known that in graphene the capability to prepare high quality and high mobility samples has enabled the experimental observations of many interesting phenomena that arises from its 2D Dirac fermions. The large Fermi velocity and high mobility in Cd$_3$As$_2$ are among the important experimental criteria to explore the 3D relativistic physics in various Hall phenomena in tailored Cd$_3$As$_2$.
We compare ARPES observations with our theoretical calculations which is qualitatively consistent with previous calculations [@Dai]. The reason for the use of our calculations is two fold: first, our calculations are fine tuned based on the characterization of samples used in the present ARPES study, second, sufficiently detailed cuts are not readily available from ref [@Dai] which is necessary for a detailed comparison of ARPES data with theory. In theory, there are two 3D Dirac nodes that are expected at two special $\mathbf{k}$ points along the $\Gamma-Z$ momentum space cut-direction, as shown by the red crossings in Fig. 1d. At the (001) surface, these two $\mathbf{k}$ points along the $\Gamma-Z$ axis project on to the $\bar{\Gamma}$ point of the (001) surface BZ (Fig. 1d). Therefore, at the (001) surface, theory predicts one 3D Dirac cone at the BZ center $\bar{\Gamma}$ point, as shown in Fig. 2a. These results are in qualitative agreement with our data, which supports our experimental observation of the 3D BDS phase in Cd$_3$As$_2$. We also study the ARPES measured constant energy contour maps (Fig. 2c and d). At the Fermi level, the constant energy contour consists of a single pocket centered at the $\bar{\Gamma}$ point. With increasing binding energy, the size of the pocket decreases and eventually shrinks to a point (the 3D Dirac point) near $E_{\textrm{B}}\simeq0.2$ eV. The observed anisotropies in the iso-energetic contours are likely due to matrix element effects associated with the standard p-polarization geometry used in our measurements.
**Three-dimensional dispersive nature**
A 3D Dirac semimetal is expected to feature nearly linear dispersion along all three momentum space directions close to the crossing point, even though the Fermi/Dirac velocity can vary significantly along different directions. It is well known that in real materials such as pure Bi or graphene or topological insulators the Dirac cones are never perfectly linear over a large energy window yet they can be approximated to be so within a narrow energy window and in comparison to the large effective mass of conventional band electrons in many other materials. In order to probe the 3D nature of the observed low-energy Dirac-like bands in Cd$_3$As$_2$, we performed ARPES measurements as a function of incident photon energy to study the out-of-plane dispersion perpendicular to the (001) surface. Upon varying the photon energy, one can effectively probe the electronic structure at different out-of-plane momentum $k_z$ values in a three-dimensional Brillouin zone and compare with band calculations. In Cd$_3$As$_2$, the electronic structure or band dispersions in the vicinity of its 3D Dirac-like node can be approximated as : $v_{\|}^2(k_x^2+k_y^2)+v_{\perp}^2(k_z-k_0)^2=E^2$, where $k_0$ is the out-of-plane momentum value of the 3D Dirac point. Thus at a fixed $k_z$ value (which is determined by the incident photon energy value), the in-plane electronic dispersion takes the form: $v_{\|}^2(k_x^2+k_y^2)=E^2-v_{\perp}^2(k_z-k_0)^2$. It can be seen that only at $k_z=k_0$ the in-plane dispersion is a gapless Dirac cone, whereas in the case for $k_z\neq{k_0}$ the nonzero $k_z-{k_0}$ term acts as an effective mass term and opens up a gap in the in-plane dispersion relation. Fig. 3a shows the ARPES measured in-plane electronic dispersion at various photon energies. At a photon energy of $102$ eV, a gapless Dirac-like cone is observed, which shows that photon energy $h\nu=102$ eV corresponds to a $k_z$ value that is close to the out-of-plane momentum value of the 3D Dirac node $k_0$. As photon energy is changed away from $102$ eV in either direction, the bulk conduction and valence bands are observed within experimental resolution to be separated along the energy axis and a gap opens in the in-plane dispersion. At photon energies sufficiently away from $102$ eV, such as 90 eV or 114 eV in Fig. 3a, the in-plane gap is large enough so that the bottom of the upper Dirac cone (bulk conduction band) is moved above the Fermi level, and therefore only the lower Dirac cone is observed. We now fix the in-plane momenta at 0 and plot the ARPES data at $k_x=k_y=0$ as a function of incidence photon energy. As shown in Fig. 3b, a $E-k_z$ dispersion is observed in the out-of-plane momentum space cut direction, which is in qualitative agreement with the theoretical calculations (Fig. 3c). The Fermi velocity in the z-direction can be estimated (only at the order of magnitude level) to be about 10$^{5}$ ms$^{-1}$. We note that the sample we used for $k_z$ dispersion measurements (Figs. 3a-c) is relatively $p-$type (Fermi velocity is about 80 meV from the Dirac point) as compared to the sample we used to measure the in-plane dispersion and Fermi surfaces (Figs. 1-2) where chemical potential is about 200 meV from the Dirac point. It is important to note that the magnitude of Fermi velocity anisotropy strongly depends on the position of the sample chemical potential ($n-$type sample leads to weaker anisotropy), and therefore the direct comparison between our results and previous transport data in terms of this anisotropy is not applicable. These systematic incident photon energy dependent measurements show that the observed Dirac-like band disperses along both the in-plane and the out-of-plane directions suggesting its three-dimensional or bulk nature consistent with theory.
In order to further understand the nature of the observed Dirac band, we study the spin polarization or spin texture properties of Cd$_3$As$_2$. As shown in Fig. 3f, spin-resolved ARPES measurements are performed on a relatively $p-$type sample. Two spin-resolved energy-dispersive curve (EDC) cuts are shown at momenta of $\pm0.1$ $\AA^{-1}$ on the opposite sides of the Fermi surface. The obtained spin data shown in Figs. 3g and h show no observable net spin polarization or texture behavior within our experimental resolution, which is in remarkable contrast with the clear spin texture in 2D Dirac fermions on the surfaces of topological insulators. The absence of spin texture in our observed Dirac fermion in Cd$_3$As$_2$ bands is consistent with their bulk origin, which agrees with the theoretical prediction. It also provides a strong evidence that our ARPES signal is mainly due to the bulk Dirac bands on the surface of Cd$_3$As$_2$, whereas the predicted surface (resonance) states [@Dai] that lie along the boundary of the bulk Dirac cone projection has a small spectral weight (intensity) contribution to the photoemission signal. In other words, according to our experimental data, the surface electronic structure of Cd$_3$As$_2$ is dominated by the spin-degenerate bulk bands, which is very different from that of the 3D topological insulators.
**Discussion** The distinct semimetal nature of Cd$_3$As$_2$ is better understood from ARPES data if we compare our results with that of the prototype TI, Bi$_2$Se$_3$. In Bi$_2$Se$_3$ as shown in Fig. 4b, the bulk conduction and valence bands are fully separated (gapped), and a linearly dispersive topological surface state is observed that connect across the bulk band-gap. In the case of Cd$_3$As$_2$ (Fig. 4a), there does not exist a full bulk energy gap. On the other hand, the bulk conduction and valence bands “touch” (and only “touch”) at one specific location in the momentum space, which is the 3D band-touching node, thus realizing a 3D BDS. For comparison, we further show that a similar BDS state is also realized by tuning the chemical composition $\delta$ (effectively the spin-orbit coupling strength) to the critical point of a topological phase transition between a normal insulator and a topological insulator. Figs. 4c and d present the surface electronic structure of two other BDS phases in the BiTl(S$_{1-\delta}$Se$_{\delta}$)$_2$ and (Bi$_{1-\delta}$In$_{\delta}$)$_2$Se$_3$ systems. In both systems, it has been shown that tuning the chemical composition $\delta$ can drive the system from a normal insulator state to a topological insulator state [@Suyang; @Ando; @Oh]. The critical compositions for the two topological phase transitions are approximately near $\delta=0.5$ and $\delta=0.04$, respectively. Figs. 4c and d show the ARPES measured surface electronic structure of the critical compositions for both BiTl(S$_{1-\delta}$Se$_{\delta}$)$_2$ and (Bi$_{1-\delta}$In$_{\delta}$)$_2$Se$_3$ systems, which are expected to exhibit the BDS phase. Indeed, the bulk critical compositions where bulk and surface Dirac bands collapse also show Dirac cones with intensities filled inside the cones, which is qualitatively similar to the case in Cd$_3$As$_2$. Currently, the origin of the filling behavior is not fully understood irrespective of the bulk (out-of-plane dispersive behavior) nature of the overall band dispersion interpreted in connection to band calculations (see Fig. 2). Based on the ARPES data in Figs. 4c and d, the Fermi velocity is estimated to be $\sim4$ eV$\cdot\textrm{\AA}$ and $\sim2$ eV$\cdot\textrm{\AA}$ for the 3D Dirac fermions in BiTl(S$_{1-\delta}$Se$_{\delta}$)$_2$ and (Bi$_{1-\delta}$In$_{\delta}$)$_2$Se$_3$ respectively, which is much lower than that of what we observe in Cd$_3$As$_2$, thus likely limiting the carrier mobility. The mobility is also limited by the disorder due to strong chemical alloying. More importantly, the fine control of doping/alloying $\delta$ value and keeping the composition exactly at the bulk critical composition is difficult to achieve [@Suyang], especially while considering the chemical inhomogeneity introduced by the dopants. For example, although similarly high electron mobility on the order of $10^5$ cm$^2$V$^{-1}$s$^{-1}$ has been reported in the bulk states of Pb$_{1-x}$Sn$_x$Se ($x=0.23$) [@Ong], the bulk Dirac fermions there are in fact massive due to the difficulty of controlling the composition exactly at the critical point. These facts taken together exclude the possibility of realizing proposed topological physics including the Weyl semimetal and quantum spin Hall phases using the bulk Dirac states in the Pb$_{1-x}$Sn$_x$Se. These issues do not arise in the stoichiometric Cd$_3$As$_2$ system since its BDS phase is protected by the crystal symmetry, which does not require chemical doping and therefore the natural high electron mobility is retained (not diminished). We note that our crystals of Cd$_3$As$_2$ are nearly stoichiometric within the resolution of electron probe micro-analyzer (EPMA) and X-ray diffraction (XRD) analysis. The existence of some low level defects is not ruled out. However, these defects do not affect the main conclusion regarding the 3D Dirac band structure ground state of this compound. Beside Cd$_3$As$_2$ and the topological phase transition critical composition samples as discussed above, we also note that bulk Dirac semimetals unrelated to the combination of $C_4$ symmetry and band-inverted spin-orbit coupling (combination of which has been termed “topological” in theory [@Dai]) have been studied previously in pnictide BaFe$_2$As$_2$ [@Ding], heavy fermion LaRhIn$_5$ [@LaRhIn5], and organic compound $\alpha$-(BEDT-TTF)$_2$I$_3$ [@Organic]. The recent interest is actually focused on spin-orbit based 3D bulk Dirac semimetal phase since the spin-orbit coupling can drive exotic topological phenomena and quantum transport in such materials as the Weyl phases, high temperature linear quantum magnetoresistance and topological magnetic phases [@Fang; @Ashvin; @Balent; @3D_Dirac; @Volovik; @Dirac_3D; @Dirac_semi; @Dai]. Our observation of the bulk Dirac states in Cd$_3$As$_2$ provides a unique combination of physical properties, including high spin-orbit coupling strength, high electron mobility, massless nature guaranteed by the crystal symmetry protection without compositional tuning, making it an ideal and unique platform to realize many of the proposed exciting new topological physics [@Fang; @Ashvin; @Balent; @3D_Dirac; @Volovik; @Dirac_3D; @Dirac_semi; @Dai].
In conclusion, we have experimentally discovered the crystalline-symmetry-protected 3D spin-orbit BDS phase in a stoichiometric system Cd$_3$As$_2$ (see Fig. 5). The combination of a large Fermi velocity and very high electron mobility of the 3D carriers with nearly linear dispersion at the crossing point makes it a promising platform to explore novel 3D relativistic physics in various types of quantum Hall phenomena. Our band structure study of the predicted 3D BDS phase also paves the way for designing and realizing a number of related exotic topological phenomena in future experiments. For example, if the $C_4$ crystalline symmetry is broken, the 3D Dirac cone in Cd$_3$As$_2$ can open up a gap and therefore a topological insulator phase is realized in a high mobility setting (current Bi-based TIs feature low carrier mobility). Furthermore upon doping magnetic elements or fabricating superlattice hetero-structures, the 3D Dirac node in Cd$_3$As$_2$ can be split into two topologically protected Weyl nodes, realizing the much sought out Fermi arcs phases in solid-state setting.
Concurrently posted preprints (refs [@MN] (ours) and [@Cava]) report ARPES studies of experimental realization of 3D topological Dirac semimetal phase in Cd$_3$As$_2$, however, many of the experimental details and interpretations of the data differ from ours. Later, two other preprints (refs [@Chen] and [@SYX]) report experimental realization of the 3D Dirac phase in a metastable low mobility compound, Na$_3$Bi.
**Methods**
**Sample growth and characterization**
Single crystalline samples of Cd$_3$As$_2$ were grown using the standard method, which is described elsewhere [@crys_str]. The Cd$_3$As$_2$ samples used for our ARPES studies show carrier density of $5.2\times10^{18}$ cm$^{-3}$ and mobility up to $42850$ cm$^2$V$^{-1}$s$^{-1}$ at temperature of 130 K, which is consistent with the mobility of $10^4$ cm$^2$V$^{-1}$s$^{-1}-10^5$ cm$^2$V$^{-1}$s$^{-1}$ reported elsewhere [@Mobi; @Mobi2]. A slight variation of the value of carrier density and mobility is observed for different growth batch samples. We note that our samples show different chemical potential position (measured by ARPES) and different carrier density (measured by transport) depending on the detailed growth conditions. Moreover, our crystals of Cd$_3$As$_2$ are nearly stoichiometric within the resolution of electron probe micro-analyzer (EPMA) and X-ray diffraction (XRD) analysis. The existence of some low level defects is not ruled out. **Spectroscopic measurements**
ARPES measurements for the low energy electronic structure were performed at the PGM beamline in Synchrotron Radiation Center (SRC) in Wisconsin, and at the beamlines 4.0.3, 10.0.1 and 12.0.1 at the Advanced Light Source (ALS) in Berkeley California, equipped with high efficiency VG-Scienta R4000 or R8000 electron analyzers. Spin-resolved ARPES measurements were performed at the ESPRESSO endstation at HiSOR. Photoelectrons are excited by an unpolarized He-I$\alpha$ light (21.21 eV). The spin polarization is detected by state-of-the-art very low energy electron diffraction (VLEED) spin detectors utilizing preoxidized Fe(001)-p($1 \times 1$)-O targets [@Okuda_BL9B]. The two spin detectors are placed at an angle of 90$^\circ$ and are directly attached to a VG-Scienta R4000 hemispheric analyzer, enabling simultaneous spin-resolved ARPES measurements for all three spin components as well as high resolution spin integrated ARPES experiments. The energy and momentum resolution was better than 40 meV and $1\%$ of the surface BZ for spin-integrated ARPES measurements at the SRC and the ALS, and 80 meV and $3\%$ of the surface BZ for spin-resolved ARPES measurements at ESPRESSO endstation at HiSOR. Samples were cleaved *in situ* and measured at $10-80$ K in a vacuum better than $1\times10^{-10}$ torr. They were found to be very stable and without degradation for the typical measurement period of 20 hours.
**Theoretical calculations**
The first-principles calculations are based on the generalized gradient approximation (GGA) [@Perdew] using the projector augmented wave method [@Blochl; @Blochl_1] as implemented in the VASP package [@Kress; @Kress_1]. The experimental crystal structure was used [@crys_str]. The electronic structure calculations were performed over $4\times4\times2$ Monkhorst-Pack k-mesh with the spin-orbit coupling included self-consistently.
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**Acknowledgements** The work at Princeton and Princeton-led synchrotron X-ray-based measurements and the related theory at Northeastern University are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DE-FG-02-05ER46200, AC03-76SF00098 and DE-FG02-07ER46352). We thank J. Denlinger, S.-K. Mo and A. Fedorov for beamline assistance at the DOE supported Advanced Light Source (ALS-LBNL) in Berkeley. We also thank M. Bissen and M. Severson for beamline assistance at SRC, WI. M.Z.H. acknowledges Visiting Scientist support from LBNL, Princeton University and the A. P. Sloan Foundation.
**Author contributions** M.N., and S.-Y.X. performed the experiments with assistance from N.A., G.B., C.L., I. B. and M.Z.H.; M.N., and M.Z.H. performed data analysis, figure planning and draft preparation; R. S. and F.-C. C. provided the single-crystal samples and performed sample characterization; T.R.C., H.T.J., H.L., and A.B. carried out calculations; M.Z.H. was responsible for the conception and the overall direction, planning and integration among different research units.
**Additional information** Competing financial interests: The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to M.Z.H. (Email: mzhasan@princeton.edu).
{width="15cm"}
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![**Surface electronic structure of 2D and 3D Dirac fermions.** **a,** ARPES measured surface electronic structure dispersion map of Cd$_3$As$_2$ and its corresponding momentum distribution curves (MDCs). **b,** ARPES measured surface dispersion map of the prototype TI Bi$_2$Se$_3$ and its corresponding momentum distribution curves. Both spectra are measured with photon energy of 22 eV and at a sample temperature of 15 K. The black arrows show the ARPES intensity peaks in the MDC plots. **c and d** ARPES spectra of two Bi-based 3D Dirac semimetals, which are realized by fine tuning the chemical composition to the critical point of a topological phase transition between a normal insulator and a TI: **c,** TlBi(S$_{1-\delta}$Se$_\delta$)$_2$ ($\delta=0.5$) (see Xu $et$ $al.$ [@Suyang]), and (Bi$_{1-\delta}$In$_{\delta}$)$_2$Se$_3$ ($\delta=0.04$) (see Brahlek $et$ $al.$ [@Oh]). **d,**. Spectrum in panel **c** is measured with photon energy of 16 eV and spectrum in panel **d** is measured with photon energy of 41 eV. For the 2D topological surface Dirac cone in Bi$_2$Se$_3$, a distinct in-plane ($E_{\textrm{B}}-k_x$) dispersion is observed in ARPES, whereas for the 3D bulk Dirac cones in Cd$_3$As$_2$, TlBi(S$_{0.5}$Se$_{0.5}$)$_2$, and (Bi$_{0.96}$In$_{0.04}$)$_{2}$Se$_3$, a Dirac-cone-like intensity continuum is also observed. ](Fig4){width="16cm"}
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---
abstract: 'We study the influence of a thermal environment on a non-adiabatic spin-flip driving protocol of spin-orbit qubits. The driving protocol operates by moving the qubit, trapped in a harmonic potential, along a nanowire in the presence of a time-dependent spin-orbit interaction. We consider the harmonic degrees of freedom to be weakly coupled to a thermal bath. We find an analytical expression for the Floquet states and derive the Lindblad equation for a strongly non-adiabatically driven qubit. The Lindblad equation corrects the dynamics of an isolated qubit with Lamb shift terms and a dissipative behaviour. Using the Lindblad equation, the influence of a thermal environment on the spin-flip protocol is analysed.'
author:
- Brecht Donvil
- Lara Ulčakar
- Tomaž Rejec
- Anton Ramšak
bibliography:
- 'bibliography\_qubit\_list.bib'
title: 'Thermal effects on a nonadiabatic spin-flip protocol of spin-orbit qubits'
---
Introduction
============
Electron-spin qubits are promising candidates as building blocks of quantum computers. They can be realized in gated semiconductor devices based on quantum dots and quantum wires [@wolf01; @hanson07] and their state can be manipulated via magnetic fields [@dresselhaus55; @bychkov84] or the spin-orbit interaction, which is easily controlled with electrostatic gates [@stepanenko04; @flindt06; @coish06; @sanjose08; @golovach10; @bednarek08; @fan16; @gomez12; @pawlowski16; @pawlowski16b; @Pawlowski2017; @Pawlowski2018]. Such systems were already experimentally realized in various semiconducting devices [@nadjperge12; @nadjperge10; @fasth05; @fasth07; @shin12].
Recent studies proposed non-adiabatic protocols, where spin-qubit manipulation is achieved by translating a spin-qubit in one dimension [@cadez13; @cadez14; @Veszeli2018] in the presence of a time-dependent Rashba interaction [@nitta97; @liang12; @Yang2015]. Refs. [@cadez13; @cadez14] give exact analytical solutions for the time-dependent Schrödinger equation of such a driven spin-qubit, confined in a harmonic potential, and express the spin rotation in terms of the non-adiabatic non-Abelian Anandan phase [@anandan88]. While qubit transformations in linear systems are limited to spin rotations around a fixed axis, this limitation can be eliminated on a ring structure [@kregar16; @kregar16b].
Manipulations of quantum systems are inevitably accompanied by external noise, coming from fluctuating electric fields [created]{} by the piezoelectric phonons [@sanjose08; @sanjose06; @huang13; @echeveria13], for example, or due to phonon-mediated instabilities in molecular systems with phonon-assisted potential barriers [@mravlje06; @mravlje08]. Flying qubits could be carried by surface acoustic waves, where the noise can [arise]{} due to time dependence in the electron-electron interaction effects [@giavaras06; @rejec00; @jefferson06]. Recent related studies [@Gefen05; @lara17; @lara18; @Pyshkin2018; @Li2018; @Lu2018] considered the effects of additive noise present in the driving functions of the qubit.
In order to study effects of a thermal environment on the qubit manipulation, we aim to derive an effective dynamics for the spin-qubit from a full microscopic model of the qubit weakly interacting with a thermal bath. In the case of weak interactions, there exists a well-known approximation scheme to integrate out the dynamics of the bath and obtain the effective dynamics for the system, which is given by the Lindblad equation [@BreuerBook]. For adiabatic and weakly non-adiabatic driving of the system, the aforementioned weak-coupling scheme still holds and leads to a slightly modified Lindblad equation. However, for strongly non-adiabatic driving, which we consider in the present work, the necessary assumptions for this approximation scheme break down.
In a recent work[@Dann2018], the authors showed how a modified weak-coupling scheme can be performed in order to derive the Lindblad equation for an arbitrarily driven weakly-coupled system. In the case of periodic driving, this result reduces to the earlier derived Floquet-Lindblad form [@BlBu1991; @BPfloquet]. Although these results show how in principle a Lindblad equation can be derived, employing these methods essentially requires one to solve the evolution of the driven system in absence of the bath. In general, obtaining an analytic expression for the evolution of a driven system is non trivial and one has to resort to numerical approximations. In the present work, however, we consider periodic driving for which the analytic solution is known [@cadez13], giving us access to the exact Floquet-Lindblad equation.
The paper is structured as follows: in Section \[sec:model\] we introduce the model and the coupling to the thermal bath. In Section \[sec:Floquet\] the Floquet formalism and the corresponding exact solutions are given, which, in Section \[sec:Lind\], serves as a basis for an exact derivation of the Lindblad operators and the Floquet-Lindblad equation. The formalism is then applied to a simple example of a non-adiabatic driving and spin rotation in Section \[sec:example\]. In Section \[sec:conclusion\] we give conclusions and in Appendices A and B we present derivations of individual terms of the Floquet-Lindblad equation.
Model {#sec:model}
=====
Our system of interest is a spin-qubit represented as an electron confined in a quantum wire with a harmonic potential.[@cadez13; @cadez14] The centre of the trap, $\xi(t)$, can be arbitrarily translated along the wire by means of time-dependent external electric fields.
The spin-orbit Rashba interaction couples the electron’s spin with its orbital motion, resulting in the system Hamiltonian $$\label{H}
H(t)=\frac{p^{2}}{2m^{*}}+\frac{m^{*}\omega_0^{2}}{2}\big(x-\xi(t)\big)^{2}+\alpha(t)\,p\, \sigma_y,$$ where $m^{*}$ is the effective electron mass, $\omega_0$ is the frequency of the harmonic trap, and $p$ and $\sigma_y$ are the momentum and spin operators, respectively. The strength of the spin-orbit interaction $\alpha(t)$ is time dependent due to time dependent external electric fields and the spin rotation axis is fixed along the $y$-direction [@nadjperge12]. Throughout the paper we set $\hbar=1$ and initial time $t=0$. The exact time dependent solution of the Schr[ö]{}dinger equation corresponding to the Hamiltonian Eq. is given by[@cadez14] $$\label{eq:psi}
|\Psi_{}(t)\rangle=U(t,0)|\Psi_{}(0)\rangle,$$ where, in the time evolution operator $$\label{eq:ev}
U(t,0)=\mathcal{U}^\dagger(t)e^{-i H_{0} t}\mathcal{U}(0),$$ $H_0$ represents the time independent harmonic oscillator, i.e., Eq. (\[H\]) with $\xi(t)=\alpha(t)=0$, and
$$\begin{aligned}
\label{psi}\label{eq:transf}
\mathcal{U}^\dagger(t)&=&e^{-{i }(\varphi_0(t)+\varphi(t) \sigma_y)}\mathcal{A}_{\alpha}(t)\mathcal{X}_{\xi}(t),\\
\mathcal{A}_{\alpha}(t)&=&e^{-i\dot{a}_{c}(t)p\sigma_y/\omega_0^2}
e^{-im^{*}a_{c}(t){x}\sigma_y},\\
\mathcal{X}_{\xi}(t)&=&e^{im^{*}\big(x-x_{c}(t)\big)\dot{x}_{c}(t)}e^{-ix_{c}(t)p}.\end{aligned}$$
The unitary transformations $\mathcal{X}_{\xi}(t)$ and $\mathcal{A}_{\alpha}(t)$ are completely determined by the classical responses $x_c(t)$ and $a_c(t)$ to the driving. The responses solve the differential equations
\[xcac\] $$\begin{aligned}
\ddot{x}_c(t)+\omega_0^{2}x_{c}(t)&=&\omega_0^{2}\xi(t),\\
\ddot{a}_c(t)+\omega_0^{2}a_{c}(t)&=&\omega_0^{2}\alpha(t).\end{aligned}$$
$\varphi(t)=-m^{*}\int_{0}^{t}\dot{a}_{c}(\tau)\xi(\tau)\mathrm{d}\tau$ while the time dependent phase $\varphi_0(t)$, given in Ref. , is irrelevant for the time evolution operator Eq. (\[eq:ev\]) and will be omitted. Also note that the time evolution operator is invariant with respect to the gauge transformation $$\label{gauge}
\mathcal{U}^\dagger(t) \to e^{i(\delta_1+\delta_2 \sigma_y)} \mathcal{U}^\dagger(t),$$ where $\delta_1$ and $\delta_2$ are real constants.
The system described by the time dependent $H(t)$ is coupled to a bosonic thermal bath. The total Hamiltonian of the spin-qubit interacting with the bath is $$\label{eq:Hamiltonian}
H_{\rm tot}(t)=H(t)+H_B+H_I(t),$$ where the bath Hamiltonian $H_B$ represents a set of oscillators $$H_B=\sum_{\mathbf{k}} \omega_{\mathbf{k}}b^\dagger_{\mathbf{k}} b_{\mathbf{k}},$$ where $b^\dagger_{\mathbf{k}}$ ($ b_{\mathbf{k}}$) are creation (annihilation) operators and the oscillators have a linear dispersion relation $\omega_{\mathbf{k}}=c|\mathbf{k}|$, $\mathbf{k}\in \mathbb{R}^3$. We consider only states with energies below a cut-off energy $\omega_c$.
The spin-qubit is coupled to the bath through the interaction Hamiltonian $H_I(t)\propto\left(x-\xi(t)\right)\sum_{\mathbf{k}} x_{\mathbf{k}}$ which couples the respective position operators, $$\label{eq:intHam}
H_I(t)=g\sqrt{\frac{\pi c^3}{\omega_0 v}}\left(a^\dagger+a-\sqrt{2m^*\omega_0}\xi(t)\right)\sum_{\mathbf{k}}(b_{\mathbf{k}}+b^\dagger_{\mathbf{k}}),$$ where $g$ is a dimensionless coupling strength, $a^\dagger$ ($ a$) is a bosonic creation (annihilation) operator of the harmonic trap and $v$ is the volume of the bath.
Floquet Theory for quantum systems {#sec:Floquet}
==================================
Before deriving the Lindblad equation for the driven spin-qubit, it is instructive to briefly discuss the Floquet theory for periodically driven quantum systems. For systems with a Hamiltonian periodic in time, $H(t+T_d)=H(t)$, it is possible to describe the time-evolution in terms of periodic eigenvectors of the Schrödinger equation called the Floquet states [@Shirley; @Zeldovich]. The Floquet states $|\phi_q(t)\rangle$ form a complete basis and are defined as solutions of the eigenvalue problem
\[eq:Flset\] $$\begin{aligned}
&(H(t)-i\partial_t)|\phi_q(t)\rangle=\epsilon_q|\phi_q(t)\rangle,\\
&|\phi_q(t+T_d)\rangle=|\phi_q(t)\rangle,\end{aligned}$$
where $\epsilon_q$ is called a quasi-energy. The evolution of an arbitrary initial state $|\psi_0\rangle$ from an initial time $0$ to time $t$, expressed in terms of the Floquet states, is $$|\psi(t)\rangle=\sum_q e^{-i\epsilon_q t}|\phi_q(t)\rangle \langle\phi(0)|\psi_0\rangle.$$
In practice, solving Eq. proves to be non-trivial. However, with a gauge transformation $V(t)$ such that $$\label{eq:flfl}
G=V(t)H(t)V(t)^\dagger+i(\partial_t V(t))V(t)^\dagger$$ is time independent, the Floquet states and quasi-energies can be found in terms of eigenvectors and eigenvalues of $G$. Let $|q\rangle$ be an eigenvector of $G$ with the eigenvalue $\epsilon_q$. One can check that the state $$\label{eq:fl}
|\phi_q(t)\rangle= V(t)^\dagger |q\rangle$$ is a Floquet state with the quasi-energy $\epsilon_q$. Note that $\mathcal{U}(t)$, as defined in Eq. , has exactly the property Eq. with $G=H_0$. Therefore, the Floquet states of the driven qubit are $|\phi_q(t)\rangle= \mathcal{U}^\dagger(t) |\psi_q\rangle$, where $|\psi_q\rangle$ is an eigenstate of $H_0$ with the energy $\epsilon_q=\omega_0(q+\frac{1}{2})$.
Derivation of the Lindblad equation {#sec:Lind}
===================================
The reduced density matrix of the spin-qubit at time $t$ in the Schrödinger picture is $$\bar{\rho}(t)=\operatorname{tr}_B\left( U_{\textrm{tot}}(t,0)\rho_{\textrm{tot}}(0)U_{\textrm{tot}}^\dagger(t,0)\right)$$ where $\mathrm{tr}_B$ denotes the trace over bath degrees of freedom, $U_\textrm{tot}(t,0)$ the time evolution operator of the whole system, and $$\rho_\textrm{tot}(0)=\rho(0)\otimes \frac{e^{-\beta H_B}}{\operatorname{tr}_B e^{-\beta H_B}},
\label{rhotot}$$ is an initially separable density matrix consisting of the qubit in the state $\rho(0)$ and the bath at the inverse temperature $\beta=\frac{1}{k_BT}$. In the interaction picture, $$\rho(t)=U^\dagger(t,0)\bar{\rho}(t)U(t,0)$$ where $U(t,0)$ is the time evolution operator of the qubit, the Floquet-Lindblad equation for the qubit interacting with the bath via Eq. is of the form[@BreuerBook] $$\begin{aligned}
\label{eq:lind}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\rho(t)=-{i}\big[H_{LS},\rho(t)\big]+\mathcal{D}\big(\rho(t)\big).\end{aligned}$$
In what follows we assume that the driving frequency $\omega_d=\frac{2\pi}{T_d}$ is $\omega_d=\frac{\omega_0}{n_d}$ with $n_d\in\mathbb{N}$, as appropriate for the spin-flip protocol studied in this paper. The first term on the right hand side of Eq. (\[eq:lind\]) contains the Lamb shift Hamiltonian $$\label{Lambshift}
H_{LS}=\sum_{n\in \mathbb{Z}} S(n\omega_d) A_n^\dagger A_n,$$ where $A_n$ are the Lindblad operators (to be defined below) and $$S(\omega)=\frac{g^2}{2\pi\omega_0}{\cal P}\!\!\int_0^{\omega_c} {\mathop{}\!\mathrm{d}}\omega' \omega'^2\left(\frac{1+N(\omega')}{\omega-\omega'}+\frac{N(\omega')}{\omega+\omega'}\right)$$ with $\cal P $ denoting the principal value and $$N(\omega)=\frac{1}{\exp (\beta\omega)-1}$$ being the Bose occupation numbers of the bath degrees of freedom. Throughout the paper we express the temperature in terms of the average occupation of the system oscillator $n_T=N(\omega_0)$. In Appendix \[secapp:ladder1\] we show the Lamb shift Hamiltonian in an explicit form.
The second term on the right hand side of Eq. is the dissipator term $$\label{eq:diss}
\mathcal{D}\big(\rho(t)\big)=\sum_{n\in \mathbb{Z}} \gamma(n\omega_d)\left(A_n\rho(t)A_n^\dagger -\tfrac{1}{2}\{A_n^\dagger A_n,\rho(t)\}\right),$$ with rates $$\gamma(\omega)=\begin{cases}g^2\frac{\omega^2}{\omega_0}(1+N(\omega)), &\omega\ge0,\\g^2\frac{\omega^2}{\omega_0}N(|\omega|),&\omega<0.\end{cases}$$
The Lindblad operators are obtained by finding $A_n$ such that [@BPfloquet; @BreuerBook] $$\begin{aligned}
\label{eq:ladders}
U^\dagger(t,0)\left(a^\dagger+a-\sqrt{2m^*\omega_0}\xi(t)\right)U(t,0)\nonumber\\
=\sum_{n\in\mathbb{Z}} A_n e^{-i n\omega_d t}.\end{aligned}$$ We present the actual calculation of the Lindblad operators in Appendix \[secapp:ladder\]. The result for $n\ge0$ is $$\begin{aligned}
\label{eq:ladder1}
A_n&=&\delta_{n,n_d}\,a\nonumber
+\sqrt{{ 2m^* \omega_0}}\bigg(\hat{x}_{c,n}-\hat{\xi}_n+\frac{\hat{\dot{a}}_{c,n}}{\omega^2_0}\sigma_y+
\nonumber\\
& &+\delta_{n,n_d}\frac{1}{2}\left(-\frac{\dot{a}_c(0)}{\omega_0^2}+i\frac{a_c(0)}{\omega_0}\right)\sigma_y+
\nonumber\\
& &+\delta_{n,n_d}\frac{1}{2}\left(-x_c(0)-i\frac{\dot{x}_c(0)}{\omega_0}\right) \bigg),\end{aligned}$$ and $A_{-n}=A_n^\dagger$. Here by $\hat{f}_n$ we denote a Fourier component of the function $f$. The Lindblad operators are completely determined by the solutions of the response Eqs. .
We can greatly simplify the form of the dissipator Eq. by absorbing the constant terms in the Lindblad operators into the Lamb shift. This procedure is outlined in Appendix \[secapp:ladder1\]. Let us define a rate $$\bar{\gamma}=\frac{2 m^* }{\omega_0^3}\sum_{\substack{n\in \mathbb{Z}\\ n\ne\pm n_d}} \gamma(n\omega_d)|\hat{\dot{a}}_{c,n}|^2,$$ and a new Lindblad operator $$\bar{A}=a+\sqrt{\frac{m^* }{2\omega_0}}\bigg(2\frac{\hat{\dot{a}}_{c,n_d}}{\omega_0}-\frac{\dot{a}_c(0)}{\omega_0}+ia_c(0)\bigg)\sigma_y.$$ With these definitions, Eq. reduces to a much simpler form $$\begin{aligned}
\label{eq:Lindf}
\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\rho(t)=&-{i}\left[\bar{H}_{LS},\rho(t)\right]+\bar{\gamma} \left(\sigma_y\rho(t)\sigma_y-\rho(t)\right)+
\nonumber\\&+ \gamma(\omega_0)\left(\bar{A}\rho(t)\bar{A}^\dagger-\tfrac{1}{2}\left\{\bar{A}^\dagger\bar{A},\rho(t)\right\}\right)+\nonumber\\&+ \gamma(-\omega_0)\left(\bar{A}^\dagger\rho(t)\bar{A}-\tfrac{1}{2}\left\{\bar{A}\bar{A}^\dagger,\rho(t)\right\}\right).\end{aligned}$$ The redefined Lamb shift Hamiltonian $\bar{H}_{LS}$ is shown explicitly in Appendix \[secapp:ladder1\]. The dissipator in the above equation consists of two types of terms: the term proportional to $\bar{\gamma}$ is a dephasing term and causes a decay of spin size. The other terms lead to thermal activation in the oscillator component.
Example {#sec:example}
=======
At the initial time $t=0$, let the electron be in the ground state manifold of $H(0)$, spanned by a Kramers doublet. In particular, we choose the qubit to be in the spin-up state, i.e., with $\langle\Psi(0)| \sigma_x |\Psi(0)\rangle=\langle\Psi(0)| \sigma_y |\Psi(0)\rangle=0$. Using Eq. (\[gauge\]), such a state can be constructed as $$|\Psi(0)\rangle=e^{i m^*\big({a}_{c}(0){x}_{c}(0)+\frac{\dot{a}_{c}(0)\dot{x}_{c}(0)}{\omega_0^2}\big)\sigma_y}
\mathcal{U}^\dagger(0)|\psi_{0}\rangle|\chi_{\uparrow}\rangle.$$ Here $|\psi_{0}\rangle$ is the ground state of the harmonic oscillator Hamiltonian $H_0$ and $|\chi_{\uparrow}\rangle$ is the up-state spinor in the eigenbasis of $\sigma_z$. The initial density matrix of the qubit Eq. (\[rhotot\]) is $\rho(0)=|\Psi(0)\rangle\langle\Psi(0)|$.
As a simple example of the theory outlined in the previous section, we consider periodic spin transformations following an elliptic path in the parametric space $[\xi(t),\alpha(t)]$ with
\[driving\] $$\begin{aligned}
&\xi(t) = \xi_0 \cos\frac{\omega_0 t}{2},\\
&\alpha(t)=\alpha_0-\alpha_0 \sin\frac{\omega_0 t}{2},
\end{aligned}$$
i.e, $n_d=2$. This choice of driving, together with initial conditions $x_c(0)=\xi(0)$, $a_c(0)=\alpha(0)$ and $\dot{x}_c(0)=0$, $\dot{a}_c(0)=0$ for differential equations , leads to classical responses
\[eq:exresp\] $$\begin{aligned}
x_c(t)&=&\frac{\xi_0}{3} \left(4 \cos\frac{ \omega_0 t}{2}-\cos\omega_0 t \right),\\
a_c(t)&=&\alpha_0-\frac{\alpha_0}{3} \left(4 \sin\frac{\omega_0 t}{2} -2\sin \omega_0 t\right).
\label{response}
\end{aligned}$$
The driving guarantees that after a completed cycle the state $|\Psi(T_d)\rangle$ returns to the ground state – with spin rotated around the $y$-axis by the Anandan quantum phase $\phi_A=2\varphi(T_d)$ determined solely by the contour $\mathcal{C}_{\xi}$ in the parametric space $[\xi(t),a_c(t)]$ or, equivalently, $\mathcal{C}_{\alpha}$ in the space $[x_c(t),\alpha(t)]$, $$\label{anandan}
\phi_A=2m^{*}\oint_{\mathcal{C}_{\xi}} a_{c}[\xi]{\rm d}\xi=2m^{*}\oint_{\mathcal{C}_{\alpha}}x_{c}[\alpha]{\rm d}\alpha.$$ The corresponding contour ${\cal{C}}_{\alpha}$ is shown in Fig. \[fig:Bloch\](a) with a full black line and the dashed line indicates the driving protocol $[\xi(t),\alpha(t)]$. Note that the area of the shaded region equals $\phi_A/(2m^*)$.
During the motion of the system, the electron’s spatial wave function is a superposition containing also the oscillator’s excited states and the spin of the electron is rotated around the $y$-axis by the angle $\phi(t)$. After a completed driving cycle the electron’s spin has, according to Eq. (\[anandan\]), rotated by the angle $\phi(T_d)=\phi_A =\frac{8}{3} \pi m^*\alpha_0\xi_0$.
In the upcoming numerical studies we take $\gamma/\omega_0\lesssim \frac{1}{10}$. This is to ensure that the effect of the bath remains a perturbation to the free dynamics of the spin-qubit, which is the underlying assumption for the Lindblad equation Eq. . We also fix the Rashba coupling $\alpha_0=\frac{3}{16} (\omega_0/m^*)^{1/2}$ and $\xi_0=(m^*\omega_0)^{-1/2}$ such that $\phi_A={\pi \over 2}$, i.e., two driving cycles are needed for a spin-flip. The cut-off frequency is set to $\omega_c=2 \omega_0$ and all numerical calculations were carried out using the QuTiP framework [@qutip12; @qutip13].
Spatial position and spin properties
------------------------------------
In Fig. \[fig:Bloch\](b) we show the time dependence of the expectation value of the spin. Since we are considering rotations around the $y$-axis, $\langle \sigma_y \rangle=0$, i.e., the expectation value is confined to the $x$-$z$ plane and [*within*]{} the Bloch sphere. Bullets represent the values at equal time steps for the total time duration of two cycles. Note that in the absence of the interaction, $g=0$, the spin transformation is exactly one spin-flip (black). The red bullets represent the corresponding result for $g=0.2$, $n_T=1$ and $\omega_c=2\omega_0$. The orange square indicates the spin at $t=T_d$.
![(a) Contour ${\cal{C}}_{\alpha}$ of the driving and the response in the parametric space spanned by the spatial position and the spin-orbit driving. The red bullet represents the starting point and the orange square the response $[\langle x\rangle,\alpha]$ after one cycle for $g=0.3$, $n_T=1$ and $\omega_c=2\omega_0$. (b) Two cycles of the qubit rotation within the Bloch sphere: the black bullets and the black arrow represent the noninteracting result, $g=0$, with spin rotation around the $y$-axis for an angle $\phi=\pi$. The red bullets and the red arrow show the rotation for $g=0.2$, $n_T=1$ and $\omega_c=2\omega_0$. Bullets show positions at equal time intervals.[]{data-label="fig:Bloch"}](Fig1.pdf){width="5"}
The interaction with the bath influences the spin-flip protocol, namely: the angle of the spin rotation, the size of the spin, the expectation value of the position $\langle x \rangle$ and the oscillator part of the wave function, which does not return to the ground state at the end of the transformation. Below we discuss these effects.
The influence of the interaction with the bath on the expected position of the oscillator $\langle x\rangle$ can be observed from Figs. \[fig:Bloch\](a) and \[fig:x\](a) by comparing the result for the non-interacting pure dynamics (black line) to the interacting result (red line). In the non-interacting case, the expected position is equal to the classical response, $\langle x\rangle=x_c(t)$, as can easily be shown from Eq. . In the presence of the bath $\langle x\rangle$ deviates from $x_c(t)$ and is shifted from the starting point (red bullet) after one cycle (orange square). Fig. \[fig:x\](a) shows that the shift of $\langle x\rangle$ to larger values is increased after the second cycle, $t=2T_d$, and is further increased until the driven system reaches a steady state. This is in agreement with the classical result for a driven weakly damped harmonic oscillator, $x_c \to {4 \over 3}\xi_0$, after a large number of cycles \[the first term in Eq. (\[eq:exresp\]a)\]. The approach to the steady state can be observed in Fig. \[fig:x\](b) which shows $\langle x\rangle$ in terms of the number of driving cycles (needed to half-flip the spin), for different values of the coupling strength $g$ at $T=0$. Due to the low frequencies introduced by the Lamb shift in Eq. , $\langle x\rangle$ exhibits slow damped oscillations. The inset of Fig. \[fig:x\](b) shows $\langle x\rangle$ after one cycle as a function of $g$ for various values of the cut-off energy $\omega_c$. Note the absence of oscillations for $\omega_c=1.16\, \omega_0$, dotted line. At this cut-off energy $S(\omega_0)+S(-\omega_0)\to 0$ and consequently spatial parts of $H_{LS}\to 0$, see Eqs. (\[HLS\]) and (\[Spm\]), which leads to results qualitatively close to the classical result for a damped oscillator. At higher cut-off energies quantum terms in $H_{LS}$ leads to a much richer dynamics. The orange squares and circles indicate the spatial position after the first cycle, Figs. \[fig:Bloch\](a) and \[fig:x\]. The shift of the position and consequently the changed contour in the parametric space affect the spin behaviour, as will be discussed below.
![(a) Driving $\xi(t)$ as a function of time for two cycles (dotted line) and the expected position of the electron $\langle x \rangle$ in the case of no interaction (black) and for $g=0.3$, $n_T=1$ (red). The orange square corresponds to the square in Fig. \[fig:Bloch\]. (b) $\langle x \rangle$ as a function of the number of qubit transformation cycles, for different values of the coupling $g$ at $T=0$. Inset: $\langle x \rangle$ after one cycle as a function of $g$ for $\omega_c/\omega_0=1.16$, 2, and 3. The orange circle corresponds to $g=0.3$.[]{data-label="fig:x"}](Fig2.pdf){width="6.5"}
The spin response of the system is analysed in Fig. \[fig:spin\]. First we concentrate on the size of the spin, which for $g=0$ does not depend on the position coordinates regardless of the driving, as can be checked by the application of the unitary transformation Eq. (\[psi\]) and a direct evaluation of the spin expectation values. The exact result is $$\label{eq:sigmana}
\sqrt{\langle \sigma_x \rangle^2+\langle \sigma_z \rangle^2}=e^{- {m^*} \big(a^2_c(t)/\omega_0 +{\dot a}_c^2(t)/\omega_0^3\big)},$$ which reduces to $e^{- {m^*} \alpha(t)^2/\omega_0 }$ for the case of slow (adiabatic) driving. In the case of a time independent Rashba coupling this is a known result for Kramers doublets [@cadez13]. Fig. \[fig:spin\](a) shows the size of the spin during the spin-flip protocol in absence of the environment (full black line), the zero temperature result at $g=0.3$ (red line) and the result at a finite temperature $n_T=1$ (dashed line). The orange square indicates the result after the first cycle, as in previous figures. Effects of the coupling to the bath are more pronounced at an elevated temperature.
![(a) Time dependence of the spin size $\sqrt{\langle \sigma_z \rangle^2+\langle \sigma_x \rangle^2}$ during two cycles for $g=0$ (black line), $g=0.3$, $T=0$ (red) and $g=0.3$, $n_T=1$ (dashed). (b) Angle $\phi$ as a function of time. Inset: Angle error $\delta \phi=\phi-{\pi \over 2}$ at the final time of one cycle $t=T_d$ as a function of $g$. In (a) and (b) the red bullets and the orange squares indicate the one cycle starting and final values, correspondingly.[]{data-label="fig:spin"}](Fig3.pdf){width="8.5"}
The spin rotation, the most relevant property for the qubit manipulation, is shown in Fig. \[fig:spin\](b) for $g=0$ (black) and $g=0.3$, $n_T=0$ (red). Due to the interaction with the bath the rotation angle is slightly increased with respect to the non-interacting value $\phi={\pi \over 2}$ \[the orange square marks the value after one completed cycle\]. The deviation $\delta\phi=\phi-{\pi \over 2}$ is additionally presented as a function of $g$ in the inset. Comparing the finite temperature result $n_T=1$ (blue dashed line) with $n_T=0$ (full red) shows that the effect is further increased at finite temperatures.
![(a) Deviation of the angle $\phi$ from $\frac{\pi}{2}$ after $N_{\rm c}$ cycles using renormalised Rashba coupling $\frac{\alpha_0}{N_c}$ for various $g$ at $T=0$. (b) Number of cycles $N_g$ from (a) as a function of $g$. Note the orange symbol relating (a) and (b). The dotted line represents the approximation $N_g\approx 0.36 \,g^{-2}$.[]{data-label="fig:angle"}](Fig4.pdf){width="8"}
Error analysis and fidelity
---------------------------
Deviations of $\phi$ from the target value ${\pi \over 2}$ are additionally explored and presented in Fig. \[fig:angle\]. Here we performed $N_{\rm c}$ consequent cycles, with renormalised value of the Rashba coupling $\alpha_0(N_{\rm c}) = \alpha_0 / N_{\rm c}$ such that after $N_{\rm c}$ cycles at $g=0$ exactly one half spin-flip is performed. Fig. \[fig:angle\](a) shows that initially the error decreases with an increasing number of cycles. For a large number of cycles, however, the error approaches a constant value. The typical transition number of cycles $N_g$ is for $g=0.3$ indicated by an orange square and an arrow. In Fig. \[fig:angle\](b), $N_g$ is shown as a function of $g$. It clearly exhibits a $g^{-2}$ scaling as expected for the typical relaxation time scale (measured by the number of cycles) for error generating Lamb shift Hamiltonian and Lindblad terms, Eq. (\[eq:lind\]).
Let us discuss the effects of the environment also in the framework of fidelity of the spin-qubit transformation. In particular, we consider the Uhlman-Josza fidelity $$F=\operatorname{tr}\sqrt{\sqrt{\rho_0}\rho_g\sqrt{\rho_0}},$$ where $\rho_0$ and $\rho_g$ represent the density matrix for the non-interacting and the interacting regimes of the model, respectively. At the initial time $t=0$ (or in the absence of interaction) $F=1$, but with increasing time $F$ progressively diminishes due to error generating processes in the Lindblad equation. Fig. \[fig:fidelity\] shows the fidelity calculated at the end of each cycle (red dots) as a function of the number of cycles $N_{\rm c}$ for $g=0.2$ and $T=0$. Here, the Rashba coupling $\alpha_0$ is independent of the number of cycles $N_{\rm c}$.
The structure of $F$ exhibits different short time and long time behaviours. To analyse these behaviours, we make a simple estimate of fidelity at the end of the driving cycle, $$F\approx |\langle\psi_0|\psi_{\delta x}\rangle|^2 |\langle\chi_0|\chi_{\delta \phi}\rangle|^2,
\label{F}$$ where $|\psi_0\rangle$ is the target final harmonic oscillator ground state and $|\psi_{\delta x}\rangle$ is the ground state of the harmonic oscillator with the potential displaced by $\langle x\rangle-\xi_0=\delta x\xi_0$, giving $$\label{eq:ov1}
|\langle\psi_0|\psi_{\delta x}\rangle|^2=e^{-{1 \over 2}\delta x^2}.$$ Similarly, $|\chi_{0}\rangle$ is the target final spin state and $|\chi_{\delta \phi}\rangle$ is the spin state with the angle $\delta \phi=\phi-N_{\rm c} {\pi \over 2}$ off from the target state, so that $$\label{eq:ov2}
|\langle\chi_0|\chi_{\delta \phi}\rangle|^2=\cos^2\frac{\delta \phi}{2}.$$ The fidelity (red line and bullets) and its estimate Eq. (black), shown in Fig. \[fig:fidelity\] for $g=0.2$, behave qualitatively similar. At zero temperature considered here the two overlaps Eqs. and represent major sources of the fidelity reduction. The remaining contributions, much more pronounced at finite temperatures (not shown), are mainly due to the fact that the system is not in a pure state and $F$ simply cannot be expressed solely in terms of wave function overlaps.
The separate curves $|\langle\psi_0|\psi_{\delta x}\rangle|^2$ (green dots) and $\langle\chi_0|\chi_{\delta \phi}\rangle|^2$ (blue dots) allow us to analyse the short and long-time behaviour of the fidelity. For small number of cycles, $N_{\rm c} \lesssim 10 $, the fidelity is mainly reduced due to the shift of the electron position $\delta x>0$ after a completed cycle. This affects the fidelity due to the reduced overlap of the target spatial wave function and the actual result in the presence of interaction. As discussed before, $\delta x$ exhibits oscillatory behaviour which is damped out (see Fig. \[fig:x\]) and $\delta x \to {1 \over 3}$ at larger times $t \gtrsim 10 T_d$, hence the overlap approaches $|\langle\psi_0|\psi_{\delta x}\rangle|^2=e^{-1/18} = 0.95$.
The spin contribution to the fidelity reduction due to the error in the angle of rotation, $\cos^2\frac{\delta \phi}{2}$, is at short times also generated due to oscillations of the orbit $[\langle x \rangle,\alpha]$ and the corresponding deviations from the noninteracting contour ${\cal C}_\alpha$, see Fig. \[fig:Bloch\](a). For $N_{\rm c} \gtrsim 10 $ the orbit in the parametric space progressively relaxes to the steady state contour and then the error $\delta\phi$ increases monotonously, similar to the recent study of adiabatic non-Abelian dephasing [@gefen19a; @gefen19b]. There are several competing error-generating sources also in the Lamb shift Hamiltonian $H_{LS}$, Eq. (\[HLS\]): the spin rotation terms are of the Rashba coupling form $\propto p\, \sigma_y$, a space dependent magnetic field $\propto x\,\sigma_y$ and a constant magnetic field term $\propto \sigma_y$. Dissipative terms in the Lindblad equation are another important source of the fidelity reduction at larger times. They additionally contribute to spin errors and most importantly, to the size of the spin, shown in Fig. \[fig:spin\](a). At elevated temperatures dephasing effects discussed above amplify due to the increase of coupling factors $S(\omega)$ and $\gamma(\omega)$.
![Uhlmann-Josza fidelity $F$ as a function of the number of cycles for $g=0.2$ and $T=0$ (red). The green squares represent $|\langle\psi_0|\psi_{\delta x}\rangle|^2$, the overlap of two ground states of a harmonic oscillator, one at the origin and the other displaced by $\delta x$. The blue squares represent the overlap $|\langle \chi_0|\chi_{\delta\phi} \rangle |^2$ of two spin states with relative spin angle $\delta \phi$. The black bullets represent the combined overlaps as an estimate of the fidelity.[]{data-label="fig:fidelity"}](Fig5.pdf){width="6"}
Conclusions {#sec:conclusion}
===========
In this paper we have studied the effects of a thermal environment on a non-adiabatic spin-flip protocol. The protocol is based on confining an electron in a harmonic trap and simultaneously manipulating the position of the centre of the trap and the Rashba interaction. For arbitrary driving protocols, assuming weak coupling between the system and its environment, the effective dynamics of the system can be obtained in terms of the Lindblad equation [@Dann2018]. In the case of periodic driving it reduces to the Floquet-Lindblad equation[@BlBu1991; @BPfloquet]. However, to obtain the explicit form of the Floquet-Lindblad equation one needs to solve the closed system dynamics, which in the case of periodic driving translates to solving the eigenvalue problem Eqs. . The protocol we are considering is exactly solvable[@cadez13], and the resulting Floquet-Lindblad equation is fully determined by two classical responses Eqs. to the driving.
Access to the Floquet-Lindblad equation allows us to study the effects of the thermal environment on the driving protocol. The Lindblad equation modifies the free spin-qubit Hamiltonian by introducing low frequencies, an effect called the Lamb shift. Additionally, there are dissipative terms giving rise to spin-dephasing effects as well as thermal activation of the oscillator.
As an example we consider a specific driving protocol with classical response functions Eq. . Interestingly, we find that the low frequencies and other terms introduced by the Lamb shift Hamiltonian result in an optimal number of driving cycles to complete the protocol. Figure \[fig:angle\] displays this result, where depending on the coupling strength to the bath, the optimal number of driving cycles allows us to minimise the error in the final angle of the spin. The Lamb shift Hamiltonian is a time independent shifted harmonic oscillator with spin dependent terms, and the exact solution is a coherent state analogous to Eq. (\[eq:psi\]). This allows for an analytical analysis and a deeper understanding of the decoherence dynamics, enabling further possibilities of protocol optimisation at zero and finite temperatures.
The exact study of the considered spin-qubit interacting with the environment can be extended to any driving protocol and any kind of a bath as long as the interaction is weak.
Calculation of the Lindblad operators {#secapp:ladder}
=====================================
The time evolution operator Eq. of the free spin-qubit consists of three terms. We calculate $U^\dagger (t,0) (a+a^\dagger-\sqrt{{2m^*\omega_0}}\xi(t) )U(t,0)$ by first applying $\mathcal{U}^\dagger(t)$, resulting in the expression $$a+a^\dagger+\sqrt{{2m^* \omega_0 }}\left(x_c(t)-\xi(t)+\frac{\dot{a}_c(t)}{\omega^2_0}\sigma_y\right).$$ Applying the time independent harmonic oscillator term $e^{-iH_0 t}$ transforms this into $$a e^{-i\omega_0 t}+a^\dagger e^{i\omega_0 t}+\sqrt{{2m^* \omega_0 }}\left(x_c(t)-\xi(t)+\frac{\dot{a}_c(t)}{\omega^2_0}\sigma_y\right).$$ Finally, we apply $\mathcal{U}(0)$ and thus obtain $$\begin{split}
&a e^{-i\omega_0 t}+a^\dagger e^{i\omega_0 t}+\sqrt{{2m^* \omega_0 }}\bigg(x_c(t)-\xi(t)+\frac{\dot{a}_c(t)}{\omega^2_0}\sigma_y-\\
&-x_c(0)\cos(\omega_0t)-\frac{\dot{a}_c(0)}{\omega_0^2}\cos(\omega_0t)\sigma_y+\frac{a_c(0)}{\omega_0}\sin(\omega_0t)\sigma_y-\frac{\dot{x}_c(0)}{\omega_0}\sin(\omega_0t)\bigg).
\end{split}$$ Expressing $x_c(t)$ and $a_c(t)$ in terms of their Fourier components $\hat{x}_{c,n}$ and $\hat{a}_{c,n}$ results in the form Eq. , $$\begin{split}
&a e^{-i\omega_0 t}+a^\dagger e^{i\omega_0 t}+\sqrt{{2m^* \omega_0 }}\sum_{\substack{n\in \mathbb{Z}\\n\ne\pm n_d}}
\left(\hat{x}_{c,n}-\hat{\xi}_n+\frac{\hat{\dot{a}}_{c,n}}{\omega^2_0}\sigma_y\right)e^{-in \omega_d t}+\\
&+\sqrt{{\frac{m^* \omega_0}{2} }}\left(-x_c(0)-i\frac{\dot{x}_c(0)}{\omega_0}-\frac{\dot{a}_c(0)}{\omega_0^2}\sigma_y+i\frac{a_c(0)}{\omega_0}\sigma_y\right){e^{-i\omega_0 t}}+\\
&+\sqrt{{\frac{m^* \omega_0}{2} }}\left(-x_c(0)+i\frac{\dot{x}_c(0)}{\omega_0}-\frac{\dot{a}_c(0)}{\omega_0^2}\sigma_y-i\frac{a_c(0)}{\omega_0}\sigma_y\right){e^{i\omega_0 t}},
\end{split}$$ from which we read the jump operators of Eq. . The Fourier components $\hat {f}_n$ are defined by $f(t)=\sum_{n\in\mathbb{Z}} {\hat f}_n e^{-i n \omega_d t}$ and are interconnected by useful relations $\hat{\dot{a}}_{c,n}=-i {n \over n_d} \omega_0 \hat{a}_{c,n}$ and, for $n\ne\pm n_d$, $\hat{a}_{c,n}={n_d^2 /( {n_d^2-n^2})}\hat{\alpha}_n$ and $\hat{x}_{c,n}-\hat{\xi}_n={n^2 /( {n_d^2-n^2})}\hat{\xi}_n$.
Lamb shift Hamiltonian {#secapp:ladder1}
======================
The Lindblad equation is invariant under inhomogeneous transformations
\[B1\] $$\begin{aligned}
& A_n\rightarrow \bar{A}_n=A_n+z_n,\\
& H_{LS}\rightarrow \bar{H}_{LS}=H_{LS} +\frac{1}{2i}\sum_{n\in\mathbb{Z}} \gamma(n\omega_d)\left(z_n^\ast A_n-z_nA_n^\dagger\right)+c\end{aligned}$$
where $z_n\in \mathbb{C}$ and $c\in \mathbb{R}$. The Lindblad operators Eq. consist of two parts, one proportional to $\sigma_y$, $a$ and $a^\dagger$ and the other proportional to the identity. The latter can be eliminated using the above transformation, leading to Eq. with the transformed Lamb shift Hamiltonian $$\label{hlsbar}
\bar{H}_{LS}=H_{LS}
-\frac{1}{2i}g^2\omega_0\sqrt{2 m^* \omega_0}\left(\big(\hat{x}_{c,-n_d}-\hat{\xi}_{-n_d}-\frac{1}{2}x_c(0)+i{\dot{x}_c(0) \over 2\omega_0}\big)A_{n_d} - \mathrm{h.c.}\right)+\bar{B}\sigma_y$$ where $$\bar{B}=2g^2 m^*\sum_{\substack{n\in \mathbb{N}\\ n\ne n_d}}
{n^5 \omega_0\over n_d\left( {n_d^2}-n^2\right)^2} \mathrm{Re}\left\{ \hat{\alpha}_n \hat{\xi}_n^\ast\right\}.
$$
In the particular case of an even periodic driving function $\xi(t)$ and an odd periodic driving function $\alpha(t)$, i.e., when $\xi(-t)=\xi(t)$ and $\alpha(-t)-\alpha(0)=-\left(\alpha(t)-\alpha(0)\right)$ \[as is the case in the example studied in Section \[sec:example\]\], $x_c(-t)=x_c(t)$ and $\dot{a}_c(-t)=\dot{a}_c(t)$ and the Lamb shift Hamiltonian Eq. (\[Lambshift\]) simplifies. It can be represented as a shifted harmonic oscillator in the presence of the Rashba interaction and an inhomogeneous magnetic field,
\[HLS\] $$\begin{aligned}
H_{LS}&=&\frac{p^2}{2m_{LS}} + \frac{m_{LS}\omega_{LS}^2}{2}(x-x_{LS})^2
+\big(\alpha_{LS}p +b_{LS}x+B_{LS}\big)\sigma_y,\\
B_{LS}&=&-b_{LS} x_{LS}-
4\zeta m_{LS}\sum_{\substack{n\in \mathbb{Z}\\n\ne\pm n_d}}
{n_d n^3 \over ({n_d^2-n^2})^2}S(n\omega_d)
\;i\hat{{\alpha}}_n\hat{\xi}_n.\end{aligned}$$
Here $\zeta=\left(S(\omega_0)+S(-\omega_0)\right)/\omega_0$, $m_{LS}=\zeta^{-1}m^*$, $\omega_{LS}=\zeta \omega_0$, $x_{LS}=x_c(0)-2\big(\hat{x}_{c,n_d}-\hat{\xi}_{n_d}\big)$, $\alpha_{LS}=\zeta a_c(0)$, $b_{LS}=\zeta^2m_{LS}\big(2\hat{\dot{a}}_{c,n_d}-\dot{a}_c(0)\big)$ and, at $T=0$, $$\label{Spm}
\zeta=-\frac{g^2}{2\pi}\left(\frac{\omega_c (\omega_c+2\omega_0)}{2\omega_0^2}+\log \frac{\left|\omega_c-\omega_0\right|}{\omega_0}\right),$$ which is zero at $\omega_c=1.16\,\omega_0$ and at $\omega_c=2\omega_0$, as used throughout the paper, $\zeta=-{1 \over \pi}g^2$. Note that the resonant frequency components of driving, $\hat{\xi}_{\pm n_d}$, should vanish if the steady state regime of driving and response is to be studied. Note also that $\hat{\xi}_{\pm n_d}=0$ does not imply $\hat{x}_{c,\pm n_d}=0$, respectively. Applying the transformation Eqs. (\[B1\]) results in $\bar{B}=0$ and $$\bar{H}_{LS}=H_{LS}+\frac{1}{2}g^2\omega_0x_{LS}\left(p+m_{LS}\alpha_{LS}\sigma_y\right).$$
|
---
abstract: 'We study the probability distribution of a current flowing through a diffusive system connected to a pair of reservoirs at its two ends. Sufficient conditions for the occurrence of a host of possible phase transitions both in and out of equilibrium are derived. These transitions manifest themselves as singularities in the large deviation function, resulting in enhanced current fluctuations. Microscopic models which implement each of the scenarios are presented, with possible experimental realizations. Depending on the model, the singularity is associated either with a particle–hole symmetry breaking, which leads to a continuous transition, or in the absence of the symmetry with a first-order phase transition. An exact Landau theory which captures the different singular behaviors is derived.'
author:
- Yongjoo Baek
- Yariv Kafri
- Vivien Lecomte
bibliography:
- 'symmetry\_breaking\_BKL.bib'
title: |
Dynamical symmetry breaking and phase transitions\
in driven diffusive systems
---
In recent years there has been much activity focused on understanding probability distributions in systems which are far from thermal equilibrium. In particular, the probability of observing a current flowing between two reservoirs, through an interacting channel, was studied in many works for both quantum [@levitov1993pis; @levitov_electron_1996; @pilgram_stochastic_2003; @jordan_fluctuation_2004; @flindt_trajectory_2013] (in the context of ‘full counting statistics’) and classical systems [@derrida_exact_1998; @derrida_universal_1999; @derrida_current_2004; @bodineau_current_2004; @bertini_current_2005; @bodineau_distribution_2005; @bertini_non_2006; @Bodineau:2007iq; @prolhac_current_2008; @appert-rolland_universal_2008; @Imparato2009; @prolhac_cumulants_2009; @baek_large_n_2016; @hurtado_test_2009; @prados_large_2011; @de_gier_large_2011; @lazarescu_exact_2011; @hurtado_spontaneous_2011; @derrida_microscopic_2011; @gorissen_current_2012; @gorissen_exact_2012; @krapivsky_fluctuations_2012; @meerson_extreme_2013; @meerson_extreme_2014; @znidaric_exact_2014; @Hurtado:2014bn; @Lazarescu2015; @Shpielberg2016; @Zarfaty:2016dv; @hirschberg_zrp_2015]. The properties of the distribution encode much information about the interactions in the channel.
One of the most dramatic consequences of such interactions is the occurrence of dynamical phase transitions (DPT) [@derrida_lebowitz_speer_2002; @hirschberg_zrp_2015; @bertini_current_2005; @bodineau_distribution_2005; @Bodineau:2007iq; @lecomte_thermodynamic_2007; @garrahan_dynamical_2007; @bunin_non-differentiable_2012; @baek_singularities_2015; @tizon-escamilla_order_2016], which are the focus of this Letter [^1]. They imply an enhanced probability of observing certain current fluctuations. Beyond certain current thresholds, the mode of transport through the channel changes abruptly. These DPTs manifest themselves as singularities in a *large deviation function* (LDF) that characterizes the probability distribution of the time-averaged current $J$ in the limit of a large observation time. The function plays, for time-integrated observables, like $J$, the same role as the equilibrium free energy for static observables [@touchette_large_2009]. For classical interacting particles systems, it can be computed using exact microscopic solutions [@derrida_exact_1998; @derrida_universal_1999; @derrida_current_2004; @prolhac_current_2008; @prolhac_cumulants_2009; @lazarescu_exact_2011; @de_gier_large_2011; @derrida_microscopic_2011; @gorissen_exact_2012; @Lazarescu2015] or macroscopic approaches (see [@bertini_macroscopic_2015] for a review).
So far, for current large deviations in driven diffusive systems, only one class of DPTs with concrete microscopic models has been observed; these occur solely for periodic systems which are not connected to reservoirs [@bertini_current_2005; @bodineau_distribution_2005; @Bodineau:2007iq; @appert-rolland_universal_2008; @tizon-escamilla_order_2016]. There one finds that, for currents close to the mean value, the fluctuation manifests itself through a [*time-independent*]{} density profile. The DPT occurs at a critical value of the current beyond which the fluctuation is realized through a [*time-dependent*]{} density profile. Such transitions are referred to as resulting from a failure of the ‘additivity principle’ [@bodineau_current_2004]. Another scenario which involves a ‘first-order’ transition between two distinct time-independent density profiles was suggested in [@bertini_non_2006]. However, lacking any concrete microscopic model, the scenario remains speculative.
In this Letter we study current large deviations in one-dimensional diffusive systems coupled to two reservoirs. Based on an exact Landau theory for the DPTs derived using the Macroscopic Fluctuation Theory (MFT) [@bertini_fluctuations_2001; @*bertini_macroscopic_2002; @*bertini_minimum_2004; @*bertini_stochastic_2007; @bertini_macroscopic_2015], we obtain the following new results: first, we identify DPTs that are not associated with a breaking of the additivity principle, along with sufficient conditions for their existence in terms of transport coefficients; second, we describe a new type of ‘second-order’ DPTs associated with a *symmetry breaking* in the density profiles which realize the current fluctuations; third, we show that well-studied microscopic models, namely the Katz–Lebowitz–Spohn (KLS) [@katz_nonequilibrium_1983] model and the weakly asymmetric simple exclusion process (WASEP) [@WASEP; @gartner_convergence_1987], implement both the first- and the second-order DPTs described above; finally, possible experimental realizations are discussed.
[*Settings*]{} — We consider a one-dimensional driven diffusive system connecting two particle reservoirs using the standard approach of fluctuating hydrodynamics [@spohn_long_1983; @Spohn_Book; @jordan_fluctuation_2004; @bertini_macroscopic_2015]. The particle density profile $\rho(x,t)$ evolves according to a continuity equation $$\begin{aligned}
\label{eq:continuity}
\partial_t \rho(x,t) + \partial_x j(x,t) = 0\,,\end{aligned}$$ where the spatial coordinate $x$ is rescaled by the system size $L$ so that $x \in [0,1]$, $t$ denotes time measured in units of $L^2$, and $j(x,t)$ is the fluctuating current given by $$\begin{aligned}
\label{eq:current}
j(x,t) = -D(\rho)\partial_{x}\rho + \sigma(\rho)E + \sqrt{\sigma(\rho)}\eta(x,t) \, .\end{aligned}$$ The current consists of contributions from Fick’s law, the response to a bulk field $E$, and a noise term. The diffusivity $D(\rho)$ and the mobility $\sigma(\rho)$ are in general density-dependent and connected by the Einstein relation, $2D(\rho)/\sigma(\rho)=\partial_\rho^2 f(\rho)$, with $f(\rho)$ the free energy density of the system at equilibrium. The noise $\eta(x,t)$ satisfies $\langle \eta(x,t) \rangle = 0$ and $$\begin{aligned}
\label{eq:noise}
\langle \eta(x,t)\eta(x',t') \rangle = L^{-1}\delta(x-x')\delta(t-t') \,,\end{aligned}$$ where $\langle \cdot \rangle$ denotes an average over all realizations of the noise. The spatial boundary conditions are fixed as $\rho(0,t) = \bar\rho_a$ and $\rho(1,t) = \bar\rho_b$, where $\bar\rho_a$ and $\bar\rho_b$ are time-independent densities imposed by the reservoirs. We are interested in phase transitions [@bertini_current_2005; @bodineau_distribution_2005; @Bodineau:2007iq] associated with the time-averaged current $$\begin{aligned}
\label{eq:J_def}
J \equiv \frac{1}{T} \int_0^T \mathrm{d}t \int_0^1 \mathrm{d}x \, j(x,t)\,,\end{aligned}$$ whose statistics obey a large deviation principle [@bertini_macroscopic_2015] $$\begin{aligned}
\label{eq:J_ldp}
P(J) \sim \exp \left[-TL\Phi(J)\right]\end{aligned}$$ for $T \gg 1$. A singularity in the LDF $\Phi(J)$ marks a DPT. It proves to be convenient to change ensembles and work with the scaled cumulant generating function (CGF) $$\begin{aligned}
\label{eq:scgf_def}
\Psi(\lambda) \equiv \lim_{T \to \infty} \frac{1}{TL} \ln \left\langle e^{TL \lambda J} \right\rangle\,.\end{aligned}$$ Standard saddle-point arguments [@touchette_large_2009] show that the scaled CGF is related to the LDF by a Legendre transform $\Psi(\lambda) = \sup_J \left[\lambda J - \Phi(J) \right]$. To calculate $\Psi(\lambda)$, we rewrite Eq. in a path integral form using the Martin–Siggia–Rose formalism [@martin_statistical_1973; @*dominicis_technics_1976; @*janssen_lagrangean_1976], which gives $$\begin{aligned}
\Psi(\lambda) = \lim_{T\to\infty}\frac{1}{TL} \ln \int \mathcal{D}\rho \mathcal{D}\hat\rho \, e^{-L\int_0^T \mathrm{d}t \int_0^1 \mathrm{d}x \, \left[\hat\rho\partial_t\rho - H(\rho,\hat\rho)\right]} \,,\end{aligned}$$ with the Hamiltonian density $H(\rho,\hat\rho)$ defined as $$\begin{aligned}
\hspace*{-1.5mm}
H(\rho,\hat\rho) \equiv -D(\rho)(\partial_x\rho)(\partial_x\hat\rho) + \tfrac{\sigma(\rho)}{2}(\partial_x\hat\rho)(2E+\partial_x\hat\rho) .\!
\label{eq:hamiltonian}\end{aligned}$$ The ‘momentum’ variable $\hat\rho$ satisfies the boundary conditions (see Appendix \[sec:saddle\_point\_eqns\]) $\hat\rho(0) = 0$ and $\hat\rho(1) = \lambda$. The scaled CGF $\Psi(\lambda)$ can then be obtained using a saddle-point method. For our cases of interest, we argue that the saddle-point solutions are time-independent, so that the additivity principle is satisfied. The calculations are detailed in Appendices \[sec:eq\_bcs\] and \[sec:ueq\_bcs\], and yield profiles $\rho^*(x)$ and $\hat\rho^*(x)$ which minimize the action $\int_0^T \mathrm{d}t \int_0^1 \mathrm{d}x \, \left[\hat\rho\partial_t\rho - H(\rho,\hat\rho)\right]$. These profiles, which are called the *optimal profiles*, represent the dominant realizations of current fluctuations at a given value of $\lambda$. As we will see, phase transitions are associated with abrupt changes in the shape of the optimal profile as $\lambda$ is varied.
[*Results*]{} — In what follows, we first consider systems with equal boundary densities $\bar\rho_a = \bar\rho_b = \bar\rho$ with $\bar\rho$ very close to an [*extremum*]{} of $\sigma(\rho)$. Already in this case, depending on $D(\rho)$ and $\sigma(\rho)$, all singular behaviors described above are observed. Interestingly, this includes systems which are [*in equilibrium*]{}. Then, for more general boundary conditions given by $\bar\rho_a = \bar\rho - \delta\rho$ and $\bar\rho_b = \bar\rho + \delta\rho$, we argue perturbatively to the leading order in $\delta\rho$ that the behaviors are unchanged up to a shift of the transition point.
As shown in Appendix \[sec:eq\_bcs\], the problem of minimizing over profiles can be reexpressed as $$\begin{aligned}
\label{eq:psi}
\Psi(\lambda) = \frac{\bar\sigma}{2}\,\lambda\,(\lambda+2E) - \inf_m \mathcal{L}(m) \,,\end{aligned}$$ where the Landau-like function $\mathcal{L}(m)$ of the parameter $m\in \mathbb R$, which captures the singular behaviors of $\Psi(\lambda)$, can be written in a truncated form $$\begin{aligned}
\label{eq:landau_sym}
\mathcal{L}(m) &\simeq
-\frac{2\pi \bar D^2}{\bar\sigma\bar\sigma''} \, \bar\sigma' \, m
-\frac{(\lambda_{\mathrm{c}}+ E) \bar\sigma''}{4} \,(\lambda - \lambda_{\mathrm{c}}) \, m^2 \nonumber\\
& \quad-\frac{2\pi \bar D(\bar D \bar\sigma^{(3)}-3\bar D'\bar\sigma'')}{9\bar\sigma\bar\sigma''}
\,m^3 \nonumber\\
&\quad + \left[\frac{\pi^2\bar D\left(4\bar D'' \bar\sigma'' - \bar D \bar\sigma^{(4)}\right)}{64\bar\sigma\bar\sigma''}
+ \frac{\bar\sigma''^2 E^2}{64\bar\sigma}
\right] m^4 \, .\end{aligned}$$ Here $\lambda_{\mathrm{c}}$ is equal to one of the two values [^2] $$\begin{aligned}
\label{eq:lc_def}
\lambda_{\mathrm{c}}^\pm \equiv -E \pm \sqrt{E^2 + \frac{2\pi^2 \bar D^2}{\bar\sigma \bar\sigma''}} \,,\end{aligned}$$ and we use the shorthand notations $\bar g \equiv g(\bar\rho)$, $\bar g' \equiv g'(\bar\rho)$, $\bar g'' \equiv g''(\bar\rho)$, and $\bar g^{(n)} \equiv g^{(n)}(\bar\rho)$ for derivatives of any function $g(\rho)$ evaluated at $\rho = \bar\rho$. The optimal value of the order parameter $m$ in Eq. , which we denote by $m^*$, measures the deviation of the optimal profile from the flat reference profile of density $\bar\rho$ (similar to the zero magnetization in the Landau theory for the Ising model): $$\begin{aligned}
\rho^*(x) = \bar\rho + m^* \sin (\pi x) + O\left[(m^*)^2\right] \, .
\label{eq:defrhostar} \end{aligned}$$ The scaled CGF $\Psi(\lambda)$ has a singularity when $m^*$ changes in a singular manner as $\lambda$ is varied [^3].
Clearly, $\mathcal{L}(m)$ can be truncated as in Eq. only if the coefficient of $m^4$ is positive. For the microscopic models we study below, this is always the case. While there could be other models for which higher-order terms in $m$ need to be considered, these are beyond the scope of this Letter. Moreover, for a transition to occur as $\lambda$ is varied, we need $\bar\sigma' = 0$, and $\lambda_{\mathrm{c}}$ defined in Eq. has to be real-valued. This is the case if $\sigma(\rho)$ has a local minimum at $\rho = \bar\rho$, so that $\bar\sigma'' > 0$; otherwise, if $\bar\sigma'' < 0$, the bulk field has to be sufficiently strong so that $$\begin{aligned}
\label{eq:e_cond}
E^2 > \frac{2\pi^2\bar D^2}{\bar\sigma |\bar\sigma''|} \,.\end{aligned}$$ We observe different transition behaviors depending on the sign of $\bar\sigma''$, each of which we discuss in the following.
[*Case 1a: $\bar\sigma'' > 0$, symmetry breaking*]{} — Consider a particle–hole symmetric system, whose Hamiltonian density shown in Eq. is invariant under the transformation defined by $x \to 1-x$, $\rho(x,t) \to 2\bar\rho - \rho(1-x,t)$, and $\hat\rho(x,t) \to \lambda - \hat\rho(1-x,t)$. Assuming that $D(\rho)$ and $\sigma(\rho)$ are analytic, their odd-order derivatives vanish at $\rho = \bar\rho$, i.e. $\bar D^{(2n+1)} = \bar\sigma^{(2n+1)} = 0$ for $n = 0,\,1,\,\ldots$ Then only the $m^2$ and $m^4$ terms survive in Eq. , turning $\mathcal{L}(m)$ into the form of a Landau free energy of Ising-like systems. For $\lambda_{\mathrm{c}}^- < \lambda < \lambda_{\mathrm{c}}^+$, $\mathcal{L}(m)$ is minimized at $m^* = 0$, and $\Psi(\lambda)$ has a quadratic form corresponding to Gaussian fluctuations. For $\lambda > \lambda_{\mathrm{c}}^+$ or $\lambda < \lambda_{\mathrm{c}}^-$, we have $m^* \sim \pm |\lambda-\lambda_{\mathrm{c}}|^{1/2}$, corresponding to a pair of symmetry-breaking profiles given by Eq. which are mutually related by a particle–hole transformation defined above. This implies that for each instance of a current fluctuation $J$ in this regime, there is a symmetry breaking so that one of the two optimal profiles is observed with equal probability (see Fig. \[fig:fig1\]). Near the transition points, the scaled CGF $\Psi(\lambda)$ has singularities which behave as $\lim_{\lambda\downarrow\lambda_{\mathrm{c}}}\Psi(\lambda) - \lim_{\lambda\uparrow\lambda_{\mathrm{c}}}\Psi(\lambda) \sim |\lambda - \lambda_{\mathrm{c}}|^2$, implying second-order transitions. Clearly, the same critical scaling behavior is observed if $\bar D^{(3)}$, $\bar\sigma^{(5)}$ or higher-order derivatives are nonzero, although in such cases only one of the two density profiles is optimal.
[*Case 1b: $\bar\sigma'' > 0$, first-order transition*]{} — Now consider the case when $\bar D'$ and $\bar\sigma^{(3)}$ have nonzero values. For a consistent Landau theory, we assume that $\bar D'$ and $\bar\sigma^{(3)}$ scale as $m^*$. Then the $m^3$ term induces a weak first-order singularity of the scaled CGF [^4]. On general grounds, similar results will be obtained even if $\bar D'$ and $\bar\sigma^{(3)}$ are larger. The transition shows up as jumps of $m^*$ at transition points $\lambda_{\mathrm{d}}^\pm$ which are slightly shifted from $\lambda_{\mathrm{c}}^\pm$, respectively (see Appendix \[ssec:landau\_eq\_asym\]). In a manner similar to Case 1a, the fluctuations are Gaussian for $\lambda_{\mathrm{d}}^- < \lambda < \lambda_{\mathrm{d}}^+$ and non-Gaussian otherwise (see Fig. \[fig:fig1\]). This behavior corresponds to a scenario discussed in [@bertini_non_2006]: when a current fluctuation $J$ occurs within the intervals $[J_1^\pm,J_2^\pm]$ defined by $J_1^\pm \equiv \lim_{\lambda \uparrow \lambda_{\mathrm{d}}^\pm}\Psi'(\lambda)$ and $J_2^\pm \equiv \lim_{\lambda \downarrow \lambda_{\mathrm{d}}^\pm}\Psi'(\lambda)$, we observe $J_1^\pm$ and $J_2^\pm$ with probability $p_1^\pm$ and $1-p_1^\pm$, respectively, such that $J = p_1^\pm J_1^\pm + (1-p_1^\pm) J_2^\pm$. This is a direct analog of phase coexistence in equilibrium first-order transitions.
{width="95.00000%"}
[*Case 2: $\bar\sigma'' < 0$*]{} — For Case 1, the bulk field $E$ is not essential for the existence of a DPT: it only shifts the location of the transition point according to Eq. . In contrast, for Case 2 phase transitions occur only when the bulk field $E$ is strong enough to satisfy Eq. . Since the form of $\mathcal{L}(m)$ remains the same, the system again exhibits symmetry breaking transitions for fully particle–hole symmetric systems, and first-order transitions in the absence of symmetry due to nonzero $\bar D'$ and $\bar\sigma^{(3)}$. Note that while the regions of non-Gaussian fluctuations were unbounded in Case 1, here they are bounded. This is because for $\bar\sigma'' < 0$ both transition points $\lambda_{\mathrm{c}}^\pm$ have the same sign, as implied by Eq. (see Fig. \[fig:fig1\]).
[*Generalization to $\rho_a \neq \rho_b$*]{} — We now turn to the case of unequal boundary densities given by $\bar\rho_a = \bar\rho - \delta\rho$ and $\bar\rho_b = \bar\rho + \delta\rho$. Treating $\delta\rho$ as a perturbation, we find to linear order in $\delta\rho$ that (see Appendix \[sec:ueq\_bcs\]) $$\begin{aligned}
\label{eq:psi}
\Psi(\lambda) = \frac{\bar\sigma}{2}\,\lambda\,(\lambda+2E) - 2\,\delta\rho \, \bar D \,\lambda - \inf_m \mathcal{L}(m) \,,\end{aligned}$$ with only the quadratic term in $\mathcal{L}(m)$ modified as $$\begin{aligned}
(\lambda - \lambda_{\mathrm{c}})m^2 \to \left(\lambda - \lambda_{\mathrm{c}}- \frac{2\bar D}{\bar\sigma}\delta\rho\right)m^2,\end{aligned}$$ which implies that the transition point is shifted but the other properties of phase transitions are unchanged. If $\bar\rho_a - \bar\rho \neq \bar\rho - \bar\rho_b$, we can use $(\bar\rho_a + \bar\rho_b)/2$ as the new value of $\bar\rho$. Provided that the odd-order derivatives of $D(\rho)$ and $\sigma(\rho)$ evaluated at the new $\bar\rho$ remain small, all results presented above are still valid.
[*Microscopic models*]{} — We now present two lattice gas models, each of which exhibits one of the two cases of phase transitions described above.
[*Case 1: $\bar\sigma'' > 0$*]{} — We consider a KLS [@katz_nonequilibrium_1983] model with zero bulk bias, which features on-site exclusion and nearest-neighbor interactions. It is defined on a one-dimensional lattice, each site of which can be either occupied (“1”) or empty (“0”). The model is characterized by two parameters $\delta$ and $\varepsilon$, which govern the hopping dynamics according to the following transition rates (in arbitrary units): $$\begin{aligned}
& 0100\xrightarrow{1+\delta}0010 \,,\ 1101\xrightarrow{1-\delta
}1011\, ,\\
& 1100\xrightarrow{1+\varepsilon}1010 \,,\ 1010\xrightarrow{1-\varepsilon
}0110\, .\end{aligned}$$ Spatially inverted versions of these transitions occur with identical rates. Using the methods of [@Spohn_Book; @hager_minimal_2001; @krapivsky_unpub], $D(\rho)$ and $\sigma(\rho)$ of the model can be derived exactly as functions of $\rho \in [0,1]$ (see Appendix \[sec:kls\] for their explicit forms). If $\delta = 0$, the model possesses a particle–hole symmetry, so that all odd-order derivatives of $D(\rho)$ and $\sigma(\rho)$ with respect to $\rho$ vanish at $\rho = 1/2$. More interestingly, for $\epsilon > 4/5$, one finds that $\sigma(1/2)$ becomes a local minimum. Thus, all results of Case 1a can be applied to this model by setting $\bar\rho = 1/2$. On the other hand, if $\delta \neq 0$, the system does not have a particle–hole symmetry. Then, for $\epsilon$ greater than some $\delta$-dependent threshold, $\sigma(\rho)$ has a local minimum at some $\delta$-dependent $\bar\rho$. All results of Case 1b are then applicable to this system.
[*Case 2: $\bar\sigma'' < 0$*]{} — Consider a WASEP on a one-dimensional lattice of $L$ sites, whose hopping rates (in arbitrary units) are given by $10\xrightarrow{1+\delta}01$, $01\xrightarrow{1-\delta}10$. If $\delta = E/L$, it is well known [@WASEP; @gartner_convergence_1987] that the system is characterized by $D(\rho) = 1$ and $\sigma(\rho) = 2\rho(1-\rho)$, so that $\sigma''(\rho) < 0$ for any $\rho \in [0,1]$ with the maximum of $\sigma(\rho)$ located at $\rho = 1/2$. Applying the results of Case 2, Eq. implies that the system exhibits singularities of LDFs when $|E| > \pi$.
[*Mechanism for symmetry breaking*]{} — To gain more intuition into the origin of the DPT, it is helpful to examine the Lagrangian formulation of the LDF [@bertini_macroscopic_2015] $$\begin{aligned}
\label{eq:phi_lagrangian}
\Phi(J) = \inf_{\rho} \int_0^1 \mathrm{d}x \, \frac{\left[J+D(\rho)\partial_x \rho - \sigma(\rho)E \right]^2}{2\sigma(\rho)} \,.\end{aligned}$$ Close to the transition point, $\Phi(J)$ is minimized by an optimal profile of the form $\rho(x) = \bar\rho + m \sin(\pi x)$. Keeping the leading-order corrections in $m$, we obtain $$\begin{aligned}
\Phi(J) &\simeq \frac{\delta J^2}{2\bar\sigma} \\
& + \inf_m \left[\left(\frac{\bar D^2}{2} - \frac{\bar\sigma''E \, \delta J}{2} - \frac{\bar\sigma'' \, \delta J^2}{4\bar\sigma}\right)m^2 +O (m^4)\right] \,, \nonumber\end{aligned}$$ where $\delta J \equiv J - \bar\sigma E$. The occurrence of symmetry breaking is controlled by the sign of the coefficient in front of $m^2$, whose three terms represent contributions from diffusion, bulk field $E$, and noise amplitude. The first two originate from the numerator of Eq. and the last one comes from the denominator. The competition between these factors dictate whether it is beneficial to break the symmetry by density modulations. Depending on the sign of $\bar\sigma''$, there are two possible scenarios.
If $\bar\sigma'' > 0$, the coefficient of $m^2$ is positive for $\delta J$ close to zero and becomes negative for sufficiently large $\delta J$, signaling the symmetry breaking transition — for large enough $\delta J$, the gain in action from the denominator overwhelms the cost of density modulations in the numerator.
On the other hand, if $\bar\sigma'' < 0$, both the diffusion and the noise lead to a positive cost for density modulations. Negative contributions arise only from the field term. A large enough $E$ can make density modulations favorable for an intermediate range of $\delta J$, inducing a transition.
The origins of DPTs in these two cases are different. For $\bar\sigma'' > 0$ the transitions are due to the competition between the diffusion, which favors a flat profile, and the noise, which favors modulations. In contrast, for $\bar\sigma'' < 0$ the transitions are ruled by the contribution of the bulk field, which favors modulations, competing against the diffusion and the noise, both of which favor a flat profile. Similar arguments also apply to first-order transitions.
Comparisons with previous studies are in order. A recent study [@Shpielberg2016] proposed a criterion which forbids DPTs of Case 2; but our results explicitly show that the WASEP is a counterexample to this criterion [^5]. We also note that the asymmetric simple exclusion process (ASEP), which is non-diffusive, also exhibits DPTs in current fluctuations [@dudzinski_relaxation_2000; @deGier2005; @gorissen_exact_2012; @Lazarescu2015]. While these DPTs are remnants of the well-known boundary-induced transitions in mean behaviors, the DPTs of diffusive systems discussed above are very different.
There remains the question of how the DPTs discussed so far can be experimentally observed. Recently, the LDF for heat current in an RC circuit was empirically measured in [@Ciliberto2013a; @*Ciliberto2013b], where the fast electronic dynamics allows the current LDF to be measured over a wide range [^6]. To observe the DPTs discussed here in a similar experiment, one has to look at diffusive electronic transport with an extremum in $\sigma(\rho)$. These are common, resulting from non-monotonic changes in the electronic density of states. For example, minima of $\sigma(\rho)$ were observed in graphene transport [@Adam2007; @*Tan2007; @*Chen2008] and maxima in fullerene peapods [@Utko2010]. Using these systems, both cases of DPTs discussed above can in principle be observed.
In summary, we have studied a general one-dimensional diffusive transport through a channel connecting two reservoirs. Using a perturbative approach for general $D(\rho)$ and $\sigma(\rho)$, we find a large class of new DPTs which are not associated with the breaking of the additivity principle in the sense that the optimal profiles remain time-independent. For some of these DPTs we can explicitly prove the validity of the additivity principle, which we expect to hold for all cases (see Appendix \[ssec:additivity\]). It would be interesting to check whether other kinds of DPTs occur at larger values of $J$ or $\delta\rho$, and how the results can be generalized to higher dimensions.
We are grateful to Paul Krapivsky for his collaboration at early stages of this work. We also thank Giovanni Jona-Lasinio, Kirone Mallick, Ohad Shpielberg, Daniel Podolsky, and Michael Reznikov for helpful comments. YB and YK are supported by an ISF grant, and VL is supported by the ANR-15-CE40-0020-03 Grant LSD.
Derivation of the saddle-point equations {#sec:saddle_point_eqns}
========================================
For completeness we outline the derivation of the saddle-point equations which are used to obtain the scaled cumulant generating function (CGF) for the time-averaged current. Similar derivations can also be found elsewhere in the literature (see, for example, [@Imparato2009]).
From Eqs. and of the main text, the scaled CGF is given by $$\begin{aligned}
\label{eq:cgf_path_avg}
\Psi(\lambda) = \lim_{T \to \infty} \frac{1}{T L} \ln \left\langle e^{\lambda L \int_0^T \mathrm{d}t\, \int_0^1\mathrm{d}x\, j(x,t)} \right\rangle \,,\end{aligned}$$ where $\langle\cdot\rangle$ denotes an average over the noise realizations. Using the Langevin equation $$\begin{aligned}
\label{eq:langevin}
\partial_t \rho(x,t) + \partial_x j(x,t) = 0 \,, \quad
j(x,t)=-D(\rho)\partial_{x}\rho+\sigma(\rho)E+\sqrt{\sigma(\rho)}\eta(x,t)\end{aligned}$$ with the spatial boundary conditions $$\begin{aligned}
\label{eq:r_bcs}
\rho(0) = \bar\rho_a \,, \quad \rho(1) = \bar\rho_b \,,\end{aligned}$$ the average on the r.h.s. of Eq. can be written as $$\begin{aligned}
\label{eq:integ_rje}
\left\langle e^{\lambda L \int_0^{T}\mathrm{d}t\, \int_0^1\mathrm{d}x\, j} \right\rangle
= \left\langle \int \mathcal{D}\rho\, \mathcal{D}j\,
e^{\lambda L \int_0^{T}\mathrm{d}t\, \int_0^1\mathrm{d}x\, j}\,
\delta\left[\dot{\rho}+\nabla j\right] \,
\delta\left[j + D(\rho)\nabla \rho - \sigma(\rho) E - \sqrt{\sigma(\rho)} \eta \right] \right\rangle\,.\end{aligned}$$ The two delta functionals in the path integral make sure that the integration is carried out only over the paths governed by Eq. . The functional $\delta\left[\dot{\rho}+\nabla j\right]$ can be rewritten in terms of its Fourier representation $$\begin{aligned}
\delta\left[\dot{\rho}+\nabla j\right]
= \int \mathcal{D}\hat\rho \, e^{-L \int_0^T \mathrm{d}t \, \int_0^1 \mathrm{d}x \, \hat\rho(\dot{\rho}+\nabla j)} \,,\end{aligned}$$ which introduces an auxiliary field $\hat{\rho}(x,t)$. Then Eq. can be integrated over the current $j(x,t)$ and the noise $\eta(x,t)$ to yield $$\begin{aligned}
\left\langle e^{\lambda L \int_0^{T}\mathrm{d}t\, \int_0^1\mathrm{d}x\, j} \right\rangle
&= \int \mathcal{D}\rho\, \mathcal{D}\hat\rho
\exp \left\{L \int_0^{T}\mathrm{d}t\,
\int_0^1\mathrm{d}x\,
\left[-\hat\rho \dot{\rho} - D(\rho)(\nabla \rho) (\lambda + \nabla\hat\rho)
+\frac{\sigma(\rho)}{2}(\lambda + \nabla \hat\rho)(\lambda + \nabla\hat\rho + 2E) \right] \right\} \,.\end{aligned}$$ Here the auxiliary field variable $\hat\rho(x,t)$ satisfies the boundary conditions $$\begin{aligned}
\label{eq:rh_bcs}
\hat\rho(0,t) = 0,\quad \hat\rho(1,t) = 0,\end{aligned}$$ which accounts for the absence of fluctuations at the boundaries [@Tailleur2008]. It is useful to introduce a change of variables $$\begin{aligned}
\label{eq:rh2rhl}
\hat\rho(x,t) \to \hat\rho_\lambda(x,t) - \lambda x,\end{aligned}$$ which gives $$\begin{aligned}
\label{eq:integ_rrh}
\left\langle e^{\lambda L \int_0^{T}\mathrm{d}t\, \int_0^1\mathrm{d}x\, j} \right\rangle
&= \int \mathcal{D}\rho\, \mathcal{D}\hat\rho_\lambda
\exp \left\{- L \int_0^{T}\mathrm{d}t\,
\int_0^1\mathrm{d}x\,
\left[\hat\rho_\lambda \dot{\rho} + D(\rho)(\nabla \rho) (\nabla\hat\rho_\lambda)
-\frac{\sigma(\rho)}{2}(\nabla \hat\rho_\lambda)(\nabla\hat\rho_\lambda + 2E) \right] \right\}.\end{aligned}$$ Here a temporal boundary term $L\int_0^1\mathrm{d}x\, \{\lambda x [\rho(x,T)-\rho(x,0)]\}$ in the exponent is neglected as it becomes negligible for $T \gg 1$.
Since we are interested in the large $L$ limit, the scaled CGF can be evaluated using a saddle point so that $$\begin{aligned}
\label{eq:psi_least_action}
\Psi(\lambda) = - \lim_{T \to \infty} \frac{1}{T} \inf_{\rho,\,\hat\rho_\lambda} \int_0^{T}\mathrm{d}t\,
\int_0^1\mathrm{d}x\, \left[ \hat\rho_\lambda \dot{\rho} - H(\rho,\hat\rho_\lambda)\right],\end{aligned}$$ with the Hamiltonian density $H$ defined as $$\begin{aligned}
H(\rho,\hat\rho_\lambda) \equiv -D(\rho)(\nabla \rho) (\nabla\hat\rho_\lambda)
+\frac{\sigma(\rho)}{2}(\nabla \hat\rho_\lambda)(\nabla\hat\rho_\lambda + 2E) \,.\end{aligned}$$ Here $\rho$ and $\hat\rho_\lambda$ can be interpreted as position and momentum variables, respectively. Thus, the saddle-point solutions are obtained by solving the equations $$\begin{aligned}
\label{eq:saddle_traj}
\dot{\rho} = \nabla\left[D(\rho)\nabla\rho - \sigma(\rho)(\nabla\hat\rho_\lambda +E)\right], \quad
\dot{\hat\rho}_\lambda = - D(\rho) \nabla^2 \hat\rho_\lambda - \frac{1}{2}\sigma'(\rho)(\nabla\hat\rho_\lambda)(\nabla\hat\rho_\lambda + 2E)\end{aligned}$$ with the spatial boundary conditions $$\begin{aligned}
\rho(0,t) = \bar\rho_a,\quad \rho(1,t) = \bar\rho_b,\quad \hat\rho_\lambda(0,t) = 0,\quad \hat\rho_\lambda(1,t) = \lambda \;.\end{aligned}$$ These are consistent with Eqs. , , and . With the understanding that these boundary conditions are always assumed, for brevity in what follows we drop the subscript $\lambda$ from $\hat\rho_\lambda$.
Equal boundary densities {#sec:eq_bcs}
========================
In what follows, we first consider the case when the two particle reservoirs have equal densities $\bar\rho_a = \bar\rho_b = \bar\rho$. The case $E=0$ then corresponds to an equilibrium system, while when $E \neq 0$ the system is out of equilibrium. We identify the symmetry-breaking transition point $\lambda_{\mathrm{c}}$ discussed in the main text, and then construct from first principles a Landau theory for the transition. Finally, we prove that in this case the additivity principle (assumed to hold throughout the derivation) is valid.
Symmetry breaking in particle–hole symmetric systems {#ssec:sym_break_eq}
----------------------------------------------------
We start by analyzing systems which are particle–hole symmetric about $\rho = \bar\rho$, so that all odd-order derivatives of the transport coefficients at $\rho = \bar\rho$ vanish: $$\begin{aligned}
\label{eq:odd_deriv_zero}
\bar D^{(2n+1)} \equiv D^{(2n+1)}(\bar\rho) = 0 \,, \quad
\bar\sigma^{(2n+1)} \equiv \sigma^{(2n+1)}(\bar\rho) = 0 \quad
\text{for $n = 0$, $1$, $2$, $\ldots$}\end{aligned}$$ Assuming that the additivity principle holds, the saddle-point equations reduce to their time-independent forms $$\begin{aligned}
\label{eq:saddle_traj_time_indept}
\nabla\left[D(\rho)\nabla\rho - \sigma(\rho)(\nabla\hat\rho +E)\right] = 0 \,, \quad
D(\rho) \nabla^2 \hat\rho + \frac{1}{2}\sigma'(\rho)(\nabla\hat\rho)(\nabla\hat\rho+ 2E) = 0 \,.\end{aligned}$$ As discussed in the main text, such system may exhibit a dynamical phase transition at $\lambda=\lambda_{\mathrm{c}}$. For $\lambda<\lambda_{\mathrm{c}}$ the density and momentum profiles $$\begin{aligned}
\label{eq:rho_sym_eq}
\rho^\text{sym}(x) \equiv \bar\rho \,, \quad
\hat\rho^\text{sym}(x) \equiv \lambda x \,,\end{aligned}$$ which are symmetric in the sense that they are invariant under $$\begin{aligned}
\label{eq:par_hole_mapping}
\rho(x) \to 2\bar\rho - \rho(1-x) \,, \quad \hat\rho(x) \to \lambda - \hat\rho(1-x)\,,\end{aligned}$$ are the only solution to the saddle-point equation. In contrast, for $\lambda>\lambda_{\mathrm{c}}$ two additional symmetry-breaking solutions appear and become more dominant than the symmetric profile. In what follows we prove the existence of this transition.
Note that near a transition point $\lambda = \lambda_{\mathrm{c}}$, the symmetry of the optimal profile is weakly broken by small deviations from Eq. . In other words, $$\begin{aligned}
\label{eq:another_saddle}
\rho(x) = \rho^\text{sym}(x) + \varphi(x) \,, \quad
\hat\rho(x) = \hat\rho^\text{sym}(x) + \hat\varphi(x)\end{aligned}$$ with small but nonzero $\varphi$ and $\hat\varphi$ satisfying the boundary conditions $$\begin{aligned}
\label{eq:phi_zero_bcs}
\varphi(0) = \varphi(1) = \hat\varphi(0) = \hat\varphi(1) = 0\end{aligned}$$ will be another solution of Eq. . Linearizing Eq. with respect to $\varphi$ and $\hat\varphi$, we obtain a system of linear differential equations $$\begin{aligned}
\label{eq:phi_saddle_time_indept}
\bar D \nabla^2 \varphi - \bar\sigma \nabla^2 \hat\varphi = 0, \quad
\bar D \nabla^2 \hat\varphi + \frac{\bar\sigma''}{2}\lambda(\lambda+2E) \varphi = 0.\end{aligned}$$ Using the Fourier transforms $$\begin{aligned}
\label{eq:fourier_time_indept}
\varphi(x) = \sum_{n = 1}^\infty \psi_n \sin(n\pi x), \quad
\hat\varphi(x) = \sum_{n = 1}^\infty \hat\psi_n \sin(n\pi x),\end{aligned}$$ we can rewrite Eq. as $$\begin{aligned}
\label{eq:psi_n}
\bar D \psi_n - \bar\sigma \hat\psi_n = 0,
\quad
n^2 \pi^2 \bar D \hat\psi_n - \frac{\bar\sigma''}{2}\lambda(\lambda+2E)\psi_n = 0.\end{aligned}$$ These linear equations have nonzero solutions for $\psi_n$ and $\hat\psi_n$ if and only if $$\begin{aligned}
\lambda(\lambda+2E) = \frac{2 n^2 \pi^2 \bar D^2}{\bar\sigma \bar\sigma''} \,,\end{aligned}$$ i.e. when $\lambda$ is equal to $$\begin{aligned}
\label{eq:lambda_cn}
\lambda_{\text{c},n}^\pm \equiv -E \pm \sqrt{E^2 + \frac{2n^2 \pi^2 \bar D^2}{\bar\sigma \bar\sigma''}} \,.\end{aligned}$$ From Eq. it is clear that if $\psi_n$ and $\hat\psi_n$ are solutions, so are $-\psi_n$ and $-\hat\psi_n$. This implies that Eq. allows a symmetry breaking at critical values $\lambda_{\text{c},n}^\pm$. The symmetry breaking transition will clearly occur for the $n$ with a minimal $|\lambda_{\text{c},n}^\pm|$. For the case $\bar\sigma'' > 0$, we always have $\lambda_{\text{c},n}^- < 0 < \lambda_{\text{c},n}^+$, so that the symmetry breaking occurs on both sides of $\lambda = 0$. It is clear then that $|\lambda_{\text{c},n}^\pm|$ is minimized for $n = 1$.
For the case $\bar\sigma'' < 0$, we assume that $|E|$ is large enough to keep $\lambda_{\text{c},n}^\pm$ real-valued, as discussed in the main text. If $E > 0$ ($E < 0$), we have $\lambda_{\text{c},n}^- < \lambda_{\text{c},n}^+ < 0$ ($0 < \lambda_{\text{c},n}^- < \lambda_{\text{c},n}^+$), which means that the symmetry breaking occurs as $\lambda$ is decreased from zero to $\lambda_{\text{c},n}^+$ (increased from zero to $\lambda_{\text{c},n}^-$). Again, $|\lambda_{\text{c},n}^+|$ ($|\lambda_{\text{c},n}^-|$) is minimized at $n = 1$.
Hence, regardless of the sign of $\bar\sigma''$, only a deviation of the form $\varphi(x) \sim \sin(\pi x)$ is relevant to the symmetry-breaking transition. Moreover, due to the Gallavotti–Cohen symmetry [@Gallavotti1995a; @*Gallavotti1995b] $$\begin{aligned}
\Psi(\lambda) = \Psi(- 2E - \lambda),\end{aligned}$$ if a dynamical phase transition occurs at $\lambda = \lambda_{\text{c},1}^+$, the same kind of transition occurs at $\lambda = \lambda_{\text{c},1}^-$, and vice versa. Thus the transition points are always at $$\begin{aligned}
\label{eq:lambda_c_eq}
\lambda_{\mathrm{c}}^\pm \equiv \lambda_{\text{c},1}^\pm = -E \pm \sqrt{E^2 + \frac{2 \pi^2 \bar D^2}{\bar\sigma \bar\sigma''}} \,.\end{aligned}$$ In the following, for simplicity we will use $\lambda_{\mathrm{c}}$ when we are referring to one of the two transition points.
While we have shown that another solution appears at $\lambda_{\mathrm{c}}$ we have not shown that it dominates to CGF thus leading to a transition. This is done next by deriving a Landau theory for the transition from first principles.
Landau theory for symmetry-breaking transitions {#ssec:landau_eq_sym}
-----------------------------------------------
Here we give a detailed derivation of the Landau theory describing the symmetry-breaking transitions. For this purpose, the system is again assumed to be particle–hole symmetric, so that Eqs. and of Sec. \[ssec:sym\_break\_eq\] are still valid.
Under the assumption of the additivity principle, the scaled CGF can be obtained from a time-independent version of Eq. : $$\begin{aligned}
\label{eq:psi_least_action_time_indept}
\Psi(\lambda) = \sup_{\rho,\,\hat\rho} \int_0^1 \mathrm{d}x\, H(\rho,\hat\rho).\end{aligned}$$ To construct a Landau theory of the transition, for $\lambda$ close to $\lambda_{\mathrm{c}}$ we can use an expansion $$\begin{aligned}
\label{eq:phi_opt}
\varphi(x) &= m^* \sin (\pi x) + (m^*)^2 \varphi_2(x)
+ (m^*)^3 \varphi_3(x) + O\left[(m^*)^4\right]\,, \nonumber\\
\hat\varphi(x) &= m^* \frac{\bar D}{\bar\sigma}\sin (\pi x) + (m^*)^2 \hat\varphi_2(x)
+ (m^*)^3 \hat\varphi_3(x) + O\left[(m^*)^4\right] \,,\end{aligned}$$ where $m^*$ measures the contribution of $\sin(\pi x)$ to the symmetry breaking. The functions $\varphi_2$, $\varphi_3$, $\ldots$ and $\hat\varphi_2$, $\hat\varphi_3$, $\ldots$ are zero at the boundaries ($x = 0$ and $x = 1$) and orthogonal to $\sin (\pi x)$; they are to be determined by solving Eq. perturbatively (see below). Then, if we define $$\begin{aligned}
\label{eq:phi_m}
\varphi^m(x) &\equiv m \sin (\pi x) + m^2 \varphi_2(x)
+ m^3 \varphi_3(x) + O(m^4) \,, \nonumber\\
\hat\varphi^m(x) &\equiv m \frac{\bar D}{\bar\sigma}\sin (\pi x) + m^2 \hat\varphi_2(x)
+ m^3 \hat\varphi_3(x) + O(m^4) \,,\end{aligned}$$ a Landau function can be written as $$\begin{aligned}
\label{eq:landau_def}
{\cal L}(m) \equiv \int_0^1 \mathrm{d}x\, \left[
H(\rho^\text{sym},\hat\rho^\text{sym})
- H(\rho^\text{sym} + \varphi^m,\hat\rho^\text{sym} + \hat\varphi^m) \right] \,,\end{aligned}$$ and the minimization problem associated with the scaled CGF can be cast as $$\begin{aligned}
\Psi(\lambda) = \int_0^1 \mathrm{d}x\, H(\rho^\text{sym},\hat\rho^\text{sym})
- \inf_m {\cal L}(m) \,.\end{aligned}$$ Thus here a minimization over profiles in Eq. is simplified to that over a single parameter $m$.
To calculate $\varphi_2$, $\varphi_3$, $\ldots$ and $\hat\varphi_2$, $\hat\varphi_3$, $\ldots$, we substitute $\rho = \rho^\text{sym} + \varphi$ and $\hat\rho = \hat\rho^\text{sym} + \hat\varphi$ into Eq. and expand the equations with respect to $m^*$. This allows us to solve the differential equations order by order. The perturbation analysis can be carried out in a well-defined way if the distance from the symmetry-breaking transition point $\delta\lambda \equiv \lambda - \lambda_{\mathrm{c}}$ satisfies a scaling relation with $m^*$. Inspired by an Ising Landau theory, we use the scaling ansatz $$\begin{aligned}
\label{eq:scalings}
\delta\lambda \simeq c^{\delta\lambda} (m^*)^2,\end{aligned}$$ where the value of the coefficient $c^{\delta\lambda}$ is determined below. Carrying out this procedure to order $(m^*)^2$, we obtain $$\begin{aligned}
\label{eq:ode_20}
\nabla^2 \varphi_2 = -\pi^2 \varphi_2, \quad
\nabla^2 \hat\varphi_2 = \frac{\bar D}{\bar\sigma} \nabla^2 \varphi_2 - \frac{\pi\bar\sigma''(\lambda_{\mathrm{c}}+E)}{2\bar\sigma}\sin(2\pi x).\end{aligned}$$ Keeping in mind that $\varphi_2$ must be orthogonal to $\sin(\pi x)$ gives $$\begin{aligned}
\varphi_2(x) = 0, \quad
\hat\varphi_2(x) = \frac{\bar\sigma''(\lambda_{\mathrm{c}}+E)}{8\pi \bar\sigma}\sin(2\pi x).\end{aligned}$$ To order $(m^*)^3$, we obtain $$\begin{aligned}
\label{eq:ode_30_1}
\nabla^2 \varphi_3 &= -\pi^2 \varphi_3 + \left[\frac{1}{8}\left(\frac{4\pi^2 \bar D''}{\bar D} + \frac{\bar\sigma''^2 E^2}{\bar D^2} - \frac{\pi^2 \bar\sigma^{(4)}}{\bar\sigma''}\right) - \frac{(\lambda_{\mathrm{c}}+E) \bar\sigma\bar\sigma'' c^{\delta\lambda}}{\bar D^2}\right] \sin (\pi x) \nonumber\\
&\quad -\frac{1}{24} \left(\frac{12 \pi ^2 \bar D''}{\bar D}+\frac{3 \bar\sigma''^2 E^2}{\bar D^2}-\frac{\pi ^2 \bar\sigma^{(4)}}{\bar\sigma''}\right) \sin (3 \pi x)\end{aligned}$$ and $$\begin{aligned}
\label{eq:ode_30_2}
\nabla^2 \hat\varphi_3 &= \frac{\bar D}{\bar\sigma} \nabla^2 \varphi_3
+\frac{\pi ^2 \left(\bar D \bar\sigma''-\bar\sigma \bar D''\right)}{8 \bar\sigma^2} \left[\sin (\pi x) - 3\sin (3\pi x)\right].\end{aligned}$$ The differential equation has terms of the form $$\begin{aligned}
\label{eq:ode_f}
\nabla^2 f(x) = -\pi^2 f(x) + a \sin(\pi x) \,.\end{aligned}$$ This equation has a solution with $f(0) = f(1) = 0$ if and only if $a = 0$. This condition fixes the coefficient $$\begin{aligned}
\label{eq:ode_30_const}
c^{\delta\lambda} = \frac{\bar D^2}{8 (\lambda_{\mathrm{c}}+ E) \bar\sigma \bar\sigma''^2}\left(\frac{4\pi^2 \bar D''}{\bar D} + \frac{\bar\sigma''^2 E^2}{\bar D^2} - \frac{\pi^2 \bar\sigma^{(4)}}{\bar\sigma''}\right),\end{aligned}$$ with which we obtain the solutions $$\begin{aligned}
\varphi_3(x) &= \frac{1}{192\pi^2}\left(\frac{12\pi^2\bar D''}{\bar D}+\frac{3\bar\sigma''^2 E^2}{\bar D^2}-\frac{\pi^2\bar\sigma^{(4)}}{\bar\sigma''}\right)\sin (3 \pi x)\end{aligned}$$ and $$\begin{aligned}
\hat\varphi_3(x) &= - \frac{\bar D \bar\sigma''-\bar D''\bar\sigma}{8\bar\sigma^2}\sin (\pi x)
+\left[\frac{\bar D}{192\pi^2\bar\sigma}\left(\frac{12\pi^2\bar D''}{\bar D}+\frac{3\bar\sigma''^2 E^2}{\bar D^2}-\frac{\pi^2\bar\sigma^{(4)}}{\bar\sigma''}\right) + \frac{\bar D \bar\sigma''-\bar D'' \bar\sigma}{24\bar\sigma^2}\right]\sin (3 \pi x) \,.\end{aligned}$$
Using $\varphi^m$ and $\hat\varphi^m$ to order $m^3$, we finally obtain $$\begin{aligned}
\label{eq:landau_eq_sym}
{\cal L}(m) &=
-\frac{(\lambda_{\mathrm{c}}+ E) \bar\sigma''}{4} \,\delta\lambda \, m^2
+ \left[\frac{\pi^2\bar D\left(4\bar D'' \bar\sigma'' - \bar D \bar\sigma^{(4)}\right)}{64\bar\sigma\bar\sigma''} + \frac{\bar\sigma''^2 E^2}{64\bar\sigma}
\right] m^4 + O(m^5) \,,\end{aligned}$$ which indeed has the form of a Landau function describing a symmetry-breaking transition at $\delta\lambda = 0$. We note that Eqs. and guarantee ${\cal L}'(m^*) = 0$, which is indeed a condition required for the optimal value of the order parameter $m$.
Systems without particle–hole symmetry {#ssec:landau_eq_asym}
--------------------------------------
The derivation of the Landau theory described above can be generalized to systems with a weak particle–hole asymmetry. To construct a consistent perturbative Landau function, we have to take the odd-order derivatives $\bar D'$, $\bar\sigma'$, and $\bar\sigma^{(3)}$ (which contribute to the asymmetry) to scale with $m^*$ (as was done for $\delta\lambda$ in the previous discussion). More specifically, we now assume $$\begin{aligned}
\label{eq:scalings_asym}
\delta\lambda \simeq c^{\delta\lambda} (m^*)^2 \,, \quad
\bar \sigma' \simeq c^{\bar\sigma'} (m^*)^3 \,, \quad
\bar D' \simeq c^{\bar D'} m^* \,, \quad
\bar \sigma^{(3)} \simeq c^{\bar\sigma^{(3)}} m^* \,.\end{aligned}$$ Then we can again put $\rho = \rho^\text{sym} + \varphi$ and $\hat\rho = \hat\rho^\text{sym} + \hat\varphi$ into Eq. , expand the equations, and solve them with the boundary conditions order by order. The equations can be solved only if $$\begin{aligned}
\label{eq:ode_30_const_b}
c^{\delta\lambda} = \frac{4 \pi\bar D^2}{(\lambda_{\mathrm{c}}+ E) \bar\sigma \bar\sigma''}\left(\frac{c^{\bar D'}}{\bar D}-\frac{3 c^{\bar\sigma'} + c^{\bar\sigma^{(3)}}}{3\bar\sigma''}\right)+\frac{\bar D^2}{8 (\lambda_{\mathrm{c}}+ E) \bar\sigma \bar\sigma''^2}\left(\frac{4\pi^2 \bar D''}{\bar D} + \frac{\bar\sigma''^2 E^2}{\bar D^2} - \frac{\pi^2 \bar\sigma^{(4)}}{\bar\sigma''}\right).\end{aligned}$$ This relation does not mean that only three parameters among $\delta\lambda$, $\bar D'$, $\bar\sigma'$, and $\bar\sigma^{(3)}$ are mutually independent. The degree of freedom which is actually lost is $m^*$, whose value is obtained by combining Eqs. and .
Finally, using the solutions for $\varphi^m$ and $\hat\varphi^m$ up to the order of $m^3$, we can combine Eqs. and to obtain $$\begin{aligned}
\label{eq:landau_eq_asym}
{\cal L}(m) &=
-\frac{2\pi \bar D^2}{\bar\sigma\bar\sigma''} \, \bar\sigma' \, m
-\frac{(\lambda_{\mathrm{c}}+ E) \bar\sigma''}{4} \,\delta\lambda \, m^2
-\frac{2\pi \bar D(\bar D \bar\sigma^{(3)}-3\bar D'\bar\sigma'')}{9\bar\sigma\bar\sigma''}
\,m^3 \nonumber\\
&\quad + \left[\frac{\pi^2\bar D\left(4\bar D'' \bar\sigma'' - \bar D \bar\sigma^{(4)}\right)}{64\bar\sigma\bar\sigma''} + \frac{\bar\sigma''^2 E^2}{64\bar\sigma}
\right] m^4 + O(m^5),\end{aligned}$$ which is the expression for the Landau function presented in Eq. (14) of the main text. If $\bar\sigma' = 0$, $\bar D\bar\sigma^{(3)} \neq 3\bar D'\bar\sigma''$, and the coefficient of $m^4$ is positive, this Landau function implies discontinuous transitions at $$\begin{aligned}
\lambda_{\mathrm{d}}^\pm = \lambda_{\mathrm{c}}^\pm - \frac{128(\lambda_{\mathrm{c}}^\pm + E)}{27\bar\sigma''}\frac{(3\bar D'\bar\sigma''-\bar D\bar\sigma^{(3)})^2}{\pi^2 \bar D(4\bar D''\bar\sigma'' - \bar D\bar \sigma^{(4)}) + \bar\sigma''^3 E^2}\,.\end{aligned}$$ If $\bar D\bar\sigma^{(3)} = 3\bar D'\bar\sigma''$, the Landau function becomes identical to Eq. up to order $m^4$. This means that the system exhibits continuous transitions at $\lambda = \lambda_{\mathrm{c}}^\pm$, which are no longer symmetry-breaking transitions because the particle–hole symmetry is already broken by higher-order terms. Finally, we note that Eqs. and guarantee ${\cal L}'(m^*) = 0$, which is indeed a condition required for $m^*$.
Validity of the additivity principle {#ssec:additivity}
------------------------------------
So far we assumed that the additivity principle holds. One might be worried about possible time-dependent saddle-point solutions with a lower action. Here we prove that this is not the case for systems with equal boundary densities and a particle–hole symmetry. This is done by studying time-dependent perturbations of the symmetric profile given by $$\begin{aligned}
\rho(x,t) = \rho^\text{sym}(x) + \varphi(x,t) \,, \quad \hat\rho(x,t) = \hat\rho^\text{sym}(x) + \hat\varphi(x,t)\,,\end{aligned}$$ with the boundary conditions $$\begin{aligned}
\varphi(0,t) = \varphi(1,t) = \hat\varphi(0,t) = \hat\varphi(1,t) = 0 \,.\end{aligned}$$
As the first step, we linearize the time-dependent saddle-point equations with respect to these perturbations, which gives $$\begin{aligned}
\label{eq:phi_saddle_traj_eq}
\dot{\varphi} = \bar D \nabla^2 \varphi - \bar\sigma \nabla^2 \hat\varphi \,, \quad
\dot{\hat\varphi} = - \bar D \nabla^2 \hat\varphi - \frac{\bar\sigma''}{2}\lambda(\lambda+2E) \varphi \,.\end{aligned}$$ Using the Fourier transforms $$\begin{aligned}
\label{eq:phi_fourier}
\varphi(x,t) = \sum_{n = 1}^\infty \int_{-\infty}^\infty \frac{\mathrm{d}\omega}{2\pi} \,
\psi_n(\omega) \, e^{i \omega t} \sin(n\pi x) \,, \quad
\hat\varphi(x,t) = \sum_{n = 1}^\infty \int_{-\infty}^\infty \frac{\mathrm{d}\omega}{2\pi} \,
\hat{\psi}_n(\omega) \, e^{i \omega t} \sin(n\pi x) \,,\end{aligned}$$ we can rewrite Eq. as $$\begin{aligned}
i\omega\psi_n(\omega) = - n^2 \pi^2 \bar D \psi_n(\omega) + n^2 \pi^2 \bar\sigma \hat\psi_n(\omega) \,,
\quad
i\omega\hat\psi_n(\omega) = n^2 \pi^2 \bar D \hat\psi_n(\omega) - \frac{\bar\sigma''}{2}\lambda(\lambda+2E)\psi_n(\omega) \,.\end{aligned}$$ These linear equations have nonzero solutions for $\psi_n$ and $\hat\psi_n$ if and only if $$\begin{aligned}
\label{eq:lambda_sym_break}
\lambda(\lambda+2E) = \frac{2 \left(n^4 \pi^4 \bar D^2 + \omega^2\right)}{n^2 \pi^2 \bar\sigma \bar\sigma''} \,.\end{aligned}$$ This implies that Eq. allows a symmetry breaking by $\varphi(x,t) \sim e^{i\omega t}\sin(n\pi x)$ if $\lambda$ is equal to $$\begin{aligned}
\label{eq:lambda_cn_omega}
\lambda_{\text{c},n}^\pm(\omega) \equiv -E \pm \sqrt{E^2 + \frac{2 \left(n^4 \pi^4 \bar D^2 + \omega^2\right)}{n^2 \pi^2 \bar\sigma \bar\sigma''}} \,,\end{aligned}$$ with depends on both $n$ and $\omega$.
The rest of the proof is a repetition of the argument by which we identified the symmetry-breaking transition point in Sec. \[ssec:sym\_break\_eq\]. The transition occurs for the values of $n$ and $\omega$ which minimize $|\lambda_{\text{c},n}^\pm(\omega)|$. Since increasing $|\omega|$ has the same effect on $|\lambda_{\text{c},n}^\pm(\omega)|$ as increasing $n$ does, both parameters have the smallest possible value at the transition point, so that $n = 1$ and $\omega = 0$. This implies that the symmetry-breaking profile has the longest possible wavelength ($n = 1$) and zero frequency ($\omega =0$). The result is consistent with the value of $\lambda_{\mathrm{c}}$ obtained in Eq. . This shows that the additivity principle is valid at the transition point.
Unequal boundary densities {#sec:ueq_bcs}
==========================
We now discuss the case when the two particle reservoirs have unequal densities $\bar\rho_a = \bar\rho - \delta\rho$ and $\bar\rho_b = \bar\rho + \delta\rho$, so that the system has a boundary driving in addition to the possible bulk driving. Assuming the boundary driving to be small ($\delta\rho \ll 1$), we perturbatively obtain the linear corrections to the results obtained in Sec. \[sec:eq\_bcs\].
Symmetry-breaking transition point and the additivity principle
---------------------------------------------------------------
We first discuss how the transition point $\lambda_{\mathrm{c}}$ and the validity of the additivity principle are affected by the boundary driving $\delta\rho$. If $\delta\rho \neq 0$, the symmetric density and momentum profiles $\rho^\text{sym}(x)$ and $\hat\rho^\text{sym}(x)$ given by Eqs. and are no longer valid saddle-point solutions, because they are inconsistent with the boundary conditions for $\bar\rho_a$ and $\bar\rho_b$. Thus there must be corrections which alter the symmetric profiles as $$\begin{aligned}
\label{eq:rho_sym_ueq}
\rho^\text{sym}(x) \equiv \bar\rho + \delta\rho\,\rho_1(x) + O(\delta\rho^2) \,, \quad
\hat\rho^\text{sym}(x) \equiv \lambda x + \delta\rho\,\hat\rho_1(x) + O(\delta\rho^2) \,.\end{aligned}$$ Solving the saddle-point equations perturbatively, the linear corrections are obtained as $$\begin{aligned}
\label{eq:rho_sym_ueq_corr}
\rho_1(x) = \csc \frac{\alpha(\lambda)}{2} \sin \left[\alpha(\lambda)\left(x-\frac{1}{2}\right)\right] \,, \quad
\hat\rho_1(x) = \frac{\bar D}{\bar \sigma} \rho_1(x) - \frac{2\bar D}{\bar\sigma} \left( x - \frac{1}{2}\right) \,,\end{aligned}$$ with $\alpha(\lambda)$ denoting $$\begin{aligned}
\label{eq:alpha}
\alpha(\lambda) \equiv \sqrt{\frac{\lambda(\lambda+2E)\bar\sigma\bar\sigma''}{2\bar D^2}} \,.\end{aligned}$$ It is easy to verify that the profiles given by Eqs. and are indeed symmetric under Eq. .
Based on the modified symmetric profiles obtained above, we identify the critical $\lambda$ at which symmetry-breaking saddle-point solutions are allowed. This can be done by repeating the procedure described in Sec. \[ssec:additivity\] while keeping track of the linear corrections in $\delta\rho$. After some algebra, we find that Eq. is modified to $$\begin{aligned}
\lambda_{\text{c},n}^\pm(\omega) \simeq -E \pm \sqrt{E^2 + \frac{2 \left(n^4 \pi^4 \bar D^2 + \omega^2\right)}{n^2 \pi^2 \bar\sigma \bar\sigma''}} + \frac{2 \bar D}{\bar \sigma}\delta\rho \,,\end{aligned}$$ which shows that up to order $\delta\rho$ the threshold is shifted by the same amount for each value of $n$ and $\omega$. As already discussed, the actual symmetry-breaking transition occurs for the values of $n$ and $\omega$ which minimize $|\lambda_{\text{c},n}^\pm(\omega)|$. Thus the transition occurs at the critical point given by the longest wavelength time-independent deviation ($n = 1$ and $\omega = 0$), as in the case of $\delta\rho = 0$. Thus the additivity principle remains valid up to order $\delta\rho$, and the transition point is shifted by $$\begin{aligned}
\label{eq:lambda_c_ueq}
\lambda_{\mathrm{c}}\to \lambda_{\mathrm{c}}+ \frac{2 \bar D}{\bar \sigma}\delta\rho \,.\end{aligned}$$
Derivation of the Landau theory
-------------------------------
The Landau theory for $\delta\rho \neq 0$ can be derived through a procedure which is almost the same as the one for $\delta\rho = 0$ described in Sec. \[ssec:landau\_eq\_sym\] and \[ssec:landau\_eq\_asym\], except that we need to keep track of the linear corrections in $\delta\rho$. These corrections appear in the symmetric profiles $\rho^\text{sym}$ and $\hat\rho^\text{sym}$ as obtained in Eq. , the deviations of the optimal profiles $\varphi$ and $\hat\varphi$ introduced in Eq. , and the amplitudes $c^{\delta\lambda}$, $c^{\bar\sigma'}$, $c^{\bar D'}$, and $c^{\bar\sigma^{(3)}}$ introduced in Eq. . After calculating $\varphi^m$ and $\hat\varphi^m$ up to order $m^3$, the Landau function is again obtained in the form of Eq. , with the only change being that $\delta\lambda$ is modified to $$\begin{aligned}
\delta\lambda \equiv \lambda - \lambda_{\mathrm{c}}- \frac{2\bar D}{\bar\sigma} \delta\rho\end{aligned}$$ for $\lambda_{\mathrm{c}}$ given by the unshifted form Eq. .
![\[fig:kls\] Density dependence of transport coefficients of the KLS model. The mobility coefficient $\sigma(\rho)$ is shown in solid lines, and the diffusion coefficient $D(\rho)$ in dashed lines for (a) particle–hole symmetric and (b) asymmetric systems.](kls_a.pdf "fig:"){width="49.00000%"} ![\[fig:kls\] Density dependence of transport coefficients of the KLS model. The mobility coefficient $\sigma(\rho)$ is shown in solid lines, and the diffusion coefficient $D(\rho)$ in dashed lines for (a) particle–hole symmetric and (b) asymmetric systems.](kls_b.pdf "fig:"){width="49.00000%"}
Transport coefficients of the Katz–Lebowitz–Spohn model {#sec:kls}
=======================================================
In the following we present explicit formulas for the transport coefficients of the Katz–Lebowitz–Spohn (KLS) model, which can be obtained by the methods of [@hager_minimal_2001; @Krapivsky_2013; @krapivsky_unpub]. The diffusion coefficient is given by $$\begin{aligned}
D(\rho) = \frac{\mathcal{J}(\rho)}{\chi(\rho)} \,,\end{aligned}$$ where $\mathcal{J}(\rho)$ is the average current of the totally asymmetric version of the model satisfying $$\label{current}
\mathcal{J}(\rho) = \frac{\nu [1+\delta(1-2\rho)]-\epsilon\sqrt{4\rho(1-\rho)}}{\nu^3} \,,$$ and $\chi(\rho)$ is the compressibility given by $$\label{compress:def}
\chi(\rho) = \rho(1-\rho)\sqrt{(2\rho-1)^2 + 4\rho(1-\rho)e^{-4\beta}} \,,$$ with $$\label{lambda:Krug}
\nu \equiv \frac{1+ \sqrt{(2\rho-1)^2 + 4\rho(1-\rho)e^{-4\beta}}}{\sqrt{4\rho(1-\rho)}} \,,
\quad e^{4\beta} \equiv \frac{1+\epsilon}{1-\epsilon} \,.$$ Then the mobility coefficient $\sigma(\rho)$ is obtained from the Einstein relation $$\begin{aligned}
\sigma(\rho) = 2 D(\rho) \, \chi(\rho) \,.\end{aligned}$$
Behaviors of the transport coefficients obtained from the above results are illustrated in Fig. \[fig:kls\]. When the system has a full particle–hole symmetry ($\delta = 0$), $\sigma(\rho)$ has a local extremum at $\rho = 1/2$, which becomes a local minimum for sufficiently strong repulsion ($\epsilon > 4/5$), as shown in Fig. \[fig:kls\](a). In the absence of the symmetry ($\delta \neq 0$), $\sigma(\rho)$ has a local extremum at a different value of $\rho$, which again becomes a local minimum for sufficiently large $\epsilon$ (see Fig. \[fig:kls\](b)).
Saddle point stability in the presence of bulk field {#sec:stability_field}
====================================================
An argument proposed by [@Shpielberg2016] states that a time-independent saddle-point solution $\rho_0(x)$ satisfying $$\begin{aligned}
\label{eq:shpielberg_condition}
D'(\rho_0(x))\sigma'(\rho_0(x)) \ge D(\rho_0(x))\sigma''(\rho_0(x))\end{aligned}$$ across the system ($0 \le x \le 1$) is stable regardless of the bulk field $E$. If true, the argument forbids symmetry-breaking transitions in bulk-driven systems with a local maximum of $\sigma(\rho)$ (e.g. in the WASEP), which are predicted by our study. In the following we address this apparent contradiction.
As stated in Eq. , the scaled CGF is obtained from minimization of the action $$\begin{aligned}
S \equiv \int_0^T \mathrm{d}t \, \int_0^1 \mathrm{d}x \,
\left[\hat\rho\dot\rho - H(\rho,\hat\rho)\right].\end{aligned}$$ Suppose that a time-independent saddle-point solution is given by $\rho(x,t) = \rho_0(x)$ and $\hat\rho(x,t) = \hat\rho_0(x)$. For this solution to be unstable, in the vicinity there must be another saddle-point solution $\rho(x,t) = \rho_0(x) + \varphi(x,t)$ and $\hat\rho(x,t) = \hat\rho_0(x) + \hat\varphi(x,t)$ whose value of $S$ is smaller. In [@Shpielberg2016] it was shown that the change of action due to $\varphi(x,t)$ and $\hat\varphi(x,t)$ is given by $$\begin{aligned}
\Delta S \simeq \int_0^T \mathrm{d}t \, \int_0^1 \mathrm{d}x \,
\left[\frac{D'(\rho_0)\sigma'(\rho_0)-D(\rho_0)\sigma''(\rho_0)}{4D(\rho_0)}\left(\nabla\hat\rho_0\right)\left(\nabla\hat\rho_0+2E\right)\varphi^2 + \frac{\sigma(\rho_0)}{2}\left(\nabla\hat\varphi\right)^2\right]\end{aligned}$$ up to the leading-order contributions. In the absence of the bulk field ($E = 0$), it is clear that Eq. implies $\Delta S \ge 0$, implying the stability of $\rho_0(x)$ and $\hat\rho_0(x)$. If $E \neq 0$, the sign of $\left(\nabla\hat\rho_0\right)\left(\nabla\hat\rho_0+2E\right)$ determines whether Eq. remains the sufficient condition for stability.
Defining $u \equiv \nabla\hat\rho_0 + E$, from the second equation of Eq. one finds [@Shpielberg2016]: $$\begin{aligned}
\frac{\nabla u}{u^2 - E^2} = -\frac{\sigma'(\rho_0)}{2D(\rho_0)}\, .\end{aligned}$$ The l.h.s. of this equation can be written as $$\begin{aligned}
\frac{\nabla u}{u^2 - E^2} = \begin{cases}
-\frac{1}{E}\nabla \mathrm{arctanh} \frac{u}{E} &\text{ if $|u| < |E|$,} \\
-\frac{1}{E}\nabla \mathrm{arccoth} \frac{u}{E} &\text{ if $|u| > |E|$.}
\end{cases}
\label{eq:diff_u}\end{aligned}$$ In [@Shpielberg2016], only the latter case is considered, so that one can write $u = E \coth (Eh)$ where $h$ satisfies $\nabla h = \sigma'(\rho)/\left[2D(\rho)\right]$. Then we obtain $$\begin{aligned}
\left(\nabla\hat\rho_0\right)\left(\nabla\hat\rho_0+2E\right)
= u^2 - E^2 = \frac{E^2}{\sinh^2 (Eh)} > 0 \,,\end{aligned}$$ which ensures that Eq. is still a sufficient condition for the stability of $\rho_0(x)$ and $\hat\rho_0(x)$.
However, close to the symmetry-breaking transition points of the WASEP, one can show that $|u| < |E|$ is satisfied across the system. In this case, the first case of Eq. should be used. This implies $u = E \tanh (Eh)$, from which we obtain $$\begin{aligned}
\left(\nabla\hat\rho_0\right)\left(\nabla\hat\rho_0+2E\right)
= u^2 - E^2 = -\frac{E^2}{\cosh^2 (Eh)} < 0 \,.\end{aligned}$$ Since the sign of $\left(\nabla\hat\rho_0\right)\left(\nabla\hat\rho_0+2E\right)$ is inverted, Eq. is no longer a sufficient condition for the stability of $\rho_0(x)$ and $\hat\rho_0(x)$. Thus, while Eq. gives the correct sufficient condition for stability in the absence of the bulk field $E$, it does not apply to the case when $\bar\sigma'' < 0$ and $E \neq 0$.
[^1]: We also note that there are DPTs associated with changes of mean behaviors as boundary conditions are varied, most notably those of the ASEP (see [@Blythe2007] for a review). These are phenomena of a very different origin.
[^2]: This expression for $\lambda_{\mathrm{c}}$ is consistent with [@Imparato2009], in which a condition for $\lambda_{\mathrm{c}}$ was derived for the special case of constant $D(\rho)$, quadratic $\sigma(\rho)$, and $E = 0$.
[^3]: We note that a similar Landau theory for DPTs in periodic systems has been constructed in [@Bodineau:2007iq].
[^4]: An exception could occur if the system is fine-tuned to satisfy $\bar D\bar\sigma^{(3)} = 3\bar D'\bar\sigma''$, see Appendix \[ssec:landau\_eq\_asym\].
[^5]: The argument for the criterion proposed by [@Shpielberg2016] does not apply to the case when $|\partial_x \hat\rho + E| < |E|$, which is true near the DPTs of Case 2. See Appendix \[sec:stability\_field\] for more details.
[^6]: Unlike the continuous media studied here, the systems studied in [@Ciliberto2013a; @*Ciliberto2013b] are described by only a few degrees of freedom. Even for such finite-dimensional systems, the DPTs similar to those described in this Letter can still occur, as is to be discussed in [@baek_unpub].
|
---
author:
- Kristina Frantzen
date: Oktober 2008
nocite: '[@atlas; @jamesliebeck]'
title: 'K3-surfaces with special symmetry'
---
Introduction {#introduction .unnumbered}
============
K3-surfaces are special two-dimensional holomorphic symplectic manifolds. They come equipped with a symplectic form $\omega$, which is unique up to a scalar factor, and their symmetries are naturally partitioned into symplectic and nonsymplectic transformations. An important class of K3-surfaces consists of those possessing an *antisymplectic involution*, i.e., a holomorphic involution $\sigma$ such that $\sigma^* \omega = - \omega$.
K3-surfaces with antisymplectic involution occur classically as branched double covers of the projective plane, or more generally of Del Pezzo surfaces. This construction is a prominent source of examples and plays a significant role in the classification of log Del Pezzo surfaces of index two (see the works of Alexeev and Nikulin e.g.in [@AlexNikulin] and the classification by Nakayama [@Nakayama]). Moduli spaces of K3-surfaces with antisymplectic involution are studied by Yoshikawa in [@yoshikawaInvent], [@yoshikawaPreprint], and lead to new developments in the area of automorphic forms.
In this monograph we study K3-surfaces with antisymplectic involution from the point of view of symmetry. On a K3-surface $X$ with antisymplectic involution it is natural the consider those holomorphic symmetries of $X$ compatible with the given structure $(X,\omega, \sigma)$. These are symplectic automorphisms of $X$ commuting with $\sigma$.
Given a finite group $G$ one wishes to understand if it can act in the above fashion on a K3-surface $X$ with antisymplectic involution $\sigma$. If this is the case, i.e., if there exists a holomorphic action of $G$ on $X$ such that $g^* \omega = \omega$ and $g \circ
\sigma = \sigma \circ g$ for all $g \in G$, then the structure of $G$ can yield strong constraints on the geometry of $X$. More precisely, if the group $G$ has rich structure or large order, it is possible to obtain a precise description of $X$. This can be considered the guiding classification problem of this monograph.
In Chapter \[chapterlarge\] we derive a classification of K3-surfaces with antisymplectic involution centralized by a group of symplectic automorphisms of order greater than or equal to 96. We prove (cf. Theorem \[roughclassi\]):
\[1\] Let $X$ be a K3-surface with a symplectic action of $G$ centralized by an antisymplectic involution $\sigma$ such that $\mathrm{Fix}(\sigma)\neq \emptyset$. If $|G|>96$, then $X/\sigma$ is a Del Pezzo surface and $\mathrm{Fix}(\sigma)$ is a smooth connected curve $C$ with $ g(C)\geq
3$.
By a theorem due to Mukai [@mukai] finite groups of symplectic transformations on K3-surfaces are characterized by the existence of a certain embedding into a particular Mathieu group and are subgroups of eleven specified finite groups of maximal symplectic symmetry. This result naturally limits our considerations and has led us to consider the above classification problem for a group $G$ from this list of eleven *Mukai groups*.
Theorem \[1\] above can be refined to obtain a complete classification of K3-surfaces with a symplectic action of a Mukai group centralized by an antisymplectic involution with fixed points (cf. Theorem \[thm mukai times invol\]).
Let $G$ be a Mukai group acting on a K3-surface $X$ by symplectic transformations. Let $\sigma $ be an antisymplectic involution on $X$ centralizing $G$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then the pair $(X,G)$ can be found in Table \[Mukai times invol\].
In addition to a number of examples presented by Mukai we find new examples of K3-surfaces with maximal symplectic symmetry as equivariant double covers of Del Pezzo surfaces.
It should be emphasized that the description of K3-surfaces with given symmetry does however not necessary rely on the size of the group or its maximality and a classification can also be obtained for rather small subgroups of the Mukai groups. In order to illustrate that the approach does rather depend on the structure of the group, we prove a classification of K3-surfaces with a symplectic action of the group $C_3 \ltimes C_7$ centralized by an antisymplectic involution in Chapter \[chapterC3C7\]. The surfaces with this given symmetry are characterized as double covers of $\mathbb P_2$ branched along invariant sextics in a precisely described one-dimensional family $\mathcal M$ (Theorem \[mainthmc3c7\]).
The K3-surfaces with a symplectic action of $G = C_3 \ltimes C_7$ centralized by an antisymplectic involution $\sigma$ are parametrized by the space $\mathcal M$ of equivalence classes of sextic branch curves in $\mathbb P_2$.
The group $C_3 \ltimes C_7$ is a subgroup of the simple group $L_2(7)$ of order 168 which is among the Mukai groups. The actions of $L_2(7)$ on K3-surfaces have been studied by Oguiso and Zhang [@OZ168] in an a priori more general setup. Namely, they consider finite groups containing $L_2(7)$ as a proper subgroup and obtain lattice theoretic classification results using the Torelli theorem. Since a finite group containing $L_2(7)$ as a proper subgroup posseses, in the cases considered, an antisymplectic involution centralizing $L_2(7)$, we can apply Theorem \[thm mukai times invol\] and improve the existing result (cf. Theorem \[improve OZ\]).
All classification results summarized above are proved by applying the following general strategy.
The quotient of a K3-surface by an antisymplectic involution $\sigma$ with fixed points centralized by a finite group $G$ is a rational $G$-surface $Y$. We apply an equivariant version of the minimal model program respecting finite symmetry groups to the surface $Y$. Chapter \[chapter mmp\] is dedicated to a detailed derivation of this method, a brief outline of which can also be found in the book of Kollár and Mori ([@kollarmori] Example 2.18, see also Section 2.3 in [@Mori]). In the setup of rational surfaces it leads to the well-known classification of $G$-minimal rational surfaces ([@maninminimal], [@isk]).
Equivariant Mori reduction and the theory of $G$-minimal models have applications in various different context and can also be generalized to higher dimensions. Initiated by Bayle and Beauville in [@Bayle], the methods have been employed in the classification of subgroups of the Cremona group $\mathrm{Bir}(\mathbb P_2)$ of the plane for example by Beauville and Blanc ([@Beauville], [@BeauBlancPrime], [@PhDBlanc]), [@Blanc1], etc.), de Fernex [@fernex], Dolgachev and Iskovskikh [@DolgIsk], and Zhang [@ZhangRational].
The equivariant minimal model $Y_\mathrm{min}$ of $Y$ is obtained from $Y$ by a finite number of blow-downs of (-1)-curves. Since individual (-1)-curves are not necessarily invariant, each reduction step blows down a number of disjoint (-1)-curves. The surface $Y_\mathrm{min}$ is, in all cases considered, a Del Pezzo surface.
Using detailed knowledge of the equivariant reduction map $Y \to Y_\mathrm{min}$, the shape of the invariant set $\mathrm{Fix}_X(\sigma)$, and the equivariant geometry of Del Pezzo surfaces, we classify $Y$, $Y_\mathrm{min}$ and $\mathrm{Fix}_X(\sigma)$ and can describe $X$ as an equivariant double cover of a possibly blown-up Del Pezzo surface. Besides the book of Manin, [@manin], our analysis relies, to a certain extend, on Dolgachev’s discussion of automorphism groups of Del Pezzo surfaces in [@dolgachev], Chapter 10.
In addition to classification, this method yields a multitude of new examples of K3-surfaces with given symmetry and a more geometric understanding of existing examples. It should be remarked that a number of these arise when the reduction $Y \to Y_{\mathrm{min}}$ is nontrivial.
In the last two chapters we present two different generalizations of our classification strategy for K3-surfaces with antisymplectic involution.
One of our starting points has been the study of K3-surfaces with $L_2(7)$-symmetry by Oguiso and Zhang mentioned above. Apart from a classification result for K3-surfaces with an action the group $L_2(7)\times C_4$, they also show that there does not exist a K3-surface with an action of a the group $L_2(7)\times C_3$. We give an independent proof of this result in Chapter \[chapter non exist\]. Assuming the existence of such a surface and following the strategy above, we consider the quotient by the nonsymplectic action of $C_3$ and apply the equivariant minimal model program to its desingularization. Combining this with additional geometric consideration we reach a contradiction.
In the last chapter we consider K3-surfaces $X$ with an action of a finite group $\tilde G$ which contains an antisymplectic involution $\sigma$ but is not of the form $\tilde G_\mathrm{symp} \times \langle
\sigma \rangle$. Since the action of $\tilde G_\mathrm{symp}$ does not descend to the quotient $X/\sigma$ we need to restrict our considerations to the centralizer of $\sigma$ inside $\tilde G$. This strategy is exemplified for a finite group $\tilde A_6$ characterized by the short exact sequence $\{\mathrm{id}\} \to A_6 \to \tilde A_6
\to C_4 \to \{\mathrm{id}\}$. In analogy to the $L_2(7)$-case, the action of $\tilde A_6$ on K3-surfaces has been studied by Keum, Oguiso, and Zhang ([@KOZLeech], [@KOZExten]), and a characterization of $X$ using lattice theory and the Torelli theorem has been derived. Since the existing realization of $X$ does however not reveal its equivariant geometry, we reconsider the problem and, though lacking the ultimate classification, find families of K3-surfaces with $D_{16}$-symmetry, in which the $\tilde A_6$-surface is to be found, as branched double covers. These families are of independent interest and should be studied further. In particular, it remains to find criteria to identify the $\tilde A_6$-surface inside these families. Possible approaches are outlined at the end of Chapter \[chapterA6\].
Since none of our results depends on the Torelli theorem, our approach to the classification problem allows generalization to fields of appropriate positive characteristic. This possible direction of further research was proposed to the author by Prof.Keiji Oguiso. Another potential further development would be the adaptation of the methods involved in the present work to related questions in higher dimensions.
*Research supported by Studienstiftung des deutschen Volkes and Deutsche Forschungsgemeinschaft*.
Finite group actions on K3-surfaces
===================================
This chapter is devoted to a brief introduction to finite groups actions on K3-surfaces and presents a number of basic, well-known results: We consider quotients of K3-surfaces by finite groups of symplectic or nonsymplectic automorphisms. It is shown that the quotient of a K3-surface by a finite group of symplectic automorphisms is a K3-surface, whereas the quotient by a finite group containing nonsymplectic transformations is either rational or an Enriques surface. Our attention concerning nonsymplectic automorphisms is then focussed on antisymplectic involutions and the description of their fixed point set. The chapter concludes with Mukai’s classification of finite groups of symplectic automorphisms on K3-surfaces and a discussion of basic examples.
Basic notation and definitions
------------------------------
Let $X$ be a n-dimensional compact complex manifold. We denote by $\mathcal O_X$ the sheaf of holomorphic functions on $X$ and by $\mathcal K_X$ its canonical line bundle. The *$i^{\text{th}}$ Betti number of $X$* is the rank of the free part of $H_i(X)$ and denoted by $b_i(X)$.
A *surface* is a compact connected complex manifold of complex dimension two. A *curve* on a surface $X$ is an irreducible 1-dimensional closed subspace of $X$. The (arithmetic) genus of a curve $C$ is denoted by $g(C)$ .
A *K3-surface* is a surface $X$ with trivial canonical bundle $\mathcal K_X$ and $b_1(X)=0$.
Note that a K3-surface is equivalently characterized if the condition $b_1(X)=0$ is replaced by $q(X) = \mathrm{dim}_\mathbb C H^1(X, \mathcal O_X) =0$ or $\pi_1(X) =\{\mathrm{id}\}$, i.e., $X$ is simply-connected. Examples of K3-surfaces arise as Kummer surfaces, quartic surfaces in $\mathbb P_3$ or double coverings of $\mathbb P_2$ branched along smooth curves of degree six.
Let $X$ be a K3-surface. Triviality of $\mathcal K _X$ is equivalent to the existence of a nowhere vanishing holomorphic 2-form $\omega$ on $X$. Any 2-form on $X$ can be expressed as a complex multiple of $\omega$. We will therefore mostly refer to $\omega$ (or $\omega _X$) as “the” holomorphic 2-form on $X$ . We denote by $\mathrm{Aut}_\mathcal O (X)=\mathrm{Aut}(X)$ the group of holomorphic automorphisms of $X$ and consider a (finite) subgroup $G \hookrightarrow \mathrm{Aut}(X)$. If the context is clear, the abstract finite group $G$ is identified with its image in $\mathrm{Aut}(X)$. The group $G$ is referred to as a transformation group, symmetry group or automorphism group of $X$. Note that our considerations are independent of the question whether the group $\mathrm{Aut}(X)$ is finite or not. The order of $G$ is denoted by $|G|$.
The action of $G$ on $X$ is called *symplectic* if $\omega$ is $G$-invariant, i.e., $g^*\omega = \omega$ for all $g \in G$.
For a finite group $G < \mathrm{Aut}(X)$ we denote by $G_\mathrm{symp}$ the subgroup of symplectic transformations in $G$ . This group is the kernel of the homomorphism $\chi : G \to \mathbb{C}^*$ defined by the action of $G$ on the space of holomorphic 2-forms $\Omega^2(X) \cong \mathbb C \omega$. It follows that $G$ fits into the short exact sequence $$\{\mathrm{id}\} \to G_\mathrm{symp} \to G \to C_n \to \{\mathrm{id}\}$$ for some cyclic group $C_n$. If both $G_\text{symp}$ and $C_n \cong G/G_\text{symp}$ are nontrivial, then $G$ is called a symmetry group of *mixed type* .
Quotients of K3-surfaces
------------------------
Let $X$ be a surface and let $G < \mathrm{Aut}(X)$ be a finite subgroup of the group of holomorphic automorphisms of $X$. The orbit space $X/G$ carries the structure of a reduced, irreducible, normal complex space of dimension 2 where the sheaf of holomorphic functions is given by the sheaf $G$-invariant functions on $X$. In many cases, the quotient is a singular space. The map $X \to X/G$ is referred to as a quotient map or a covering (map).
For reduced, irreducible complex spaces $X,Y$ of dimension 2 a proper holomorphic map $f: X \to Y$ is called *bimeromorphic* if there exist proper analytic subsets $A \subset X$ and $B \subset Y$ such that $f: X \backslash A \to Y \backslash B$ is biholomorphic. A holomorphic, bimeromorphic map $f: X \to Y$ with $X$ smooth is a *resolution of singularities of $Y$* .
A resolution of singularities $f: X \to Y$ is called *minimal* if it does not contract any (-1)-curves, i.e., there is no curve $E \subset X$ with $E \cong \mathbb{P}_1$ and $E^2=-1$ such that $f(E) = \{\text{point}\}$.
Every normal surface $Y$ admits a minimal resolution of singularities $f: X \to Y$ which is uniquely determined by $Y$. In particular, this resolution is equivariant.
### Quotients by finite groups of symplectic transformations
In the study and classification of finite groups of symplectic transformations on K3-surfaces, the following well-known result has proved to be very useful (see e.g. [@NikulinFinite])
\[K3quotsymp\] Let $X$ be a K3-surface, $G$ be a finite group of automorphisms of $X$ and $f: Y \to X/G$ be the minimal resolution of $X/G$. Then $Y$ is a K3-surface if and only if $G$ acts by symplectic transformations.
For the reader’s convenience we give a detailed proof of this theorem. We begin with the following lemma.
\[betti\] Let $X$ be a simply-connected surface, $G$ be a finite group of automorphisms and $f: Y \to X/G$ be an arbitrary resolution of singularities of $X/G$. Then $b_1(Y) =0$.
We denote by $\pi_1(Y)$ the fundamental group of $Y$ and by $[\gamma] \in \pi_1(Y)$ the homotopy equivalence class of a closed continuous path $\gamma$. The first Betti number is the rank of the free part of $$H_1(Y) = \pi_1(Y) / [\pi_1(Y), \pi_1(Y)].$$ We show that for each $ [\gamma] \in \pi_1(Y)$ there exists $N \in \mathbb N$ such that $[\gamma]^N =0$, i.e., $\gamma ^N$ is homotopic to zero for some $N \in \mathbb N$ . It then follows that $H_1(Y)$ is a torsion group and $b_1(Y) =0$.
Let $C \subset X/G$ be the union of branch curves of the covering $q: X \to X/G$, let $P \subset X/G$ be the set of isolated singularities of $X/G$, and $E \subset Y$ be the exceptional locus of $f$. Let $\gamma: [0,1] \to Y$ be a closed path in $Y$. By choosing a path homotopic to $\gamma$ which does not intersect $E \cup f^{-1}(C)$ we may assume without loss of generality that $\gamma \cap (E \cup f^{-1}(C)) =
\emptyset$.
The path $\gamma$ is mapped to a closed path in $(X/G)\backslash (C \cup P)$ which we denote also by $\gamma$. The quotient $q:X \to X/G$ is unbranched outside $C \cup P$ and we can lift $\gamma$ to a path $\widetilde{\gamma}$ in $X$. Let $\widetilde{\gamma}(0) = x \in X$, then $\widetilde{\gamma}(1) = g.x$ for some $g
\in G$. Since $G$ is a finite group, it follows that $\widetilde{\gamma^N}$ is closed for some $N \in \mathbb N$.
As $X$ is simply-connected, we know that also $X \backslash q^{-1}(P)$ is simply-connected. So $\widetilde{\gamma^N }$ is homotopic to zero in $X\backslash q^{-1}(P)$. We can map the corresponding homotopy to $(X/G)\backslash P$ and conclude that $\gamma^N$ is homotopic to zero in $(X/G)\backslash P$. It follows that $\gamma
^N$ is homotopic to zero in $Y\backslash E$ and therefore in $Y$.
We let $E \subset Y$ denote the exceptional locus of the map $f:Y \to X/G$. If $Y$ is a K3-surface, let $\omega_Y$ denote the nowhere vanishing holomorphic 2-form on $Y$. Let $(X/G)_\text{reg}$ denote the regular part of $X/G$. Since $f|_{Y\backslash E}: Y\backslash E \to (X/G)_\text{reg}$ is biholomorphic, this defines a holomorphic 2-form $\omega_{(X/G)_\text{reg}}$ on $(X/G)_\text{reg}$. Pulling this form back to $X$, we obtain a $G$-invariant holomorphic 2-form on $\pi^{-1}((X/G)_\text{reg}) = X \backslash \{p_1,\dots p_k\}$. This extends to a nonzero, i.e., not identically zero, $G$-invariant holomorphic 2-form on $X$. In particular, any holomorphic 2-form on $X$ is $G$-invariant and the action of $G$ is by symplectic transformations.
Conversely, if $G$ acts by symplectic transformations on $X$, then $\omega_X$ defines a nowhere vanishing holomorphic 2-form on $(X/G)_\text{reg}$ and on $ Y \backslash E$ . Our aim is to show that it extends to a nowhere vanishing holomorphic 2-form on $Y$. In combination with Lemma \[betti\] this yields that $Y$ is a K3-surface.
Locally at $p \in X$ the action of $G_p$ can be linearized. I.e., there exist a neighbourhood of $p$ in $X$ which is $G_p$-equivariantly isomorphic to a neighbourhood of $0 \in \mathbb C^2$ with a linear action of $G_p$. A neighbourhood of $\pi(p) \in X/G$ is isomorphic to a neighbourhood of the origin in $\mathbb C^2 / \Gamma$ for some finite subgroup $\Gamma < \mathrm{SL}(2,\mathbb C)$. In particular, the points with nontrivial isotropy are isolated. The singularities of $X/G$ are called simple singularities, Kleinian singularities, Du Val singularities or rational double points. Following [@shafarevic] IV.4.3, we sketch an argument which yields the desired extension result.
Let $X \times_{(X/G)} Y = \{ (x,y) \in X\times Y \, | \, \pi(x) = f(y) \}$ and let $ N$ be its normalization. Consider the diagram $$\begin{xymatrix}
{
X\ar[d]_{\pi} & \ar[l]_{p_X}\ar[d]^{p_Y}N \\
X/G & \ar[l]^f Y.
}
\end{xymatrix}$$
We let $\omega_X$ denote the nowhere vanishing holomorphic 2-form on $X$. Its pullback $p_X^*\omega_X$ defines a nowhere vanishing holomorphic 2-form on $N_\text{reg}$. Simultaneously, we consider the meromorphic 2-form $\omega _Y$ on $Y$ obtained by pulling back the 2-form on $X/G$ induced by the $G$-invariant 2-form $\omega_X$. By contruction, the pullback $p_Y^*\omega_Y $ coincides with the pullback $p_X^*\omega_X$ on $N_\text{reg}$.
Consider the finite holomorphic map $p_Y|_{N_\text{reg}}: N_\text{reg} \to p_Y(N_\text{reg}) \subset Y$. Since $p_Y^*\omega_Y $ is holomorphic on $N_\text{reg}$, one checks (by a calculation in local coordinates) that $\omega_Y$ is holomorphic on $p_Y(N_\text{reg}) = Y \backslash \{y_1, \dots y_k\}$ and consequently extends to a holomorphic 2-form on $Y$. Since $p_X^*\omega_X =p_Y^*\omega_Y $ is nowhere vanishing on $N_\text{reg}$, it follows that $\omega_Y$ defines a global, nowhere vanishing holomorphic 2-form on $Y$.
Let $g$ be a symplectic automorphism of finite order on a K3-surface $X$. Since K3-surfaces are simply-connected, the covering $X \to X/\langle g \rangle$ can never be unbranched. It follows that $g$ must have fixed points.
Using Theorem \[K3quotsymp\] we give an outline of Nikulin’s classification of finite Abelian groups of symplectic transformations on a K3-surface [@NikulinFinite]. Let $C_p$ be a cyclic group of prime order acting on a K3-surface $X$ by symplectic transformations and $Y$ be the minimal desingularization of the quotient $X/C_p$.
Notice that by adjunction the self-intersection number of a curve $D$ of genus $g(D)$ on a K3-surface is given by $D^2 = 2g(D)-2$. In particular, if $D$ is smooth, then $D^2 = -e(D)$.
The exceptional locus of the map $Y \to X/G$ is a union of (-2)-curves and one can calculate their contribution to the topological Euler characteristic $e(Y)$ in relation to $e(X/C_p)$. Let $n_p$ denote the number of fixed point of $C_p$ on $X$. Then $$\begin{aligned}
24 &= e(X) = p\cdot e(X/G) - n_p \\
24 &= e(Y) = e(X/G) + n_p \cdot p.\end{aligned}$$ Combining these formulas gives $n_p = 24/(p+1)$. For a general finite Abelian group $G$ acting symplectically on a K3-surface $X$, one needs to consider all possible isotropy groups $G_x$ for $x \in X$. By linearization, $G_x < \mathrm{SL}_2(\mathbb C)$. Since $G$ is Abelian, it follows that $G_x$ is cyclic and an analoguous formula relating the Euler characteristic of $X$, $X/G$, and $Y$ can be derived. A case by case study then yields Nikulin’s classification. In particular, we emphasize the following remark.
\[order of symp aut\] If $g \in \mathrm{Aut}(X)$ is a symplectic automorphism of finite order $n(g)$ on a K3-surface $X$, then $n(g)$ is bounded by eight and the number of fixed points of $g$ is given by the following table:
$n(g)$ 2 3 4 5 6 7 8
----------------------- --- --- --- --- --- --- ---
$|\mathrm{Fix}_X(g)|$ 8 6 4 4 2 3 2
: Fixed points of symplectic automorphisms on K3-surfaces[]{data-label="fix points symplectic"}
### Quotients by finite groups of nonsymplectic transformations
In this subsection we consider the quotient of a K3-surface $X$ by a finite group $G$ such that $G \neq G_\text{symp}$, i.e., there exists $g \in G$ such that $g^* \omega \neq \omega$. We prove
\[K3quotnonsymp\] Let $X$ be a K3-surface and let $G < \mathrm{Aut}(X)$ be a finite group such that $g^* \omega \ne \omega$ for some $g \in G $. Then either
- $X/G$ is rational, i.e., bimeromorphically equivalent to $\mathbb P_2$, or
- the minimal desingularisation of $X/G$ is a minimal Enriques surface and $$G/G_\text{symp} \cong C_2.$$ In this case, $\pi: X \to X/G$ is unbranched if and only if $G_\text{symp} = \{ \mathrm{id}\}$.
Before giving the proof, we establish the necessary notation and state two useful lemmata. We denote by $\pi: X \to X/G$ the quotient map. This map can be ramified at isolated points and along curves. Let $P=
\{p_1, \dots, p_n\}$ denote the set of singularities of $X/G$. For simplicity, the denote the correspondig subset $\pi^{-1}(P)$ of $X$ also by $P$. Outside $P$, the map $\pi$ is ramified along curves $C_i$ of ramification order $c_i+1$. We write $C = \sum c_i C_i$.
Let $r: Y \to X/G$ denote a minimal resolution of singularities of $X/G$. The exceptional locus of $r$ in $Y$ is denoted by $D$. As $Y$ is not necessarily a minimal surface, we denote by $p: Y \to Y_ \text{min}$ the sucessive blow-down of (-1)-curves. The union of exceptional curves of $p$ is denoted by $E$. $$\begin{xymatrix}{
C \subset \mathbf{X}\supset P \ar[d]^\pi \\
\pi(C) \subset \mathbf{X/G} \supset P & D \subset \mathbf{Y} \supset E \ar[l]_>>>>>>{r} \ar[d]^p\\
& \mathbf{Y_\text{min}}
}
\end{xymatrix}$$ The following two lemmata (cf. e.g. [@BPV] I.16 and Thm. I.9.1) will be useful in order to relate the canonical bundles of the spaces $X$, $(X/G)_\text{reg}$, $Y$ and $Y_ \text{min}$. For a divisor $D$ on a manifold $X$ we denote by $\mathcal O_X(D)$ the line bundle associated to $D$ .
\[adj1\] Let $X,Y$ be surfaces and let $\varphi:X \to Y$ be a surjective finite proper holomorphic map ramified along a curve $C$ in $X$ of ramification order $k$. Then $$\mathcal{K}_X = \varphi^*( \mathcal K_Y ) \otimes \mathcal O_X (C)^{\otimes(k-1)}.$$ More generally, if $\pi$ is ramified along a ramification divisor $R = \sum_i r_i R_i$, where $R_i$ is an irreducible curve and $r_i + 1$ is the ramification order of $\pi$ along $R_i$, then $$\mathcal{K}_X = \pi^*( \mathcal K_Y ) \otimes \mathcal O_X (R).$$
\[adj2\] Let $X$ be a surface and let $b: X
\to Y$ be the blow-down of a (-1)-curve $E \subset X$. Then $$\mathcal{K}_X = b^* ( \mathcal K_Y ) \otimes \mathcal O_X (E).$$
We present a proof of Theorem \[K3quotnonsymp\] using the Enriques Kodaira classification of surfaces.
The Kodaira dimension of the K3-surface $X$ is $\mathrm{kod}(X)=0$. The Kodaira dimension of $X/G$, which is defined as the Kodaira dimension of some resolution of $X/G$, is less than or equal to the Kodaira dimension of $X$. (c.f. Theorem 6.10 in [@ueno]), $$0=\mathrm{kod}(X) \geq \mathrm{kod}(X/G) = \mathrm{kod}(Y) =
\mathrm{kod}(Y_\text{min}) \in \{0, -\infty\}.$$ By Lemma \[betti\], the first Betti number of $Y$ and $Y_\text{min}$ is zero. If $\mathrm{kod}(Y) = - \infty$, then $Y$ is a smooth rational surface. If $\mathrm{kod}(Y) =\mathrm{kod}(Y_\text{min})= 0$, then, since $Y$ is not a K3-surface by Theorem \[K3quotsymp\], it follows that $Y_\text{min}$ is an Enriques surface.
If $Y_\text{min}$ is an Enriques surface, then $\mathcal K_{Y_\text{min}}^{\otimes 2}$ is trivial. Let $ s \in
\Gamma(Y_\text{min}, \mathcal{K}_{Y_\text{min}}^{\otimes 2})$ be a nowhere vanishing section. Consecutive application of Lemma \[adj2\] yields the following formula $$\mathcal{K}_Y^{\otimes 2} = (p^* \mathcal{K}_{Y_\mathrm{min}})^{\otimes 2} \otimes \mathcal O_Y(E)^{\otimes 2}= p^* (\mathcal{K}_{Y_\mathrm{min}}^{\otimes 2}) \otimes \mathcal O_Y(E)^{\otimes 2}.$$ Let $e \in \Gamma(Y, \mathcal O_Y(E)^{\otimes 2})$ and write $\tilde{s}= p^*(s) \cdot e$. This global section of $\mathcal{K}_Y^{\otimes 2}$ vanishes along $E$ and is nowhere vanishing outside $E$. By restricting $\tilde{s}$ to $Y\backslash D$ we obtain a section of $\mathcal{K}_{Y \backslash D}^{\otimes 2}$. Since $\pi$ is biholomorphic outside $D$, we can map the restricted section to $(X/G)\backslash P =
(X/G)_\text{reg}$ and obtain a section $\hat{s}$ of $\mathcal{K}_{(X/G)_\text{reg}}^{\otimes 2}$. Note that $\hat{s}$ is not the zero-section. If $E \neq \emptyset$, i.e., $Y$ is not minimal, let $E_1 \subset E$ be a (-1)-curve. The minimality of the resolution $r: Y \to X/G$ implies $E_1 \nsubseteq D$. It follows that $\hat{s}$ vanishes along the image of $E_1$ in $(X/G)_\text{reg}$
We may now apply Lemma \[adj1\] to the map $\pi|_{X\backslash P}$ to see $$\begin{aligned}
\mathcal{K}_{X \backslash P}^{\otimes 2} &= (\pi^*
\mathcal{K}_{(X/G)_\text{reg}})^{\otimes 2} \otimes \mathcal O_{X\backslash P}(C)^{\otimes 2}\\
&=\pi^* (\mathcal{K}_{(X/G)_\text{reg}}^{\otimes 2}) \otimes \mathcal O_{X\backslash P}(C)^{\otimes 2}.\end{aligned}$$ Let $c \in \Gamma(X\backslash P,
\mathcal O_{X\backslash P}(C))^{\otimes 2}$. Then $t := \pi^*
\hat{s} \cdot c \in \Gamma(X\backslash P, \mathcal{K}_{X\backslash P}^{\otimes2})$ is not the zero-section but vanishes along $C$ and along the preimage of the zeroes of $\hat{s}$.
Now $t$ extends to a holomorphic section $\tilde{t} \in \Gamma(X,
\mathcal{K}_X^{\otimes 2})$. Since $X$ is K3, it follows that both $\mathcal{K}_X$ and $\mathcal{K}_X^{\otimes 2}$ are trivial and $\tilde t$ must be nowhere vanishing. Consequently, both $E$ and $C$ must be empty. It follows that the map $\pi$ is at worst branched at points $P$ (not along curves) and the minimal resolution $Y$ of $X/G$ is a minimal surface. $$\begin{xymatrix}{
P \subset \mathbf{X} \ar[d]^\pi \\
P \subset \mathbf{X/G} & \mathbf{Y} \supset D \ar[l]_>>>>>>{r}
}
\end{xymatrix}$$ The section $\tilde{t}$ on $X$ is $G$-invariant by construction. Let $\omega$ be a nonzero section of the trivial bundle $\mathcal{K}_X$ such that $\tilde t = \omega ^2$. The action of $G$ on $X$ is nonsymplectic, therefore $\omega$ is not invariant but $\tilde{t}$ is. Hence $G$ acts on $\omega$ by multiplication with $\{1,-1\}$ and $G/G_\text{symp} \cong C_2$.
If $\pi: X \to X/G$ is unbranched, it follows that $\mathrm{Fix}_X(g) = \emptyset$ for all $g \in G \backslash \{\mathrm{id}\}$. Since symplectic automorphisms of finite order necessarily have fixed points, this implies $G_\text{symp} = \{\mathrm{id}\}$.
Conversely, if $G$ is isomorphic to $C_2$, it remains to show that the set $P=\{p_1,\dots,p_n\}$ is empty. Our argument uses the Euler characteristic $e$ of $X$, $X/G$, and $Y$. By chosing a triangulation of $X/G$ such that all points $p_i$ lie on vertices we calculate $24= e(X) = 2e(X/G)-n$. Blowing up the $C_2$-quotient singularities of $X/G$ we obtain $12=e(Y) = e(X/G) +n$. This implies $e(X/G) =12$ and $n=0$ and completes the proof of the theorem.
Antisymplectic involutions on K3-surfaces
-----------------------------------------
As a special case of the theorem above we consider the quotient of a K3-surface $X$ by an involution $\sigma \in \mathrm{Aut}(X)$ which acts on the 2-form $\omega$ by multiplication by $-1$ and is therefore called *antisymplectic involution*.
\[K3quotnonsympinvo\] Let $\pi:X \to X/\sigma$ be the quotient of a K3-surface by an antisymplectic involution $\sigma$. If $\mathrm{Fix}_X(\sigma) \neq \emptyset$, then $\mathrm{Fix}_X(\sigma)$ is a disjoint union of smooth curves and $X/\sigma$ is a smooth rational surface. Furthermore, $\mathrm{Fix}_X(\sigma) = \emptyset$ if and only if $X/\sigma$ is an Enriques surfaces.
If $\mathrm{Fix}_X(\sigma) \neq \emptyset$, then Theorem \[K3quotnonsymp\] and linearization of the $\sigma$-action at its fixed points yields the proposition. If $\mathrm{Fix}_X(\sigma) = \emptyset$, then $X \to X/\sigma$ is unbranched and $\mathrm{kod}(X) = \mathrm{kod}(X/G)$. It follows that $X/G$ is an Enriques surface.
In order to sketch Nikulin’s description of the fixed point set of an antisymplectic involution we summarize some information about the Picard lattice of a K3-surface.
### Picard lattices of K3-surfaces
Let $X$ be a complex manifold. The *Picard group of $X$* is the group of isomorphism classes of line bundles on $X$ and denoted by $\mathrm{Pic}(X)$. It is isomorphic to $H^1(X, \mathcal O_X^*)$. Let $\mathbb Z_X$ denote the constant sheaf on $X$ corresponding to $\mathbb Z$, then the exponential sequence $0 \to \mathbb Z_X \to \mathcal O_X \to \mathcal O_X^* \to 0$ induces a map $$\delta: H^1(X, \mathcal O_X^*) \to H^2(X, \mathbb Z).$$ Its kernel is the identity component $\mathrm{Pic}^0(X)$ of the Picard group. The quotient $\mathrm{Pic}(X) / \mathrm{Pic}^0(X)$ is isomorphic to a subgroup of $ H^2(X, \mathbb Z)$ and referred to as the *Néron-Severi group $NS(X)$ of $X$* . On the space $ H^2(X, \mathbb Z)$ there is the natural intersection or cupproduct pairing. The rank of the Néron-Severi group of $X$ is denoted by $\rho(X)$ and referred to as the *Picard number of $X$*
If $X$ is a K3-surface, then $H^1(X, \mathcal O_X)=\{0\}$ and $\mathrm{Pic}(X)$ is isomorphic to $NS(X)$. In particular, the Picard group carries the structure of a lattice, the *Picard lattice* of $X$, and is regarded as a sublattice of $ H^2(X, \mathbb Z)$, which is known to have signature $(3,19)$ (cf. VIII.3 in [@BPV]).
If $X$ is an algebraic K3-surface, i.e., the transcendence degree of the field of meromorphic functions on $X$ equals 2, then $\mathrm{Pic}(X)$ is a nondegenerate lattice of signature $(1, \rho -1)$ (cf. §3.2 in [@NikulinFinite]).
### The fixed point set of an antisymplectic involution
We can now present Nikulin’s classification of the fixed point set of an antisymplectic involution on a K3-surface [@NikulinFix].
\[FixSigma\] The fixed point set of an antisymplectic involution $\sigma$ on a K3-surface $X$ is one of the following three types: $$\text{1.)\ }\ \mathrm{Fix}(\sigma) = D_g \cup \bigcup_{i=1}^n R_i,
\quad \quad
\text{2.)\ }\ \mathrm{Fix}(\sigma) = D_1 \cup D'_1,
\quad ¸\quad
\text{3.)\ }\ \mathrm{Fix}(\sigma)= \emptyset,$$ where $D_g$ denotes a smooth curve of genus $g \geq 0$ and $\bigcup_{i=1}^n R_i$ is a possibly empty union of smooth disjoint rational curves. In case 2.), $D_1$ and $D'_1$ denote disjoint elliptic curves.
Assume there exists a curve $D_g$ of genus $g \geq 2$ in $\mathrm{Fix}(\sigma)$. By adjunction, this curve has positive self-intersection. We claim that each curve $D$ in $\mathrm{Fix}(\sigma)$ disjoint from $D_g$ is rational.
First note that the existence of an antisymplectic automorphism on $X$ implies that $X$ is algebraic (cf. Thm. 3.1 in [@NikulinFinite]) and therefore $\mathrm{Pic}(X)$ is a nondegenerate lattice of signature $(1, \rho -1)$.
If $D$ is elliptic, then $D^2 =0$, $D_g^2>0$ and $D \cdot D_g=0$ is contrary to the fact that $\mathrm{Pic}(X)$ has signature $(1,\rho-1)$. If $D$ is of genus $\geq 2$, then $D^2 >0$ and we obtain the same contradiction.
Now assume that there exists an elliptic curve $D_1$ in $\mathrm{Fix}(\sigma)$. By the considerations above, there may not be curves of genus $\geq 2$ in $\mathrm{Fix}(\sigma)$. If there are no further elliptic curves in $\mathrm{Fix}(\sigma)$, we are in case 1) of the classification. If there is another elliptic curve $D'_1$ in $\mathrm{Fix}(\sigma)$, this has to be linearly equivalent to $D_1$, as otherwise the intersection form of $\mathrm{Pic}(X)$ would degenerate on the span of $D_1$ and $D'_1$. The linear system of $D_1$ defines an elliptic fibration $X \to \mathbb{P}_1$. The induced action of $\sigma$ on the base may not be trivial since this would force $\sigma$ to act trivially in a neighbourhood of $D_1$ in $X$. It follows that the induced action of $\sigma$ on $\mathbb P_1$ has precisely two fixed points and that $\mathrm{Fix}(\sigma)$ contains no other curves than $D_1$ and $D'_1$. This completes the proof of the theorem.
Finite groups of symplectic automorphisms
-----------------------------------------
In preparation for stating Mukai’s classification of finite groups of symplectic automorphisms on K3-surfaces we present his list [@mukai] of symplectic actions of finite groups $G$ on K3-surfaces $X$. It is an important source of examples, many of these will occur in our later discussion. For the sake of brevity, at this point we do not introduce the notation of groups used in this table.
$G$ $|G|$ **K3-surface** $X$
---- ----------- ------- ---------------------------------------------------------------------------------------------------
1 $L_2(7)$ 168 $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$
2 $A_6$ 360 $\{\sum_{i=1}^6 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^6 x_i^3=0\} \subset \mathbb P_5$
3 $S_5$ 120 $\{\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0\} \subset \mathbb P_5$
4 $M_{20}$ 960 $\{ x_1^4+ x_2 ^4 + x_3^4 +x_4^4 +12 x_1x_2x_3x_4 = 0\} \subset \mathbb P_3$
5 $F_{384}$ 384 $\{ x_1^4+ x_2 ^4 + x_3^4 +x_4^4 = 0\} \subset \mathbb P_3$
6 $A_{4,4}$ 288 $\{x_1^2+x_2^2 +x_3^2 = \sqrt{3}x_4^2\} \cap $
$ \{x_1^2+ \omega x_2^2 +\omega^2 x_3^2 = \sqrt{3}x_5^2\} \cap $
$ \{x_1^2+\omega^2 x_2^2 +\omega x_3^2 = \sqrt{3}x_6^2\} \subset \mathbb P_5$
7 $T_{192}$ 192 $\{ x_1^4+ x_2 ^4 + x_3^4 +x_4^4 - 2 \sqrt{-3}(x_1^2x_2^2 + x_3^2x_4^2 = 0\} \subset \mathbb P_3$
8 $H_{192}$ 192 $ \{x_1^2+x_3^2+x_5^2 = x_2^2 + x_4^2 + x_6^2\} \cap $
$ \{x_1^2+x_4^2 = x_2^2+x_5^2=x_3^2+x_6^2\} \subset \mathbb P_5$
9 $N_{72}$ 72 $\{ x_1^3+ x_2 ^3 + x_3^3 +x_4^3= x_1x_2 + x_3x_4+ x_5^2 = 0 \} \subset \mathbb P_4$
10 $M_9$ 72 Double cover of $\mathbb P_2$ branched along
$\{x_1^6+y_2^6 +x_3^6 -10(x_1^3x_2^3 + x_2^3x_3^3 +x_3^3x_1^3) =0\}$
11 $ T_{48}$ 48 Double cover of $\mathbb P_2$ branched along
$\{x_1x_2(x_1^4-x_2^4)+ x_3^6 =0\}$
: Finite groups of symplectic automorphisms on K3-surfaces[]{data-label="TableMukai"}
The following theorem (Theorem 0.6 in [@mukai]) characterizes finite groups of symplectic automorphisms on K3-surfaces.
\[mukaithm\] A finite group $G$ has an effective sympletic actions on a K3-surface if and only if it is isomorphic to a subgroup of one of the eleven groups in Table \[TableMukai\].
The “only if”-implication of this theorem follows from the list of eleven examples summarized in Table \[TableMukai\]. This list of examples is, however, far from being exhaustive. It is therefore desirable to find further examples of K3-surfaces where the groups from this list occur and describe or classify these surfaces with maximal symplectic symmetry..
\[maxsymp\] By Proposition 8.8 in [@mukai] there are no subgroup relations among the eleven groups in Mukai’s list. Therefore, the groups are *maximal finite groups of symplectic transformations*. We refer to the groups in this list also as *Mukai groups*.
### Examples of K3-surfaces with symplectic symmetry
We conclude this chapter by presenting two typical examples of K3-surface with symplectic symmetry.
\[L2(7)example\] The group $L_2(7) = \mathrm{PSL}(2, \mathbb F_7)=\mathrm{GL}_3(\mathbb F_2)$ is a simple group of order 168. It is generated by the three projective transformations $\alpha, \beta, \gamma$ of $\mathbb P_1( \mathbb F_7)$ given by $$\alpha(x) = x+1; \quad \beta(x) =2x; \quad \gamma(x) = -x^{-1}.$$ In terms of these generators, we define a three-dimensional representation of $L_2(7)$ by $$\alpha \mapsto
\begin{pmatrix}
\xi & 0 & 0\\
0 & \xi^2 & 0\\
0 & 0 & \xi^4
\end{pmatrix};\quad
\beta \mapsto
\begin{pmatrix}
0 & 0 & 1\\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix};\,
\gamma \mapsto \frac{-1}{\sqrt{-7}}
\begin{pmatrix}
a&b&c\\
b&c&a\\
c&a&b
\end{pmatrix}$$ where $
\xi=e^{\frac{2\pi i }{7}},\,
a=\xi^2-\xi^5,\,
b=\xi-\xi^6,\,
c=\xi^4-\xi^3,
$ and $\sqrt{-7}= \xi+\xi^2+\xi^4-\xi^3-\xi^5-\xi^6$. Klein’s quartic curve $$C_\text{Klein}= \{x_1x_2^3 + x_2x_3^3 + x_3x_1^3=0\} \subset \mathbb P_2$$ is invariant with respect to induced action of $L_2(7)$ on $\mathbb P_2$. Mukai’s example of a K3-surface with symplectic $L_2(7)$-symmetry is the smooth quartic hypersurface in $\mathbb P_3$ defined by $$X_\text{KM} = \{x_1x_2^3 + x_2x_3^3 + x_3x_1^3 + x_4^4=0\} \subset \mathbb P_3,$$ where the action of $L_2(7)$ is defined to be trivial on the coordinate $x_4$ and defined as above on $x_1,x_2,x_3$. Since $L_2(7)$ is a simple group, it follows that the action is effective and symplectic. The surface $ X_\text{KM}$ is called the *Klein-Mukai surface*. By construction, it is a cyclic degree four cover of $\mathbb P_2$ branched along Klein’s quartic curve. In fact, there is an action of the group $L_2(7) \times C_4$ on $X_\text{KM}$, where the action of $C_4$ is by nonsymplectic transformations. The Klein-Mukai surface will play an important role in Sections \[KMsurface\] and \[168\].
#### Cyclic coverings
Since many examples of K3-surfaces are constructed as double covers we discuss the construction of branched cyclic covers with emphasis on group actions induced on the covering space.
Let $Y$ be a surface such that Picard group of $Y$ has no torsion, i.e., there does not exist a nontrivial line bundle $E$ on $Y$ such that $E^{\otimes n}$ is trivial for some $n \in \mathbb N$.
Let $B$ be an effective and reduced divisor on $Y$ and suppose there exists a line bundle $L$ on $Y$ such that $\mathcal O_Y(B) = L^{\otimes n}$ and a section $s \in \Gamma( Y , L^{\otimes n})$ whose zero-divisor is $B$. Let $p: L \to L^{\otimes n}$ denote the bundle homomorphism mapping each element $(y,z) \in L$ for $y \in Y$ to $(y,z^n ) \in L^{\otimes n}$. The preimage $X = p^{-1}(\mathrm{Im}(s))$ of the image of $s$ is an analytic subspace of $L$. The bundle projection $L \to Y$ restricted to $X$ defines surjective holomorphic map $X \to Y$ of degree $n$. $$\begin{xymatrix}{
X \subset L \ar[r]^p \ar[d]& L^{\otimes n} \supset \mathrm{Im}(s) \ar[d]\\
Y \ar[r]_{\mathrm{id}}& Y \ar@/_1pc/[u]_s
}
\end{xymatrix}$$ Since $\mathrm{Pic}(Y)$ is torsion free, the line bundle $L$ is uniquely determined by $B$. It follows than $X$ is uniquely determined and we refer to $X$ as *the* cyclic degree $n$ covering of $Y$ branched along $B$. We note that $X$ is normal and irreducible. It is smooth if the divisor $B$ is smooth. (cf. I.17 in [@BPV])
Let $G$ be a finite group in $\mathrm{Aut}(Y)$ and assume that the divisor $B$ is invariant, i.e., $gB =B$ for all $g \in G$. Then the pull-back bundle $g^* L^{\otimes n}$ is isomorphic to $L^{\otimes n}$. We consider the group $\mathrm{BAut}(L^{\otimes n})$ of bundle maps of $ L^{\otimes n}$ and the homomorphism $\mathrm{BAut}(L^{\otimes n}) \to \mathrm{Aut}(Y)$ mapping each bundle map to the corresponding automorphism of the base. Its kernel is isomorphic to $\mathbb C^*$. The observation $g^* L^{\otimes n} \cong L^{\otimes n}$ implies that the group $G$ is contained in the image of $\mathrm{BAut}(L^{\otimes n})$ in $\mathrm{Aut}(Y)$.
By assumption, the zero set of the section $s$ is $G$-invariant. The bundle map induced by $g^*$ maps the section $s$ to a multiple $\chi(g) s$ of $s$ for some character $\chi: G \to \mathbb C^*$. It follows that the bundle map $\tilde g$ induced by $\chi(g)^{-1} g^*$ stabilizes the section. The group $\tilde G = \{ \tilde g \, | \, g \in G \} \subset \mathrm{BAut}(L^{\otimes n})$ is isomorphic to $G$ and stabilizes $\mathrm{Im}(s) \subset L^{\otimes n} $.
In order to define a corresponding action on $X$, first observe that $g^* L \cong L$ for all $g \in G$. This follows from the observation that $ g^* L \otimes L^{-1}$ is a torsion bundle and the assumption that $\mathrm{Pic}(Y)$ has no torsion. As above, we deduce that the group $G$ is contained in the image of $\mathrm{BAut}(L)$ in $\mathrm{Aut}(Y)$. Let $\overline G$ be the preimage of $G$ in $\mathrm{BAut}(L)$. Then $\overline G$ is a central $\mathbb C^*$-extension of $G$, $$\{\mathrm{id}\} \to \mathbb C^* \to \overline G \to G \to \{\mathrm{id}\}.$$ The map $p: L \to L^{\otimes n}$ induces a homomorphism $p_*:\mathrm{BAut}(L) \to \mathrm{BAut}(L^{\otimes n})$. Its kernel is isomorphic to $C_n < \mathbb C^*$ and we consider the preimage $H = p_*^{-1}(\tilde G)$ in $\mathrm{BAut}(L)$. The group $H < \overline G$ is a central $C_n$-extension of $\tilde G \cong G$, $$\{\mathrm{id}\} \to C_n \to H \to G \to \{\mathrm{id}\}.$$ By construction, the subset $X \subset L$ is invariant with respect to $H$. This discussion proves the following proposition.
Let $Y$ by a surface such that $\mathrm{Pic}(Y)$ is torsion free and $G < \mathrm{Aut}(Y)$ be a finite group. If $B \subset Y$ is an effective, reduced, $G$-invariant divisor defined by a section $s \in \Gamma( Y, L^{\otimes n})$ for some line bundle $L$, then the cyclic degree $n$ covering $X$ of $Y$ branched along $B$ carries the induced action of a central $C_n$-extension $H$ of $G$ such that the covering map $\pi: X \to Y$ is equivariant.
For any finite subgroup $G < \mathrm{PSL}(3,\mathbb C)$ and any $G$-invariant smooth curve $C \subset \mathbb P_2$ of degree six, the double cover $X$ of $\mathbb P_2$ branched along $C$ is a K3-surface with an induced action of a degree two central extension of the group $G$. Many interesting examples (no. 10 and 11 in Mukai’s table) can be contructed this way. For example, the Hessian of Klein’s curve $\mathrm{Hess}(C_\text{Klein})$ is an $L_2(7)$-invariant sextic curve and the double cover of $\mathbb{P}_2$ branched along $\mathrm{Hess}(C_\text{Klein})$ is a K3-surface with a symplectic action of $L_2(7)$ centralized by the antisymplectic covering involution (cf. Section \[168\]).
Equivariant Mori reduction {#chapter mmp}
==========================
This chapter deals with a detailed discussion of Example 2.18 in [@kollarmori] (see also Section 2.3 in [@Mori]) and introduces a minimal model program for surfaces respecting finite groups of symmetries. Given a projective algebraic surface $X$ with $G$-action, in analogy to the usual minimal model program, one obtains from $X$ a $G$-minimal model $X_{G\text{-min}}$ by a finite number of $G$-equivariant blow-downs, each contracting a finite number of disjoint (-1)-curves. The surface $X_{G\text{-min}}$ is either a conic bundle over a smooth curve, a Del Pezzo surface or has nef canonical bundle. The case $G \cong C_2$ is also discussed in [@Bayle], the case $G \cong C_p$ for $p$ prime in [@fernex]. As indicated in the introduction, applications can be found throughout the literature.
The cone of curves and the cone theorem
---------------------------------------
Throughout this chapter we let $X$ be a smooth projective algebraic surface and let $\mathrm{Pic}(X)$ denote the group of isomorphism classes of line bundles on $X$.
A *divisor* on $X$ is a formal linear combination of irreducible curves $D = \sum a_i C_i$ with $a_i \in \mathbb Z$. A *1-cycle* on $X$ is a formal linear combination of irreducible curves $C = \sum b_i C_i$ with $b_i \in \mathbb R$. A 1-cycle is *effective* if $b_i \geq 0$ for all $i$.
We define a pairing $\mathrm{Pic}(X) \times \{\text{divisors}\} \to \mathbb Z$ by $(L,D) \mapsto L \cdot D = \deg(L|_D)$. Extending by linearity, this defines a pairing $\mathrm{Pic}(X) \times \{\text{1-cycles}\} \to \mathbb R$. We use this notation for the intersection number also for pairs of divisors $C$ and $D$ and write $C\cdot D = \deg(\mathcal O_X(D)|_C)$. Two 1-cycles $C,C'$ are called *numerically equivalent* if $L\cdot C = L \cdot C'$ for all $L \in \mathrm{Pic}(X)$. We write $C \equiv C'$. The numerical equivalence class of a 1-cycle $C$ is denoted by $[C]$. The space of all 1-cycles with real coefficients modulo numerical equivalence is a real vector space denoted by $N_1(X)$. Note that $N_1(X)$ is finite-dimensional.
Let $L$ be a line bundle on $X$ and let $L^{-1}$ denote its dual bundle. Then $L^{-1} \cdot C = -L \cdot C$ for all $[C]\in N_1(X)$. We therefore write $L^{-1} = -L$ in the following.
A line bundle $L$ is called *nef* if $L \cdot C \geq 0$ for all irreducible curves $C$.
We set $$NE(X) = \{ \sum a_i[C_i] \ | \ C_i \subset X \text{ irreducible curve},\, 0 \leq a_i \in \mathbb R\} \subset N_1(X).$$ The closure $\overline{NE}(X)$ of $NE(X)$ in $N_1(X)$ is called *Kleiman-Mori cone* or *cone of curves* on $X$.
For a line bundle $L$, we write $\overline{NE}(X)_{L\geq 0} = \{ [C]\in N_1(X) \ |\ L \cdot C\geq 0 \} \cap \overline{NE}(X)$. Analogously, we define $\overline{NE}(X)_{L\leq 0}$, $\overline{NE}(X)_{L > 0}$, and $\overline{NE}(X)_{L < 0}$.
Using this notation we phrase Kleiman’s ampleness criterion (cf. Theorem 1.18 in [@kollarmori])
A line bundle $L$ on $X$ is ample if and only if $\overline{NE}(X)_{L>0} = \overline{NE}(X)\backslash \{0\}$.
Let $V$ be a finite-dimensional real vector space . A subset $N \subset V$ is called *cone* if $0\in N$ and $N$ is closed under multiplication by positive real numbers. A subcone $M \subset N$ is called *extremal* if $u,v \in N $ satisfy $u,v \in M$ whenever $u+v \in M$. An extremal subcone is also referred to as an *extremal face*. A 1-dimensional extremal face is called *extremal ray*. For subsets $A, B \subset V$ we define $A+B := \{ a+b \,|\, a\in A, b\in B\}$.
The cone of curves $\overline{NE}(X)$ is a convex cone in $N_1(X)$ and the following cone theorem, which is stated here only for surfaces, describes its geometry (cf. Theorem 1.24 in [@kollarmori]).
\[conethm\] Let $X$ be a smooth projective surface and let $\mathcal{K}_X$ denote the canonical line bundle on $X$. There are countably many rational curves $C_i \in X$ such that $0 < -\mathcal{K}_X \cdot C_i \leq \mathrm{dim}(X) +1 $ and $$\overline{NE}(X) = \overline{NE}(X) _{\mathcal{K}_X \geq 0} + \sum_i \mathbb R_{\geq 0} [C_i].$$ For any $\varepsilon>0$ and any ample line bundle $L$ $$\overline{NE}(X) = \overline{NE}(X) _{(\mathcal{K}_X+\varepsilon L) \geq 0} + \sum_{\text{finite}} \mathbb R_{\geq 0} [C_i].$$
Surfaces with group action and the cone of invariant curves
-----------------------------------------------------------
Let $X$ be a smooth projective surface and let $G< \mathrm{Aut}_{\mathcal{O}}(X)$ be a group of holomorphic transformations of $X$. We consider the induced action on the space of 1-cycles on $X$. For $g \in G$ and an irreducible curve $C_i$ we denote by $g C_i$ the image of $C_i$ under $g$. For a 1-cycle $C = \sum a_i C_i$ we define $gC = \sum a_i (g C_i)$. This defines a $G$-action on the space of 1-cycles.
Let $C_1, C_2$ be 1-cycles and $C_1 \equiv C_2$. Then $gC_1 \equiv gC_2$ for any $g \in G$.
The 1-cycle $gC_1$ is numerically equivalent to $g C_2$ if and only if $L \cdot (gC_1) = L \cdot (gC_2)$ for all $L \in \mathrm{Pic}(X)$. For $g \in G$ and $L \in \mathrm{Pic}(X)$ let $g^*L$ denote the pullback of $L$ by $g$. The claim above is equivalent to $((g^{-1})^*L) \cdot (gC_1) = ((g^{-1})^*L) \cdot (gC_2)$ for all $L \in \mathrm{Pic}(X)$. Now $$((g^{-1})^*L) \cdot (gC_1) = \mathrm{deg}((g^{-1})^* L|_{gC_1}) = \mathrm{deg}(L|_{C_1}) = L \cdot C_1 = L \cdot C_2 = (g^{-1})^*L(gC_2)$$ for all $L \in\mathrm{Pic}(X)$.
This lemma allows us to define a $G$-action on $N_1(X)$ by setting $g[C] := [gC]$ and extending by linearity. We write $N_1(X)^G = \{ [C] \in N_1(X) \ | \ [C]=[gC] \text{ for all } g \in G\}$, the set of invariant 1-cycles modulo numerical equivalence. This space is a linear subspace of $N_1(X)$.
Since the cone $NE(X)$ is a $G$-invariant set it follows that its closure $\overline{NE}(X)$ is $G$-invariant. The subset of invariant elements in $\overline{NE}(X)$ is denoted by $\overline{NE}(X)^G$.
$
\overline{NE}(X)^G = \overline{NE(X) \cap N_1(X)^G}=\overline{NE}(X) \cap N_1(X)^G .
$
The subcone $\overline{NE}(X)^G$ of $\overline{NE}(X)$ is seen to inherit the geometric properties of $\overline{NE}(X)$ established by the cone theorem. Note however that the extremal rays of $\overline{NE}(X)^G$ are in general neither extremal in $\overline{NE}(X)$ (cf. Figure \[moribild\]) nor generated by classes of curves but by classes of 1-cycles.
The extremal rays of $\overline{NE}(X)^G$ are called *$G$-extremal rays* .
\[Gextremalray\] Let $G$ be a finite group and let $R$ be a $G$-extremal ray with $\mathcal{K}_X \cdot R<0$. Then there exists a rational curve $C_0$ such that $R$ is generated by the class of the 1-cycle $C = \sum_{g \in G} gC_0$.
Consider an $G$-extremal ray $R = \mathbb{R }_{\geq 0}[E]$ where $[E] \in \overline{NE}(X)^G \subset \overline{NE}(X)$. By the cone theorem (Theorem \[conethm\]) it can be written as $[E] = [\sum_i a_i C_i] +[F]$, where $ \mathcal{K}_X \cdot F \geq 0$, $a_i \geq 0$ and $C_i$ are rational curves. Let $|G|$ denote the order of $G$ and let $[GF] = G[F] = \sum_{g\in G} g[F]$. Since $g[E]=[E]$ for all $g\in G$ we can write $$|G| [E] = \sum_{g\in G} g[E] = \sum_{g\in G}([\sum_i a_igC_i] + g[F]) = \sum_i a_i G[C_i] + G[F].$$ The element $[\sum a_i (GC_i)] + [GF]$ of the extremal ray $\mathbb{R }_{\geq 0}[E]$ is decomposed as the sum of two elements in $\overline{NE}(X)^G$. Since $R$ is extremal in $\overline{NE}(X)^G$ both must lie in $R=\mathbb{R }_{\geq 0}[E]$ .
Consider $[GF] \in R$. Since $g^*\mathcal{K}_X \equiv \mathcal{K}_X$ for all $g \in G$, we obtain $$\mathcal{K}_X \cdot (GF) = \sum_{g\in G} \mathcal{K}_X \cdot (gF) = \sum_{g \in G}(g^*\mathcal{K}_X) \cdot F= |G|\mathcal{K}_X \cdot F \geq 0.$$ Since $\mathcal{K}_X \cdot R < 0$ by assumption this implies $[F]=0$ and $\mathbb{R }_{\geq 0}[E] = \mathbb{R }_{\geq 0}[\sum a_i (GCi)]$. Again using the fact that $R$ is extremal in $\overline{NE}(X)^G$, we conclude that each summand of $[\sum a_i (GC_i)]$ must be contained in $R=\mathbb{R }_{\geq 0}[E]$ and the extremal ray $\mathbb{R }_{\geq 0}[E]$ is therefore generated by $[GC_i]$ for some $C_i$ chosen such that $[GC_i] \neq 0$. This completes the proof of the lemma.
The contraction theorem and minimal models of surfaces {#mmp}
------------------------------------------------------
In this section, we state the contraction theorem for smooth projective surfaces. The proof of this theorem can be found e.g. in [@kollarmori] and needs to be modified slightly in order to give an equivariant contraction theorem in the next section.
Let $X$ be a smooth projective surface and let $F \subset \overline{NE}(X)$ be an extremal face. A morphism $\mathrm{cont}_F: X \to Z$ is called the *contraction of $F$* if
- $(\mathrm{cont}_F)_*\mathcal{O}_X = \mathcal{O}_Z$ and
- $\mathrm{cont}_F(C) = \{\text{point}\}$ for an irreducible curve $C\subset X$ if and only if $[C] \in F$.
The following result is known as the contraction theorem (cf. Theorem 1.28 in [@kollarmori]).
\[contractionthm\] Let $X$ be a smooth projective surface and $R \subset \overline{NE}(X)$ an extremal ray such that $\mathcal{K}_X \cdot R<0$. Then the contraction morphism $\mathrm{cont}_R: X \to Z$ exists and is one of the following types:
1. [$Z$ is a smooth surface and $X$ is obtained from $Z$ by blowing up a point. ]{}
2. [$Z$ is a smooth curve and $\mathrm{cont}_R:X \to Z $ is a minimal ruled surface over $Z$.]{}
3. [$Z$ is a point and $-\mathcal{K}_X$ is ample. ]{}
The contraction theorem leads to the minimal model program for surfaces: Starting from $X$, if $\mathcal{K}_X$ is not nef, i.e, there exists an irreducible curve $C$ such that $\mathcal{K}_X C < 0$, then $\overline{NE}(X)_{\mathcal{K}_X <0}$ is nonempty and there exists an extremal ray $R$ which can be contracted. The contraction morphisms either gives a new surface $Z$ (in case 1) or provides a structure theorem for $X$ which is then either a minimal ruled surface over a smooth curve (case 2) or isomorphic to $\mathbb P^2$ (case 3). Note that the contraction theorem as stated above only implies $-\mathcal{K}_X$ ample in case 3. It can be shown that $X$ is in fact $\mathbb P^2$. This is omitted here since the statement does not transfer to the equivariant setup. In case 1, we can repeat the procedure if $K_Z$ is not nef. Since the Picard number drops with each blow down, this process terminates after a finite number of steps. The surface obtained from $X$ at the end of this program is called a *minimal model* of $X$.
Let $E$ be a (-1)-curve on $X$. If $C$ is any irreducible curve on $X$, then $E \cdot C < 0$ if and only if $C =E$. It follows that $\overline{NE}(X) = \mathrm{span}(\mathbb R_{\geq 0}[E], \overline{NE}(X)_{E \geq 0})$. Now $E^2 = -1$ implies $ E \not\in \overline{NE}(X)_{E \geq 0}$ and $E$ is seen to generate an extremal ray in $\overline{NE}(X)$. By adjunction, $\mathcal K_X \cdot E < 0$. The contraction of the extremal ray $R = \mathbb R_{\geq 0}[E]$ is precisely the contraction of the (-1)-curve $E$. Conversely, each extremal contraction of type 1 above is the contraction of a (-1)-curve generating the extremal ray $R$.
Equivariant contraction theorem and $G$-minimal models
------------------------------------------------------
We state and prove an equivariant contraction theorem for smooth projective surfaces with finite groups of symmetries. Most steps in the proof are carried out in analogy to the proof of the standard contraction theorem.
\[equiContraction\] Let $G$ be a finite group, let $X$ be a smooth projective surface with $G$-action and let $R \subset \overline{NE}(X)^G$ be $G$-extremal ray. A morphism $\mathrm{cont}_R^G: X \to Z$ is called the *$G$-equivariant contraction of $R$* if
- $\mathrm{cont}_R^G$ is equivariant with respect to $G$
- $(\mathrm{cont}_R^G)_*\mathcal{O}_X = \mathcal{O}_Z$ and
- $\mathrm{cont}_R(C) = \{\text{point}\}$ for an irreducible curve $C\subset X$ if and only if $[GC] \in R$.
Let $G$ be a finite group, let $X$ be a smooth projective surface with $G$-action and let $R$ be a $G$-extremal ray with $ \mathcal{K}_X \cdot R <0$. Then $R$ can be spanned by the class of $C= \sum_{g \in G} gC_0$ for a rational curve $C_0$, the equivariant contraction morphism $\mathrm{cont}_R^G: X \to Z$ exists and is one of the following three types:
1. $C^2 <0$ and $gC_0$ are smooth disjoint (-1)-curves. The map $\mathrm{cont}_R^G: X \to Z$ is the equivariant blow down of the disjoint union $\bigcup_{g \in G} gC_0$.
2. $C^2 =0$ and any connected component of $C$ is either irreducible or the union of two (-1)-curves intersecting transversally at a single point. The map $\mathrm{cont}_R^G: X \to Z$ defines an equivariant conic bundle over a smooth curve .
3. $C^2 >0$ , $N_1(X)^G = \mathbb{R }$ and $\mathcal{K}_X^{-1}$ is ample, i.e., $X$ is a Del Pezzo surface. The map $\mathrm{cont}_R^G: X \to Z$ is constant, $Z$ is a point.
Let $R$ be a $G$-extremal ray with $\mathcal{K}_X \cdot R <0$. It follows from Lemma \[Gextremalray\] that the ray $R$ can be spanned by a 1-cycle of the form $C = GC_0$ for a rational curve $C_0$. Let $n = |GC_0|$ and write $C = \sum_{i=1}^n C_i$ where the $C_i$ correspond to $gC_0$ for some $g \in G$. We distinguish three cases according to the sign of the self-intersection of $C$.
**The case $C^2 <0$**
We write $0 > C^2 = \sum_i C_i^2 + \sum_{i\neq j}C_i \cdot C_j$. Since $C_i$ are effective curves we know $C_i \cdot C_j \geq 0$ for all $i \neq j$. Since all curves $C_i$ have the same negative self-intersection and by assumption, $\mathcal{K}_X \cdot C = \sum_i \mathcal{K}_X \cdot C_i = n(\mathcal{K}_X \cdot C_i) <0$ the adjunction formula reads $
2g(C_i) -2 = -2 = \mathcal{K}_X \cdot C_i + C_i^2
$. Consequently, $\mathcal{K}_X \cdot C_i = -1$ and $C_i^2 = -1$. It remains to show that all $C_i$ are disjoint. We assume the contrary and without loss of generality $C_1 \cap C_2 \neq \emptyset$. Now $gC_1 \cap g C_2 \neq \emptyset $ for all $g \in G$ and $\sum_{i \neq j} C_i \cdot C_j \geq n$. This is however contrary to $0>C^2 = \sum_i C_i^2 + \sum_{i\neq j}C_i \cdot C_j = -n + \sum_{i\neq j}C_i \cdot C_j$.
We let $\mathrm{cont}_R^G: X \to Z$ be the blow-down of $\bigcup_{g \in G} gC_0$ which is equivariant with respect to the induced action on $Z$ and fulfills $(\mathrm{cont}_R^G)_*\mathcal{O}_X = \mathcal{O}_Z$. If $D$ is an irreducible curve such that $\mathrm{cont}_R^G(D)= \{\text{point}\}$, then $D= gC_0$ for some $g \in G$. In particular, $GD= GC_0 =C$ and $[GD] \in R$. Conversely, if $[GD] \in R$ for some irreducible curve $D$, then $[GD]= \lambda [C]$ for some $\lambda \in \mathbb R _{\geq 0}$. Now $(GD)\cdot C = \lambda C^2 <0$. It follows that $D$ is an irreducible component of $C$.
**The case $C^2 >0$**
This case is treated in precisely the same way as the corresponding case in the standard contraction theorem. Our aim is to show that $[C]$ is in the interior of $\overline{NE}(X)^G$. This is a consequence of the following lemma.
Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. Then the set $Q = \{[E] \in N_1(X) \ |\ E^2 >0\}$ has two connected components $Q^+=\{ [E] \in Q \ |L \cdot E >0\}$ and $Q^-=\{ [E] \in Q \ |L \cdot E <0\}$. Moreover, $Q^+ \subset \overline{NE}(X)$.
This result follows from the Hodge Index Theorem (cf. Theorem IV.2.14 in [@BPV]) and the fact, that $E^2 >0$ implies that either $E$ or $-E$ is effective. For a proof of this lemma, we refer the reader to Corollary 1.21 in [@kollarmori].
We consider an effective cycle $C = \sum C_i$ with $C^2 >0$. By the above lemma, $[C]$ is contained in $Q^+$ which is an open subset of $N_1(X)$ contained in $\overline{NE}(X)$. It follows that $[C]$ lies in the interior of $\overline{NE}(X)$. The $G$-extremal ray $R = \mathbb{R }_{\geq 0} [C]$ can only lie in the interior if $\overline{NE}(X)^G= R$. By assumption $\mathcal{K}_X \cdot R <0$, so that $\mathcal{K}_X$ is negative on $\overline{NE}(X)^G \backslash \{0\}$ and therefore on $\overline{NE}(X) \backslash \{0\}$. The anticanonical bundle $\mathcal{K}_X^{-1}$ is ample by Kleiman ampleness criterion and $X$ is a Del Pezzo surface.
We can define a constant map $\mathrm{cont}_R^G$ mapping $X$ to a point $Z$ which is the equivariant contraction of $R = \overline{NE}(X)$ in the sense of Definition \[equiContraction\].
**The case $C^2 =0$**
Our aim is to show that for some $m >0$ the linear system $|mC|$ defines a conic bundle structure on $X$. The argument is seperated into a number of lemmata. For the convenience of the reader, we include also the proofs of well-known preparatory lemmata which do not involve group actions. Recall that $\mathcal{O}(D)$ denotes the line bundle associated to the divisor $D$ on $X$.
$H^2(X, \mathcal{O}(mC)) =0$ for $m \gg 0$.
By Serre’s duality (cf. Theorem I.5.3 in [@BPV]) $$h^2(X, \mathcal{O}(mC))= h^0(\mathcal{O}(-mC)\otimes \mathcal K_X).$$ Since $C$ is an effective divisor on $X$, it follows that $h^0(\mathcal{O}(-mC)\otimes \mathcal K_X)=0$ for $m \gg 0$.
For $m \gg 0$ the dimension $h^0(X,\mathcal{O}(mC))$ of $H^0(X,\mathcal{O}(mC))$ is at least two.
Let $m$ be such that $h^2(X, \mathcal{O}(mC)) =0$. For a line bundle $L$ on $X$ we denote by $\chi(L) = \sum_i(-1)^i h^i(X,L)$ the Euler characteristic of $L$. Using the theorem of Riemann-Roch (cf. Theorem V.1.6 in [@hartshorne]), $$\begin{aligned}
h^0(X, \mathcal{O}(mC)) &\geq h^0(X, \mathcal{O}(mC))- h^1(X, \mathcal{O}(mC))\\
& = h^0(X, \mathcal{O}(mC)) - h^1(X, \mathcal{O}(mC)) + h^2(X, \mathcal{O}(mC))\\
& = \chi(\mathcal{O}(mC))\\
& = \chi (\mathcal{O}) + \frac{1}{2}(\mathcal{O}(mC)\otimes\mathcal{K}_X^{-1})\cdot(mC)\\
& \overset{C^2=0}{=} \chi (\mathcal{O}) - \frac{m}{2}\mathcal{K}_X \cdot C.\end{aligned}$$ Now $\mathcal{K}_XC<0$ implies the desired behaviour of $h^0(X, \mathcal{O}(mC))$.
For a divisor $D$ on $X$ we denote by $|D|$ the complete *linear system of $D$*, i.e., the set of all effective divisors linearly equivalent to $D$. A point $ p \in X$ is called a *base point* of $|D|$ if $p \in \mathrm{support}(C)$ for all $C \in |D|$.
There exists $m' >0$ such that the linear system $|m'C|$ is base point free.
We first exclude a positive dimensional set of base points. Let $m$ be chosen such that $h^0(X, \mathcal{O}(mC)) \geq 2$. We denote by $B$ the *fixed part* of the linear system $|mC|$, i.e., the biggest divisor $B$ such that each $D \in |mC|$ can be decomposed as $D = B + E_D$ for some effective divisor $E_D$. The support of $B$ is the union of all positive dimensional components of the set of base points of $|mC|$. We assume that $B$ is nonempty. The choice of $m$ guarantees that $|mC|$ is not fixed, i.e., there exists $D \in |mC|$ with $D \neq B$. Since $\mathrm{supp}(B) \subset \{s=0\}$ for all $ s \in \Gamma(X, \mathcal O(mC))$, each irreducible component of $\mathrm{supp}(B)$ is an irreducible component of $C$ and $G$-invariance of $C$ implies $G$-invariance of the fixed part of $|mC|$. It follows that $B=m_0 C$ for some $m_0 < m$. Decomposing $|mC|$ into the fixed part $B = m_0C$ and the remaining *free part* $|(m-m_0)C|$ shows that some multiple $|m'C|$ for $m' >0$ has no fixed components. The linear system $|m'C|$ has no isolated base points since these would correspond to isolated points of intersection of divisors linearly equivalent to $m'C$. Such intersections are excluded by $C^2=0$.
We consider the base point free linear system $|m'C|$ and the induced morphism $$\begin{aligned}
\varphi =\varphi_{|m'C|}: &X \to \varphi(X) \subset \mathbb P(\Gamma(X,\mathcal{O}(m'C))^*)\\
&x \mapsto \{ s \in \Gamma(X,\mathcal{O}(m'C)) \, | \, s(x)=0 \}\end{aligned}$$ Since $C$ is $G$-invariant, it follows that $\varphi$ is an equivariant map with respect to action of $G$ on $\mathbb P(\Gamma(X,\mathcal{O}(m'C))^*)$ induced by pullback of sections.
Let us study the fibers of $\varphi$. Let $z$ be a linear hyperplane in $\Gamma(X,\mathcal{O}(m'C))$. By definition, $\varphi^{-1}(z)= \bigcap _{s\in z}(s)_0$ where $(s)_0$ denotes the zero set of the section $s$. Since $(s)_0$ is linearly equivalent to $m'C$ and $C^2=0$, the intersection $\bigcap _{s\in z}(s)_0$ does not consist of isolated points but all $(s)_0$ with $s \in z$ have a common component. In particular, each fiber is one-dimensional.
Let $f: X \to Z$ be the Stein factorization of $\varphi: X \to \varphi(X)$. The space $Z$ is normal and 1-dimensional, i.e., $Z$ is a smooth curve. Note that there is a $G$-action on the smooth curve $Z$ such that $f$ is equivariant.
The map $f: X \to Z$ defines an equivariant conic bundle, i.e., an equivariant fibration with general fiber isomorphic to $\mathbb{P}_1$.
Let $F$ be a smooth fiber of $f$. By construction, $F$ is a component of $(s)_0$ for some $s \in \Gamma(X,\mathcal{O}(m'C))$. We can find an effective 1-cycle $D$ such that $(s)_0 = F+D$. Averaging over the group $G$ we obtain $
\sum_{g\in G}gF + \sum_{g\in G}gD = \sum_{g\in G}g(s)_0
$. Recalling $(s)_0 \sim m'C $ and $[C] \in \overline{NE}(X)^G$ we deduce $$[\sum_{g\in G}gF + \sum_{g\in G}gD] = [\sum_{g\in G}g(s)_0] = m'[\sum_{g\in G}gC]= m |G| [C]$$ showing that $[\sum_{g\in G}gF + \sum_{g\in G}gD]$ in contained in the $G$-extremal ray generated by $[C]$. Now by the definition of extremality $[\sum_{g\in G}gF] = \lambda [C] \in \mathbb{R }^{>0}[C]$ and therefore $\mathcal{K}_X \cdot (\sum_{g\in G}gF) <0$. This implies $\mathcal{K}_X F<0$.
In order to determine the self-intersection of $F$, we first observe $(\sum_{g\in G}gF)^2= \lambda^2 C^2 =0$. Since $F$ is a fiber of a $G$-equivariant fibration, we know that $\sum_{g\in G}gF = kF + kF_1 + \dots + kF_l$ where $F, F_1, \dots F_l$ are distinct fibers of $f$ and $k \in \mathbb N ^{>0}$. Now $0=(\sum_{g\in G}gF)^2 = (l+1)k^2F^2$ shows $F^2=0$. The adjunction formula then implies $g(F)=0$ and $F$ is isomorphic to $\mathbb P_1$.
The map $\mathrm{cont}_R^G:=f$ is equivariant and fulfills $f_* \mathcal{O}_X = \mathcal{O}_Z$ by Stein’s factorization theorem. Let $D$ be an irreducible curve in $X$ such that $f$ maps $D$ to a point, i.e., $D$ is contained in a fiber of $f$. Going through the same arguments as above one checks that $[GD] \in R$. Conversely, if $D$ is an irreducible curve in $X$ such that $[GD] \in R$ it follows that $(GD) \cdot C=0$. If $D$ is not contracted by $f$, then $f(D) = Z$ and $D$ meets every fiber of $f$. In particular, $D \cdot C >0$, a contradiction. It follows that $D$ must be contracted by $f$.
This completes the proof of the equivariant contraction theorem.
The singular fibers of the conic bundle in case 2 of the theorem above are characterized by the following lemma.
\[singular fibers of conic bundle\] Let $R = \mathbb R ^{>0} [C]$ be a $\mathcal K_X$-negative $G$-extremal ray with $C^2=0$. Let $\mathrm{cont}_R^G:=f: X \to Z$ be the equivariant contraction of $R$ defining a conic bundle structure on $X$. Then every singular fiber of $f$ is a union of two (-1)-curves intersecting transversally.
Let $F$ be a singular fiber of $f$. The same argument as in the previous lemma yields that $\mathcal{K}_X \cdot F<0$ and $F^2 =0$. Since $F$ is connected, adjunction implies that the arithmetic genus of $F$ is zero and $\mathcal{K}_X \cdot F = -2$. It follows from the assumption on $F$ being singular that $F$ must be reducible. Let $F = \sum F_i$ be the decomposition into irreducible components. Now $g(F)=0$ implies $g(F_i)=0$ for all $i$.
We apply the same argument as above to the component $F_i$ of $F$: after averaging over $G$ we deduce that $GF_i$ is in the $G$-extremal ray $R$ and $ \mathcal K _X \cdot F_i <0$. Since $-2 = \mathcal K _X \cdot F = \sum \mathcal K _X \cdot F_i$, we may conclude that $F = F_1 +F_2$ and $F_i^2=-1$. The desired result follows.
$G$-minimal models of surfaces {#g-minimal-models-of-surfaces .unnumbered}
------------------------------
Let $X$ be a surface with $G$-action such that $\mathcal{K}_X$ is not nef, i.e., $\overline{NE}(X)_{\mathcal{K}_X<0}$ is nonempty.
There exists a $G$-extremal ray $R$ such that $\mathcal{K}_X \cdot R<0$.
Let $[C] \in \overline{NE}(X)_{\mathcal{K}_X<0}\neq \emptyset$ and consider $[GC] \in \overline{NE}(X)^G$. The $G$-orbit or $G$-average of a $\mathcal{K}_X$-negative effective curve is again $\mathcal{K}_X$-negative. It follows that $\overline{NE}(X)^G_{\mathcal{K}_X<0}$ is nonempty. Let $L$ be a $G$-invariant ample line bundle on $X$. By the cone theorem, for any $\varepsilon>0$ $$\overline{NE}(X)^G = \overline{NE}(X)^G _{(\mathcal{K}_X+\varepsilon L) \geq 0} + \sum_{\text{finite}} \mathbb R_{\geq 0} G[C_i].$$ where $\mathcal K _X \cdot C_i < 0$ for all $i$. Since $\overline{NE}(X)^G_{\mathcal{K}_X<0}$ is nonempty, we may choose $\varepsilon>0$ such that $\overline{NE}(X)^G \neq \overline{NE}(X)^G _{(\mathcal{K}_X+\varepsilon L) \geq 0}$. If the ray $R_1 = \mathbb R_{\geq 0} G[C_1]$ is not extremal in $\overline{NE}(X)^G$, then its generator $G[C_1]$ can be decomposed as a sum of elements of $\overline{NE}(X)^G$ not contained in $R_1$. It follows that $$\overline{NE}(X)^G = \overline{NE}(X)^G _{(\mathcal{K}_X+\varepsilon L) \geq 0} + \underset{\text{finite}}{\sum_{i\neq 1}} \mathbb R_{\geq 0} G[C_i],$$ i.e., the ray $R_1$ is superfluous in the formula. By assumption $\overline{NE}(X)^G \neq \overline{NE}(X)^G _{(\mathcal{K}_X+\varepsilon L) \geq 0}$ and we may therefore not remove all rays $R_i$ from the formula and at least one ray $R_i = \mathbb R_{\geq 0} G[C_i]$ is $G$-extremal.
We apply the equivariant contraction theorem to $X$: In case 1 we obtain from $X$ a new surface $Z$ by blowing down a $G$-orbit of disjoint (-1)-curves. There is a canonically defined holomorphic $G$-action on $Z$ such that the blow-down is equivariant. If $K_Z$ is not nef, we repeat the procedure which will stop after a finite number of steps. In case 2 we obtain an equivariant conic bundle structure on $X$. In case 3 we conclude that $X$ is a Del Pezzo surface with $G$-action. We call the $G$-surface obtained from $X$ at the end of this procedure a *$G$-minimal model of $X$* .
As a special case, we consider a rational surface $X$ with $G$-action. Since the canonical bundle $\mathcal{K}_X$ of a rational surface $X$ is never nef (cf. Theorem VI.2.1 in [@BPV]), a $G$-minimal model of $X$ is an equivariant conic bundle over $Z$ or a Del Pezzo surface with $G$-action. Note that the base curve $Z$ must be rational: if $Z$ is not rational, one finds nonzero holomorphic one-forms on $Z$. Pulling these back to $X$ gives rise to nonzero holomorphic one-forms on the rational surface $X$, a contradiction.
This proves the well-known classification of $G$-minimal models of rational surfaces (cf. [@maninminimal], [@isk]). Although this classification does classically not rely on Mori theory, the proof given above is based on Mori’s approach. We therefore refer to an equivariant reduction $Y \to Y_\mathrm{min}$ as an *equivariant Mori reduction*.
In the following chapters we will apply the equivariant minimal model program to quotients of K3-surfaces by nonsymplectic automorphisms.
Centralizers of antisymplectic involutions {#chapterlarge}
==========================================
This chapter is dedicated to a rough classification of K3-surfaces with antisymplectic involutions centralized by large groups of symplectic transformations (Theorem \[roughclassi\]).
We consider a K3-surface $X$ with an action of a finite group $G
\times C_2 < \mathrm{Aut}(X)$ and assume that the action of $G$ is by symplectic transformations whereas $C_2$ is generated by an antisymplectic involution $\sigma$ centralizing $G$. Furthermore, we assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Let $\pi: X \to X/ \sigma = Y$ denote the quotient map. The quotient surface $Y$ is a smooth rational $G$-surface to which we apply the equivariant minimal model program developed in the previous chapter. A $G$-minimal model of $Y$ can either be a Del Pezzo surface or an equivariant conic bundle over $\mathbb P_1$. In the later case, the possibilities for $G$ are limited by the classification of finite groups with an effective action on $\mathbb P_1$
\[autP\_1\] The classification of finite subgroups of $\mathrm{SU}(2, \mathbb C)$ (or $\mathrm{SO}(3, \mathbb R)$) yields the following list of finite groups with an effective action on $\mathbb P_1$:
- cyclic groups $C_n$
- dihedral groups $D_{2n}$
- the tetrahedral group $T_{12} \cong A_4$
- the octahedral group $O_{24} \cong S_4$
- the icosahedral group $I_{60} \cong A_5$
If $G$ is any finite group acting on a space $X$, we refer to the number of elements in an orbit $G.x = \{ g.x \, | \, g \in G\}$ as the *length of the $G$-orbit $G.x$*. Note that the length of a $T_{12}$-orbit in $\mathbb P_1$ is at least four, the length of an $O_{24}$-orbit in $\mathbb P_1$ is at least six, and the length of an $I_{60}$-orbit in $\mathbb P_1$ is at least twelve.
\[conicbundle\] If a $G$-minimal model $Y_\mathrm{min}$ of $Y$ is an equivariant conic bundle, then $|G| \leq 96$.
Let $\varphi: Y_\mathrm{min} \to \mathbb{P}_1$ be an equivariant conic bundle structure on $Y_\mathrm{min}$. By definition, the general fiber of $\varphi$ is isomorphic to $\mathbb{P}_1$. We consider the induced action of $G$ on the base $\mathbb P_1$. If this action is effective, then $G$ is among the groups specified in the remark above. Since the maximal order of an element in $G$ is eight (cf. Remark \[order of symp aut\]), it follows that the order $G$ is bounded by 60.
If the action of $G$ on the base $\mathbb P_1$ is not effective, every element $n$ of the ineffectivity $N < G$ has two fixed points in the general fiber. This gives rise to a positive-dimensional $n$-fixed point set in $Y_\mathrm{min}$ and $Y$. A symplectic automorphism however has only isolated fixed points. It follows that the action of $n$ on $X$ coincides with the action of $\sigma $ on $\pi^{-1}(\mathrm{Fix}_Y(N))$. In particular, the order of $n$ is two. Since $N$ acts effectively on the general fiber, it follows that $N$ is isomorphic to either $C_2$ or $C_2 \times C_2$.
If $G/N$ is isomorphic to the icosahedral group $I_{60}= A_5$, then $G$ fits into the exact sequence $ 1 \to N \to G \to A_5 \to 1$ for $N=C_2$ or $C_2 \times C_2$. Let $\eta$ be an element of order five inside $A_5$. One can find an element $\xi$ of order five in $G$ which is mapped to $\eta$. Since neither $C_2$ nor $C_2 \times C_2$ has automorphisms of order five it follows that $\xi$ centralizes the normal subgroup $N$. In particular, there is a subgroup $C_2 \times C_5 \cong C_{10}$ in $G$ which is contrary to the assumption that $G$ is a group of symplectic transformations and therefore its elements have order at most eight.
If $G/N$ is cyclic or dihedral, we again use the fact that the order of elements in $G$ is bounded by $8$ and conclude $|G/N| \leq 16$. It follows that the maximal possible order of $G/N$ is 24. Using $|N| \leq 4$ we obtain $|G| \leq 96$.
If $| G | >96$, the lemma above allows us to restrict our classification to the case where a $G$-minimal model $Y_\mathrm{min}$ of $Y$ is a Del Pezzo surface. The next section is devoted to a brief introduction to Del Pezzo surfaces and their automorphisms groups.
Del Pezzo surfaces
------------------
A *Del Pezzo surface* is a smooth surface $Z$ such that the anticanonical bundle $\mathcal K_Z^{-1} = \mathcal O _Z(-K_Z)$ is ample. The self-intersection number of the canonical divisor $d:= K_Z^2$ is referred to as the *degree* of the Del Pezzo surface and $1 \leq d
\leq 9$ (cf. Theorem 24.3 in [@manin]).
Let $Z = \{f_3=0\} \subset \mathbb P_3$ be a smooth cubic surface. The anticanonical bundle $\mathcal K_Z^{-1}$ of $Z$ is given by the restriction of the hyperplane bundle $\mathcal O_{\mathbb P_3}(1)$ to $Z$ and therefore ample.
As a consequence of the adjunction formula, an irreducible curve with negative self-intersection on a Del Pezzo surface is a (-1)-curve. The following theorem (cf. Theorem 24.4 in [@manin]) gives a classification of Del Pezzo surfaces according to their degree.
\[classiDelPezzo\] Let $Z$ be a Del Pezzo surface of degree $d$.
- If $d=9$, then $Z$ is isomorphic to $\mathbb P_2$.
- If $d=8$, then $Z$ is isomorphic to either $\mathbb P_1 \times \mathbb
P_1$ or the blow-up of $\mathbb P_2$ in one point.
- If $1 \leq d \leq 7$, then $Z$ is isomorphic to the blow-up of $\mathbb P_2$ in $9-d$ points in general position, i.e., no three points lie on one line and no six points lie on one conic.
In our later considerations of Del Pezzo surfaces Table \[minus one curves\] below (cf. Theorem 26.2 in [@manin]) specifying the number of (-1)-curves on a Del Pezzo surface of degree $d$ will be very useful.
degree $d$ 1 2 3 4 5 6 7
----------------------- ----- ---- ---- ---- ---- --- ---
number of (-1)-curves 240 56 27 16 10 6 3
: (-1)-curves on Del Pezzo surfaces[]{data-label="minus one curves"}
Let $Z$ be a Del Pezzo surface of degree 5. It follows from the theorem above that $Z$ is isomorphic to the blow-up of $\mathbb P_2$ in four points $p_1, \dots, p_4$ in general position. We denote by $E_i$ the preimage of $p_i$ is $Z$. Let $L_{ij}$ denote the line in $\mathbb P_2$ joining $p_i$ and $p_j$ and note that there are precisely six lines of this type. The proper transform of $L_{ij}$ is a (-1)-curve in $Z$. We have thereby specified all ten (-1)-curves in $Z$. Their incidence graph is known as the *Petersen graph* .
The following theorem summarizes properties of the anticanonical map, i.e., the map associated to the linear system $|-K_Z|$ of the anticanonical divisor (Theorem 24.5 in [@manin] and Theorem 8.3.2 in [@dolgachev])
\[antican models of del pezzo\] Let $Z$ be a Del Pezzo surface of degree $d$. If $d \geq 3 $, then $\mathcal K_Z^{-1}$ is very ample and the anticanonical map is a holomorphic embedding of $Z$ into $\mathbb P_d$ such that the image of $Z$ in $\mathbb P_d$ is of degree $d$.
If $d=2$, then the anticanonical map is a holomorphic degree two cover $\varphi: Z \to \mathbb P_2$ branched along a smooth quartic curve.
If $d=1$, then the linear system $|-K_Z|$ has exactly one base point $p$. Let $ Z' \to Z$ be the blow-up of $p$. Then the pull-back of $-K_Z$ to $Z'$ defines an elliptic fibration $f: Z' \to \mathbb P_1$. The linear system $|-2K_Z|$ defines a finite map of degree two onto a quadric cone $Q$ in $\mathbb P_3$. Its branch locus is given by the intersection of $Q$ with a cubic surface.
Our understanding of Del Pezzo surfaces as surfaces obtained by blowing-up points in $\mathbb P_ 2$ in general position or as degree $d$ subvarieties of $\mathbb P_d$ enables us the decide whether certain finite groups $G$ can occur as subgroups of the automorphisms group $\mathrm{Aut}(Z)$ of a Del Pezzo surface $Z$.
\[DelPezzoC3C7\] Consider the semi-direct product $G= C_3 \ltimes C_7$ where the action of $C_3$ on $C_7$ is defined by the embedding of $C_3$ into $\mathrm{Aut}(C_7) \cong C_6$. The group $G$ is a maximal subgroup of the simple group $L_2(7)$ which is discussed below. Let $Z$ be a Del Pezzo surface of degree $d$ with an effective action of $G$. Since $G$ does not admit a two-dimensional representation, it follows that $G$ does not have fixed points in $Z$. In particular, $d \neq
1$. For the same reason, $Z$ is not the blow-up of $\mathbb P_2$ in one or two points. Since there is no nontrivial homomorphisms $G \to C_2$ and no injective homomorphism $ G \to \mathrm{PSL}(2, \mathbb C)$ it follows that $G \not\hookrightarrow \mathrm{Aut}(\mathbb P_1 \times \mathbb
P_1) = (\mathrm{PSL}_2(\mathbb C) \times \mathrm{PSL}_2(\mathbb C)) \rtimes C_2$.
In many cases it can be useful to consider possible actions of a finite group $G$ on the union of (-1)-curves on a Del Pezzo surfaces.
\[DelPezzoL2(7)\] We consider $G= L_2(7)$, the simple group of order 168. Its maximal subgroups are $C_3 \ltimes C_7$ and $S_4$. Assume $G$ acts effectively on a Del Pezzo surface $Z$ of degree $d$. Since $L_2(7)$ does not stabilize any smooth rational curve, the $G$-orbit of a (-1)-curve $E
\subset Z$ consists of 7, 8, 14, 24 or more curves. It now follows from Table \[minus one curves\] that $d \neq 3, 5, 6$.
If $d=4$, then the union of (-1)-curves on $Z$ would consist of two $G$-orbits of length 8. In particular, $\mathrm{Stab}_G(E)\cong C_3 \ltimes C_7$ for any (-1)-curve $E \subset Z$. Blowing down $E$ to a point $p \in Z'$ induces an action of $C_3 \ltimes C_7$ on $Z'$ fixing $p$. Since $C_3 \ltimes C_7$ does not admit a two-dimensional representation, it follows that the normal subgroup $C_7$ acts trivially on $Z'$ and therefore on $Z$. This is a contradiction.
Using the result of the previous example, it follows that $Z$ is either a Del Pezzo surface of degree 2 or isomorphic to $\mathbb P_2
$. Both cases will play a role in our discussion of K3-surfaces with an action of $L_2(7)$.
Let be the Del Pezzo surface obtained by blowing up one point $p$ in $\mathbb P_2$. Then its automorphims group is the subgroup of $\mathrm{Aut}(\mathbb P_2)$ fixing the point $p$. Similarly, if $Z$ is the Del Pezzo surface obtained by blowing up two points $p,q$ in $\mathbb P_2$, then $\mathrm{Aut}(Z) = G \rtimes C_2$ where $G$ is the subgroup of $\mathrm{Aut}(\mathbb P_2)$ fixing the two points $p, q$ and $C_2$ acts by switching the exceptional curves $E_p,E_q$.
In the previous chapter we have shown that Del Pezzo surfaces can occur as equivariant minimal models. It should be remarked that the blow-up of $\mathbb P_2$ in one or two points is never equivariantly minimal: Let $Z$ be the surface obtained by blowing up one or two points in $\mathbb P_2$. Then $Z$ contains an $\mathrm{Aut}(Z)$-invariant (-1)-curve, namely the curve $E_p$ in the first case and the proper transform of the line joining $p$ and $q$ in the second case. This curve can always be blown down equivariantly. Using the language of equivariant Mori theory introduced in the previous chapter, the $\mathrm{Aut}(Z)$-invariant (-1)-curve spans a $\mathrm{Aut}(Z)$-extremal ray $R$ of the cone of invariant curves $\overline{NE}(X)^{\mathrm{Aut}(Z)}$ with $\mathcal K_Z \cdot R <0$. Its contraction defines an $\mathrm{Aut}(Z)$-equivariant map to $\mathbb P_2$. In particular, $Z$ is not equivariantly minimal.
A complete classification of automorphisms groups of Del Pezzo surfaces can be found in [@dolgachev].
Branch curves and Mori fibers {#branch curves mori fibers}
-----------------------------
We return to the initial setup where $X$ is a K3-surface with an action of $G \times \langle \sigma \rangle$ and $\pi: X \to X/\sigma =Y$ denotes the quotient map, and fix an equivariant Mori reduction $M: Y \to Y_\mathrm{min}$.
A rational curve $E \subset Y$ is called a *Mori fiber* if it is contracted in some step of the equivariant Mori reduction $Y \to Y_\mathrm{min}$. The set of all Mori fibers is denoted by $\mathcal E$. Its cardinality $|\mathcal E|$ is denoted by $m$. We let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$.
The total number $m$ of Mori fibers in $Y$ is bounded by $m \leq n+ 12 - e(Y_\mathrm{min}) \leq n+9$.
\[moribound\] Recall that $\mathrm{Fix}_X(\sigma)$ is a disjoint union of smooth curves. We choose a triangulation of $\mathrm{Fix}_X(\sigma)$ and extend it to a triangulation of the surface $X$. The topological Euler characteristic of the double cover is $$\begin{aligned}
e(X) = 24 &= 2e(Y) - \sum_{C \subset \mathrm{Fix}_X(\sigma)} e(C)\\
&= 2e(Y) - \sum_{C \subset \mathrm{Fix}_X(\sigma)} (2-2g(C))\\
&= 2e(Y) - 2n + \underset{\ g(C) \geq 1}{\sum_{C \subset \mathrm{Fix}_X(\sigma)}} (2g(C)-2)\\
& \geq 2e(Y) -2n\\
&= 2(e(Y_\mathrm{min}) +m) -2n\end{aligned}$$ This yields $m \leq n+12 - e(Y_\mathrm{min})$, and $e(Y_{min}) \geq 3$ completes the proof of the lemma.
Let $R:= \mathrm{Fix}_X(\sigma) \subset X$ denote the ramification locus of $\pi$ and let $B:=\pi(R) \subset Y$ be its branch locus. In the following, we repeatedly use the fact that for a finite proper surjective holomorphic map of complex manifolds (spaces) $\pi: X \to Y$ of degree $d$, the intersection number of pullback divisors fulfills $(\pi^*D_1 \cdot \pi^*D_2) = d(D_1 \cdot D_2)$.
\[preimage of E in X\] Let $E \in \mathcal{E}$ be a Mori fiber such that $E \not\subset B$ and $|E\cap B| \geq 2$ or $E \cdot B \geq 3$. Then $E^2 = -1$ and $\pi^{-1}(E)$ is a smooth rational curve in $X$. Furthermore, $E \cdot B = |E \cap B| =2$.
Let $k < 0$ denote self-intersection number of $E$. By the remark above, the divisor $\pi^{-1}(E) = \pi^* E$ has self-intersection $2k$. Assume that $\pi^{-1}(E)$ is reducible and let $\tilde E_1, \tilde E_2$ denote its irreducible components. They are rational and therefore, by adjunction on the K3-surface $X$, have self-intersection number $-2$. Write $$0 > 2k = (\pi^{-1}(E))^2 = \tilde E_1^2 + \tilde E_2^2 + 2 (\tilde E_1 \cdot \tilde E_2) = -4 + 2 (\tilde E_1 \cdot \tilde E_2).$$ Since $\tilde E_1$ and $\tilde E_2$ intersect at points in the preimage of $E \cap B$, we obtain $\tilde E_1 \cdot \tilde E_2 \geq 2$, a contradiction. It follows that $\pi^{-1}(E)$ is irreducible. Consequently, $k=-1$ and $\pi^{-1}(E)$ is a smooth rational curve with two $\sigma$-fixed points .
\[self-int of Mori-fibers\] Let $E \in \mathcal E$ be a Mori fiber.
- If $E \subset B$, then $E$ is the image of a rational curve in $X$ and $E^2 = -4$. (cf. Corollary \[minusfour\] below).
- If $E \not\subset B$ and $\pi^{-1}(E)$ is irreducible, then $2E^2 = (\pi^{-1}(E))^2 <0$. Adjunction on $X$ implies that $(\pi^{-1}(E))^2=-2$ and that $\pi^{-1}(E)$ is a smooth rational curve in $X$. The action of $\sigma$ has two fixed points on $\pi^{-1}(E)$ and the restricted degree two map $\pi|_{\pi^{-1}(E)}: \pi^{-1}(E) \to E$ is necessarily branched, i.e., $E \cap B \neq \emptyset$.
- If $E \not\subset B$ and $\pi^{-1}(E)= \tilde E_1 + \tilde E_2$ is reducible, then $$2E^2 = \underset{\geq -2}{\underbrace{\tilde E_1^2}} + \underset{\geq 0}{\underbrace{2(\tilde E_1 \cdot \tilde E_2)}} + \underset{\geq -2}{\underbrace{\tilde E_2^2}} \geq -4.$$ In particular, $E^2 \in \{-1,-2\}$.
- If $E^2=-1$, then $\tilde E_1 \cdot \tilde E_2 =1$ and $E\cap B \neq \emptyset$.
- If $E^2=-2$, then $\tilde E_1\cdot \tilde E_2 =0$ and $E\cap B = \emptyset$.
In summary, a Mori fiber $E \not\subset B$ has self-intersection -1 if and only if $E\cap B \neq \emptyset$ and self-intersection -2 if and only if $E\cap B = \emptyset$. A Mori fiber $E$ has self-intersection -4 if and only if $E \subset B$.
More generally, any (-1)-curve $E$ on $Y$ meets $B$ in either one or two points. If $|E \cap B |=1$, then $\pi^{-1}(E) = E_1 \cup E_2$ is reducible. If $|E \cap B |=2$, then $\pi^{-1}(E)$ is irreducible and meets $\mathrm{Fix}_X(\sigma)=R = \pi^{-1}(B)$ in two points.
\[at most two\] Every Mori fiber $E \in \mathcal{E}$, $ E \not\subset B$ meets the branch locus $B$ in at most two points. If $E$ and $B$ are tangent at $p$, then $E\cap B = \{p\}$ and $(E \cdot B)_p =2$.
Let $E \in \mathcal E$, $ E \not\subset B$ and assume $|E\cap B| \geq 2$ or $E \cdot B \geq
3$. Then by the lemma above, $\tilde E = \pi^{-1}(E)$ is a smooth rational curve in $X$. Since $\tilde E \not\subset \mathrm{Fix}_X(\sigma)$, the involution $\sigma$ has exactly two fixed points on $\tilde E$ showing $|E\cap B| = 2$. It remains to show that the intersection is transversal.
To see this, let $N_{\tilde{E}}$ denote the normal bundle of $\tilde{E}$ in $X$. We consider the induced action of $\sigma$ on $N_{\tilde{E}}$ by a bundle automorphism. Using an equivariant tubular neighbourhood theorem we may equivariantly identify a neighbourhood of $\tilde E$ in $X$ with $N_{\tilde E}$ via a $C^{\infty}$-diffeomorphism. The $\sigma$-fixed point curves intersecting $\tilde{E}$ map to curves of $\sigma$-fixed points in $N_{\tilde{E}}$ intersecting the zero-section and vice versa. Let $D$ be a curve of $\sigma$-fixed point in $N_{\tilde{E}}$. If $D$ is not a fiber of $N_{\tilde E}$, it follows that $\sigma$ stabilizes all fibers intersecting $D$ and the induced action of $\sigma$ on the base must be trivial, a contradiction. It follows that the $\sigma$-fixed point curves correspond to fibers of $N_{\tilde{E}}$, and $E$ and $B$ meet transversally.
By negation of the implication above, if $E$ and $B$ are tangent at $p$, then $|E \cap B |=1$ and $E \cdot B=2$.
### Rational branch curves
In this section we find conditions on $G$, in particular conditions on the order of $G$, guaranteeing the absence of rational curves in $\mathrm{Fix}_X(\sigma)$.
\[selfintbranch\] Let $\pi:X \to Y$ be a cyclic degree two cover of surfaces and let $C \subset X$ be a smooth curve contained in the ramification locus of $\pi$. Then the image of $C$ in $Y$ has self-intersection $(\pi(C))^2= 2 C^2$.
We recall that the intersection of pullback divisors fulfills $\pi^*D_1 \cdot \pi^*D_2 = 2(D_1 \cdot D_2)$. In the setup of the lemma, $(\pi^* \pi(C))^2 = 2 (\pi(C))^2$. Now $\pi^* \pi(C) \sim 2C$ implies the desired result.
Note that the lemma above can also be proved by considering the normal bundle $N_C$ of $C$ and the induced action of $\sigma$ on it. The normal bundle $N_{\pi(C)}$ is isomorphic to $N_C^2$. Since the self-intersection of a curve is the degree of the normal bundle restricted to the curve, the formula follows.
\[minusfour\] Let $X$ be a K3-surface and let $\pi: X \to Y$ be a cyclic degree two cover. Then a rational branch curve of $\pi $ has self-intersection -4.
Let $C$ be a rational curve on the K3-surface $X$. Then by adjunction $C^2 =-2$ and the image $\pi(C)$ in $Y$ is a (-4)-curve by Lemma \[selfintbranch\] above.
On a Del Pezzo surface a curve with negative self-intersection necessarily has self-intersection -1. So if $Y_\mathrm{min}$ is a Del Pezzo surface, all rational branch curves of $\pi$, which have self-intersection -4 by Corollary \[minusfour\], need to be modified by the Mori reduction when passing to $Y_{\mathrm{min}}$ and therefore have nonempty intersection with the union of Mori fibers.
An important tool in the study of rational branch curves is provided by the following lemma which describes the behaviour of self-intersection numbers under monoidal transformations.
\[selfintblowdown\] Let $\tilde X$ and $X$ be smooth projective surfaces and let $b: \tilde X \to X$ be the blow-down of a (-1)-curve $ E \subset \tilde X$. For a curve $B \subset \tilde X$ having no common component with $ E$ the self-intersection of its image in $X$ is given by $$b( B)^2 = B^2 + ( E \cdot B)^2.$$
We may choose an ample divisor $H$ in $X$ with $p \not\in \mathrm{supp}(H)$ and $D$ linearly equivalent to $ b(B) +H$ such that $p \not\in \mathrm{supp}(D)$. Since $b$ is biholomorphic away from $p$, we know $$(b(B) +H)^2 = D^2 =(b^*D)^2 = (b^*((b(B) +H))^2.$$ Using $(b^*H)^2 = H^2$ and $b^*(b(B))\cdot b^*H = b(B) \cdot H$ we find $b(B)^2 = (b^*B)^2$. Now $b^*B = B + \mu E$ where $\mu$ denotes the multiplicity of the point $p \in b(B)$. This multiplicity equals the intersection multiplicity $E \cdot B$. Therefore, $$b( B)^2 = (b^*B)^2= (B + \mu E)^2 = B^2 + 2 \mu^2 -\mu^2= B^2 + \mu^2.$$ and the lemma follows.
We denote by $\mathcal{C}$ the set of rational branch curves of $\pi$. The total number $|\mathcal{C}|$ of these curves is denoted by $n$. The union of all Mori fibers not contained in the branch locus $B$ is denoted by $\bigcup E_i$.
Let $\mathcal{C}_{\geq k}= \{ C \in \mathcal C \, | \, |C \cap \bigcup E_i | \geq k\} $ be the set of those rational branch curves $C$ which meet $\bigcup E_i$ in at least $k$ distinct points and let $| \mathcal{C}_{\geq k}| = r_k$. We let $\mathcal{E}_{\geq k}$ denote the set of Mori fibers $E \not\subset B$ which intersect some $C$ in $\mathcal{C}_{\geq k}$ and define $$P_k = \{ (p,E) \, |\, p \in C \cap E,\, E \in \mathcal{E}_{\geq k},\, C
\in \mathcal{C}_{\geq k}\} \subseteq Y \times \mathcal{E}_{\geq k}$$ and the projection map $\mathrm{pr}_k: P_k \to \mathcal{E}_{\geq k}$ mapping $(p,E)$ to $E$. This map is surjective by definition of $\mathcal{E}_{\geq k}$ and its fibers consist of $\leq 2$ points by Proposition \[at most two\]. Using $|P_k| \geq k r_k$ we see $$\label{boundforE_k}
|\mathcal{E}_{\geq k}| \geq \frac{k}{2}r_k.$$ Let $N$ be the largest positive integer such that $\mathcal{C}_{\geq N} =
\mathcal{C}$, i.e., each rational ramification curve is intersected at least $N$ times by Mori fibers. A curve $C \in \mathcal{C}$ which is intersected precisely $N$ times by Mori fibers is referred to as a *minimizing curve*. In the following, let $C$ be a minimizing curve and let $ H =
\mathrm{Stab}_G(C) < G$ be the stabilizer of $C$ in $G$.
The index of $H$ in $G$ is bounded by $n= r_N$.
### Bounds for $n$ {#bounds-for-n .unnumbered}
A smooth rational curve on a K3-surface has self-intersection -2 and all curves in $\mathrm{Fix}_X(\sigma)$ are disjoint. Therefore, the rational curves in $\mathrm{Fix}_X(\sigma)$ generate a sublattice of $\mathrm{Pic}(X)$ of signature $(0,n)$. It follows immediately that $n
\leq 19$.
A sharper bound $n \leq 16$ for the number of disjoint (-2)-curves on a K3-surface has been obtained by Nikulin [@NikulinKummer] and the following optimal bound in our setup is due to Zhang [@ZhangInvolutions], Theorem 3.
The total number of connected curves in the fixed point set of an antisymplectic involution on a K3-surface is bounded by 10.
\[atmostten\] The number $n$ of rational curves in $\mathrm{Fix}_X(\sigma)$ is at most 10. If $n=10$, then $\mathrm{Fix}_X(\sigma)$ is a union of rational curves.
In the following, we use Zhang’s bound $n \leq 10$. Note, however, that all results can likewise be obtained by using the weakest bound $n \leq 19$.
For $N \geq 4 $ Zhang’s bound can be sharpened using the notion of Mori fibers and minimizing curves.
\[boundforn\] $\frac{N}{2}n \leq n +12 - e(Y_\mathrm{min}) \leq n+9$.
Using Lemma \[moribound\] and inequality $
\frac{N}{2}n = \frac{N}{2}r_N \leq |\mathcal{E}_{\geq N}|\leq
|\mathcal{E}| \leq n +12 - e(Y_\mathrm{min}) \leq n+9
$.
In the following we consider the stabilizer $H$ of a minimizing curve $C$ and using the above bounds for $n$, we obtain bounds for $|G|$.
### A bound for $|G|$ {#a-bound-for-g .unnumbered}
Let $X$ be a K3-surface with an action of a finite group $G \times \langle \sigma
\rangle$ such that $ G < \mathrm{Aut}_\mathrm{symp}(X)$ and $\sigma$ is an antisymplectic involution with fixed points. If $| G | > 108$, then $\mathrm{Fix}_X(\sigma)$ contains no rational curves.
Assume that $\mathrm{Fix}_X(\sigma)$ contains rational curves and consider a minimizing curve $C \subset B$ and its stabilizer $\mathrm{Stab}_G(C) =:H$. Since a symplectic automorphism on $X$ does not admit a one-dimensional set of fixed points, it follows that the action of $H$ on $C$ is effective and $H$ is among the groups discussed in Remark \[autP\_1\]. We recall the possible lengths of $H$-orbits in $C$: the length of an orbit of a dihedral group is at least two, the length of a $T_{12}$-orbit in $\mathbb P_1$ is at least four, the length of an $O_{24}$-orbit in $\mathbb P_1$ is at least six, and the length of an $I_{60}$-orbit in $\mathbb P_1$ is at least twelve.
Let $Y_\mathrm{min}$ be a $G$-minimal model of $X/\sigma =Y$. Recall that by Lemma \[conicbundle\] $Y_\mathrm{min}$ is a Del Pezzo surface. Each rational branch curve is a (-4)-curve in $Y$. Since its image in $Y_\mathrm{min}$ has self-intersection $\geq -1$, it must intersect Mori fibers.
- If $N=1$, i.e., the rational curve $C$ meets the union of Mori fibers in exactly one point $p$, then $p$ is a fixed point of the $H$-action on $C$. In particular, $H$ is a cyclic group $C_k$. By Remark \[order of symp aut\] $k \leq 8 $. Since the index of $H$ in $G$ is bounded by $n \leq 10$, it follows that $|G | \leq 80$.
- If $N=2$, then $H$ is either a cyclic or a dihedral group. By Proposition 3.10 in [@mukai] the maximal order of a dihedral group of symplectic automorphisms on a K3-surface is 12. We first assume $H \cong D_{2m}$ and that the $G$-orbit $G.C$ of the rational branch curve $C$ has the maximal length $n=|G.C|=10$, i.e., $B = G \cdot C$. Each curve in $G \cdot C$ meets the union of Mori fibers in precisely two points forming an $D_{2m}$-orbit. If a Mori fiber $E_C$ meets the curve $C$ twice, then it follows from Proposition \[at most two\] that $E$ meets no other curve in $B$. The contraction of $E$ transforms $C$ into a singular curve of self-intersection zero. The Del Pezzo surface $Y_\mathrm{min}$ does however not admit a curve of this type. It follows, that $E$ meets a Mori fiber $E'$ which is contracted in a later step of the Mori reduction and meets no other Mori fiber than $E'$. The described configuration $G E \cup G E'$ requires a total number of at least 20 Mori fibers and therefore contradicts Lemma \[moribound\]. If $C$ meets two distinct Mori fibers $E_1, E_2$, each of these two can meet at most one further curve in $B$. The contraction of $E_1$ and $E_2$ transforms $C$ into a (-2)-curve. As above, the existence of further Mori fibers meeting $E_i$ follows. Again, by invariance, the total number of Mori fibers exceeds 20, a contradiction. It follows that either $H$ is cyclic or $ |G.C|\leq 9$. Both imply $|G| \leq 108$.
- If $N=3$, let $S=\{p_1,p_2,p_3\}$ be the points of intersection of $C$ with the union $\bigcup E_i$ of Mori fibers. The set $S$ is $H$-invariant. It follows that $H$ is either trivial or isomorphic to $C_2$, $C_3$ or $D_6$ and that $|G| \leq 60$
- If $N = 4$, it follows from Lemma \[boundforn\] that $n \leq 9 $. Now $|H| \leq 12$ implies $|G| \leq 108$. The bound for the order of $H$ is attained by the tetrahedral group $T_{12}$. If the group $G$ does not contain a tetrahedral group, then $|H| \leq 8$ and $|G| \leq
72$.
- If $N=5$, the largest possible group acting on $C$ such that there is an invariant subset of cardinality 5 is the dihedral group $D_{10}$. Since \[boundforn\] implies $n \leq 6$, we conclude $|G| \leq 60$.
- If $N=6$, then $n \leq 4 $ and $|H| \leq 24$ implies $|G| \leq
96$. This bound is attained if and only if $H \cong O_{24}$. If there is no octahedral group in $G$, then $|H| \leq 12$ and $|G|\leq 48$.
- If $N\geq 12$, then $n=1$ and $H =G$. The maximal order 60 is attained by the icosahedral group.
- If $6 < N <12$, we combine $n \leq 4$ and $|H| \leq 24$ to obtain $|G| \leq 96$. If $H$ is not the octahedral group, then $|H| \leq 16$ and $|G| \leq
64$.
The case by case discussion shows that the existence of a rational curve in $B$ implies $| G|\leq 108$ and the proposition follows.
If the group $G$ under consideration does not contain certain subgroups (such as large dihedral groups or $T_{12}$, $O_{24}$ or $I_{60}$) then the condition $|G| >108$ in the proposition above can be improved and non-existence of rational ramification curves also follows for smaller $G$.
### Elliptic branch curves
The aim of this section is to find conditions on the order of $G$ which allow us to exclude elliptic curves in $\mathrm{Fix}_X(\sigma)$. We prove:
\[elliptic branch\] Let $X$ be a K3-surface with an action of a finite group $G \times \langle \sigma
\rangle$ such that $ G < \mathrm{Aut}_\mathrm{symp}(X)$ and $\sigma$ is an antisymplectic involution with fixed points. If $| G | > 108$, then $\mathrm{Fix}_X(\sigma)$ contains neither rational nor elliptic ramification curves.
By the previous proposition $\mathrm{Fix}_X(\sigma)$ contains no rational curves. It follows from Nikulin’s description of $\mathrm{Fix}_X(\sigma)$ (cf. Theorem \[FixSigma\]) that it is either a single curve of genus $g \geq 1$ or the disjoint union of two elliptic curves.
Let $T \subset B$ be an elliptic branch curve and let $H:= \mathrm{Stab}_G(T)$. If $H \neq G$, then $H$ has index two in $G$. The action of $H$ on $T$ is effective. The automorphism group $\mathrm{Aut}(T)$ of $T$ is a semidirect product $L \ltimes T$, where $L$ is a linear cyclic group of order at most 6. We consider the projection $\mathrm{pr}_L: \mathrm{Aut}(T) \to
L$ and let $\lambda \in \mathrm{Pr}_L(H)$ be a generating root of unity. We consider $T$ as a quotient $\mathbb C / \Gamma$ and choose $h \in H$ with $h(z) = \lambda z + \omega$ and $t \in T$ such that $\omega +(1-\lambda)t = 0$. After conjugation with the translation $z \mapsto z+t$ the group $H < \mathrm{Aut}(T)$ inherits the semidirect product structure of $\mathrm{Aut}(T)$, i.e., $$H = (H \cap L) \ltimes (H \cap T) .$$ We refer to this decomposition as the *normal form* of $H$. By Lemma \[conicbundle\] a $G$-minimal model of $Y$ is a Del Pezzo surface and therefore does not admit elliptic curves with self-intersection zero. It follows that $T$ meets the union $\bigcup E_i$ of Mori fibers. Let $E$ be a Mori fiber meeting $T$. By Proposition \[at most two\] $|T \cap E | \in \{1,2\}$. The stabilzer of $E$ in $H$ is denoted by $\mathrm{Stab}_H(E)$. Since the total number of Mori fibers is bounded by 9 (cf. Lemma \[moribound\]), the index of $\mathrm{Stab}_H(E)$ in $H$ is bounded by 9.
If $T \cap E =\{p\}$, then $\mathrm{Stab}_H(E)$ is a cyclic group of order less than or equal to six. It follows that $|G | \leq 6\cdot 9 \cdot 2 = 108$.
If $T \cap E =\{p_1, p_2\}$, then $B \cap E = T \cap E$ and the stabilizer $\mathrm{Stab}_G(E)$ of $E$ in $G$ is contained in $H$. If both points $p_1,p_2$ are fixed by $\mathrm{Stab}_G(E)$, then $|\mathrm{Stab}_G(E)| \leq 6$. If $p_1,p_2$ form a $\mathrm{Stab}_G(E)$-orbit, then in the normal form $|\mathrm{Stab}_G(E) \cap T | =2$. It follows that $\mathrm{Stab}_G(E)$ is either $C_2$ or $D_4= C_2 \times C_2$. The index of $\mathrm{Stab}_G(E)$ in $G$ is bounded by 9 and $|G| \leq 54$.
In summary, the existence of an elliptic curve in $B$ implies $|G| \leq 108$ and the proposition follows.
Rough classification
--------------------
With the preparations of the previous sections we may now turn to a classification result for K3-surfaces with antisymplectic involution centralized by a large group.
\[roughclassi\] Let $X$ be a K3-surface with a symplectic action of $G$ centralized by an antisymplectic involution $\sigma$ such that $\mathrm{Fix}(\sigma)\neq \emptyset$. If $|G|>96$, then $Y$ is a $G$-minimal Del Pezzo surface and there are no rational or elliptic curves in $\mathrm{Fix}(\sigma)$. In particular, $\mathrm{Fix}(\sigma)$ is a single smooth curve $C$ with $ g(C)\geq
3$ and $\pi(C) \sim -2K_Y$, where $K_Y$ denotes the canonical divisor on $Y$.
The group $G$ is a subgroup of one of the eleven groups on Mukai’s list [@mukai] (cf. Theorem \[mukaithm\] and Table \[TableMukai\]). The orders of these Mukai groups are 48, 72, 120, 168, 192, 288, 360, 384, 960. None of these groups can have a subgroup $G$ with $96 < |G| < 120$. In particular, the order of $G$ is at least 120.
We may therefore apply the results of the previous two sections and conclude that $\pi: X \to Y$ is branched along a single smooth curve $C$ of general type. Its genus $g(C)$ must be $\geq 3$ by Hurwitz’ formula. It remains to show that $Y$ is $G$-minimal.
Assume the contrary and let $E\subset Y$ be a Mori fiber with $E^2 = -1$. As before we let $B \subset Y$ denote the branch locus of $\pi: X \to Y$. By Remark \[self-int of Mori-fibers\] $E \cap B \neq \emptyset$. It follows that $|E \cap B| \in \{1,2\}$. Let $\mathrm{Stab}_G(E)$ denote the stabilizer of $E$ in $G$.
If $\pi^{-1}(E)$ is reducible its two irreducible components meet transversally in one point corresponding to $\{p\} =E \cap B$. The curve $E$ is tangent to $B$ at $p$ and we consider the linearization of the action of $\mathrm{Stab}_G(E)$ at $p$. If the action of $\mathrm{Stab}_G(E)$ on $E$ is not effective, the linearization of the ineffectivity $I < \mathrm{Stab}_G(E)$ yields a trivial action of $I$ on the tangent line of $B$ at $p$. It follows that the action of $I$ is trivial in a neighbourhood of $\pi^{-1}(p) \in R =\pi^{-1}(B)$. This is contrary to the assumption that $G$ acts symplectically on $X$. Consequently, the action of $\mathrm{Stab}_G(E)$ on $E$ is effective and in particular, $\mathrm{Stab}_G(E)$ is a cyclic group.
If $\pi^{-1}(E)$ is irreducible, then it is a smooth rational curve with an effective action of $\mathrm{Stab}_G(E)$. It follows that $\mathrm{Stab}_G(E)$ is either cyclic or dihedral. The largest dihedral group with a symplectic action on a K3-surface is $D_{12}$ (Proposition 3.10 in [@mukai]).
We conclude that the order of $\mathrm{Stab}_G(E)$ is bounded 12 and the index of $G_E$ in $G$ is $> 9 $. By Lemma \[moribound\] the total number $m$ of Mori fibers however satifies $m \leq 9$. This contradiction shows that $Y$ is $G$-minimal and, in particular, a Del Pezzo surface.
\[stab of minus one curve\] Let $X$ be a K3-surface with a symplectic action of $G$ centralized by an antisymplectic involution $\sigma $ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$ and let $E$ be a (-1)-curve on $Y = X/\sigma$. Then the argument above can be applied to see that the stabilizer of $E$ in $G$ is cyclic or dihedral and therefore has order at most 12.
In the following chapter, the classification above is applied and extended to the case where $G$ is a maximal group of symplectic transformations on a K3-surface.
Mukai groups centralized by antisymplectic involutions {#chaptermukai}
======================================================
In this chapter we consider K3-surfaces with a symplectic action of one of the eleven groups from Mukai’s list (Table \[TableMukai\]) and assume that it is centralized by an antisymplectic involution. We prove the following classification result.
\[thm mukai times invol\] Let $G$ be a Mukai group acting on a K3-surface $X$ by symplectic transformations and $\sigma $ be an antisymplectic involution on $X$ centralizing $G$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then the pair $(X,G)$ is in Table \[Mukai times invol\] below. In particular, for groups $G$ numbered 4-8 on Mukai’s list, there does not exist a K3-surface with an action of $G \times C_2$ with the properties above.
$G$ $|G|$ **K3-surface** $X$
----- ----------- ------- ------------------------------------------------------------------------------------------------------------------
1a $L_2(7)$ 168 $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$
1b $L_2(7)$ 168 Double cover of $\mathbb P_2$ branched along
$\{x_1^5x_2+x_3^5x_1+x_2^5x_3-5x_1^2 x_2^2 x_3^2 =0\}$
2 $A_6$ 360 Double cover of $\mathbb P_2$ branched along
$\{10 x_1^3x_2^3+ 9 x_1^5x_3 + 9 x_2^3x_3^3-45 x_1^2 x_2^2 x_3^2-135 x_1 x_2 x_3^4 + 27 x_3^6 =0\}$
3a $S_5$ 120 $\{\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0\} \subset \mathbb P_5$
3b $S_5$ 120 Double cover of $\mathbb P_2$ branched along
$\{ F_{S_5} =0\}$
9 $N_{72}$ 72 $\{ x_1^3+ x_2 ^3 + x_3^3 +x_4^3= x_1x_2 + x_3x_4+ x_5^2 = 0 \} \subset \mathbb P_4$
10 $M_9$ 72 Double cover of $\mathbb P_2$ branched along
$\{x_1^6+x_2^6 +x_3^6 -10(x_1^3x_2^3 + x_2^3x_3^3 +x_3^3x_1^3) =0\}$
11a $ T_{48}$ 48 Double cover of $\mathbb P_2$ branched along
$\{x_1x_2(x_1^4-x_2^4)+ x_3^6 =0\}$
11b $ T_{48}$ 48 Double cover of $\{ x_0x_1(x_0^4-x_1^4)+ x_2^3+x_3^2=0 \} \subset \mathbb P(1,1,2,3)$
branched along $\{x_2=0\} $
: K3-surfaces with $G \times C_2$-symmetry[]{data-label="Mukai times invol"}
\
The polynomial $F_{S_5}$ in case 3b) is given by $$\begin{aligned}
&2(x^4yz+xy^4z+xyz^4)-2(x^4y^2+x^4z^2+x^2y^4+x^2z^4+y^4z^2+y^2z^4)\\
+&2(x^3y^3+x^3z^3+y^3z^3)+x^3y^2z+x^3yz^2+x^2y^3z+x^2yz^3+xy^3z^2+xy^2z^3-6x^2y^2z^2.\end{aligned}$$
The examples 1a, 3a, 9, 10, and 11a appaer in Mukai’s list, the remaining cases 1b, 2, 3b, and 11b provide additional examples of K3-surfaces with maximal symplectic symmetry.
For the proof of this theorem we consider each group separately and apply the following general strategy.
For a K3-surface $X$ with $G \times C_2$-symmetry we consider the quotient $Y= X/C_2$ and a $G$-minimal model $Y_\mathrm{min}$ of the rational surface $Y$. We show that $Y_\mathrm{min}$ is a Del Pezzo surface and investigate which Del Pezzo surfaces admit an action of the group $G$.
It is then essential to study the branch locus $B$ of the covering $X \to Y$. As a first step, we exclude rational and elliptic curves in $B$. In order to exclude rational branch curves, we study their images in $Y_\mathrm{min}$ and their intersection with the union of Mori fibers.
We then deduce that $B$ consists of a single curve of genus $\geq 2$ with an effective action of the group $G$. The possible genera of $B$ are restricted by the nature of the group $G$ and the Riemann-Hurwitz formula for the quotient of $B$ by an appropriate normal subgroup $N$ of $G$. The equations of $B$ or $X$ given in Table \[Mukai times invol\] are derived using invariant theory.
Throughout the remainder of this chapter, the Euler characteristic formula $$24 = e(X) = 2 e(Y_\mathrm{min}) + 2 m -2n +\underset{\text{branch curve exists}}{\underset{\text{if non-rational}}{\underbrace{(2g-2)}}}$$ is exploited various times. Here $m$ denotes the total number of Mori contractions of the reduction $Y \to Y_ \mathrm{min}$, the total number of rational branch curves is denoted by $n$ and $g$ is the genus of a non-rational branch curve.
All classification results are up to *equivariant equivalence*:
\[equivariantequivalence\] Let $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ be K3-surfaces with antisymplectic involution and let $G$ be a finite group acting on $X_1$ and $X_2$ by $$\alpha_i: G \to \mathrm{Aut}_\mathrm{symp}(X_i),$$ such that $\alpha_i(g) \circ \sigma_i = \sigma_i \circ \alpha_i(g)$ for $i =1,2$ and all $g \in G$. Then the surfaces $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ are considered *equivariantly equivalent* if there exist a biholomorphic map $\varphi: X_1 \to X_2$ and a group automorphism $\psi \in \mathrm{Aut}(G)$ such that $$\alpha_2(g) \varphi (x) = \varphi ( \alpha_1(\psi(g))x) \quad \text{and} \quad \sigma_2(\varphi (x)) = \varphi ( \sigma_1(x)).$$ for all $x \in X_1$ and all $g \in G$.
More generally, two surfaces $Y_1$ and $Y_2$, without additional structure such as a symplectic form or an involution, with actions of a finite group $G$ $$\alpha_i: G \to \mathrm{Aut}(Y_i)$$ are considered equivariantly equivalent if there exist a biholomorphic map $\varphi: Y_1 \to Y_2$ and a group automorphism $\psi \in \mathrm{Aut}(G)$ such that $$\alpha_2(g) \varphi (y) = \varphi ( \alpha_1(\psi(g))y)$$ for all $y \in Y_1$ and all $g \in G$.
This notion differs from the notion of equivalence in representation theory. Two non-equivalent linear represenations of a group $G$ can induce equivalent actions on the projective plane if they differ by an outer automorphism of the group.
If two K3-surfaces $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ are $G$-equivariantly equivalent, then the quotient surfaces $X_i / \sigma_i$ are equivariantly equivalent with respect to the induced action of $G$.
Conversely, let $Y$ be a rational surface with two action of a finite group $G$ which are equivalent in the above sense and let $\varphi \in \mathrm{Aut}(Y)$ be the isomorphims identifying these two actions. We consider a smooth $G$-invariant curve $B$ linearly equivalent to $-2K_Y$ and the K3-surfaces $X_B$ and $X_{\varphi(B)}$ obtained as double covers branched along $B$ and $\varphi(B)$ equipped with their respective antisymplectic covering involution.
Note that $X_B$ and $X_{\varphi(B)}$ are constructed as subsets of the anticanonical line bundle where the involution $\sigma$ is canonically defined. The induced biholomorphic map $\varphi_X: X_B \to X_{\varphi(B)}$ fulfills $\sigma \circ \varphi_X = \varphi_X \circ \sigma$ by construction.
If all elements of the group $G$ can be lifted to symplectic transformations on $X_B$ and $X_{\varphi(B)}$, then the central degree two extensions $E$ of $G$ acting on $X_B$, $X_{\varphi(B)}$, respectively, split as $E = E_\mathrm{symp} \times C_2$ with $E_\mathrm{symp} =G$. In this case the group $G$ acts by symplectic transformations on $X_B$ and $X_{\varphi(B)}$ and these are $G$-equivariantly equivalent in strong sense introduced above. This follows from the assumption that the corresponding $G$-actions on the base $Y$ are equivalent and the fact that for each $g \in G \subset \mathrm{Aut}(Y)$ there is only one choice of symplectic lifting $\tilde g \in \mathrm{Aut}(X_B)$ and $\mathrm{Aut}(X_{\varphi(B)})$.
In the following sections we will go through the lists of Mukai groups and for each group we prove the classification claimed in Theorem \[thm mukai times invol\].
The group $L_2(7)$ {#mukaiL2(7)}
------------------
Let $G \cong L_2(7)$ be the finite simple group of order 168. If $G$ acts on a K3-surface $X$, then the kernels of the homomorphism $G \to \mathrm{Aut}(X)$ and the homomorphism $ G \to \Omega^2(X)$ are trivial and the action is effective and symplectic. Let $\sigma$ be an antisymplectic involution on $X$ centralizing $G$. Since $G$ has an element of order seven which is known to have exactly three fixed points $p_1, p_2,p_3$ and $\sigma$ acts on this set of three points, we know that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. By Theorem \[roughclassi\], the K3-surface $X$ is a double cover of a Del Pezzo surface $Y$. Our study of Del Pezzo surfaces with an action of $L_2(7)$ in Example \[DelPezzoL2(7)\] has revealed that $Y$ is either $\mathbb P_2$ or a Del Pezzo surface of degree 2. In the first case, $\pi: X \to Y$ is branched along a curve of genus 10, in the second case $\pi$ is branched along a curve of genus 3. Section \[168\] in the next chapter is devoted to an inspection of K3-surfaces with an action of $L_2(7) \times C_2$ and a precise classification result in the setup above will be obtained. The pair $(X,G)$ is equivariantly isomorphic to either the surface 1a) or 1b).
The group $A_6$ {#A6Valentiner}
---------------
Let $G \cong A_6$ be the alternating group degree 6. It is a simple group and if it acts on a K3-surface $X$, then this action effective and symplectic. Let $\sigma$ be an antisymplectic involution on $X$ centralizing $G$ and assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. By Theorem \[roughclassi\], the K3-surface $X$ is a double cover of a Del Pezzo surface $Y$ with an effective action of $A_6$.
The Del Pezzo surface $Y$ is isomorphic to $\mathbb P_2$ with a uniquely determined action of $A_6$ given by a nontrivial central extension $V=3.A_6$ of degree three known as *Valentiner’s group*.
We go through the list of Del Pezzo surfaces.
- If $Y$ has degree one, then $| -K_Y|$ has precisely one base point which would have to be an $A_6$-fixed point. This is contrary to the fact that $A_6$ has no faithful two-dimensional representation.
- We recall that the stabilizer of a (-1)-curve $E$ in $Y$ is either cyclic or dihedral (Remark \[stab of minus one curve\]). In particular, its order is at most 12 and therefore its index in $A_6$ is at least 30. Using Table \[minus one curves\] we see that $Y$ can not be a Del Pezzo surface of degree 2,3,4,5,6.
- Since the blow-up of $\mathbb P_2$ in one point is never $G$-minimal, it remains to exclude $Y \cong \mathbb P_1 \times \mathbb P_1$. Assume there is an action of $A_6$ on $\mathbb P_1 \times \mathbb P_1$. Since $A_6$ has no subgroups of index two, it follows that $A_6 < \mathrm{PSL}(2, \mathbb C) \times \mathrm{PSL}(2, \mathbb C)$ and both canonical projections are $A_6$-equivariant. Since $A_6$ has neither an effective action on $\mathbb P_1$ nor nontrivial normal subgroups of ineffectivity, it follows that $A_6$ acts trivially on $Y$.
It follows that $Y \cong \mathbb P_2$. The action of $A_6$ on $\mathbb P_2$ is given by a degree three central extension of $A_6$. Since $A_6$ has no faithful three-dimensional representation, this extension is nontrivial and isomorphic the unique nontrivial degree three extension $V=3.A_6$ known as Valentiner’s group. Up to equivariant equivalence, there is a unique action of $A_6$ on $\mathbb P_2$. This follows from the classification of finite subgroup of $\mathrm{SL}_3(\mathbb C)$ (cf. [@blichfeldtbook], [@blichfeldt], and [@YauYu]) and can also be derived as follows: An action of $A_6$ on $\mathbb P_2$ is given by a threedimensional projective representation. We wish to show that any two actions induced by $\rho_1, \rho_2$ are equivalent. We restrict the projective representations $\rho_1$ and $\rho_2$ to the subgroup $A_5$. The restricted representations are linear and after a change of coordinates $\rho_1(A_5) = \rho_2(A_5) \subset \mathrm{SL}_3(\mathbb C)$. We fix a subgroup $A_4$ in $A_5$ and consider its normalizer $N$ in $A_6$. The groups $N$ and $A_4$ generate the full group $A_6$ and it suffices to prove that $\rho_1(N) = \rho_2(N)$. This is shown by considering an explicit three-dimensional representation of $A_4 < A_5$ and the normalizer $\mathcal N$ of $A_4$ inside $\mathrm{PSL}_3(\mathbb C)$. The group $A_4$ has index two in $\mathcal N$ and therefore $\mathcal N = \rho_1(N)= \rho_2(N)$..
The covering $X \to Y$ is branched along an invariant curve $C$ of degree six. This curve is defined by an invariant polynomial $F_{A_6}$ of degree six, which is unique by Molien’s formula. Its explicit equation is derived in [@crass]. In appropriately chosen coordinates, $$F_{A_6}(x_1,x_2,x_3) = 10 x_1^3x_2^3+ 9 x_1^5x_3 + 9 x_2^3x_3^3-45 x_1^2 x_2^2 x_3^2-135 x_1 x_2 x_3^4 + 27 x_3^6.$$ If a K3-surface with $A_6 \times C_2$-symmetry exists, then it must be the double cover of $\mathbb P_2$ branched along $\{F_{A_6}=0\}$.
The action of $A_6$ on $\mathbb P_2$ induces an action of a central degree two extension of $E$ on the double cover branched along $\{F_{A_6}=0\}$, $$\{\mathrm{id}\} \to C_2 \to E \to A_6 \to \{\mathrm{id}\}.$$
Let $E_\mathrm{symp} \neq E$ be the normal subgroup of symplectic automorphisms in $E$. Since $A_6$ is simple, it follows that $E_\mathrm{symp}$ is mapped surjectively to $A_6$ and $E_\mathrm{symp} \cong A_6$. In particular, the group $E$ splits as $E_\mathrm{symp} \times C_2$ where $C_2$ is generated by the antisymplectic covering involution. This proves the existence of a unique K3-surface with $A_6 \times C_2$-symmetry. We refer to this K3-surface as the *Valentiner surface*.
The group $S_5$
---------------
In this section we study K3-surfaces with an symplectic action of the symmetric group $S_5$ centralized by an antisymplectic involution.
Let $X$ be a K3-surface with a symplectic action of $G = S_5$ and let $\sigma$ denote an antisymplectic involution centralizing $G$. We assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. We may apply Theorem \[roughclassi\] which yields that $X/ \sigma =Y$ is a $G$-minimal Del Pezzo surface and $\pi: X \to Y$ is branched along a smooth connected curve $B$ of genus $$g(B) = 13-e(Y).$$ We will see in the following that only very few Del Pezzo surfaces admit an effective action of $S_5$ or a smooth $S_5$-invariant curve of appropriate genus.
The degree $d(Y)$ of the Del Pezzo surface $Y$ is either three or five.
We prove the statement by excluding Del Pezzo surfaces of degree $\neq 3,5$.
- Assume $Y \cong \mathbb P_2$. Then $G =S_5$ is acting effectively on $\mathbb P_2$, i.e., $S_5 \hookrightarrow \mathrm{PSL}_3(\mathbb C)$. Let $\tilde G$ denote the preimage of $G$ in $\mathrm{SL}_3(\mathbb C)$. Since $A_5$ has no nontrivial central extension of degree three, it follows that the preimage of $A_5 < S_5$ in $\tilde G$ splits as $\tilde A_5 = A_5 \times C_3$. It has index two in $\tilde G$ and therefore is a normal subgroup of $\tilde G$. Let $g \in S_5$ be any transposition and pick $\tilde g$ in its preimage with $\tilde g^2 = \mathrm{id}$. Now $\tilde g$ and $ A_5$ generate a copy of $S_5$ in $\mathrm{SL}_3(\mathbb C)$. The action of $S_5$ is given by a three-dimensional representation. The irreducible representations of $S_5$ have dimensions $1,4,5$ or $6$ and it follows that there is no faithful three-dimensional represenation of $S_5$ and therefore no effective $S_5$-action on $\mathbb P_2$.
- Assume that $Y$ is isomorphic to $\mathbb P_1 \times \mathbb P_1$. We investigate the action of $S_5 = A_5 \rtimes C_2$ and note that $A_5$ is a simple group. The automorphism group $\mathrm{Aut}(Y)$ is given by $$(\mathrm{PSL}_2( \mathbb C) \times \mathrm{PSL}_2(\mathbb C)) \rtimes C_2.$$ It follows that $A_5 < \mathrm{PSL}_2( \mathbb C) \times \mathrm{PSL}_2( \mathbb C)$, and the action of $A_5$ respects the product structure, i.e, the canonical projections onto the factors are $A_5$-equivariant. If $A_5$ acts trivially on one of the factors, then the generator $\tau$ of the outer $C_2$ stabilizes this factor because $A_5$ must act nontrivially on the second factor. It follows that $S_5$ stablizes the second factor which is impossible since there is no effective action of $S_5$ on $\mathbb P_1$. It follows that $A_5$ acts effectively on both factors and $\tau$ exchanges them. We consider an element $\lambda$ of order five in $A_5$ and chose coordinates on $\mathbb P_1 \times \mathbb P_1$ such that $\lambda$ acts by $$([z_1:z_2],[w_1:w_2]) \mapsto ([\xi z_1:z_2],[\xi ^a w_1:w_2])$$ for some $a \in \{1,2,3,4\}$ and $\xi^5 =1$. The automorphism $\lambda$ has four fixed points $$\begin{aligned}
p_1 = ([1:0],[1:0]),\,\,
p_2 = ([1:0],[0:1]),\, \,
p_3 = ([0:1],[1:0]),\,\,
p_4 = ([0:1],[0:1]).\,\,\end{aligned}$$ Since it lifts to a symplectic automorphism on the K3-surface $X$ with four fixed points, all fixed points must lie on the branch curve. The branch curve $B \subset Y$ is a smooth invariant curve linearly equivalent to $-2K_Y$ and is therefore given by an $S_5$-semi-invariant polynomial $f$ of bidegree $(4,4)$. Since $f$ must be invariant with respect to the subgroup $A_5$, it is a linear combination of $\lambda$-invariant monomials of bidegree $(4,4)$. For each choice of $a$ one lists all $\lambda$-invariant monomials of bidegree $(4,4)$. For $a=1$ these are $$z_1 z_2^3 w_1^4,\,\, z_1^2 z_2^2 w_1^3 w_2 ,\,\, z_1^3 z_2 w_1^2 w_2^2,\,\, z_1^4 w_1 w_2^3 , \,\,z_2^4 w_2^4.$$ Since $f$ must vanish at $p_1 \dots p_4$, one sees that $f$ may not contain $z_2^4 w_2^4$. The remaining monomials have a common component $z_1 w_1$ such that $f$ factorizes and $C$ must be reducible, a contradiction. The same argument can be carried out for each choice of $a$. It follows that the action of $S_5$ on $\mathbb P_1 \times \mathbb P_1$ does not admit irreducble curves of bidegree $(4,4)$. This eliminates the case $Y \cong \mathbb P_1 \times \mathbb P_1$.
- Again using the fact that the largest subgroup of $S_5$ which can stabilize a (-1)-curve in $Y$ is the group $D_{12}$ of index 10, it follows that the number of (-1)-curves in a $G$-orbit is at least 10. A Del Pezzo surface of degree six has six (-1)-curves and therefore $d(Y) \neq 6$. A Del Pezzo surface of degree four contains sixteen (-1)-curves. Since 16 does not divide the the order of $S_5$, the set of these curves is not a single $G$-orbit. As it cannot be the union of $G$-orbits either, we conclude $d(Y) \neq 4$.
- If $d(Y) =2$, then the anticanonical map defines an $\mathrm{Aut}(Y)$-equivariant double cover of $\mathbb P_2$. The induced action of $S_5$ on $\mathbb P_2$ would have to be effective and therefore we obtain a contradiction as in the case $Y \cong \mathbb P_2$.
- If $d(Y)=1$ then the anticanonical system $|-K_Y|$ is known to have precisely one base point which has to be fixed point of the action of $S_5$. Since $S_5$ has no faithful two-dimensional representation, this is a contradiction.
Since we have considered all possible $G$-minimal Del Pezzo surfaces the proof of the lemma is completed.
### Double covers of Del Pezzo surfaces of degree three
The following example of a K3-surface $X$ with an action of $S_5 \times C_2$ such that $X/\sigma$ is a Del Pezzo surface of degree three can be found in Mukai’s list [@mukai] (cf. also Table \[TableMukai\]).
\[MukaiS5\] Let $X$ be the K3-surface in $\mathbb P_5$ given by $$\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0$$ and let $S_5$ act on $\mathbb P_5$ by permuting the first five variables and by the character $\mathrm{sgn}$ on the last variable. This induces an action on $X$.
The commutator subgroup $ S_5' = A_5 < S_5$ acts by symplectic transformations. In order to show that the full group acts symplectically, consider the transposition $\tau = (12) \in S_5$ acting on $\mathbb P_5$ by $[x_1:x_2:x_3:x_4:x_5:x_6] \mapsto [x_2:x_1:x_3:x_4:x_5:-x_6]$. One checks that the induced involution on $X$ has isolated fixed points and is therefore symplectic. It follows that $S_5 < \mathrm{Aut}_\mathrm{symp}(X)$.
Let $\sigma : \mathbb P_5 \to \mathbb P_5$ be the involution $[x_1:x_2:x_3:x_4:x_5:x_6] \mapsto [x_1:x_2:x_3:x_4:x_5:-x_6]$. This defines an involution on $X$ with a positive-dimensional set of fixed point $\{x_6=0 \} \cap X$. Therefore $\sigma$ is an antisymplectic involution on $X$ which centralizes the action of $S_5$.
The quotient $Y$ of $X$ by $\sigma$ is given by restricting then rational map $[x_1:x_2:x_3:x_4:x_5:x_6] \mapsto [x_1:x_2:x_3:x_4:x_5]$ to $X$. The surface $Y$ is given by $$\{ \sum_{i=1}^5 y_i= \sum_{i=1}^5 y_i^3 =0\} \subset \mathbb P_4.$$ and is isomorphic to the Clebsch diagonal surface $\{z_1^2 z_2 + z_1 z_3^2 + z_3 z_4^2 + z_4 z_2^2 = 0\} \subset \mathbb P_3$ (cf. Theorem 10.3.10 in [@dolgachev]), a Del Pezzo surface of degree three. The branch set $B$ is given by $\{ \sum_{i=1}^5 y_1^2=0\} \cap Y \subset \mathbb P_4$.
By the following proposition, the example above is the unique K3-surface with $S_5 \times C_2$-symmetry such that $X/\sigma$ is a Del Pezzo surface of degree three.
\[S5 on degree three\] Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma$ is a Del Pezzo surface of degree three, then $X$ is equivariantly isomorphic to Mukai’s $S_5$-example $\{\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0\} \subset \mathbb P_5$.
We consider the $\mathrm{Aut}(Y)$-equivariant embedding of the Del Pezzo surface $Y$ into $\mathbb P_3$ given by the anticanonical map. Any automorphism of $Y$ induced by an automorphism of the ambient projective space.
It follows from the representation and invariant theory of the group $S_5$ that a Del Pezzo surface of degree three with an effective action of the group $S_5$ is equivariantly isomorphic the Clebsch cubic $\{z_1^2 z_2 + z_1 z_3^2 + z_3 z_4^2 + z_4 z_2^2 = 0\} \subset \mathbb P_3$ (cf. Theorems 10.3.9 and 10.3.10, Table 10.3 in [@dolgachev]).
The ramification curve $B\subset Y$ is linearly equivalent to $-2K_Y$. We show that $B$ is given by intersecting $Y$ with a quadric in $\mathbb P_3$.
Applying the formula $$h^0(Y, \mathcal{O}(-rK_Y))= 1+ \frac{1}{2}r(r+1)d(Y)$$ (cf. e.g. Lemma 8.3.1 in [@dolgachev]) to $d=d(Y)=3$ and $r=2$ we obtain $h^0(Y,\mathcal{O}( -2K_Y))= 10$. This is also the dimension of the space of sections of $\mathcal O_{\mathbb P_3} (2)$ in $\mathbb P_3$ (homogeneous polynomials of degree two in four variables). It follows that the restriction map $$H^0(\mathbb P_3, \mathcal O (2)) \to H^0(Y, \mathcal O(-2K_Y))$$ is surjective and $B = Y \cap Q$ for some quadric $Q = \{f=0\}$ in $\mathbb P_3$.
Since $B$ is an $S_5$-invariant curve in $Y$, it follows that for each $g \in S_5$ the intersection of $gQ = \{ f \circ g^{-1} =0\}$ with Y coincides with $B$. It follows that $f|_Y$ is a multiple of $(f\circ g^{-1})|_Y$, i.e., there exists a constant $c \in \mathbb C^*$ such that $(f \circ g^{-1}) - cf$ vanishes identically on $Y$. Since $Y$ is irreducible, this implies $f \circ g^{-1} = cf$. It follows that the polynomial $f$ is an $S_5$- semi-invariant and therefore invariant with respect to the commutator subgroup $A_5$.
We have previously noted that after a suitable linear change of coordinates the surface $Y$ is given by $\{ \sum_{i=1}^5 y_i= \sum_{i=1}^5 y_i^3 =0\} \subset \mathbb P_4$ where $S_5$ acts by permutation. The action of any transposition on an $S_5$-semi-invariant polynomial is given by multiplication by $\pm 1$. It follows that in the coordinates $[y_1:\dots:y_5]$ the semi-invariant polynomial $f$ is given by $$a \sum_{i=1}^5 y_i^2 + b(\sum_{i=1}^5 y_i)^2 =0$$ for some $a,b \in \mathbb C$. Using the fact $Y \subset \{\sum_{i=1}^5 y_i=0\}$ it follows that $B$ is given by intersecting $Y$ with $ \{\sum_{i=1}^5 y_i^2 =0\}$ and $X$ is Mukai’s $S_5$-example discussed in Example \[MukaiS5\].
### Double covers of Del Pezzo surfaces of degree five
A second class of candidates of K3-surfaces with $S_5 \times C_2$-symmetry is given by double covers of Del Pezzo surfaces of degree five.
Any two Del Pezzo surfaces of degree five are isomorphic and the automorphisms group of a Del Pezzo surface $Y$ of degree five is $S_5$. The ten (-1)-curves on $Y$ form a graph known as the *Petersen graph*. The Petersen graph has $S_5$-symmetry and every symmetry of the abstract graph is induced by a unique automorphism of the surface $Y$.
The following proposition classifies K3-surfaces with $S_5 \times C_2$-symmetry which are double covers of Del Pezzo surfaces of degree five.
Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma$ is a Del Pezzo surface of degree five, then $X$ is equivariantly isomorphic to the minimal desingularization of the double cover of $\mathbb P_2$ branched along the sextic $$\begin{aligned}
\{&2(x^4yz+xy^4z+xyz^4)
-2(x^4y^2+x^4z^2+x^2y^4+x^2z^4+y^4z^2+y^2z^4)
+2(x^3y^3+x^3z^3+y^3z^3)\\
&+x^3y^2z+x^3yz^2+x^2y^3z+x^2yz^3+xy^3z^2+xy^2z^3
-6x^2y^2z^2
=0\}\end{aligned}$$
Let $B \subset Y$ denote the branch locus of the covering $X \to Y$. The curve $B$ is smooth, connected, invariant with respect to the full automorphism group of $Y$ and linearly equivalent to $-2K_Y$.
The Del Pezzo surface $Y$ is the blow-up of $\mathbb P_2$ in four points $p_1,p_2,p_3,p_4$ in general position. We may choose coordinates $[x:y:z]$ on $\mathbb P_2$ such that $$\begin{aligned}
p_1=[1:0:0],\quad
p_2=[0:1:0],\quad
p_3=[0:0:1],\quad
p_4=[1:1:1].\end{aligned}$$ Let $m: Y \to \mathbb P_2$ be the blow-down map and let $E_i = m^{-1}(p_i)$. Consider the $S_4$-action on $\mathbb P_2$ permuting the points $\{p_i\}$. The isotropy at the point $p_1$ is isomorphic to $S_3$ and induces an effective $S_3$-action on $E_1$.
Let $E$ be any (-1)-curve on $Y$. By adjunction $E\cdot B=2$. Since $Y$ contains precisely ten (-1)-curves forming an $S_5$-orbit, the group $H = \mathrm{Stab}_{S_5}(E)$ has order 12 and all stabilizer groups of (-1)-curves in $Y$ are conjugate. It follows that the group $H$ contains $S_3$, which is acting effectively on $E$, and therefore $H$ is isomorphic to the dihedral group of order 12. The points of intersection $B\cap E$ form an $H$-invariant subset of $E$. Since $H$ has no fixed points in $E$ and precisely one orbit $H.p = \{p,q\}$ consisting of two elements, it follows that $B$ meets $E$ transversally in $p$ and $q$.
In particular, each curve $E_i$ meets $B$ in two points and the image curve $C = m(B)$ has nodes at the four points $p_i$. By Lemma \[selfintblowdown\], the self-intersection number of $C$ is $20 + 4\cdot 4= 36$, so $C$ is a sextic curve. It is invariant with respect to the action of $S_4$ given by permutation on $p_1, \dots p_4$. For simplicity, we first only consider the action of $S_3$ permuting $p_1,p_2,p_3$ and conclude that $C$ is given by $\{f=\sum a_i f_i =0 \}$ as a linear combination of the following degree six polynomials $$\begin{aligned}
f_1&=x^6+y^6 +z^6\\
f_2&=x^5y + x^5z+ xy^5 +xz^5+y^5z+yz^6\\
f_3&=x^4yz+xy^4z+xyz^4\\
f_4&=x^4y^2+x^4z^2+x^2y^4+x^2z^4+y^4z^2+y^2z^4\\
f_5&=x^3y^3+x^3z^3+y^3z^3\\
f_6&=x^3y^2z+x^3yz^2+x^2y^3z+x^2yz^3+xy^3z^2+xy^2z^3\\
f_7&=x^2y^2z^2\end{aligned}$$ The fact that $C$ passes through $p_i$ and is singular at $p_i$ yields $a_1=a_2=0$ and $$3a_3+6a_4+3a_5+6a_6+a_7=0.$$ The two tangent lines of $C$ at the node $p_i$ correspond to the unique $\mathrm{Stab}(E_i)$-orbit of length two in $E_i$. We consider the point $p_3$ and the subgroup $S_3 < S_4$ stabilizing $p_3$. The action of $S_3$ on $E_3$ is given by the linearized $S_3$-action on the set of lines through $p_3$. One checks that in local affine coordinates $(x,y)$ the unique orbit of length two corresponds to the line pair $x^2 -xy +y^2 =0$. Dehomogenizing $f$ at $p_3$, i.e., setting $z=1$, we obtain the local equation $f_\mathrm{dehom}$ of $C$ at $p_3$. The polynomial $f_\mathrm{dehom}$ modulo terms of order three or higher must be a multiple of $x^2 -xy +y^2$. Therefore $a_3 = -a_4$.
Next we consider the intersection of $C$ with the line $L_{34} = \{x=y\}$ joining $p_3$ and $p_4$. We know that $f|_{L_{34}}$ vanishes of order two at $p_3$ and $p_4$ and at one or two further points on $L_{34}$.
Let $\widetilde L_{34}$ denote the proper transform of $L_{34}$ inside the Del Pezzo surface $Y$. The curve $\widetilde L_{34}$ is a (-1)-curve, hence its stabilizer $\mathrm{Stab}_G(\widetilde L_{34})$ is isomorphic to $D_{12} = S_3 \times C_2$. The factor $C_2$ acts trivially on $\widetilde L_{34}$. Since the intersection of $\widetilde L_{34}$ with $B$ is $\mathrm{Stab}_G(\widetilde L_{34})$ invariant, it follows that $\widetilde L_{34} \cap B$ is the unique $S_3$-orbit a length two in $\widetilde L_{34}$.
We wish to transfer our determination of the unique $S_3$-orbit of length two in $E_3$ above to the curve $\widetilde L_{34}$ using an automorphism of $Y$ mapping $E_3$ to $\widetilde L_{34}$. Consider the automorphism $\varphi$ of $Y$ induced by the birational map of $\mathbb P_2$ given by $$[x:y:z] \mapsto [x(z-y):z(x-y):xz]$$ (cf. Theorem 10.2.2 in [@dolgachev]) and let $\psi$ be the automorphism of $Y$ induced by the permutation of the points $p_2$ and $p_3$ in $\mathbb P_2$. Then $\psi \circ \varphi$ is an automorphism of $Y$ mapping $E_3$ to $\widetilde L_{34}$. If $[X:Y]$ denote homogeneous coordinates on $E_3$ induced by the affine coordinates $(x,y)$ in a neighbourhood of $p_3$, then a point $[X:Y] \in E_3$ is mapped to the point corresponding to $[X:X: X-Y] \in L_{34} \subset \mathbb P_2$. It was derived above that the unique $S_3$-orbit of length two in $E_3$ is given by $X^2-XY +Y^2$ and it follows that the unique $S_3$-orbit of length two in $\widetilde L_{34}$ corresponds to the points $[x:x:z] \in \mathbb P_2$ fulfilling $x^2 -xz +z^2 =0$.
Therefore, $f|_{L_{34}}$ is a multiple of polynomial given by $x^2(x-z)^2(x^2 -xz +z^2)$. Comparing coefficients with $f(x:x:z)$ yields $$\begin{aligned}
2a_3+2a_6 &= 2a_5 +2a_6\\
2a_4 + a_5&= 2a_4+a_3\\
8a_4+4a_5 &= 2a_4+2a_6 +a_7\\
-6a_4 -3a_5 &= 2a_5 +2a_6.\end{aligned}$$ We conclude $a_3=a_5=2= -a_4$, $a_6=1$, and $a_7=-6$. So if $X$ as in the lemma exists, it is the double cover of $Y$ branched along the proper transform of $\{f=0\}$ in $Y$, where $$\begin{aligned}
f(x,y,z)=&2(x^4yz+xy^4z+xyz^4)\\
&-2(x^4y^2+x^4z^2+x^2y^4+x^2z^4+y^4z^2+y^2z^4)\\
&+2(x^3y^3+x^3z^3+y^3z^3)\\
&+x^3y^2z+x^3yz^2+x^2y^3z+x^2yz^3+xy^3z^2+xy^2z^3\\
&-6x^2y^2z^2.\end{aligned}$$ In order to prove existence, let $X$ be the minimal desingularisation of the double cover of $\mathbb P_2$ branched along $\{f=0\}$. Then $X$ is the double cover of the Del Pezzo surface $Y$ of degree five branched along the proper transform $D$ of $\{f=0\}$ in $Y$. Since all automorphisms of $Y$ are induced by explicit biholomorphic or birational transformation of $\mathbb P_2$ one can check by direct computations that $D$ is in fact invariant with respect to the action of $\mathrm{Aut}(Y) = S_5$. The covering involution $\sigma$ is antisymplectic.
On $X$ there is an action of a central extension $E$ of $S_5$ by $C_2$. Let $E_\mathrm{symp}$ be the subgroup of symplectic automorphisms in $E$. Since $E$ contains the antisymplectic covering involution $E_\mathrm{symp} \neq E$. The image $N$ of $E_\mathrm{symp}$ in $S_5$ is normal and therefore either $N \cong S_5$ or $N \cong A_5$.
If $N \cong A_5$ and $| E_\mathrm{symp}| =60$, then $E_\mathrm{symp} \cong A_5$. Lifting any transposition from $S_5$ to an element $g$ of order two in $E$, the group generated by $g$ and $E_\mathrm{symp}$ inside $E$ is isomorphic to $S_5$. It follows that $E$ splits as $S_5 \times C_2$ and $E / E_\mathrm{symp} \cong C_2 \times C_2$. This is a contradiction.
If $N \cong A_5$ and $| E_\mathrm{symp}| =120$, then $ E = E_\mathrm{symp} \times C_2$, where the outer $C_2$ is generated by the antisymplectic covering involution $\sigma$, and $E / C_2 = S_5$ implies that $E_\mathrm{symp} \cong S_5$. This is contradictory to the assumption $N \cong A_5$.
In the last remaining case $N \cong S_5$. Since $E_\mathrm{symp} \neq E$, also $E_\mathrm{symp} \cong S_5$ and $E$ splits as $E_\mathrm{symp} \times C_2$. It follows that the action of $S_5$ on $Y$ induces an symplectic action of $S_5$ on the double cover $X$ centralized by the antisymplectic covering involution. This completes the proof of the proposition.
### Conclusion
We summarize our results of the previous subsections in the following theorem.
Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then $X$ is equivariantly isomorphic to either Mukai’s $S_5$-example or the minimal desingularization of the double cover of $\mathbb P_2$ branched along the sextic $$\begin{aligned}
\{&F_{S_5}(x_1,x_2,x_3)=\\
&2(x^4yz+xy^4z+xyz^4)-2(x^4y^2+x^4z^2+x^2y^4+x^2z^4+y^4z^2+y^2z^4)+2(x^3y^3+x^3z^3+y^3z^3)\\
&+x^3y^2z+x^3yz^2+x^2y^3z+x^2yz^3+xy^3z^2+xy^2z^3-6x^2y^2z^2=0\}.\end{aligned}$$
The group $M_{20} = C_2^4 \rtimes A_5$
--------------------------------------
There does not exist a K3-surface with a symplectic action of $M_{20}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
Assume that a K3-surface $X$ with these properties exists. Applying Theorem \[roughclassi\] we see that $X \to Y$ is branched along a single $M_{20}$-invariant smooth curve $C$ on the Del Pezzo surface $Y$. The curve $C$ is neither rational nor elliptic. By Hurtwitz’ formula, $$| \mathrm{Aut} (C) | \leq 84(g(C)-1),$$ the genus of $C$ must be at least twelve. Since $C$ is linearly equivalent to $-2K_Y$, the adjunction formula $$2g(C)-2 = (K_Y,C) + C^2 = 2K_Y^2$$ implies $\mathrm{deg}(Y) = K_Y^2 \geq 11$. This is a contradiction since the degree of a Del Pezzo surface is at most nine.
The group $F_{384} = C_2^4 \rtimes S_4$
---------------------------------------
Before we prove non-existence of K3-surfaces with $F_{384} \times C_2$-symmetry, we note the following useful fact about $S_4$-actions on Riemann surfaces.
\[S4 not on g=1,2\] The group $S_4$ does not admit an effective action on a Riemann surface of genus one or two.
The automorphism group of a Riemann surface $T$ of genus one is of the form $\mathrm{Aut}(T)= L \ltimes T$ for $L \in \{C_2, C_4, C_6\}$. We have seen before (cf. Proof of Proposition \[elliptic branch\]) that any subgroup $H$ of $\mathrm{Aut}(T)$ can be put into the form $H = (H \cap L) \ltimes (H \cap T)$. The nontrivial normal subgroups of $S_4$ are $A_4$ and $C_2 \times C_2$. Since $A_4$ is not Abelian and the quotient of $S_4$ by $S_4 \cap T = C_2 \times C_2$ is not cyclic, we conclude that $S_4$ is not a subgroup of $ \mathrm{Aut}(T)$.
Assume that $S_4$ acts effectively on a Riemann surface $H$ of genus two. Note that $H$ is hyperelliptic and the quotient of $H$ by the hyperelliptic involution is branched at six points. Since $S_4$ has no normal subgroup of order two, the induced action of $S_4$ on the quotient $\mathbb P_1$ is effective and therefore has precisely one orbit consisting of six points. The isotropy subgroup at these points is isomorphic to $C_4$. The isotropy group at the corresponding points in $H$ must be isomorphic to $C_4 \times C_2$. Since this group is not cyclic, it cannot act effectively with fixed points on a Riemann surface and we obtain a contradiction.
There does not exists a K3-surface with a symplectic action of $F_{384}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
As above, assume that a K3-surface $X$ with these properties exists and apply Theorem \[roughclassi\] to see that $X \to Y$ is branched along a single $F_{384}$-invariant smooth curve $C$ on the Del Pezzo surface $Y$. It follows from Hurwitz’ formula that the genus of $C$ is at least 6.
We use the realization of $F_{384}$ as a semi-direct product $C_4^2 \rtimes S_4$ (cf. [@mukai]) and consider the quotient $Q$ of the branch curve $C$ by the normal subgroup $N = C_4^2$. On $Q$ there is the induced action of $S_4$. It follows from the lemma above that $Q$ is either rational or $g(Q) >2$. In the second case, if we apply the Riemann-Hurwitz formula to the covering $ C \to Q$, then $$e(C) = 16 e(Q) - \text{branch point contributions} \leq -64$$ and $g(C) \geq 33$. This contradicts the adjunction formula on the Del Pezzo surface $Y$ and implies that $Q$ is a rational curve.
It follows from adjunction that $K_Y^2 = g(C)-1$. Therefore, the degree of the Del Pezzo surface $Y$ is at least five. We consider the action of $F_{384}$ on the configuration of (-1)-curves on $Y$ and recall that the order of a stabilizer of a (-1)-curve in $Y$ is at most twelve (cf. Remark \[stab of minus one curve\]) and therefore has index greater than or equal to $32$ in $G$. It follows that $Y$ is either $\mathbb P_1 \times \mathbb P_1$ or $\mathbb P_2$. In the first case, the canonical projections of $\mathbb P_1 \times \mathbb P_1$ are equivariant with respect to a subgroup of index two in $F_{384}$ and thereby contradict Lemma \[conicbundle\]. Consequently, $Y \cong \mathbb P_2$. In particular, $g(C) =10$ and $e(C) = -18$. It follows that the branch point contribution of the covering $C \to Q$ must be 50. Since isotropy groups must be cyclic, the only possible isotropy subgroups of $N = C_4^2$ at a point in $C$ are $C_2$ and $C_4$ and have index four or eight. The full branch point contribution must therefore be a multiple of four. This contradiction yields the non-existence claimed.
The group $A_{4,4} = C_2^4 \rtimes A_{3,3}$
-------------------------------------------
By $S_{p,q}$ for $p+q =n$ we denote a subgroup $S_p \times S_q$ of $S_n$ preserving a partition of the set $\{1,\dots, n\}$ into two subsets of cardinality $p$ and $q$. The intersection of $A_n$ with $S_{p.q}$ is denoted by $A_{p,q}$.
There does not exists a K3-surface with a symplectic action of $A_{4,4}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
We again assume that a K3-surface with these properties exists. Applying Theorem \[roughclassi\] we see that $X \to Y$ is branched along a single $A_{4,4}$-invariant smooth curve $C$ on the Del Pezzo surface $Y$. The group $A_{4,4}$ is a semi-direct product $C_2^4 \rtimes A_{3,3}$ (see e.g. [@mukai]). We consider the quotient $Q$ of $C$ by the normal subgroup $N \cong C_2^4$. On $Q$ there is an action of $A_{3,3}$. Since $A_{3,3}$ contains the subgroup $C_3 \times C_3$, which does not act on a rational curve, it follows that $Q$ not rational. We apply the Riemann-Hurwitz formula to the covering $C \to Q$.
If $Q$ is elliptic, then $2g(C) -2$ equals the branch point contribution of the covering $C \to Q$. As above, isotropy groups must be cyclic and the maximal possible isotropy group of the $C_2^4$-action on $C$ is $C_2$ and has index eight in $C_2^4$. Consequently, the branch point contribution at each branch point is eight. Recall that any group $H$ acting on the torus $Q$ can be put into the form $H = (H \cap L) \ltimes (H \cap Q)$ for $L \in \{C_2, C_4, C_6\}$. Since $Q$ acts freely, the action of $C_3 \times C_3 < A_{3,3}$ on the elliptic curve $Q$ has orbits of length greater than or equal to three. Therefore, the total branch point contribution must be greater than or equal to $24$. In particular, $g(C) = \mathrm{deg}(Y) +1 \geq 13$ contrary to $\mathrm{deg}(Y) \leq 9$.
If $g(Q) \geq 2$, then $g(C) \geq 17$ which is also contrary to $\mathrm{deg}(Y) \leq 9$
The groups $T_{192} = (Q_8 * Q_8) \rtimes S_3$ and $H_{192} = C_2^4 \rtimes D_{12}$
------------------------------------------------------------------------------------
By $Q_8$ we denote the quaternion group $\{+1,-1, +I,-I, +J,-J,+K,-K\}$ where $I^2= J^2= K^2 = IJK = -1$. The central product $ Q_8 * Q_8 $ is defined as the quotient of $Q_8 \times Q_8$ by the central involution $(-1, -1)$, i.e., $Q_8 * Q_8 = (Q_8 \times Q_8) / (-1,-1)$.
Note that both groups $T_{192}$ and $H_{192}$ are semi-direct products $C_2^3 \rtimes S_4$ (cf. [@mukai]).
For $G =T_{192}$ or $G = H_{192}$ there does not exists a K3-surface with a symplectic action of $G$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
Assume that a K3-surface with these properties exists. Applying Theorem \[roughclassi\] we see that $X \to Y$ is branched along a single $G$-invariant smooth curve $C$ on the Del Pezzo surface $Y$. The genus of $C$ is at least four by Hurwitz’ formula and therefore $\mathrm{deg}(Y) \geq 3$. We consider the quotient $Q$ of $C$ by the normal subgroup $N = C_2^3$. By Lemma \[S4 not on g=1,2\] the quotient $Q$ is either rational or $g(Q) >2$. In the second case $g(C) \geq 19$ and we obtain a contradiction to $\mathrm{deg}(Y) = g(C)-1 \leq 9$. It follows that $Q$ is a rational curve.
We consider the action of $G$ on the Del Pezzo surface $Y$ of degree $\geq 3$, in particular the induced action on its configuration of (-1)-curves. By Remark \[stab of minus one curve\] the stabilizer of a (-1)-curve in $Y$ has index $\geq 16$ in $G$ and we may immediately exclude the cases $\mathrm{deg}(Y) = 3,5,6,7$. The automorphism group of a Del Pezzo surface of degree four is $C_2^4 \rtimes \Gamma$ for $\Gamma \in \{C_2, C_4, S_3, D_{10}\}$ (cf. [@dolgachev]). In particular, the maximal possible order is 160 and therefore $\mathrm{deg}(Y) \neq 4$.
Assume that $Y \cong \mathbb P_1 \times \mathbb P_1$. The canonical projection $\pi_{1,2}: Y \to \mathbb P_1$ is equivariant with respect to a subgroup $H$ of $G$ of index at most two. It follows that $H$ fits into the exact sequences $$\begin{aligned}
\{\mathrm{id}\} \to I_1 \to H \overset{(\pi_1)_*}{\to} H_1\to \{\mathrm{id}\} \\
\{\mathrm{id}\} \to I_2 \to H \overset{(\pi_2)_*}{\to} H_2\to \{\mathrm{id}\}\end{aligned}$$ where $I_i \cong C_2 \times C_2$ is the ineffectivity of the induced $H$-action on the base and $H_i \cong S_4$ (cf. proof of Lemma \[conicbundle\]). Since the action of $G$ on $\mathbb P_1 \times \mathbb P_1$ is effective by assumption, it follows that $I_2$ acts effectively on $\pi_1(\mathbb P_1 \times \mathbb P_1)$. We find a set of four points in $\pi_1(\mathbb P_1 \times \mathbb P_1)$ with nontrivial isotropy with respect to $I_2 \cong C_2 \times C_2$. Since $I_2$ is a normal subgroup of $H$, this set is $H$-invariant. The action of $H_1 \cong S_4$ on $\pi_1(\mathbb P_1 \times \mathbb P_1)$ does however not admit invariant sets of cardinality four since the minimal $S_4$-orbit in $\mathbb P_1$ has length six.
We conclude that $Y$ must be isomorphic to $\mathbb P_2$. It follows that $g(C) =10$. Return to the covering $C \to Q$, $$-18 = e(C) = 8\cdot e(Q) - \text{branch point contributions}.$$ Since $Q$ is rational, the branch point contribution must $34$. The possible isotropy of $N = C_2^3$ at a point in $C$ is $C_2$ and the full branch point contribution must be divisible by four. This contradiction yields the desired non-existence.
The group $N_{72} = C_3^2 \rtimes D_8$
--------------------------------------
\[N72\] We let $X$ be a K3-surface with a symplectic action of $G=N_{72}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Note that in this case we may not apply Theorem \[roughclassi\] and therefore begin by excluding that a $G$-minimal model of $Y=X/\sigma$ is an equivariant conic bundle.
\[N72notconicbundle\] A $G$-minimal model of $Y$ is a Del Pezzo surface.
Assume the contrary and let $Y_\mathrm{min}$ be an equivariant conic bundle and a $G$-minimal model of $Y$. We consider the induced action of $G$ on the base $B=\mathbb P_1$ and denote by $I \lhd G$ the ineffectivity of the $G$-action on $B$. Arguing as in the proof of Lemma \[conicbundle\], we see that $I$ is trivial or isomorphic to either $C_2$ or $C_2 \times C_2$. In all cases the quotient $G/I$ contains the subgroup $C_3 \times C_3$, which has no effective action on the rational curve $B$.
As we will see, only very few Del Pezzo surfaces admit an effective action of the group $N_{72}$. We will explicitly use the group structure of $N_{72}= C_3^2 \rtimes D_8$: the action of $D_8 = C_2 \ltimes (C_2 \times C_2) = \langle \alpha \rangle \ltimes ( \langle \beta \rangle \times \langle \gamma \rangle) = \mathrm{Aut}(C_3 \times C_3)$ on $C_3 \times C_3$ is given by $$\alpha(a,b) = (b,a), \quad \beta(a,b)=(a^2,b), \quad \gamma(a,b) = (a,b^2).$$ As a first step we show:
The degree of a Del Pezzo surface $Y_\mathrm{min}$ is at most four.
We exclude Del Pezzo surface of degree $\geq 5$.
- A Del Pezzo surface of degree five has automorphims group $S_5$ and $N_{72} \nless S_5$.
- The automorphism group of a Del Pezzo surface of degree six is $(\mathbb C^* )^2 \rtimes (S_3 \times C_2)$ (cf. Theorem 10.2.1 in [@dolgachev]). Assume that $N_{72} = C_3^2 \rtimes D_8$ is contained in this group and consider the intersection $ A = N_{72} \cap (\mathbb C^* )^2$. The quotient of $N_{72}$ by $A$ has order at most 12 and may not contain a copy of $C_3^2$. Therefore, the order of $A$ is at least six and $A$ contains a copy of $C_3$. If $|A| =6$, then $A = C_6 = C_3 \times C_2$ and $C_2$ is central in $N_{72}$. Using the group structure of $N_{72}$ specified above one finds that there is no copy of $C_2$ in $N_{72}$ centralizing $C_3 \times C_3$ and therefore $C_2$ cannot be contained in the centre of $N_{72}$. For every choice of $C_3$ inside $C_3 \times C_3$ there is precisely one element in $\{\alpha, \beta, \gamma\}$ acting trivially on it and the centralizer of $C_3$ inside $D_8$ is isomorphic to $C_2$. If $|A| >6$, then the centralizer of $C_3$ in $D_8$ has order greater then 2, a contradiction.
- A Del Pezzo surface of degree seven is obtained by blowing-up to points $p, q$ in $\mathbb P_2$. As was mentioned before, such a surface is never $G$-minimal.
- If $G$ acts on $\mathbb P_1 \times \mathbb P_1$, then the canonical projections are equivariant with respect to a subgroup $H$ of index two in $G$. We consider one of these projections. The action of $H$ induces an effective action of $H/I$ on the base $\mathbb P_1$. The group $I$ is either trivial or isomorphic to $C_2$ or $C_2 \times C_2$. In all case we find an effective action of $C_3 ^2$ on the base, a contradiction.
- It remains to exclude $\mathbb P_2$. If $N_{72}$ acts on $\mathbb P_2$ we consider its embedding into $\mathrm{PSL}_3(\mathbb C)$, in particular the realization of the subgroup $C_3^2 = \langle a \rangle \times \langle b \rangle$ and its lifting to $\mathrm{SL}_3(\mathbb C)$.
We fix a preimage $\tilde a$ of $a$ inside $\mathrm{SL}_3(\mathbb C)$ and may assume that $\tilde a$ is diagonal. Since the action of $a$ on $\mathbb P_2$ is induced by a symplectic action on $X$, it follows that $a$ does not have a positive-dimensional set of fixed point. In appropiately chosen coordinates $$\tilde a=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \xi & 0\\
0&0& \xi^2
\end{pmatrix},$$ where $\xi$ is third root of unity. As a next step, we want to specify a preimage $\tilde b$ of $b$ inside $\mathrm{SL}_3(\mathbb C)$. Since $a$ and $b$ commute in $\mathrm{PSL}_3(\mathbb C)$, we know that $$\tilde a \tilde b \tilde a^{-1} \tilde b^{-1} = \xi^k \mathrm{id}_{\mathbb C^3}$$ for $k \in \{0, 1,2\}$. Note that $\tilde b$ is not diagonal in the coordinates chosen above since this would give rise to $C_3^2 $-fixed points in $\mathbb P_2$. As these correspond to $C_3^2$-fixed points on the double cover $X \to Y$ and a symplectic action of $C_3^2 \nless \mathrm{SL}_2(\mathbb C)$ on a K3-surface does not admit fixed points, this is a contradiction. An explicit calculation yields $$\begin{aligned}
\tilde b = \tilde b_1 =
\begin{pmatrix}
0 & 0 & * \\
* & 0 & 0\\
0 & * & 0
\end{pmatrix} \quad \text{or} \quad
\tilde b = \tilde b_2=
\begin{pmatrix}
0 & * & 0 \\
0 & 0 & *\\
* & 0 & 0
\end{pmatrix}.\end{aligned}$$ We can introduce a change of coordinates commuting with $\tilde a$ such that $$\begin{aligned}
\tilde b = \tilde b_1 =
\begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix} \quad \text{or} \quad
\tilde b = \tilde b_2=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1\\
1&0&0
\end{pmatrix}.\end{aligned}$$ Since $\tilde b_1^2 = \tilde b_2$, the two choices above correspond to the two choices of generators $b$ and $b^2$ of $\langle b \rangle$. We pick $\tilde b = \tilde b_1$.
The action of $D_8$ on $C_3^2 $ is specified above and the element $\beta \in D_8$ acts on $C_3^2 $ by $a \to a^2$ and $b \to b$. There is no element $T \in \mathrm{SL}_3(\mathbb C)$ such that (projectively) $T \tilde a T^{-1} = \tilde a^2$ and $T \tilde b T^{-1} = \tilde b$. It follows that there is no action of $N_{72}$ on $\mathbb P_2$.
This completes the proof of the lemma.
As a next step, we study the possibility of rational curves in $\mathrm{Fix}_X(\sigma)$.
There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
Let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$ and recall $n \leq 10$. If $n \neq 0$, let $C$ be a rational curve in the image of $\mathrm{Fix}_X(\sigma)$ in $Y$ and let $H = \mathrm{Stab}_G(C)$ be its stabilzer. The index of $H$ in $G$ is at most nine, therefore the order of $H$ is at least eight. The action of $H$ on $C$ is effective.
First note that $G$ does not contain $S_4 = O_{24}$ as a subgroup. If this were the case, consider the intersection $S_4 \cap C_3^2$ and the quotient $S_4 \to S_4 / (S_4 \cap C_3^2) < D_8$. Since the only nontrivial normal subgroups of $S_4$ are $A_4$ and $C_2 \times C_2$, this leads to a contradiction.
Consequently, the order of $H$ is at most twelve. In particular, $n \geq 6$. Since $C_8 \nless G$, the group $H$ is not cyclic and any $H$-orbit on $C$ consists of at least two points.
It follows from $C^2 = -4$ that $C$ must meet the union of Mori fibers and the union of Mori fibers meets the curve $C$ in at least two points. Recalling that each Mori fibers meets the branch locus $B$ in at most two points we see that at least $n$ Mori fibers meeting $B$ are required. However, no configuration of $n$ Mori fibers is sufficient to transform the curve $C$ into a curve on a Del Pezzo surface and further Mori fibers are required. By invariance, the total number $m$ of Mori fibers must be at least $2n$.
Combining the Euler-characteristic formula $$24 = 2e(Y_\mathrm{min}) +2m - 2n + \underset{\text{branch curve exists}}{\underset{\text{if non-rational }}{\underbrace{2g-2}}}$$ with our observation $\mathrm{deg}(Y_\mathrm{min}) \leq 4$, i.e., $e(Y_\mathrm{min}) \geq 8$ we see that $n \leq 4$. However, it was shown above, that if $n \neq 0$, then $n \geq 6$. It follows that $n =0$.
The quotient surface $Y$ is $G$-minimal and isomorphic to the Fermat cubic $\{x_1^3 + x_2^3 +x_3^3 +x_4^3 =0\} \subset \mathbb P_3$. Up to equivalence, there is a unique action of $G$ on $Y$ and the branch locus of $X \to Y$ is given by $\{x_1x_2 + x_3x_4 =0\}$. In particular, $X$ is equivariantly isomorphic to Mukai’s $N_{72}$-example.
We first show that the total number $m$ of Mori fibers equals zero. By the Euler-characteristic formula above, the number $m$ is bounded by four. Using the fact that the maximal order of a stabilizer group of a Mori fiber is twelve (cf. proof of Theorem \[roughclassi\]) we see that $Y$ must be $G$-minimal.
In order to conclude that $Y$ is the Fermat cubic we consult Dolgachev’s lists of automorphisms groups of Del Pezzo surfaces of degree less than or equal to four ([@dolgachev] Section 10.2.2; Tables 10.3; 10,4; and 10.5): It follows immediately from the order of $G$ that $Y$ is not of degree two or four. If $G$ were a subgroup of an automorphism group of a Del Pezzo surface of degree one, it would contain a central copy of $C_3$. The group structure of $N_{72}$ does however not allow this. After excluding the cases $\mathrm{deg}(Y) \in \{1,2,4\}$ the result now follows from the uniqueness of the cubic surface in $\mathbb P_3$ with an action of $N_{72}$ (cf. Appendix \[N72appendix\]). The action of $G$ on $Y$ is induced by a four-dimensional (projective) representation of $G$ and the branch curve $C \subset Y$ is the intersection of $Y$ with an invariant quadric (compare proof of Proposition \[S5 on degree three\]).
In the Appendix \[N72appendix\] it is shown that there is a uniquely determined action of $N_{72}$ on $\mathbb P_3$ and a unique invariant quadric hypersurface $\{x_1x_2 + x_3x_4 =0\}$. In particular, the branch curve in $Y$ is defined by $\{x_1x_2 + x_3x_4 =0\} \cap Y$.
Mukai’s $N_{72}$-example is defined by $\{ x_1^3+ x_2 ^3 + x_3^3 +x_4^3= x_1x_2 + x_3x_4+ x_5^2 = 0 \} \subset \mathbb P_4$. An anti-symplectic involution centralizing the action of $N_{72}$ is given by the map $ x_5 \mapsto -x_5$. The quotient of Mukai’s example by this involution is the Fermat cubic and the fixed point set of the involution is given by $\{x_1x_2 + x_3x_4= 0 \}$.
The group $M_9 =C_3^2 \rtimes Q_8$
----------------------------------
\[M9\] Let $G = M_9$ and let $X$ be a K3-surface with a symplectic $G$-action centralized by the antisymplectic involution $\sigma$ such that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. We proceed in analogy to the case $G=N_{72}$ above. Arguing precisely as in the proof of Lemma \[N72notconicbundle\] one shows.
A $G$-minimal model of $Y$ is a Del Pezzo surface.
We may exclude rational branch curves without studying configurations of Mori fibers.
\[subgroupsM9\] There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
Let $n$ be the total number of rational curves in $\mathrm{Fix}_X(\sigma)$. Assume $n \neq 0$, let $C$ be a rational curve in the image of $\mathrm{Fix}_X(\sigma)$ in $Y$ and let $H <G$ be its stabilizer. The action of $H$ on $C$ is effective. We go through the list of finite groups with an effective action on a rational curve.
Since $M_9$ is a group of symplectic transformations on a K3-surface, its element have order at most eight. Clearly, $A_6 \nless M_9$ and $D_{10}, \, D_{14}, \, D_{16} \nless M_9$. If $S_4 < M_9 = C_3^2 \rtimes Q_8$, then $S_4 \cap C_3^2$ is a normal subgroup of $S_4$ and it is therefore trivial. Now $ S_4 = S_4 / (S_4 \cap C_3^2) < M_9 / C_3^2 = Q_8$ yields a contradiction. The same argument can be carried out for $A_4$, $D_8$ and $C_8$. If $D_{12} < M_9 = C_3^2 \rtimes Q_8$, then either $D_{12} \cap C_3^2 = C_3$ and $C_2 \times C_2 = D_{12} / C_3 < M_9 / C_3^2 = Q_8$ or $D_{12} \cap C_3^2 = \{ \mathrm{id} \}$ and $D_{12} < Q_8$, both are impossible.
It follows that the subgroups of $M_9$ admitting an effective action on a rational curve have index greater than or equal to twelve. Therefore $n \geq 12$, contrary to the bound $n \leq 10$ obtained in Corollary \[atmostten\].
The quotient surface $Y$ is $G$-minimal and isomorphic to $\mathbb P_2$. Up to equivalence, there is a unique action of $G$ on $Y$ and the branch locus of $X \to Y$ is given by $\{x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\}$. In particular, $X$ is equivariantly isomorphic to Mukai’s $M_9$-example.
We first check that $Y$ is $G$-minimal. Again, we proceed as in the proof of Theorem \[roughclassi\] and Lemma \[subgroupsM9\] above to see that the largest possible stablizer group of a Mori fiber is $D_6 < G$. If $Y$ is not $G$-minimal, this implies that the total number of Mori fibers is $ \geq 12$, contradicting $m \leq 9$.
Note that $X \to Y$ is not branched along one or two elliptic curves as this would imply $e(Y) =12$ and contradict the fact that $Y$ is a Del Pezzo surface.
Let $D$ be the branch curve of $ X \to Y$ and consider the quotient $Q$ of $D$ by the normal subgroup $N = C_3^2$ in $G$. On $Q$ there is an action of $Q_8$ implying that $Q$ is not rational. We show that $Q_8$ does not act on an elliptic curve $Q$. If this were the case, consider the decomposition $Q_8 = (Q_8 \cap Q) \rtimes (Q_8 \cap L)$ where $(Q_8 \cap L)$ is a nontrivial cyclic group. For any choice of generator of $(Q_8 \cap L)$ the center $\{+1,-1\}$ of $Q_8$ is contained in $(Q_8 \cap L)$. Let $q: Q_8 \to Q_8 /(Q_8 \cap Q) \cong Q_8 \cap L $ denote the quotient homomorphism. The commutator subgroup $Q_8' = \{+1,-1\}$ must be contained in the kernel of $q$. This contradiction yields that $Q_8$ does not act on an elliptic curve. It follows that the genus of $Q$ is at least two and the genus of $D$ is at least ten. Adjunction on the Del Pezzo surface $Y$ now implies $g=10$ and $Y \cong \mathbb P_2$.
It is shown in Appendix \[M9 on P2\] that, up to natural equivalence, there is a unique action of $M_9$ on the projective plane. In suitably chosen coordinated the generators $a, b$ of $C_3^2$ are represented as $$\begin{aligned}
\tilde a=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \xi & 0\\
0&0& \xi^2
\end{pmatrix}, \quad
\tilde b=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1\\
1&0&0
\end{pmatrix}\end{aligned}$$ and $I,J \in Q_8$ are represented as $$\begin{aligned}
\tilde I= \frac{1}{\xi -\xi^2}
\begin{pmatrix}
1 & 1 & 1 \\
1 & \xi & \xi^2\\
1 & \xi^2& \xi
\end{pmatrix}, \quad
\tilde J= \frac{1}{\xi -\xi^2}
\begin{pmatrix}
1 & \xi & \xi \\
\xi^2 & \xi & \xi^2\\
\xi^2 & \xi^2& \xi
\end{pmatrix}.\end{aligned}$$ We study the action of $M_9$ on then space of sextic curves. By restricting our consideration to the subgroup $C_3^2$ first, we see that a polynomial defining an invariant curve must be a linear combination of the following polynomials: $$\begin{aligned}
f_1 &= x_1^6 + x_2^6+ x_3^6;\\
f_2 &= x_1^2x_2^2x_3^2;\\
f_3 &= x_1^3x_2^3 + x_1^3x_3^3+ x_2^3x_3^3;\\
f_4 &= x_1^4x_2x_3 + x_1x_2^4x_3+ x_1x_2x_3^4.\end{aligned}$$ Taking now the additional symmetries into account, we find three $M_9$-invariant sextic curves, namely $$\{f_1- 10f_3 =x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\},$$ which is the example found by Mukai, and additionally $$\{f_a=f_1 + (18-3a) f_2 +2f_3 + af_4 =0\},$$ where $a$ is a solution of the quadratic equation $a^2-6a+36$, i.e. $a= -6\xi$ or $a= -6\xi^2$. The polynomial $f_a$ is invariant with respect to the action of $M_9$ for $a= -6\xi^2$ and semi-invariant if $a= -6\xi$.
We wish to show that $X$ is not the double cover of $\mathbb P_2$ branched along $\{f_a=0\}$. If this were the case, consider the fixed point $p=[0:1:-1]$ of the automorphism $I$ and note that $f_a(p)=0$. So the $\pi^{-1}(p)$ consists of one point $x \in X$ and we linearize the $\langle I \rangle \times \langle \sigma \rangle$ at $x$. In suitably chosen coordintes the action of the symplectic automorphism $I$ of order four is of the form $(z,w) \mapsto (iz, -iw)$. Since the action of $\sigma$ commutes with $I$, the $\sigma$-quotient of $X$ is locally given by $$(z,w) \mapsto (z^2, w) \quad \text{or}\quad (z,w) \mapsto (z, w^2).$$ It follows that the action of $I$ on $Y$ is locally given by either $$\begin {pmatrix}
-1 & 0\\
0 & -i
\end {pmatrix}
\quad \text{or} \quad
\begin {pmatrix}
i & 0\\
0 & -1
\end {pmatrix}.$$ In particular, the local linearization of $I$ at $p$ has determinant $\neq 1$. By a direct computation using the explicit form of $\tilde I$ given above, in particular the facts that $\mathrm{det}(\tilde I) =1$ and $\tilde I v =v$ for $[v]=p$, we obtain a contradiction.
This completes the proof of the proposition.
\[M9 symplectic\] In the proof of the propostion above we have observed that an element of $\mathrm{SL}_3(\mathbb C)$ does not necessarily lift to a symplectic transformation on the double cover of $\mathbb P_2$ branched along a sextic given by an invariant polynomial. Mukai’s $M_9$-example $X$ is a double cover of $\mathbb P_2$ branched along the sextic curve $\{x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\}$ and for this particular example, the action of $M_9$ does lift to a group of symplectic transformation as claimed by Mukai.
To see this consider the set $\{a,b,I,J\}$ of generators of $M_9$. Since $a$ and $b$ are commutators in $M_9$, they can be lifted to symplectic transformation $\overline a, \overline b$ on $X$. For $I,J$ consider the linearization at the fixed point $[0:1:-1]$ and check that it has determinant one. Since $[0:1:-1]$ is *not* contained in the branch set of the covering, its preimage in $X$ consists of two points $p_1,p_2$. We can lift $I$ ($J$, respectively) to a transformation of $X$ fixing both $p_1,p_2$ and a neighbourhood of $p_1$ is $I$-equivariantly isomorphic to a neighbourhood of $ [0:1:-1] \in \mathbb P_2$. In particular, the action of the lifted element $\overline I$ ($\overline J$, respectively) is symplectic. On $X$ there is the action of a degree two central extension $E$ of $M_9$, $$\{\mathrm{id}\} \to C_2 \to E \to M_9 \to \{\mathrm{id}\}.$$ The elements $\overline a, \overline b, \overline I, \overline J$ generate a subgroup $\tilde M_9$ of $E_\mathrm{symp}$ mapping onto $M_9$. Since $E_\mathrm{symp} \neq E$, the order of $\tilde M_9$ is 72 and it follows that $\tilde M_9$ is isomorphic to $M_9$. In particular $E$ splits as $E_\mathrm{symp} \times C_2$ with $E_\mathrm{symp}= M_9$.
The group $T_{48} = Q_8 \rtimes S_3$
------------------------------------
We let $X$ be a K3-surface with an action of $T_{48} \times C_2$ where the action of $G = T_{48}$ is symplectic and the generator $\sigma$ of $C_2$ is antisymplectic and has fixed points. The action of $S_3$ on $Q_8$ is given as follows: The element $c$ of order three in $S_3$ acts on $Q_8$ by permuting $I,J,K$ and an element $d$ of order two acts by exchanging $I$ and $J$ and mapping $K$ to $-K$.
\[not two\] A $G$-minimal model $Y_\mathrm{min}$ of $Y$ is either $\mathbb P_2$, a Hirzebruch surface $\Sigma_n$ with $n >2$, or $e(Y_\mathrm{min}) \geq 9$.
Let us first consider the case where $ Y_\mathrm{min}$ is a Del Pezzo surface and go through the list of possibilities.
- Let $Y_\mathrm{min} \cong \mathbb P_1 \times \mathbb P_1$. Since $T_{48}$ acts on $Y_\mathrm{min}$, both canonical projections are equivariant with respect to the index two subgroup $G'= Q_8 \rtimes C_3$. Since $Q_8$ has no effective action on $\mathbb P_1$, it follows that the subgroup $Z =\{+1,-1\} < Q_8$ acts trivially on the base. Since this holds with respect to both projections, the subgroup $Z$ acts trivially on $Y_\mathrm{min}$, a contradiction.
- Using the group structure of $T_{48}$ one checks that the only nontrivial normal subgroup $N$ of $T_{48}$ such that $N \cap Q_8 \neq Q_8$ is the center $Z= \{+1,-1\}$ of $T_{48}$. It follows that $T_{48}$ is neither a subgroup of $(\mathbb C^*)^2 \rtimes (S_3 \times C_2)$ nor a subgroup of any of the automorphism groups $C_2^4 \rtimes \Gamma$ for $\Gamma \in \{C_2, C_4, S_3, D_{10}\}$ of a Del Pezzo surface of degree four. Furthermore, $T_ {48} \nless S_5$. Thus it follows that $d(Y_\mathrm{min}) \neq 4,5,6$.
So if $Y_\mathrm{min}$ is a Del Pezzo surface, then $Y_\mathrm{min} \cong \mathbb P_2$ or $e(Y_\mathrm{min}) \geq 9$ .
Let us now turn to the case where $Y_\mathrm{min}$ is an equivariant conic bundle. We first show that $Y_\mathrm{min}$ is not a conic bundle with singular fibers. We assume the contrary and let $p: Y_\mathrm{min} \to \mathbb P_1$ be an equivariant conic bundle with singular fibers. The center $Z = \{+1,-1\}$ of $G= T_{48}$ acts trivially on the base an has two fixed points in the generic fiber. Let $C_1$ and $C_2$ denote the two curves of $Z$-fixed points in $Y_\mathrm{min}$. By Lemma \[singular fibers of conic bundle\] any singular fiber $F$ is the union of two (-1)-curves $F_1,F_2$ meeting transversally in one point. We consider the action of $Z$ on this union of curves. The group $Z$ does not act trivially on either component of $F$ since linearization at a smooth point of $F$ would yield a trivial action of $Z$ on $Y_\mathrm{min}$. Consequently, it has either one or three fixed points on $F$. The first is impossible since $C_1$ and $C_2$ intersect $F$ in two points. It follows that $Z$ stabilizes each curve $F_i$. We linearize the action of $Z$ at the point of intersection $F_1 \cap F_2$. The intersection is transversal and the action of $Z$ is by $-\mathrm{Id}$ on $T_{F_1} \oplus T_{F_2}$ contradicting the fact the $Z$ acts trivially on the base. Thus $Y_{min}$ is not a conic bundle with singular fibers.
If $Y_\mathrm{min} \to \mathbb P_1$ is a Hirzebruch surface $\Sigma_n$, then the action of $T_{48}$ induces an effective action of $S_4$ on the base $\mathbb P_1$.
The action of $T_{48}$ on $\Sigma_n$ stabilizes two disjoint sections $E_\infty$ and $E_0$, the curves of $Z$-fixed points. This is only possible if $E_0^2 = -E_\infty^2 = n$. Removing the exceptional section $E_\infty$ from $\Sigma_n$, we obtain the hyperplane bundle $H^n$ of $\mathbb P_1$. Since $T_{48}$ stabilizes the section $E_0$, we chose this section to be the zero section and conclude that the action of $T_{48}$ on $H^n$ is by bundle automorphisms.
If $n=2$, then $H^n$ is the anticanonical line bundle of $\mathbb P_1$ and the action of $S_4$ on the base induces an action of $S_4$ on $H^2$ by bundle automorphisms. It follows that $T_{48}$ splits as $S_4 \times C_2$, a contradiction. Thus, if $Y_\mathrm{min}$ is a Hirzebruch surface $\Sigma_n$, then $n \neq 2$.
\[no rat T48\] There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
We let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$ and assume $n >0$. Recall $n \leq 10$, let $C$ be a rational curve in $B=\pi(\mathrm{Fix}_X(\sigma)) \subset Y$ and let $H = \mathrm{Stab}_G(C) < G$ be its stabilizer group. The action of $H$ on $C$ is effective, the index of $H$ in $G$ is at most 8. Using the quotient homomorphism $T_{48} \to T_{48}/Q_8 = S_3$ one checks that $T_{48}$ does not contain $O_{24}=S_4$ or $T_{12}= A_4$ as a subgroup. It follows that $H$ is a cyclic or a dihedral group.
If $H\in \{C_6, C_8, D_8\}$, then $H$ and all conjugates of $H$ in $G$ contain the center $Z= \{+1,-1\}$ of $G$. It follows that $Z$ has two fixed point on each curve $gC$ for $g \in G$. Since there are six (or eight) distinct curves $gC$ in $Y$, it follows that $Z$ has at least 12 fixed points in $Y$ and in $X$. This contradicts to assumption that $Z < G$ acts symplectically on $X$ and therefore has eight fixed points in the K3-surface $X$.
It remains to study the cases $H = D_{12}$ and $H = D_6$ where $n = 8$ or $n=4$.
We note that a Hirzebruch surface has precisely one curve with negative self-intersection and only fibers have self-intersection zero. A Del Pezzo surface does not contains curves of self-intersection less than $-1$. The rational branch curves must therefore meet the union of Mori fibers in $Y$.
The total number of Mori fibers is bounded by $n+9$. We study the possible stabilizer subgroups $\mathrm{Stab}_G(E) < G$ of Mori fibers. A Mori fiber $E$ with self-intersection (-1) meets the branch locus $B$ in one or two points and its stabilizer is either cyclic or dihedral. If $\mathrm{Stab}_G(E) \in \{C_4, D_8\}$, then the points of intersection of $E$ and $B$ are fixed points of the center $Z$ of $G$ and we find too many $Z$-fixed points on $X$.
Assume $n=4$ and let $R_1, \dots R_4$ be the rational curves in $B$. We denote by $\tilde R_i$ their images in $Y_\mathrm{min}$. The total number $m$ of Mori fibers is bounded by 12. We go through the list of possible configurations:
- If $m = 4$, there is no invariant configuration of Mori fibers such that the contraction maps the four rational branch curves to a configuration on the Hirzebruch or Del Pezzo surface $Y_\mathrm{min}$.
- If $m= 6$, then $\mathrm{Stab}_G(E) = C_8$ and the points of intersection of $E$ and $B$ are $Z$-fixed. Since $Z$ has at most eight fixed points on $B$, it follows that each curve $E$ meets $B$ only once. The images $\tilde R_i$ of the $R_i$ contradict our observations about curves in Del Pezzo and Hirzebruch surfaces.
- If $m=8$ and all Mori fibers have self-intersection $-1$, then each Mori fiber meets $\bigcup R_i$ in a $Z$-fixed point. Since there at at most eight such points, it follows that each Mori fibers meets $\bigcup R_i$ only once and their contractions does not transform the curves $R_i$ sufficiently.
- If $m=8$ and only four Mori fibers have self-intersection $-1$, we consider the four Mori fibers of the second reduction step. Each of these meets a Mori fiber $E$ of the first step in precisely one point. By invariance, this would have to be a fixed points of the stabilizer $ \mathrm{Stab}_G(E)= D_{12}$, a contradiction.
- If $m=12$, then either $e(Y_\mathrm{min}) = 3$ and there exist a branch curve $D_{g=2}$ of genus two or $e(Y_\mathrm{min}) = 4$ and $B = \bigcup R_i$. In the first case, $Y_\mathrm{min} \cong \mathbb P_2$ and twelve Mori fibers are not sufficient to transform $B = D_{g=2} \cup \bigcup R_i$ into a configuration of curves in the projective plane. So $Y_\mathrm{min} = \Sigma_n$ for $n > 2$.
Since $Z$ has two fixed points in each fiber of $p: \Sigma_n \to \mathbb P_1$ the $Z$-action on $\Sigma_n$ has two disjoint curves of fixed points. As was remarked above, these curves are the exceptional section $E_\infty$ of self-intersection -$n$ and a section $E_0 \sim E_\infty + n F$ of self-intersection $n$. Here $F$ denotes a fiber of $p: \Sigma_n \to \mathbb P_1$. There is no automorphisms of $\Sigma_n$ mapping $E_\infty$ to $E_0$.
Each rational branch curve $\tilde R_i$ has two $Z$-fixed points. These are exchanged by an element of $\mathrm{Stab}_G(R_i)$ and therefore both lie on either $E_\infty$ or $E_0$, i.e., $\tilde R_i$ cannot have nontrivial intersection with both $E_0$ and $E_\infty$. By invariance all curves $\tilde R_i$ either meet $E_0$ or $E_\infty$ and not both.
Using the fact that $\sum \tilde R_i$ is linearly equivalent to $-2K_{\Sigma_n} \sim 4 E_\infty +(2n +4)F$ we find that $\tilde R_i \cdot E_\infty = 0$ and $n=2$, a contradiction to Lemma \[not two\].
We have shown that all possible configurations in the case $n \neq 4$ lead to a contradiction. We now turn to the case $n=8$ and let $R_1, \dots R_8$ be the rational ramification curves. The total number of Mori fibers is bounded by 16. Note that by invariance, the orbit of a Mori fiber meets $\bigcup R_i$ in at least 16 points or not at all. In particular, Mori fibers meeting $R_i$ come in orbits of length $\geq 8$. As above, we go through the list of possible configurations.
- If $m =16$, then the set of all Mori fibers consists of two orbits of length eight. If all 16 Mori fibers meet $B$, then each meets $B$ in one point and $R_i$ is mapped to a (-2)-curve in $Y_\mathrm{min}$. If only eight Mori fibers meet $B$, then each of the eight Mori fibers of the second reduction step meets one Mori fiber $E$ of the first reduction step in one point. This point would have to be a $\mathrm{Stab}_G(E)$-fixed point. But if $\mathrm{Stab}_G(E)$ is cyclic, its fixed points coincide with the points $E\cap B$.
- If $m=12$, then the set of all Mori fibers consists of a single $G$-orbit and each curve $R_i$ meets three distinct Mori fibers. Their contraction transforms $R_i$ into a (-1)-curve on $Y_\mathrm{min}$. It follows that $Y_\mathrm{min}$ contains at least eight (-1)-curves and is a Del Pezzo surface of degree $\leq 5$. We have seen above that $d( Y_\mathrm{min}) \neq 4,5$ and therefore $e(Y_\mathrm{min}) \geq 9$. With $m=12$ and $n=8$, this contradicts the Euler characteristic formula $24 = 2 e(Y_\mathrm{min}) +2m -2n +(2g-2)$.
- If $m=8$ there is no invariant configuration of Mori fibers such that the contraction maps the eight rational branch curves to a configuration on the Hirzebruch or Del Pezzo surface $Y_\mathrm{min}$
This completes the proof of the lemma.
Since there is an effective action of $T_{48}$ on $\mathrm{Fix}_X(\sigma)$, it is neither an elliptic curve nor the union of two elliptic curves. It follows that $ X \to Y$ is branched along a single $T_{48}$-invariant curve $B$ with $g(B) \geq 2$.
The genus of $B$ is neither three nor four.
We consider the quotient $Q = B/Z$ of the curve $B$ by the center $Z$ of $G$ and apply the Euler characteristic formula, $e(B) = 2 e(Q) - |\mathrm{Fix}_B(Z)|$. On $Q$ there is an effective action of the group $G/Z = (C_2 \times C_2) \rtimes C_3 = S_4$. Using Lemma \[S4 not on g=1,2\] we see that $e(Q) \in \{2, -4, -6,-8 \dots\}$.
If $g(B)=3$, then $e(B) = -4$ and the only possibility is $Q \cong \mathbb P_1$ and $|\mathrm{Fix}_B(Z)| =8$. In particular, all $Z$-fixed points on $X$ are contained in the curve $B$. Let $A < G$ be the group generated by $I \in Q_8 = \{ \pm 1, \pm I, \pm J, \pm K \}$. The four fixed points of $A$ in $X$ are contained in $\mathrm{Fix}_X(Z) = \mathrm{Fix}_B(Z)$ and the quotient group $A/Z \cong C_2$ has four fixed points in $Q$. This is a contradiction.
If $g(B)=4$, then $e(B) = -6$ and the only possibility is $Q \cong \mathbb P_1$ and $|\mathrm{Fix}_B(Z)| =10$. This contradicts the fact that $Z$ has at most eight fixed points in $B$ since it has precisely eight fixed points in $X$.
In Lemma \[not two\] we have reduced the classification to the cases $e(Y_\mathrm{min}) \in \{3,4, 9, 10, 11\}$. In the following, we will exclude the cases $e(Y_\mathrm{min}) \in \{4, 9, 10,\}$ and describe the remaining cases more precisely. Recall that the maximal possible stabilizer subgroup of a Mori fiber is $D_{12}$, in particular, $m = 0$ or $ m \geq 4$.
If $e(Y_\mathrm{min}) =3$, then $Y_\mathrm{min} = Y = \mathbb P_2$ and $X \to Y$ is branched along the curve $\{x_1x_2(x_1^4-x_2^4) + x_3^6=0\}$. In particular, $Y$ is equivariantly isomorphic to Mukai’s $T_{48}$-example.
Let $ M: Y \to \mathbb P_2$ denote a Mori reduction of $Y$ and let $B \subset Y$ be the branch curve of the covering $X \to Y$. If $Y = Y_\mathrm{min}$, then $B= M(B)$ is a smooth sextic curve. If $Y \neq Y_\mathrm{min}$, then the Euler characteristic formula with $m \in \{4,6,8\}$ shows that $g(B) \in \{2,4,6\}$. The case $m=6$, $g(B) =4$ has been excluded by the previous lemma.
If $m=4$, then the stabilizer group of each Mori fiber is $D_{12}$ and each Mori fiber meets $B$ in two points. Furthermore, since in this case $g(B) =6$, the self-intersection of $\mathrm{Fix}_X(\sigma)$ in $X$ equals ten and therefore $B^2 = 20$. The image $M(B)$ of $B$ in $Y_\mathrm{min}$ has self-intersection $20 + 4 \cdot 4 = 36$ and follows to be an irreducible singular sextic.
If $m=8$, then $g(B) =2$ and $B^2 =4$. Since the self-intersection number $M(B)^2$ must be a square, one checks that all possible invariant configurations of Mori fibers yield $M(B)^2 =36$ and involve Mori fibers meeting $B$ is two points. In particular, $M(B)$ is a singular sextic.
We study the action of $T_{48}$ on the projective plane. As a first step, we may choose coordinates on $\mathbb P_2$ such that the automorphism $-1 \in Q_8 < T_{48}$ is represented as $$\widetilde {-1}=
\begin{pmatrix}
-1 & 0 & 0 \\
0 & -1 & 0\\
0&0& 1
\end{pmatrix}.$$ We denote by $V$ to the $-1$-eigenspace of this operator. For each element $I, J , K$ there is a unique choice $\widetilde I, \widetilde J, \widetilde K$ in $\mathrm{SL}_3(\mathbb C)$ such that $\widetilde I ^2 = \widetilde J^2 = \widetilde K^2 = \widetilde {-1}$. One checks $\widetilde I \widetilde J\widetilde K = \widetilde {-1}$. Therefore $\widetilde I, \widetilde J, \widetilde K$ generate a subgroup of $\mathrm{SL}_3(\mathbb C)$ isomorphic to $Q_8$. By construction $\widetilde I, \widetilde J, \widetilde K$ stabilze the vector space $V$. Up to isomorphisms, there is a unique faithful 2-dimensional representation of $Q_8$ and it follows that $I,J,K$ are represented as $$\begin{aligned}
\widetilde I =
\begin{pmatrix}
-i & 0 & 0 \\
0 & i & 0\\
0&0& 1
\end{pmatrix}, \quad
\widetilde J=
\begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0\\
0 & 0 & 1
\end{pmatrix}, \quad
\widetilde K=
\begin{pmatrix}
0 & i & 0 \\
i & 0 & 0\\
0 & 0 & 1
\end{pmatrix}. \end{aligned}$$ We recall that the action of $S_3$ on $Q_8$ is given as follows: The element $c$ of order three in $S_3$ acts on $Q_8$ by permuting $I,J,K$ and an element $d$ of order two acts by exchanging $I$ and $J$ and mapping $K$ to $-K$. With $\mu = \sqrt{\frac{i}{2}}$ and $ \nu = \frac{i}{\sqrt{2}}$ it follows that the elements $c$ and $d$ are represented as $$\begin{aligned}
\widetilde c =
\begin{pmatrix}
-i\mu & i\mu& 0 \\
\mu & \mu & 0\\
0&0& 1
\end{pmatrix}, \quad
\widetilde d=
\begin{pmatrix}
-i\nu & -\nu & 0 \\
\nu & i\nu & 0\\
0 & 0 & -1
\end{pmatrix}.\end{aligned}$$ In particular, there is a unique action of $T_{48}$ on $\mathbb P_2$. In the following, we denote by $[x_1:x_2:x_3]$ homogeneous coordintes such that the action of $T_{48}$ is as above. Using the explicit form of the $T_{48}$-action and the fact that the commutator subgroup of $T_{48}$ is $Q_8 \rtimes C_3$ one can check that any invariant curve of degree six is of the form $$C_\lambda = \{ x_1x_2(x_1^4-x_2^4) + \lambda x_3^6 =0\}$$ In order to avoid this calculation, one can also argue that the polynomial $x_1x_2(x_1^4-x_2^4)$ is the lowest order invariant of the octahedral group $S_4 \cong T_{48} / Z$.
The curve $C_\lambda$ is smooth and it follows that $Y = Y_\mathrm{min}$. We may adjust the coordinates equivariantly such that $\lambda =1$ and find that our surface $X$ is precisely Mukai’s $T_{48}$-example.
\[T48 symplectic\] As claimed by Mukai, the action of $T_{48}$ on $\mathbb P_2$ does indeed lift to a symplectic action of $T_{48}$ on the double cover of $\mathbb P_2$ branched along the invariant curve $\{ x_1x_2(x_1^4-x_2^4) + x_3^6 =0\}$. The elements of the commutator subgroup can be lifted to symplectic transformation on the double cover $X$.
The remaining generator $d$ is an involution fixing the point $[0:0:1]$. Any involution $\tau$ with a fixed point $p$ outside the branch locus can be lifted to a symplectic involution on the double cover $X$ as follows:
The linearized action of $\tau$ at $p$ has determinant $\pm 1$. We consider the lifting $\tilde \tau$ of $\tau$ fixing both points in the preimage of $p$. Its linearization coincides with the linearization on the base and therefore also has determinant $\pm 1$. In particular, $\tilde \tau$ is an involution. It follows that either $\tilde \tau$ or the second choice of a lifting $\sigma \tilde \tau$ acts symplectically on $X$.
The group generated by all lifted automorphisms is either isomorphic to $T_{48}$ or to the full central extension $E$ $$\{\mathrm{id}\} \to C_2 \to E \to T_{48} \to \{\mathrm{id}\}$$ acting on the double cover. Since $E_\mathrm{symp} \neq E$ the later is impossible it follows that $E$ splits as $E_\mathrm{symp} \times C_2$ with $E_\mathrm{symp} = T_{48}$.
Finally, we return to the remaining possibilities $e(Y_\mathrm{min}) \in \{4, 9, 10, 11\}$.
$e(Y_\mathrm{min}) \not\in \{4,9,10\}$.
Recalling that the genus of the branch curve $B$ is neither three nor four and that $m$ is either zero or $\geq 4$, we may exclude $e(Y_\mathrm{min}) =9,10$ using the Euler characteristic formula $12 = e(Y_\mathrm{min}) +m +g-1$. It remains to consider the case $Y_\mathrm{min} = \Sigma _n$ with $n >2$ and we claim that this is impossible.
Let $M= Y \to Y_\mathrm{min} = \Sigma_n$ denote a (possibly trivial) Mori reduction of $Y$. The image $M(B)$ of $B$ in $\Sigma _n$ is linearly equivalent to $-2K_{\Sigma _n}$. Now $M(B) \cdot E_\infty = 2(2-n) < 0$ and it follows that $M(B)$ contains the rational curve $E_\infty$. This is a contradiction since $B$ does not contain any rational curves by Lemma \[no rat T48\].
In the last remaining case, i.e., $e(Y_\mathrm{min})=11$, the quotient surface $Y$ is a $G$-minimal Del Pezzo surface of degree 1. Consulting [@dolgachev], Table 10.5, we find that $Y$ is a hypersurface in weighted projective space $\mathbb P(1,1,2,3)$ defined by the degree six equation $$x_0x_1(x_0^4-x_1^4)+ x_2^3+x_3^2.$$ This follows from the invariant theory of the group $S_4 \cong T_{48}/Z$ and fact that $Y$ is a double cover of a quadric cone $Q$ in $\mathbb P_3$ branched along the intersection of $Q$ with a cubic hypersurface (cf. Theorem \[antican models of del pezzo\]).
The linear system of the anticanonical divisor $K_Y$ has precisely one base point $p$. In coordinates $[x_0:x_1:x_2:x_3]$ this point is given as $[0:0:1:i]$. It is fixed by the action of $T_{48}$. The linearization of $T_{48}$ at $p$ is given by the unique faithful 2-dimensional represention of $T_{48}$. This represention has implicitly been discussed above as a subrepresentation $V$ of the three-dimensional representation of $T_{48}$. It follows that there is a unique action of $T_{48}$ on $Y$. The branch curve $B$ is linearly equivalent to $-2K_Y$, i.e., $B = \{s=0\}$ for a section $s \in \Gamma(Y, \mathcal O(-2K_Y))$ which is either invariant or semi-invariant.
By an adjunction formula for hypersurfaces in weighted projective space $\mathcal O (-2K_Y)) = \mathcal O _Y(2)$. The four-dimensional space of sections $\Gamma(Y, \mathcal O(-2K_Y))$ is generated by the weighted homogeneous polynomials $x_0^2, x_1^2, x_0x_1, x_2$. We consider the map $ Y \to \mathbb P (\Gamma(Y, \mathcal O(-2K_Y))^*)$ associated to $|-2K_Y|$. Since this map is equivariant with respect to $\mathrm{Aut}(Y)$, the fixed point $p$ is mapped to a fixed point in $\mathbb P (\Gamma(Y, \mathcal O(-2K_Y))^*)$. It follows that the section corresponding to the homogeneous polynomial $x_2$ is invariant or semi-invariant with respect to $T_{48}$. It is the only section of $\mathcal O(-2K_Y)$ with this property since the representation of $T_{48}$ on the span of $x_0^2, x_1^2, x_0x_1$ is irreducible.
The curve $B \subset Y$ defined by $s=0$ is connected and has arithmetic genus $2$. Since $T_{48}$ acts effectively on $B$ and does not act on $\mathbb P_1$ or a torus, it follows that $B$ is nonsingular.
It remains to check that the action of $T_{48}$ on $Y$ lifts to a group of symplectic transformation on the double cover $X$ branched along $B$. First note that $B$ does not contain the base point $p$. For $I,J,K, c \in T_{48}$ we we choose liftings $ \overline I, \overline J, \overline K , \overline c \in \mathrm{Aut}(X)$ fixing both points in $\pi^{-1}(p)= \{p_1,p_2\}$. The linearization of $ \overline I, \overline J, \overline K , \overline c$ at $p_1$ is the same as the linearization at $p$ and in particular has determinant one. By the general considerations in Remark \[T48 symplectic\] the involution $d$ can be lifted to a symplectic involution on $X$. The symplectic liftings of $I,J,K,c,d$ generate a subgroup $\tilde G$ of $\mathrm{Aut}(X)$ which is isomorphic to either $T_{48}$ or to the central degree two extension of $T_{48}$ acting on $X$. In analogy to Remarks \[M9 symplectic\] and \[T48 symplectic\] we conclude that $\tilde G \cong T_{48}$ and the action of $T_{48}$ on $Y$ induces a symplectic action of $T_{48}$ on the double cover $X$.
This completes the classification of K3-surfaces with $T_{48} \times C_2$-symmetry. We have shown:
Let $X$ be a K3-surface with a symplectic action of the group $T_{48}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then $X$ is equivariantly isomorphic either to Mukai’s $T_{48}$-example or to the double cover of $$\{x_0x_1(x_0^4-x_1^4)+ x_2^3+x_3^2=0\} \subset \mathbb P(1,1,2,3)$$ branched along $\{x_2=0\}$
The automorphism group of the Del Pezzo surface $Y = \{x_0x_1(x_0^4-x_1^4)+ x_2^3+x_3^2=0\} \subset \mathbb P(1,1,2,3)$ is the trivial central extension $C_3 \times T_{48}$. By contruction, the curve $B=\{s=0\}$ is invariant with respect to the full automorphism group. The double cover $X$ of $Y$ branched along $B$ carries the action of a finite group $\tilde G$ of order $2 \cdot 3 \cdot 48 = 288$ containing $T_{48} < \tilde G_\mathrm{symp}$. Since $T_{48}$ is a maximal group of symplectic transformations, we find $T_{48} = \tilde G_\mathrm{symp}$ and therefore $$\{\mathrm{id}\} \to T_{48} \to \tilde G \to C_6 \to \{\mathrm{id}\}.$$ In analogy to the proof of Claim 2.1 in [@OZ168], one can check that 288 is the maximal order of a finite group $H$ acting on a K3-surface with $T_{48} < H_\mathrm{symp}$. It follows that $\tilde G$ is maximal finite subgroup of $\mathrm{Aut}(X)$. For an arbitrary finite group $H$ acting on a K3-surface with $\{\mathrm{id}\} \to T_{48} \to H \to C_6 \to \{\mathrm{id}\}$, there need however not exist an involution in $H$ centralizing $T_{48}$.
K3-surfaces with an antisymplectic involution centralizing $C_3 \ltimes C_7$ {#chapterC3C7}
============================================================================
In this chapter it is illustrated that a classification of K3-surfaces with antisymplectic involution $\sigma$ can be carried out even even if the centralizer $G$ of $\sigma$ inside the group of symplectic transformations is relatively small, i.e., well below the bound 96 obtianed in Theorem \[roughclassi\], and not among the maximal groups of symplectic transformations. We consider the group $G = C_3 \ltimes C_7$, which is a subgroup of $L_2(7)$. The principles presented in Chapter \[chapterlarge\] can be transferred to this group $G$ and yield a description of K3-surfaces with $G \times \langle \sigma \rangle$-symmetry. Using this, we deduce the classification K3-surfaces with an action of $L_2(7) \times C_2$ announced in Section \[mukaiL2(7)\]. The results presented in this chapter have appeared in [@FKH1].
To begin with, we present a family of K3-surfaces with $G \times \langle \sigma \rangle$-symmetry.
We consider the action of $G$ on $\mathbb P_2$ given by one of its three-dimensional representations. After a suitable change of coordinates, the action of the commutator subgroup $G'=C_7 < G$ is given by $$[z_0:z_1:z_2] \mapsto [ \lambda z_0: \lambda^2 z_1: \lambda^4 z_2]$$ for $\lambda = \mathrm{exp}(\frac{2\pi i}{7})$ and $C_3$ is generated by the permutation $$[z_0:z_1:z_2] \mapsto [z_2:z_0:z_1].$$ The vector space of $G$-invariant homogeneous polynomials of degree six is the span of $P_1 = z_0^2 z_1^2 z_2^2$ and $P_2 = z_0^5 z_1 + z_2 ^5 z_0 + z_1^5 z_2$.
The family $\mathbb{P}(V)$ of curves defined by polynomials in $V$ contains exactly four singular curves, namely the curve defined by $z_0^2z_1^2z_2^2$ and those defined by $3z_0^2z_1^2z_2^2 -\zeta ^k(z_0^5z_1+z_2^5z_0+z_1^5z_2)$, where $\zeta $ is a nontrivial third root of unity, $k=1,2,3$. We let $\Sigma = \mathbb P(V) \backslash \{z_0^2z_1^2z_2^2=0\}$.
The double cover of $\mathbb P_2$ branched along a curve $C \in \Sigma$ is a K3-surface (singular K3-surface if $C$ is singular) with an action of $G \times C_2$ where $C_2$ acts nonsymplectically. It follows that $\Sigma$ parametrizes a family of K3-surface with $G \times C_2$-symmetry.
Let us consider the cyclic group $\Gamma$ of order three generated by the transformation $[z_0:z_1:z_2]\mapsto [z_0:\zeta z_1: \zeta ^2z_2]$ and its induced action on the space $\Sigma$. One finds that the three irreducible singular $G$-invariant curves form a $\Gamma$-orbit. Furthermore, if two curves $C_1, C_2 \in \Sigma$ are equivalent with respect to the action of $\Gamma$, then the corresponding K3-surfaces are equivariantly isomorphic (see Section \[EquiEqui\] for a detailed discussion).
The singular curve $C_\text{sing} \subset \mathbb P_2$ defined by $3z_0^2z_1^2z_2^2 -(z_0^5z_1+z_2^5z_0+z_1^5z_2)$ has exactly seven singular points $p_1, \dots p_7$ forming an $G$-orbit. Since they are in general position (cf. Proposition \[blowdownseven\]), the blow up of $\mathbb{P}_2$ in these points defines a Del Pezzo surface $Y_\text{Klein}$ of degree two with an action of $G$. It is seen to be the double cover of $\mathbb{P}_2$ branched along Klein’s quartic curve $$C_\text{Klein}:=\{z_0z_1^3+z_1z_2^3+z_2z_0^3=0\}.$$ The proper transform $B$ of $C_\text{sing}$ in $Y_\text{Klein}$ is a smooth $G$-invariant curve. It is a normalization of $C_\text{sing}$ and has genus three by the genus formula. The curve $B$ coincides with the preimage of $C_{\text{Klein}}$ in $Y_\text{Klein}$. The minimal resolution $\tilde X_\text{sing}$ of the singular surface $X_\text{sing}$ defined as the double cover of $\mathbb P_2$ branched along $C_\text{sing}$ is a K3-surface with an action of $G$. By construction, it is the double cover of $Y_\text{Klein}$ branched along $B$. In particular, $\tilde X
_\text{sing}$ is the degree four cyclic cover of $\mathbb P_2$ branched along $C_\text{Klein}$ and known as the Klein-Mukai-surface $X_{\text{KM}}$ (cf. Example \[L2(7)example\]).
In the following, the notion of “$G \times C_2$-symmetry” abbreviates a symplectic action of $G$ centralized an antisymplectic action of $C_2$.
In this chapter we will show that the space $\mathcal M = \Sigma / \Gamma$ parametrizes K3-surfaces with $G \times C_2$-symmetry up to equivariant equivalence. More precisely, we prove:
\[mainthmc3c7\] The K3-surfaces with a symplectic action of $G = C_3 \ltimes C_7$ centralized by an antisymplectic involution $\sigma$ are parametrized by the space $\mathcal M = \Sigma / \Gamma$ of equivalence classes of sextic branch curves in $\mathbb P_2$. The Klein-Mukai-surface occurs as the minimal desingularization of the double cover branched along the unique singular curve in $\mathcal M$.
Inside the family $\mathcal M$ one finds two K3-surfaces with a symplectic action of the larger group $L_2(7)$ centralized by an antisymplectic involution.
\[L2(7) times invol\] There are exactly two K3-surfaces with an action of the group $L_2(7)$ centralized by an antisymplectic involution. These are the Klein-Mukai-surface $X_\mathrm{KM}$ and the double cover of $\mathbb P_2$ branched along the curve $\mathrm{Hess}(C_\text{Klein})=\{z_0^5z_1+z_2^5z_0+z_1^5z_2-5z_0^2 z_1^2 z_2^2=0\}$.
Branch curves and Mori fibers {#branch-curves-and-mori-fibers}
-----------------------------
Let $X$ be a K3 surface with an symplectic action of $G=C_3 \ltimes C_7$ centralized by the antisymplectic involution $\sigma$. We consider the quotient $\pi: X \to X/\sigma =Y$. Since the action of $G'$ has precisely three fixed points in $X$ and $\sigma$ acts on this point set, we know that $\mathrm{Fix}_X(\sigma)$ is not empty. It follows that $Y$ is a smooth rational surface with an effective action of the group $G$ to which we apply the equivariant minimal model program. The following lemma excludes the possibility that a $G$-minimal model is a conic bundle. The argument resembles that in the proof of Lemma \[conicbundle\].
\[C3C7 conic bundle\] A $G$-minimal model of $Y$ is a Del Pezzo surface.
Assume the contrary and let $Y_\mathrm{min} \to \mathbb P_1$ be a $G$-equivariant conic bundle. Since $G$ has no effective action on the base, there must be a nontrivial normal subgroup acting trivially on the base. This subgroup must be $G'$. The action of $G'$ on the generic fiber has two fixed points and gives rise to a positive-dimensional $G'$-fixed point set in $Y_\mathrm{min}$ and $Y$. Since the action of $G'$ on $Y$ is induced by a symplectic action of $G'$ on $X$, this is a contradiction.
\[P1xP1C3C7\] Since $G$ has no subgroup of index two, the above proof also shows that $Y_\mathrm{min} \not\cong \mathbb P_1 \times \mathbb P_1$.
In analogy to the procedure of the previous chapter we exclude rational and elliptic ramification curves and show that $\pi$ is branched along a single curve of genus greater than or equal to three.
The set $\mathrm{Fix}_X(\sigma)$ consists of a single curve $C$ and $g(C) \geq 3$.
We let $\{x_1,x_2,x_3\} = \mathrm{Fix}_X(G')$. Since $G$ has no faithful two-dimensional representation, it has no fixed points in $X$ an therefore acts transitively on $\{x_1,x_2,x_3\}$. It follows that the central involution $\sigma$, which fixes at at least one point $x_i$, fixes all three points by invariance. Now $\{x_1,x_2,x_3\} \subset \mathrm{Fix}_X(\sigma)$ implies that $G'$ has precisely three fixed points in $Y$. Let $C_i$ denote the connected component of $\mathrm{Fix}_X(\sigma)$ containing $x_i$. Since $G$ acts on the set $\{C_1,C_2,C_3\}$, it follows that either $C_1=C_2=C_3$ or no two of them coincide.
In the later case, it follows from Theorem \[FixSigma\] that at least two curves $C_1,C_2$ are rational. The action of $G'$ on a rational curves $C_i$ has two fixed points. We therefore find at least five $G'$-fixed points in $X$ contradicting $|\mathrm{Fix}_X(G')|=3$.
It follows that all three points $x_1,x_2,x_3$ lie on one $G$-invariant connected component $C$ of $\mathrm{Fix}_X(\sigma)$. The action of $G$ on $C$ is effective and it follows that $C$ is not rational.
If $g(C)=1$, then an effective action of $G$ on $C$ would force $G'$ to act by translations on $C$, in particular freely, a contradiction.
If $g(C)=2$, then $C$ is hyperelliptic. The quotient $C \to \mathbb P_1$ by the hyperellitic involution is $\mathrm{Aut}(C)$-equivariant and would induce an effective action of $G$ on $\mathbb P_1$, a contradiction.
It follows that $g(C) \geq 3$ and it remains to check that there are no rational ramification curves.
We let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$. Since $G'$ acts freely on the complement of $C$ in $X$, it follows that the number $n$ must be a multiple of seven. Combining this observation with the bound $n \leq 9$ from Corollary \[atmostten\] we conclude that $n$ is either 0 or 7.
We suppose $n =7$ and let $m$ denote the total number of Mori contractions of a reduction $Y \to Y_\mathrm{min}$. The Euler characeristic formula $$13 - g(C) = e(Y_\mathrm{min}) +m -n$$ with $n=7$, $g(C) \geq 3$ and $e(Y_\mathrm{min})\geq 3$ implies $m \leq 14$.
Let us first check that no Mori fiber $E$ coincides with a rational branch curve $B$. If this was the case, then all seven rational branch curves coincide with Mori fibers. Rational branch curves have self-intersection -4 by Corollary \[minusfour\]. Before they may by contracted, they need to be transformed into (-1)-curves by earlier reduction steps. The remaining seven or less Mori contraction are not sufficient to achieve this transformation. It follows that each rational branch curve is mapped to a curve in $Y_\mathrm{min}$ and not to a point.
We now first consider the case $m=14$. The Euler characteristic formula implies $Y_\mathrm{min} \cong \mathbb P_2$ and $g(C)=3$. Using our study of Mori fibers and branch curves in Section \[branch curves mori fibers\], in particular Remark \[self-int of Mori-fibers\] and Proposition \[at most two\], we see that no configuration of 14 Mori fibers is such that the images in $Y_\mathrm{min} \cong \mathbb P_2$ of any two rational branch curves have nonempty intersection. It follows that $m \leq 13$.
Let $R_1, \dots, R_7 \subset Y$ denote the rational branch curves. Each curve $R_i$ has self-intersection -4 and therefore has nontrivial intersection with at least one Mori fiber. Let $E_1$ be a Mori fiber meeting $R_1$, let $H \cong C_3$ be the stabilizer of $R_1$ in $G$ and let $I$ be the stabilizer of $E_1$ in $G$. Since $m \leq 13$ the group $I$ is nontrivial. If $I$ does not stabilize $R_1$, then $E_1$ meets the branch locus in at least three points. This is contrary to Proposition \[at most two\]. It follows that $I=H$. If $E_1$ meets any other rational branch curve $R_2$, then it meets all curves in the $H$-orbit through $R_2$. Since $H$ acts freely on the set $\{R_2, \dots, R_7\}$, it follows that $E_1$ meets three more branch curves. This is again contradictory to Proposition \[at most two\].
Since $m \leq 13$ it follows that each rational branch curve meets exactly one Mori fiber. Their intersection can be one of the following three types:
1. $E_i \cap R_i = \{p_1,p_2\}$ or
2. $E_i \cap R_i = \{p\}$ and $ (E_i, R_i)_p =2$ or
3. $E_i \cap R_i = \{p\}$ and $ (E_i, R_i)_p =1$.
In all three cases the contraction of $E_i$ alone does not transform the curve $R_i$ into a curve on a Del Pezzo surface. So further reduction steps are needed and require the existence of Mori fibers $F_i$ disjoint from $\bigcup R_i$. Each $F_i$ is a (-2)-curve meeting $\bigcup E_i$ transversally in one point and the total number of Mori fibers exceeds our bound 13.
This contradiction yields $n=0$ and the proof of the proposition is completed.
Classification of the quotient surface $Y$
------------------------------------------
We now turn to a classification of the quotient surface $Y$.
The surface $Y$ is either $G$-minimal or the blow up of $\mathbb P_2$ in seven singularities of an irreducible $G$-invariant sextic..
Since $n=0$, the Euler characteristic formula yields $m \leq 7$. The fact that $G$ acts on the set of Mori fibers implies that $m \in \{ 0,3,6, 7\}$. If $m \in \{3, 6\}$, then $G'$ stabilizes every Mori fiber, and consequently it has more then three fixed points, a contradiction. Thus we must only consider the case $m =7$.
In this case the set of Mori fibers is a $G$-orbit and it follows that every Mori fiber has self-intersection -1 and therefore has nonempty intersection with $\pi(C)$ by Remark \[self-int of Mori-fibers\].
As before, the Euler characteristic formula implies that $g(C)=3$ and $Y_\mathrm{min}=\mathbb P_2$ and adjunction in $X$ shows that $(\pi(C))^2=8$ in $Y$. The fact that $\pi(C)$ has nonempty intersection with seven different Mori fibers implies that its image $D$ in $Y_\mathrm{min}$ has self-intersection either $15 = 8 +7$ or $36 = 8 + 4 \cdot 7$. Since the first is impossible it follows that $E \cdot \pi(C)=2$ for all Mori fibers $E$ and the $G$-invariant irreducible sextic $D$ has seven singular points corresponding to the images of $E$ in $\mathbb P_2$.
\[sing exam\] If $Y$ is not $G$-minimal, then $X$ is the minimal desingularization of a double cover of $\mathbb{P}_2$ branched along an irreducible $G$-invariant sextic with seven singular points.
We conclude this section with a classification of possible $G$-minimal models of $Y$.
The surface $Y_\mathrm{min}$ is either a Del Pezzo surface of degree two or $\mathbb P_2$.
The case $Y_\mathrm{min}=\mathbb P_1\times \mathbb P_1$ is excluded by Example \[DelPezzoC3C7\] and also by Remark \[P1xP1C3C7\].
Thus $Y_\mathrm{min}=Y_d$ is a Del Pezzo surface of degree $d=1,\ldots ,9$ which is a blowup of $\mathbb P_2$ in $9-d$ points.
If $Y_\mathrm{min}=Y_1$ the anticanonical map has exactly one base point. This point has to be $G$-fixed and since $G$ has no faithful two-dimensional representations, this case does not occur.
It remains to eliminate $d=8,\ldots ,3$. In these cases the sets $\mathcal S$ of (-1)-curves consist of 1, 2, 6, 10, 16 or 27 elements, respectively (cf. Table \[minus one curves\]). The $G$-orbits in $\mathcal S$ consist of $1$, $3$, $7$ or $21$ curves and there must be orbits of length three or one. If $G$ stablizes a curve in $\mathcal S$, then its contraction gives rise to a two-dimensional representation of $G$ which does not exist. If $G$ has an orbit consisting of three curves, then $G'$ stabilizes each of the curves in this orbit. Thus $G'$ has at least six fixed points in $Y_\mathrm{min}$ and in $Y$. This contradicts the fact that $| \mathrm{Fix}_Y(G')|=3$.
Fine classification - Computation of invariants
-----------------------------------------------
We have reduced the classification of K3-surfaces with $G \times C_2$-symmetry to the study of equivariant double covers of rational surfaces $Y$ branched along a single invariant curve of genus $g \geq 3$. Here $Y$ is either $\mathbb P_2$, the blow-up of $\mathbb P_2$ in seven singular points of an irreducible $G$-invariant sextic, or a Del Pezzo surface of degree two.
### The case $Y = Y_\mathrm{min} = \mathbb P_2$
An effective action of $G$ on $\mathbb P_2$ is given by an injective homomorphisms $G \to \mathrm {PSL}_3(\mathbb C)$. There are two central degree three extension of $G$, the trivial extension and $C_9 \ltimes C_7$. A study of their three-dimensional representation reveals that in both cases the action of $G$ on $\mathbb P_2$ is given by an irreducible representation $G \hookrightarrow \mathrm {SL}_3(\mathbb C)$. There are two isomorphism classes of irreducible 3-dimensional representations. Since these differ by a group automorphism and the corresponding actions on $\mathbb{P}_2$ are therefore equivalent, we may assume that in appropriately chosen coordinates a generator of $G'$ acts by $$\label{c7action}
[z_0:z_1:z_2]\mapsto [\lambda z_0,\lambda ^2,z_1,\lambda ^4z_2],$$ where $\lambda =\mathrm{exp}{\frac{2\pi i}{7}}$ and a generator of $C_3$ acts by the cyclic permutation $\tau $ which is defined by $$\label{tauaction}
[z_0:z_1:z_2]\mapsto [z_2:z_0:z_1].$$
A homogeneous polynomial defining an invariant curve must be a $G$-semi-invariant with $G'$ acting with eigenvalue one. The $G'$-invariant monomials of degree six are $$\mathbb C[z_0,z_1,z_2]_{(6)}^{G'}=
\mathrm {Span}\{z_0^2z_1^2z_2^2, z_0^5z_1,z_2^5z_0,z_1^5z_2\}\,.$$ Letting $P_1=z_0^2z_1^2z_2^2 $ and $P_2=z_0^5z_1+z_2^5z_0+z_1^5z_2$, it follows that $$\mathbb C[z_0,z_1,z_2]_{(6)}^{G}=
\mathrm {Span}\{P_1,P_2\}=:V\,.$$ There are two $G$-semi-invariants which are not invariant, namely $z_0^5z_1+\zeta z_2^5z_0+\zeta ^2z_1^5z_2$ for $\zeta ^3=1$ but $\zeta \not =1$. By direct computation one checks that the curves defined by these polynomials are smooth and that in both cases all $\tau $-fixed points in $\mathbb P_2$ lie on them. Thus, $\tau $ has only three fixed points on the K3-surface $X$ obtained as a double cover and therefore does not act symplectically (cf. Table \[fix points symplectic\]). Consequently, $G$ does not lift to an action by symplectic transformations on the K3-surfaces defined by these two curves. Hence it is enough to consider ramified covers $X\to Y=\mathbb P_2$, where the branch curves are defined by invariant polynomials $f \in V$.
We wish to determine which polynomials $P_{\alpha,\beta}=\alpha P_1+\beta P_2$ define singular curves. Since $\mathrm {Fix}(\tau) = \{ [1:\zeta :\zeta ^2] \, | \, \zeta ^3 =1 \}$, the curves which contain $\tau $-fixed points are defined by condition $\alpha +3\zeta \beta =0$. Let $C_{P_1} = \{ P_1 =0\} $ and let $C_\zeta $ be the curve defined by $P_{\alpha,\beta}$ for $\alpha +3\zeta \beta =0$. A direct computation shows that $C_\zeta $ is singular at the point $[1:\zeta :\zeta ^2]$. We let $\Sigma_\mathrm{reg} $ be the complement of this set of four curves, $\Sigma_\mathrm{reg} = \mathbb P(V) \backslash \{C_{P_1}; \, C_\zeta \, | \, \zeta ^3 =1 \}$.
A curve $C\in \mathbb P(V)$ is smooth if and only if $C \in \Sigma_\mathrm{reg} $.
Let $C \in \Sigma_\mathrm{reg} $. Since $\tau $ has no fixed points in $C$ by definition and every subgroup of order three in $G$ is conjugate to $\langle \tau \rangle $, it follows that any $G$-orbit $G.p$ through a point $p\in C$ has length three or 21.
The only subgroup of order seven in $G$ is the commutator group $G'$. So the $G$-orbits of length three are the orbits of the $G'$-fixed points $[1:0:0], [0:1:0],[0:0:1]$. One checks by direct computation that every $C\in \Sigma_\mathrm{reg} $ is smooth at these three points.
An irreducible curve of degree six has at most ten singular points by the genus formula. Suppose that $C$ is singular at some point $q$. Then it is singular at each of the 21 points in $G.q$ and $C$ must be reducible. Considering the $G$-action on the space of irreducible components of $C$ yields a contradiction and it follows that $C$ is smooth.
For any curve $C \in \Sigma_\mathrm{reg}$ the double cover of $\mathbb P_2$ branched along $C$ is a K3-surface $X_C$ with an action of a degree two central extension of $G$. By the following lemma, this action is always of the desired type.
\[G acts sympl\] For every $C \in \Sigma_\mathrm{reg}$ the K3-surface $X_C$ carries an action of the group $G \times \langle \sigma \rangle $. The group $G$ acts by symplectic transformations on $X_C$ and $\sigma$ denotes the covering involution.
It follows from the group structure of $G$ that the central degree two extension of $G$ acting on $X_C$ splits as $G \times C_2$. The factor $C_2$ is by construction generated by the covering involution $\sigma$. It remains to check that $G$ acts symplectically. As the commutator subgroup $G'$ acts symplectically it is sufficient to check whether $\tau$ lifts to a symplectic automorphism. Consider the $\tau$-fixed point $p=[1:1:1]$ and check that the linearization of $\tau$ at $p$ is in $\mathrm{SL}(2, \mathbb C)$. Since $p$ is not contained in $C$, it follows that the linearization of $\tau$ at a corresponding fixed point in $X_C$ is also in $\mathrm{SL}(2, \mathbb C)$. Consequently, the group $G$ acts by symplectic transformations on $X_C$.
### Equivariant equivalence {#EquiEqui}
We wish to describe the space of K3-surfaces with $G \times C_2$-symmetry modulo equivariant equivalence. For this, we study the family of K3-surfaces parametrized by the family of branch curves $\Sigma_\mathrm{reg}$. Consider the cyclic group $\Gamma$ of order three in $\mathrm{PGL}(3, \mathbb C)$ generated by $$[z_0:z_1:z_2]\mapsto [z_0:\zeta z_1: \zeta ^2z_2]$$ for $\zeta = \mathrm{exp}(\frac{2 \pi i}{3})$. The group $\Gamma$ acts on $\Sigma_\mathrm{reg}$ and by the following proposition the induced equivalence relation is precisely equivariant equivalence formulated in Definition \[equivariantequivalence\].
Two K3-surfaces $X_{C_1}$ and $X_{C_2}$ for $C_1,C_2 \in \Sigma_\mathrm{reg}$ are equivariantly equivalent if and only if $C_1 = \gamma C_2$ for some $\gamma \in \Gamma$, i.e., the quotient $\Sigma_\mathrm{reg}/ \Gamma$ parametrizes equivariant equivalence classes of K3-surfaces $X_C$ for $C \in \Sigma_\mathrm{reg}$.
If two K3-surfaces $X_{C_1}$ and $X_{C_2}$ for $C_1,C_2 \in \Sigma_\mathrm{reg}$ are equivariantly equivalent, then the isomorphism $X_{C_1} \to X_{C_2}$ induces an automorphism of $\mathbb P_2$ mapping $C_1$ to $C_2$.
Let $C\in \Sigma_\mathrm{reg}$ and for $T\in \mathrm {SL}_3(\mathbb C)$ assume that $T(C)\in \Sigma_\mathrm{reg} $. We consider the group span $S$ of $TGT^{-1}$ and $G$. By Lemma \[G acts sympl\], the group $G$ acts by symplectic transformations on $X_C$ and $X_{T(C)}$. We argue precisely as in the proof of this lemma to see that $TGT^{-1}$ also acts symplectically on the K3-surface $X_{T(C)}$. It follows that $S$ is acting as a group of symplectic transformations on this K3-surface.
If $S=G$, then $T$ normalizes $G$. The normalizer $N$ of $G$ in $\mathrm {PGL}_3(\mathbb C)$ is the product $\Gamma \times G$ and it follows that $gT$ is contained in $\Gamma $ for some $g \in G$ and $T(C) = gT(C) = \gamma C$.
Note that $L_2(7)$ is the only group in Mukai’s list which contains $G$. Therefore, $S$ is a subgroup of $L_2(7)$. The group $G$ is a maximal subgroup of $L_2(7)$ and if $S\not=G$, then it follows that $S=L_2(7)$. Any two subgroups of order 21 in $L_2(7)$ are conjugate. This implies the existence of $s \in S= L_2(7)$ such that $sTGT^{-1}s^{-1} = G$. Now $sT \in N = \Gamma \times G$ can be written as $sT = \gamma g$ for $(\gamma, g) \in \Gamma \times G$. By assumption, $s$ stabilizes $T(C)$ and $T(C) = sT(C) = \gamma g (C) = \gamma C$. This completes the proof of the proposition.
### The case $Y \neq Y_\mathrm{min}$
Let us now consider the three singular irreducible curves in our family $\mathbb P (V)$. They are identified by the action of $\Gamma$. Using Corollary \[sing exam\] we see that if $Y = X/\sigma$ is not $G$-minimal, then, up to equivariant equivalence, the K3-surface $X$ is the minimal desingularization of the double cover of $\mathbb P_2$ branched along $C_{\zeta=1} = C_\mathrm{sing}$ and $Y$ is the blow-up of $\mathbb P_2$ in the seven singular points of $C_\mathrm{sing}$. These points are the $G'$-orbit of $[1:1:1]$. In the following propostion we prove that these are in general position and therefore $Y$ is a Del Pezzo surface.
\[blowdownseven\] If $Y$ is not minimal, then it is the Del Pezzo surface of degree two which arises by blowing up the seven singular points $p_1,\ldots ,p_7$ on the curve $C_\mathrm {sing}$ in $\mathbb P_2$. The corresponding map $Y \to
\mathbb P_2$ is $G$-equivariant and therefore a Mori reduction of $Y$.
We show that the points $\{p_1, \dots, p_7\} = G'.[1:1:1]$ are in general position, i.e., no three lie on one line and no six lie on one conic. It follows from direct computation that no three points in $G'.[1:1:1]$ lie on one line. If $p_1,\dots p_6$ lie on a conic $Q$, then $g.p_1, \dots , g.p_6$ lie on $g.Q$ for every $g \in G$. Since $\{p_1, \dots, p_7\}$ is a $G$-invariant set, the conics $Q$ and $g.Q$ intersect in at least five points and therefore coincide. It follows that $Q$ is an invariant conic meeting $C_\mathrm{sing}$ at its seven singularities and $(Q,C_\mathrm{sing}) \geq 14$ implies $Q \subset C_\mathrm{sing}$, a contradiction.
Klein’s quartic and the Klein-Mukai surface {#KMsurface}
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In this section we show that the Del Pezzo surface discussed in Proposition \[blowdownseven\] above can be realized as the double cover of $\mathbb P_2$ branched along Klein’s quartic curve.
\[DelPezzoYKlein\] A Del Pezzo surface of degree two with an action of $G$ is equivariantly isomorphic to the double cover $Y_\mathrm{Klein}$ of $\mathbb P_2$ branched along Klein’s quartic curve.
Recall that the anticanonical map of a Del Pezzo surface $Y$ of degree two defines a 2:1 map to $\mathbb P_2$. This map is branched along a smooth curve of degree four and equivariant with respect to $\mathrm{Aut}(Y)$. We obtain an action of $G$ on $\mathbb P_2$ stabilizing a smooth quartic. As before, we may choose coordinates such that $G$ is acting as in equations and . Then $$\mathbb C[z_0:z_1:z_2]_{(4)}^{G'}
=\mathrm {Span}\{z_0^3z_2, z_1^3z_0,z_2^3z_1\}\,.$$ is a direct sum of $G$-eigenspaces. The eigenspace of the eigenvalue $\zeta $ is spanned by the polynomial $Q_\zeta :=z_0^3z_2+\zeta z_2^3z_1+\zeta ^2z_1^3z_0$ with $\zeta $ being a third root of unity.
In order to take into account equivariant equivalence we consider the cyclic group $\Gamma \subset \mathrm {SL}_3(\mathbb C)$ which is generated by the transformation $\gamma$, $[z_0:z_1:z_2]\mapsto [z_0:\zeta z_1:\zeta^2z_2]$. The induced action on $\mathbb C[z_0:z_1:z_2]_{(4)}^{G'}$ is transitive on the $G$-eigenspaces spanned by the $Q_\zeta $. Consequently, up to equivariant equivalence, we may assume that $Y\to \mathbb P_2$ is branched along Klein’s curve $C_\text{Klein}$ which is defined by $Q_1$.
A Del Pezzo surface of degree two with an action of $G$ is never $G$-minimal. Its Mori reduction $Y_\mathrm{Klein} \to \mathbb P_2$ is precisely the map discussed in Proposition \[blowdownseven\].
We summarize our observartions in the following proposition.
If $X$ is a K3-surface with a symplectic $G$-action centralized by an antisymplectic involution $\sigma $, then $Y_{min}=\mathbb P_2$. In all but one case $X/\sigma =Y=Y_{min}$. In the exceptional case $Y=Y_\mathrm{Klein}$, the Mori reduction $Y\to Y_{min}$ is the contraction of seven (-1)-curves to the singular points of $C_\mathrm{sing}$ and the branch set $B$ of $X\to Y$ is the proper transform of $C_\mathrm {Klein}$ in $Y$.
It remains to prove that $B$ is the proper transform of $C_\mathrm {Klein}$ in $Y$. Suppose that the branch curve of $X\to Y$ is some other curve $\widetilde B$ linearly equivalent to $-2K_Y$. Let $I:=\widetilde B\cap B$ and note that $\vert I\vert \le B \cdot \widetilde B = 4 K_Y ^2 = 8$. Since $G$ has no fixed points in $B$, it follows that $\vert I\vert =3$ and that $I$ is a $G$-orbit. Thus the intersection multiplicities at the three points in $\widetilde B\cap B$ are the same. Since 3 does not divide 8, this is a contradiction.
In order to complete the proof of Theorem \[mainthmc3c7\] it remains to show that the action of $G$ on $Y_\mathrm{Klein}$ lifts to a group of symplectic transformation on the K3-surface $X=X_{KM}$ defined as a double cover of $Y_\mathrm{Klein}$ branched along the proper transform of $C_\mathrm{sing}$.
Since $G$ stabilizes $C_\text{Klein}$ and does not admit nontrivial central extensions of degree two, it lifts to a subgroup of $\mathrm{Aut}(Y_\text{Klein})$ and subsequently to a subgroup of $\mathrm{Aut}(X)$.
The covering involution $Y_\text{Klein}\to \mathbb P_2$, lifts to a holomorphic transformation of $X$ where we also find the involution defining $X \to Y_\text{Klein}$. These two transformations generate a group $F$ of order four. The elements of $F$ all have a positive-dimensional fixed point set. It follows that $F$ acts solely by nonsymplectic transformations and is therefore isomorphic to $C_4$. The full preimage of $G$ in $\mathrm {Aut}(X)$ splits as $G\times C_4$.
Since the commutator group $G'$ automatically acts by symplectic transformations, we must only check that the lift of the cyclic permutation $\tau $, $[z_0:z_1:z_2]\mapsto [z_2:z_0:z_1]$, acts symplectically. As above, this follows from a linearization argument at a $\tau$-fixed point not in $C_\mathrm{Klein}$.
In conclusion, up to equivalence there is a unique action of $G$ by symplectic transformations on the K3-surface $X_{KM}$. It is centralized by a cyclic group of order four which acts faithfully on the symplectic form.
The Klein-Mukai-surface is the only surface with $G \times C_2$-symmetry for which $Y \not\cong \mathbb P_2$. As in the introduction of this chapter, we define $\Sigma$ as the complement of $C_{P_1}$ in $\mathbb P(V)$. Then $\Sigma = \Sigma_\mathrm{reg} \cup \{C_\zeta \, | \, \zeta^3 =1\}$. Using this notation the space $$\mathcal M = \Sigma / \Gamma$$ parametrizes the space of K3-surfaces with $G \times C_2$-symmetry up to equivariant equivalence. This completes the proof of Theorem \[mainthmc3c7\].
The group $L_2(7)$ centralized by an antisymplectic involution {#168}
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We consider the simple group of order 168. This group is $\mathrm{PSL}(2, \mathbb F_7)$ and usually denoted by $L_2(7)$. It contains our group $G = C_3 \ltimes C_7$ as a subgroup. Since $L_2(7)$ is a simple group, if it acts on a K3-surface, it automatically acts by symplectic transformations.
We wish to prove Theorem \[L2(7) times invol\] stating that there are exactly two K3-surfaces with an action of the group $L_2(7)$ centralized by an antisymplectic involution. These are the Klein-Mukai-surface $X_\mathrm{KM}$ and the double cover of $\mathbb P_2$ branched along the curve $\mathrm{Hess}(C_\text{Klein})=\{z_0^5z_1+z_2^5z_0+z_1^5z_2-5z_0^2 z_1^2 z_2^2=0\}$.
We have to check which elements of $\mathcal M$ have the symmetry of the larger group. The Klein-Mukai-surface is known to have $L_2(7) \times C_4$-symmetry (cf. Example \[L2(7)example\]). If $X \neq X_\mathrm{KM}$ has $L_2(7)$-symmetry, then it follows from the considerations of the previous sections that $X$ is an $L_2(7)$-equivariant double cover of $\mathbb P_2$ branched along a smooth $L_2(7)$-invariant sextic curve. I.e., it remains to identify the surfaces with $L_2(7)$-symmetry in the family parametrized by $\Sigma_\mathrm{reg} / \Gamma$.
\[L2(7)onP2\] The action of $L_2(7)$ on $\mathbb P_2$ is necessarily given by a three-dimensional represention.
The lemma follows from the fact that the group $L_2(7)$ does not admit nontrivial degree three central extensions. This can be derived from the cohomology group $H^2(L_2(7), \mathbb C^*) \cong C_2$ known as the Schur Multiplier.
There are two isomorphism classes of three-dimensional representations and these differ by an outer automorphism. We may therefore consider the particular representation given in Example \[L2(7)example\]. One checks that the curve $\mathrm{Hess}(C_\mathrm{Klein})$ is $L_2(7)$-invariant. The maximal possible isotropy group is $C_7$ and each $L_2(7)$-orbit in $\mathrm{Hess}(C_\mathrm{Klein})$ consists of at least 21 elements. If there was another $L_2(7)$-invariant curve $C$ in $\Sigma_\mathrm{reg}$, then the invariant set $ C \cap \mathrm{Hess}(C_\mathrm{Klein})$ consists of at most 36 points. This is a contradiction and it follows that $\mathrm{Hess}(C_\mathrm{Klein})$ is the only $L_2(7)$-invariant curve in $\Sigma_\mathrm{reg}$.
It remains to check that $L_2(7)$ lifts to a subgroup of $\mathrm{Aut}(X_{\mathrm{Hess}(C_\mathrm{Klein})})$: On $X_{\mathrm{Hess}(C_\mathrm{Klein})}$ we find an action of a central degree two extension $E$ of $L_2(7)$. Since $E \neq E_\mathrm{symp}$ and $L_2(7)$ is simple, the subgroup of symplectic transformations inside $E$ must be isomorphic to $L_2(7)$.
It follows that $X_\mathrm{KM}$ and the double cover of $\mathbb P_2$ branched along $\mathrm{Hess}(C_\mathrm{Klein})$ are the only examples of K3-surfaces with $L_2(7) \times C_2$ symmetry. This completes the proof of Theorem \[L2(7) times invol\].
If we consider the quotient $Y_\mathrm{Klein}$ of $X_\mathrm{KM}$ by the antisymplectic involution $\sigma \in C_4$, this surface was seen not to be minimal with respect to the action of $C_3 \ltimes C_7$. It is however $L_2(7)$-minimal as we cannot find a equivariant contraction morphism blowing down an orbit of disjoint (-1)-curves in $Y_\mathrm{Klein}$ . Such an orbit would have to consists of seven Mori fibers. The only subgroup of index seven is $S_4$. A Mori fiber of self-intersection (-1) does however not admit an action of the group $S_4$ (cf. Proof of Theorem \[roughclassi\]).
The simple group of order 168 {#chapter non exist}
=============================
In this chapter we consider finite groups containing $L_2(7)$, the simple group of order 168, and their actions on K3-surfaces. Based on our considerations about $L_2(7) \times C_2$-actions on K3-surfaces in Section \[168\] we derive a classification result (Theorem \[improve OZ\]). This gives a refinement of a lattice-theoretic result due to Oguiso and Zhang [@OZ168]. The main part of this chapter is dedicated to proving the non-existence of K3-surfaces with an action of the group $L_2(7) \times C_3$ (Theorem \[nonexist\]) using equivariant Mori reduction.
Finite groups containing $L_2(7)$
---------------------------------
If $H$ is a finite group acting on a K3-surface and $L_2(7) \lneqq H$, then it follows from Mukai’s theorem and the fact that $L_2(7)$ is simple, that $H$ fits into the short exact sequence $$1 \to L_2(7) = H_\mathrm{symp} \to H \to C_m \to 1$$ for some $m \in \mathbb N$. As it is noted by Oguiso and Zhang, Claim 2.1 in [@OZ168], it follows from Proposition 3.4 in [@mukai] that $m \in \{1,2,3,4,6\}$.
The action of $H$ on $L_2(7)$ by conjugation defines a homomorphism $H \to \mathrm{Aut}(L_2(7))$. Factorizing by the group of inner automorphism of $L_2(7)$ we obtain a homomorphism $$C_m \cong H/L_2(7) \to \mathrm{Out}(L_2(7)) \cong C_2.$$ If $H$ is not the nontrivial semidirect product $L_2(7) \rtimes C_2$, this homomorphism has a nontrivial kernel. In particular, we find a cyclic group $C_k < C_m$ centralizing $L_2(7)$. If $k$ is even, we may apply our results on K3-surfaces with $L_2(7) \times C_2$-symmetry from the previous chapter.
If $m = 3,6$, then $k=3$ or $k=6$. These cases may be excluded as is shown in [@OZ168], Added in proof, Proposition 1. An independent proof of this fact, i.e., the non-existence of K3-surfaces with $L_2(7) \times C_3$ symmetry, using equivariant Mori theory, in particular the classification of $L_2(7)$-minimal models, is given below (Theorem \[nonexist\]).
We summarize our observations about K3-surfaces with $L_2(7)$-symmetry in the following theorem, which improves the classification result due to Oguiso and Zhang.
\[improve OZ\] Let $H$ be finite group acting on a K3-surface $X$ with $L_2(7) \lneqq H$. Then
- $|H/L_2(7)| \in \{2,4\}$.
- If $|H/L_2(7)|= 4$, then $H = L_2(7) \times C_4$ and $X \cong X_\mathrm{KM}$.
- If $|H/L_2(7)|= 2$ and $H = L_2(7) \times C_2$, then either $X \cong X_\mathrm{KM}$ or $X \cong X_{\mathrm{Hess}(C_\mathrm{Klein})}$
The first statement follows from the non-existence of K3-surfaces with $L_2(7) \times C_3$-symmetry (Theorem \[nonexist\] below) and the third statement follows from Theorem \[L2(7) times invol\]. The remaining part ist covered in the following lemma (cf. Main Theorem in [@OZ168]).
\[OZresult\] If $X$ is a K3-surface with an action of a finite group containing the $L_2(7)$ as a subgroup of index four, then $X$ is the Klein-Mukai-surface.
We let $X$ be a K3-surface and $H$ be a finite subgroup of $\mathrm{Aut}(X)$ with $L_2(7) < H$ and $|H/L_2(7)|=4$.
Since $L_2(7)$ is simple and a maximal group of symplectic transformations, it coincides with the group of symplectic transformations in $H$. In particular, $H / L_2(7) = C_4$ and a group $\langle \sigma \rangle$ of order two is contained in the kernel of the homomorphism $H\to \mathrm {Aut}(L_2(7))$. It follows that we are in the setting of Theorem \[L2(7) times invol\] where $\Lambda :=H/\langle \sigma \rangle$ acts on $Y=X/\sigma $. If $X\not =X_{KM}$, then $Y= \mathbb P_2$. This possibility needs to be eliminated.
Let $\tau $ be any element of $\Lambda $ which is not in $L_2(7)$ and let $\Gamma = C_3 \ltimes C_7 < L_2(7)$. Since any two subgroups of order 21 in $L_2(7)$ are conjugate by an element of $L_2(7)$, it follows that there exists $h\in L_2(7)$ with $(h\tau )\Gamma(h\tau)^{-1}=\Gamma$. Thus, the normalizer $N(\Gamma)$ of $\Gamma$ in $\Lambda $ is a group of order 42 which also normalizes the commutator subgroup $\Gamma'$ and therefore stabilizes its set $F$ of fixed points.
Using coordinates $[z_0:z_1:z_2]$ of $\mathbb{P}_2$ as in Theorem \[L2(7) times invol\] one checks by direct computation that the only transformations in $\mathrm {Stab}(F)$ which stabilize the branch curve $\mathrm {Hess}(C_{\mathrm {Klein}})$ are those in $\Gamma$ itself. This contradiction shows that $Y\not =\mathbb P_2$ and therefore $X=X_{KM}$.
Non-existence of K3-surfaces with an action of $L_2(7) \times C_3$
------------------------------------------------------------------
The method of equivariant Mori reduction can be applied to obtain both classification and non-existence results. In the following, we exemplify a general approach to prove non-existence of K3-surfaces with specified symmetry by considering the group $L_2(7) \times C_3$ and give an independent proof of the following observation of Oguiso and Zhang [@OZ168]:
\[nonexist\] There does not exist a K3-surface with an action of $L_2(7) \times C_3$.
The remainder of this chapter is dedicated to the proof of this theorem.
### Global structure
Let $G \cong L_2(7)$, let $D \cong C_3$, and assume there exists a K3-surface $X$ with a holomorphic action of $G \times
D$. Since $G$ is a simple group and a maximal group of symplectic transformations on a K3-surface, it follows that $G$ acts symplectically whereas the action of $D$ is nonsymplectic. We obtain the following commuting diagram. $$\begin{xymatrix}{
X \ar[d]^{\pi} & \hat{X} \ar[d]^{\hat{\pi}} \ar[l]_{b_X}\\
X/D=Y & \hat{Y} \ar[l]_{b_Y}\ar[d]^{M_{\mathrm{red}}}\\
& \hat{Y}_\mathrm{min}=Z
}
\end{xymatrix}$$ Here $b_X$ is the blow-up of the isolated $D$-fixed points in $X$. The singularities of $X/D$ correspond to isolated $D$-fixed points. Since the linearization of the $D$-action at an isolated fixed point is locally of the form $(z,w) \mapsto (\chi z, \chi w)$ for some nontrivial character $\chi: D \to \mathbb C^*$, each singularity of $X/D$ is resolved by a single blow-up. We let $b_Y$ denote the simultanious blow-up of all singularities of $Y$. We fix a $G$-Mori reduction $M_\mathrm{red}: \hat Y \to \hat{Y}_\mathrm{min}=Z$. All maps in the diagram are $G$-equivariant. By Theorem \[K3quotnonsymp\], the surface $\hat{Y}$ is rational. As conic bundles do not admit an action of $G$ (cf. Lemma \[C3C7 conic bundle\]), we know that $\hat{Y}_\mathrm{min}$ is a Del Pezzo surface . The following lemma specifies $Z$.
The Del Pezzo surface $Z$ is either $\mathbb P_2$ or a surface obtained from $\mathbb P_2$ by blowing up 7 points in general position. In the later case, $Z$ is a $G$-equivariant double cover of $\mathbb P_2$ branched along Klein’s quartic curve. The action of $G$ on $\mathbb P_2$ is given by a three-dimensional representation.
The first part of the lemma follows from our observations in Example \[DelPezzoL2(7)\], the last part has been discussed in Lemma \[L2(7)onP2\]. If $Z$ is a Del Pezzo surface of degree two, then the anticanonical map realizes it as an equivariant double cover of $\mathbb P_2$ branched along a smooth quartic curve $C$. We choose coordinates on $\mathbb P_2$ such that the action of $G$ is given by the representation $\rho$ of Example \[L2(7)example\] (or its dual represenation $\rho^*$) and have already seen that Klein’s quartic curve $$C_\text{Klein}= \{x_1x_2^3 + x_2x_3^3 + x_3x_1^3=0\} \subset \mathbb P_2$$ is $G$-invariant. If $C \neq C_\text{Klein}$, then $C \cap C_\text{Klein}$ is a $G$-invariant subset of $\mathbb P_2$. Since the maximal cyclic subgroup of $G$ is of order seven, it follows that a $G$-orbit $G.p$ for a point $p \in C \cap C_\text{Klein}$ consists of at least 24 elements. Since $C \cap C_\text{Klein}$ however consists of at most 16 points, this is a contradiction. Therefore, $C= C_\text{Klein}$ and the lemma follows.
### $D$-fixed points {#d-fixed-points .unnumbered}
The map $\pi$ is in general ramified both at points and along curves. Let $x$ be an isolated $D$-fixed point in $X$. As was noted above, the isotropy representation of the nonsymplectic $D$-action at $x$ in local coordinates $(z,w)$ is given by $(z,w) \mapsto (\chi z, \chi w)$ for some nontrivial character $\chi: D \to \mathbb C^*$. The action of $D$ on the rational curve $\hat E$ obtained by blowing up $x$ is trivial and therefore $\hat E$ is contained in the ramification set $\mathrm{Fix}_{\hat{X}}(D)$. Let $\{\hat{E_i}\}$ denote the set of (-1)-curves in $\hat{X}$ obtained from blowing up isolated $D$-fixed points in $X$ and define $E_i = \hat{\pi}(\hat{E_i})$.
If $C$ is a curve of $D$-fixed points in $X$, it follows that $\hat \pi$ is ramified along $b_X^{-1}(C)$. Let $\{\hat {F_j}\}$ denote the set of all ramification curves of type $b_X^{-1}(C)$ and define $F_j= \hat \pi(\hat {F_j})$. The map $\hat \pi$ is a $D$-quotient and ramified along curves $$\mathrm{Fix}_{\hat{X}} (D) = \bigcup \hat{E_i} \cup
\bigcup \hat{F_j}.$$
### Mori contractions and $C_7$-fixed points
Many aspects of the group theory of $G$ can be well understood in term of its generators $\alpha, \beta, \gamma$ of order 7,3,2, respectively. Since the action of $G$ on $\mathbb{P}_2$ is given by a three-dimensional irreducible representation, the action of $G$ on $Z$ is given explicitly in terms of $\alpha, \beta, \gamma$. We let $S= \langle \alpha \rangle \cong C_7 < G$ be a cyclic subgroup of order seven in $G$.
The symplectic action of a cyclic group of order seven on an K3-surface has exactly three fixed points. Since $p_1=[1:0:0]$, $p_2=[0:1:0]$ and $p_3=[0:0:1]$ all lie on $C_\mathrm{Klein} \subset \mathbb P_2$, the action of $S$ on $Z$ has exactly three fixed points.
Let $\mathrm{Fix}_{\hat{Y}}(S) =: \{y_1, \dots,y_k\}$ and let $\mathrm{Fix}_{\hat{X}}(S) =: \{x_1, \dots,x_l\}$. Since blowing-up an $S$-fixed point in $X$ replaces the fixed point by a rational curve with two $S$-fixed points in $\hat{X}$, we find $3 \leq k \leq l
\leq 6$.
\[fixSinfixD\] The fixed points of $S$ in $\hat{X}$ are contained in the $D$-ramification set, i.e., $\mathrm{Fix}_{\hat{X}}(S) \subset
\mathrm{Fix}_{\hat{X}} (D)$.
Since $D$ centralizes $S$, the action of $D$ stabilizes the $S$-fixed point set. We first show that $\mathrm{Fix}_{{X}}(S) \subset
\mathrm{Fix}_{{X}} (D)$. Assume the contrary and let $\mathrm{Fix}_X(S)=
\{s_1,s_2,s_3\}$ be a $D$-orbit and $\pi(s_i)=y$. Then $y$ is a smooth point and fixed by the action of $S$ on $Y$. There exists a neighbourhood of $y$ in $Y$ which is biholomorphic to a neighbourhood of $b_Y^{-1}(y)= \tilde{y}$ in $\hat{Y}$. By construction, $\tilde{y}\in \mathrm{Fix}_{\hat{Y}}(S)$. Since $\mathrm{Fix}_{\hat{Y}}(S)$ consists of at least three points, we let $\tilde{\tilde{y}} \neq \tilde{y}$ be an additional $S$-fixed point on $\hat{Y}$. The fibre $\pi^{-1}(b_Y(\tilde{\tilde{y}}))$ consists of one or three points and is disjoint from $\{s_1,s_2,s_3\}$. Since the point $\tilde{\tilde{y}}$ is a fixed point of $S$, we know that $S \cong C_7$ acts on the fiber $\pi^{-1}(b_Y(\tilde{\tilde{y}}))$ and is seen to fix it pointwise. This is contrary to the fact that $\mathrm{Fix}_X(S)=\{s_1,s_2,s_3\}$. It follows that $\mathrm{Fix}_{{X}}(S) \subset
\mathrm{Fix}_{{X}} (D)$.
It remains to show the corresponding inclusion on $\hat{X}$. If the points $s_i$ do not coincide with isolated $D$-fixed points, the statement follows since $b_X$ is equivariant and biholomorphic outside the isolated $D$-fixed points. If $s_i$ is an isolated $D$-fixed point, we have seen above that the action of $D$ on the blow-up of $s_i$ is trivial. In particular, $\mathrm{Fix}_{\hat{X}}(S) \subset
\mathrm{Fix}_{\hat{X}} (D)$.
### Excluding the case $|\mathrm{Fix}_{\hat{Y}}(S)|=3$ {#excluding-the-case-mathrmfix_hatys3 .unnumbered}
If $|\mathrm{Fix}_{\hat{Y}}(S)
|=3$, then $\mathrm{Fix}_{\hat{Y}}(S) \cap \bigcup E_i = \emptyset$.
Fixed points of $S$ on a curve $\hat{E_i}$ always come in pairs: If the curve $\hat{E_i}$ contains a fixed point of $S$, then the isotropy representation of $S$ at the fixed point $b_X(\hat{E_i})$ in $X$ defines an action of the cyclic group $S$ on the rational curve $\hat{E_i}$ with exactly two fixed points. If $|\mathrm{Fix}_{\hat{Y}}(S)|=|\mathrm{Fix}_{\hat{X}}(S)
|=3$ and $\mathrm{Fix}_{\hat{Y}}(S) \cap \bigcup E_i \neq \emptyset$, then two of the $S$-fixed point lie on the same curve $\hat{ E_i}$ and $|\mathrm{Fix}_{X}(S)| \leq 2$, a contradiction.
\[fixed points on mori fibers\] If $|\mathrm{Fix}_{\hat{Y}}(S)
|=3$, then the set $\mathrm{Fix}_{\hat{Y}}(S)$ has empty intersection with the exceptional locus of the full equivariant Mori reduction $M_{\mathrm{red}}: \hat{Y} \to Z$.
Let $C$ be any exceptional curve of the Mori reduction and assume there is a fixed point of $S$ on $C$. As the point $p$ obtained from blowing down $C$ has to be a fixed point of $S$, it follows that the curve $C$ is $S$-invariant. In particular, we know that the action of $S$ on $C$ has exactly two fixed points. Now blowing down $C$ reduces the number of $S$-fixed point by 1. This contradicts the fact that $|\mathrm{Fix}_{Z}(S)|=3$.
\[shapeofSaction\] Let $|\mathrm{Fix}_{\hat{Y}}(S)|=3$ and let $p \in \mathrm{Fix}_Z (S)$. Then there exist local coordinates $(u,v)$ at $p$ and a nontrivial character $\mu: S \to \mathbb{C}^*$ such that the action of $S$ at $p$ is locally given by either $$(u,v) \mapsto (\mu^3 u, \mu^{-1} v)\quad \text{or} \quad (u,v) \mapsto (\mu u, \mu^{-3} v).$$
On the K3-surface $X$ the action of $S$ at a fixed point is in local coordinates $(z,w)$ given by $(z,w) \mapsto (\mu
z, \mu^{-1}w)$ for some nontrivial character $\mu: S \to \mathbb{C}^*$. Since $\mathrm{Fix}_{\hat{Y}}(S) \cap \bigcup E_i = \emptyset$, the map $b_X$ is biholomorphic in a neighbourhood of the fixed point. Recalling that $\mathrm{Fix}_{\hat{X}}(S)$ is contained in the ramification locus of $\hat{\pi}$ (i.e., $p \in \mathrm{Fix}_{\hat{X}}(D)$) the action of $D$ may be linearized at $p$. Since $S$ and $D$ commute, the action of $D$ is diagonal in the chosen local coordinates $(z,w)$. We conclude that $\hat \pi$ is locally of the form $(z,w) \mapsto (z^3,w)$ or $(z,w^3)$. The action of $S$ at a fixed point in $\hat{Y}$ is defined by $(\mu^3, \mu^{-1})$ or $(\mu, \mu^{-3})$, respectively. By the lemma above, the fixpoints of $S$ are not affected by the Mori reduction. The map $M_{\mathrm{red}}$ is $S$-equivariant and locally biholomorphic in a neighbourhood of a fixed point of $S$. The lemma follows.
Using our explicit knowledge of the $G$-action on $Z$ we will show in the following that the linearization of the action of $S < G$ at a fixed point in the Del Pezzo surface $Z$ is not of the type described by the lemma above. We distinguish two cases when studying $Z$.
Let $Z \cong \mathbb{P}_2$ and $[x_0:x_1:x_2]$ denote homogeneous coordinates on $\mathbb{P}_2$ such that the action of $S <G$ on $\mathbb{P}_2$ is given by $[x_0:x_1:x_2] \mapsto [\zeta x_0, \zeta^2 x_1, \zeta ^4 x_2 ]$ where $\zeta$ is a $7^\text{th}$ root of unity. Using affine coordinates $z= \frac{x_1}{x_0}, w= \frac{x_2}{x_0}$ we check that the action of $S$ at $p_1=[1:0:0]$ is locally given by $(z,w) \mapsto (\zeta z, \zeta^3 w)$. This contradicts Lemma \[shapeofSaction\].
Let $Z \overset{q}{\to} \mathbb{P}_2$ be the double cover of $\mathbb{P}_2$ branched along Klein’s quartic curve and let $[x_0:x_1:x_2]$ denote homogeneous coordinates on $\mathbb{P}_2$. As above, using affine coordinates $u= \frac{x_1}{x_0}, v= \frac{x_2}{x_0}$ we check that the action of $S$ in a neighbourhood of $[1:0:0]$ is locally given by $(u,v) \mapsto (\zeta u, \zeta^3 v)$. The branch curve $C_\mathrm{Klein} \subset \mathbb{P}_2$ is defined by the equation $u^3+uv^3+v$. In new coordinates $(\tilde u (u,v), \tilde v(u,v))= (u, u^3+uv^3+v)$ the branch curve is defined by $\tilde v = 0$ and the action of $S$ is given by $(\tilde u,\tilde v) \mapsto (\zeta \tilde u, \zeta^3 \tilde v)$. Consider the fixed point $[1:0:0] \in \mathbb P_2$ and its preimage $p \in Z$. At $p$, coordinates $(z,w)$ can be chosen such that the covering map is locally given by $(z,w) \mapsto (z,w^2) = (\tilde u, \tilde v)$. It follows that the action of $S$ at $p \in Z$ is locally given by $(z,w) \mapsto (\zeta z, \zeta^5 w)$. This is again contrary to Lemma \[shapeofSaction\].
In summary, if $|\mathrm{Fix}_{\hat{Y}}(S)|=3$, the action of $S < G$ on the Del Pezzo surface $Z$ cannot be induced by a symplectic $C_7$-action on the K3-surface $X$. This proves the following lemma.
$|\mathrm{Fix}_{\hat{Y}}(S)|\geq 4$.
### Lifting Klein’s quartic
The discussion of the previous section shows that there must be a step in the Mori reduction where the blow-down of a (-1)-curve identifies two $S$-fixed points. Let $z \in Z$ be a fixed point of $S$. Then, by equivariance, all points in the $G$-orbit of $z$ are obtained by blowing down (-1)-curves in the process of Mori reduction. If $Z \cong \mathbb{P}_2$, we denote by $C_\mathrm{Klein} \subset Z$ Klein’s quartic curve. If $Z$ is the double cover of $\mathbb{P}_2$ branched along Klein’s curve, we abuse notation and denote by $C_\mathrm{Klein}$ the ramification curve in $Z$. In the later case $C_\mathrm{Klein}$ is a $G$-invariant curve of genus 3 and self-intersection 8 by Lemma \[selfintbranch\].
Let $z \in \mathrm{Fix}_Z (S) \subset C_\mathrm{Klein}$ and consider the $G$-orbit $G\cdot z$. By invariance, $G\cdot z \subset C_\mathrm{Klein}$. The isotropy group $G_z$ must be cyclic and $G_z =S$ implies $|G\cdot z|=24$. Let $B$ denote the strict transform of $C_\mathrm{Klein}$ in $\hat Y$. The curve $B$ is a smooth $G$-invariant curve of genus 3 and meets at least 24 Mori fibers. Applying Lemma \[selfintblowdown\] to $M_\mathrm{red}(B)=C_\mathrm{Klein}$ we obtain $$B^2 \leq C_\mathrm{Klein}^2 -24 \leq -8.$$
The curve $B$ does not coincide with any of the curves of type $E$ or $F$. Its preimage $\hat B := \hat \pi^{-1}(B) \subset \hat X$ is a cyclic degree three cover of $B$ branched at $B \cap(\bigcup E_i \cup \bigcup F_j)$.
The curves $E_i \subset \hat Y$ are (-3)-curves whereas $B$ has self-intersection less than or equal to $-8$. Assume $B = F_j$ for some $j$. Then $\hat B$ is a curve of self-intersection less than or equal to $-4$ by Lemma \[selfintbranch\] which is mapped biholomorphically to the K3-surface $X$. We obtain a contradiction since K3-surfaces do not admit curves of self-intersection less than $-2$.
Since $\mathrm{Fix}_Z (S) \subset C_\mathrm{Klein}$ there are three fixed points of $S$ on $\hat B$. From $\mathrm{Fix}_{\hat{X}}(S) \subset \mathrm{Fix}_{\hat{X}} (D)$ it follows that $\hat \pi|_{\hat B}: \hat B \to B$ is branched at three or more points. In particular, the curve $\hat{B}$ is connected. In the following, we will distinguish two cases: the curve $\hat B$ being reducible or irreducible.
### Case 1: The curve $\hat{B}$ is reducible {#case-1-the-curve-hatb-is-reducible .unnumbered}
The three irreducible components $\hat{B}_i$, $i= 1,2,3$ of $\hat B$ are smooth curves which are mapped biholomorphically onto $B$. Since $B$ is exceptional, the configuration of curves $\hat{B}$ is also exceptional. It follows that the intersection matrix $(\hat{B}_i \cdot \hat{B}_j)_{ij}$ is negative definite. In the following we study the intersection matrix of $\hat B$ and will obtain a contradiction.
The restricted map $b_X: \hat B_i \to b_X(\hat B_i)$ is the normalization of $b_X(\hat B_i)$ and consequently the arithmetic genus of $b_X(\hat{B}_i)$ is given by the formula (cf. II.11 in [@BPV]) $$g(b_X(\hat{B}_i)) = g(\hat {B_i}) + \delta(b_X(\hat{B}_i)),$$ where the number $\delta$ is computed as $\delta(b_X(\hat{B}_i)) = \sum_{p \in b_X(\hat{B}_i)} \mathrm{dim}_\mathbb{C}({b_X}_*\mathcal{O}_{\hat{B}_i}/\mathcal{O}_{b_X(\hat{B}_i)})_p$. Note that the sum can also be taken over the singular points $p \in b_X(\hat{B}_i)$ only, since smooth points do not contribute to the sum. Since $X$ is a K3-surface, the adjunction formula for $b_X(\hat{B}_i)$ reads $$(b_X(\hat{B}_i))^2 = 2 g(b_X(\hat{B}_i)) -2 = 2g(\hat B_i) +2 \delta(b_X(\hat{B}_i)) -2.$$ By Lemma \[selfintblowdown\], the self-intersection number $(b_X(\hat{B}_i))^2$ can be expressed in terms of the self-intersection $\hat{B}_i^2$ and intersection multiplicities $E_j\cdot \hat{B}_i$: $$(b_X(\hat{B}_i))^2 = \hat{B}_i^2 + \sum_j (\hat{E}_j \cdot \hat{B}_i)^2.$$ It follows that the self-intersection number of $\hat{B}_i$ can be expressed as $$\label{selfintB}
\hat{B}_i^2 = 2g(\hat B_i) +2 \delta(b_X(\hat{B}_i)) -2 -\sum_j (\hat{E}_j\cdot \hat{B}_i)^2.$$ For simplicity, we first consider the case where $\hat{B}_i$ has nontrivial intersection with only one curve of type $\hat{E}$. We refer to this curve as $\hat{E}$. The general case then follows by addition over all curves $\hat E_j$, the number $\delta$ for the full contraction $b_X$ is the sum of all numbers $\delta$ obtained when blowing down disjoint curves $\hat{E}_j$ stepwise.
##### Estimating the number $\delta$
Let $C= C_1 \cup C_2$ be a connected curve consisting of two irreducible components. Then the arithmetic genus of $C$ is calculated as $g(C)= g(C_1) + g(C_2) + C_1 \cdot C_2 -1$. The normalization $ \tilde C$ of $C$ is given by the disjoint union of the normalizations $\tilde C_i$ of $C_1$ and $C_2$. In particular, $g(\tilde C) = g( \tilde C_1) + g(\tilde C_2) -1$, so that $\delta(C) = \delta(C_1) + \delta(C_2) + C_1 \cdot C_2$ (cf. II.11 in [@BPV]).
Since the number $\delta$ is a sum of contributions $ \delta_p$ at singular points $p$, we can calculate the number $\delta_p$ locally at each singularity where we decompose the germ of the curve as the union of irreducible components and use a formula generalizing the example above. We refer to an irreducible component of a curve germ realized in a open neighbourhood of the surface as a *curve segment*.
In order to study the singularities of $b_X(\hat B_i)$ one needs to consider the points of intersection $\hat E \cap \hat B_i$. These points of intersection can be of different quality:
- **Type $ m= 1$:** The intersection at $b \in \hat B_i$ is transversal and the local intersection multiplicity at $b$ is equal to 1. A neighbourhood of $b$ in $\hat B_i$ is mapped to a smooth curve segment in $b_X(\hat B_i)$.
- **Type $m>1$:** The intersection at $b \in \hat B_i$ is of higher multiplicity $m(b)$, i.e., $\hat E$ is tangent to $\hat B_i$ and in local coordinates $(z,w)$ we may write $\hat E = \{ z=0\}$ and $\hat B_i = \{ z- w^m\}$. Blowing down $\hat E$ transforms a neighbourhood of $b$ into a a curve segment isomorphic to $\{ x^{m+1} -y^m =0\}$. For the singularity $(0,0)$ of this curve we calculate $$\delta_{(0,0)}= \frac{1}{2}m(m-1).$$
Let $b_m$ denote the number of points in $\hat E \cap \hat B_i$ with local intersection multiplicity $m$. For each point of intersection of $\hat E$ and $\hat B_i$ we obtain an irreducible component of the germ of $b_X(\hat B_i)$ at $p = b_X(\hat E)$. We compute $\delta_p$ by decomposing this germ and need to determine local intersection multiplicities of all combinations of irreducible components.
\[normarg\] Two irreducible components of the germ of $b_X(\hat B_i)$ at $p$ corresponding to points in $\hat E \cap \hat B_i$ of type $m$ and $n$ meet with local intersection multiplicity greater than or equal to $mn$.
In order to determine the intersection multiplicity of two irreducible components corresponding to points of type $m$ and $n$, we write one curve as $\{ x^{m+1} -y^m =0\}$. The second curve can be expressed as $\{h_1(x,y)^{n+1} - h_2(x,y)^n=0\}$ where $(x,y) \mapsto (h_1(x,y),h_2(x,y))$ is a holomorphic change of coordinates. Now normalizing the first curve by $\xi \mapsto (\xi^m, \xi^{m+1})$ and pulling back the equation of the second curve to the normalization $\mathbb{C}$, we obtain the equation $h_1(\xi^m, \xi^{m+1})^{n+1} - h_2(\xi^m, \xi^{m+1})^n=0$ which has degree at least $mn$ in $\xi$. It follows that the local intersection multiplicity is greater than or equal to $mn$.
Counting different types of intersections of irreducible components we obtain the following estimate for $\delta_p$ $$\begin{aligned}
\delta_p &= \sum \delta_p(C_i) + \sum_{i\neq j}(C_i \cdot C_j)_p\\
&\geq \sum_{m\in\mathbb N}\frac{b_m}{2}m(m-1) + \frac{1}{2}\sum_{m \in \mathbb N}b_m(b_m-1)m^2 + \sum_{m> n} b_mb_n mn\end{aligned}$$
where $\sum_{i\neq j}(C_i \cdot C_j)_p$ decomposes into intersections $(C_i \cdot C_j)_p$ of type $mm$ and intersections of type $mn$ for $m \neq n$. The formula above applies to each curve $\hat E_j$ having nontrivial intersection with $\hat B_i$. Let $p_j$ be the point on $X$ obtained by blowing down $\hat E_j$ and let $b_m^j$ denote the number of points of type $m$ in $\hat B_i \cap \hat E_j$. Then $$\begin{aligned}
\delta ( b_X(\hat B_i)) &= \sum_j \delta_{p_j}(b_X(\hat B_i)) \\
&\geq \sum_j(\sum_{m\in\mathbb N}\frac{b_m^j}{2}m(m-1) + \frac{1}{2}\sum_{m\in \mathbb N}b_m^j(b_m^j-1)m^2 + \sum_{m> n} b_m^jb_n^j mn).\end{aligned}$$ Returning to the formula (\[selfintB\]) for $\hat B_i^2$ we obtain $$\begin{aligned}
\hat{B}_i^2 &= 2g(\hat B_i) +2 \delta(b_X(\hat{B}_i)) -2 -\sum_j (\hat{E}_j \cdot \hat{B}_i)^2\\
& \geq \sum_j(\sum_{m\in\mathbb{N}}b_m^jm(m-1) + \sum_{m\in \mathbb{N}}b_m^j(b_m^j-1)m^2 + 2\sum_{m> n} b_m^jb_n^j mn)\\
&-2 - \sum_j(\sum_m b_m^jm)^2\\
& \geq -2-\sum_j \sum_m b_m^j m.\end{aligned}$$ As a next step, we will find a bound for $(\hat B_i \cdot \hat B_k)$ in the case $i \neq k$. If a curve $\hat B_i$ intersects a ramification curve of type $\hat
E$ or $\hat F$ in a point $x$, then $(\hat B_i \cdot \hat B_k)_x \geq
1$. If $(\hat B_i \cdot \hat E_j)_x =m$, then for $k \neq i$ $$(\hat B_k \cdot\hat E_j)_x = (\varphi_D( \hat B_i) \cdot \hat E_j)_x= ( \varphi_D(\hat B_i) \cdot \varphi_D(\hat E_j))_x = (\hat B_i \cdot \hat E_j)_x =m$$ where $\varphi_D \in D$ is a biholomorphic transformation and $E_j$ is in the fixed locus of $D$.
Assume $\hat B_i$ meets a curve of type $\hat E$ or $\hat F$ in $x$ with local intersection multiplicity $m$. Then $(\hat B_i \cdot \hat B_k)_x \geq m$.
Let $\hat E$, $\hat F$ respectively, be locally given by $\{z=0\}$. Then $\hat B_i$ is locally given by $\{z-w^m=0\}$ and $\hat
B_k$ by $\{h_1(z,w) - h_2(z,w)^m =0\}$ where $(z,w) \mapsto
(h_1(z,w), h_2(z,w))$ is, as in the proof of Lemma \[normarg\], a holomorphic change of coordinates. Note that it stabilizes $\{z=0\}$, i.e., $h_1(0,w)=0$ for all $w$ and we can write $h_1(z,w) =z
\tilde{h}_1(z,w)$. The intersection of $\hat B_i$ and $\hat B_j$ corresponds to the equation $ w^m \tilde{h}_1(w^m, w) - h_2(w ^m, w)$ which is of degree greater than or equal to $m$. The lemma follows.
Summing over all points of intersection of $\hat B_i$ and $ \hat B_k$ one finds $\hat B_i \cdot \hat B_k \geq \sum_j \sum_m b_m^j m$. Recall that by Lemma \[fixSinfixD\] $\mathrm{Fix}_{\hat X}(S)$ is contained in $\mathrm{Fix}_{\hat X}(D)$ and that the curve $B$ contains three $S$-fixed points. Therefore, it intersects the ramification locus of $\hat \pi$ in at least three points. At these points the three irreducible components of $\hat B$ must meet. In particular, $(\hat B_i, \hat B_k) \geq 3 $. This yields $$\begin{aligned}
&(1,1,1) \begin{pmatrix}
\hat B_1^2 & \hat B_1 \cdot \hat B_2 &\hat B_1 \cdot \hat B_3\\
\hat B_2 \cdot \hat B_1 & \hat B_2^2 & \hat B_2 \cdot \hat B_3\\
\hat B_3 \cdot \hat B_1 & \hat B_3 \cdot \hat B_2 & \hat B_3^2
\end{pmatrix}
\begin{pmatrix}
1\\1\\1
\end{pmatrix}\\
&= \hat B_1^2 + \hat B_2^2 + \hat B_3^2 + 2(\hat B_1 \cdot \hat B_2 + \hat B_2 \cdot \hat B_3 + \hat B_1 \cdot \hat B_3)\\
& \geq -6 - 3 \sum_j \sum_m b_m^j+3\sum_j \sum_m b_m^j m +(\hat B_1 \cdot \hat B_2 + \hat B_2 \cdot \hat B_3 + \hat B_1 \cdot \hat B_3) \\
&= -6 +(\hat B_1 \cdot \hat B_2 + \hat B_2 \cdot \hat B_3 + \hat B_1 \cdot \hat B_3)\\
&\geq 3.\end{aligned}$$ Hence, the intersection matrix $(\hat B_i \cdot \hat B_j)_{ij}$ is not negative-definite contradicting the fact that $\hat B$ is exceptional. It follows that the curve $\hat B$ must be irreducible.
### Case 2: The curve $\hat B$ is irreducible {#case-2-the-curve-hat-b-is-irreducible .unnumbered}
Let $n: N \to \hat B$ be the normalization of $\hat B$. Since $b_X$ is a blow-up, $b_X \circ n: N \to b_X(\hat B)$ is the normalization of the curve $b_X(\hat B) \subset X$. It follows that $g(b_X(\hat{B})) = g(N) + \delta(b_X(\hat{B}))$. By adjunction, the self-intersection of $b_X(\hat B)$ is given by $$(b_X(\hat B))^2 = 2g(b_X(\hat{B}))- 2 = 2g(N) + 2\delta(b_X(\hat{B})) -2.$$ As above, by Lemma \[selfintblowdown\], $(b_X(\hat{B}))^2 = \hat{B}^2 + \sum_j (\hat{E}_j \cdot \hat{B})^2$. Thus, the self-intersection of $\hat B$ can be expressed as $$\hat{B}^2 = 2g(N) + 2\delta(b_X(\hat{B})) -2 - \sum_j (\hat{E}_j \cdot \hat{B})^2.$$ Since the curve $\hat B$ is exceptional, this self-intersection number must be negative. By finding a lower bound for $\hat B^2$ we will obtain a contradiction.
Let us first examine the points of intersection $\hat B \cap \hat E$ for one curve $\hat E$ among the exceptional curves of the blow-down $b_X$. We consider the corresponding points of intersection of $B$ and $E$ in $\hat Y$ and we choose coordinates $(\xi, \eta)$ such that $E$ is locally defined by $\{\xi =0\}$, the map $\hat \pi $ is locally given by $(z,w) \mapsto (z^3,w)=(\xi, \eta)$ and $B = \{f(\xi, \eta)=0\}$. It follows that $\hat B$ is locally defined by $\{ h= f\circ \hat\pi =0\}$.
If $E$ and $B$ meet transversally, we know that the function $f(\xi, \eta)$ fulfills $\frac{\partial f}{\partial \eta}|_{(0,0)} \neq 0$. It follows that $\frac{\partial h}{\partial w}|_{(0,0)} \neq 0$ and after a suitable change of coordinates $h(z,w)= z^m-w$.
If $E$ and $B$ meet tangentially, we know that the function $f(\xi, \eta)$ fulfills $\frac{\partial f}{\partial \eta}|_{(0,0)} = 0$. Since $B$ is smooth, we know $\frac{\partial f}{\partial \xi}|_{(0,0)} \neq 0$. After a suitable change of coordinates $h(z,w)= z^3-w^n$ with $n>0$. Note that in both cases the coordinate change on $\hat X$ is such that $\hat E$ is still defined by $\{z=0\}$. This will be important when describing the blow-down $b_X$ of $\hat E$.
Consider a curve segment $\{h=0\}$ in $\hat X$ and its image under the map $b_X$. If $h(z,w)= z^m-w$ then the corresponding smooth segment of $b_X(\hat B)$ is defined by $ x^{m+1} -y =0$. If $h(z,w)= z^3-w^n$ then the corresponding piece of $b_X(\hat B)$ is defined by $ x^{n+3} -y^n =0$ and has a singular point if $n>1$.
Let $p =b_X(\hat E)$. We will determine $\delta_p$ by decomposing the germ of $b_X(\hat B)$ at $p$ into its irreducible components. There are three different types of such components:
1. smooth components locally defined by $ x^{m+1} -y =0$,
2. singular components locally defined by $ x^{n+3} -y^n =0$ for $n>1$ not divisible by $3$,
3. triplets of smooth components locally defined by $ x^6 -y^3 =0$,
4. triplets of singular components locally defined by $ x^{n+3} -y^n =0$ for $n=3k$ and $k >1$.
The singularity in case 2) gives $\delta = \frac{n^2+n-2}{2}$. In case 4), each component is defined by an equation of type $x^{k+1}-y^k=0$ and the singularity of each component gives $\delta = \frac{k^2-k}{2}$.\
In order to determine $\delta_p$ we need to specify intersection multiplicities for all combinations of irreducible components.
The local intersection multiplicities of pairs of irreducible components of the germ of $b_X(\hat B)$ at $p$ in general position are given by the following table.
local equation $ x^{m_1+1} -y $ $ x^{n_1+3} -y^{n_1}$ $x^2-y$ $x^{k_1+1}-y^{k_1}$
------------------------- ------------------- ----------------------- --------- ---------------------
$ x^{m_2+1} -y $ 1 $n_1$ 1 $k_1$
$ x^{n_2+3} -y^{n_2} $ $n_2$ $n_1n_2$ $n_2$ $n_2k_1$
$x^2-y $ 1 $n_1$ 1 (2) $k_1$
$x^{k_2+1}-y^{k_2}$ $k_2$ $k_2n_1$ $k_2$ $k_1k_2$ ($k^2+k$)
Note that the local equations in the first row and column, although all written as functions of $(x,y)$, describe the curve segments in different choices of local coordinates.
As above, we rewrite one equation as $f(h_1(x,y),h_2(x,y))$ where $(h_1,h_2)$ is a holomorphic change of local coordinates. The intersection multiplicities can then be calculated by the method introduced in the proof of Lemma \[normarg\]. Two irreducible components in a triplet of type 3) meet with intersection multiplicity 2. Two irreducible components in a triplet of type 4) meet with intersection multiplicity $k^2+k$. These quantities are indicated in brackets as they differ from the intersection multiplicities of two irreducible components from different triplets.
If two irreducible components of the germ of $b_X(\hat B)$ at $p$ are in special position, their local intersection multiplicity is greater than the value specified in the above table. In particular, the table gives lower bounds for the respective intersection numbers.
Let $a$ denote the number of irreducible components of type 1), let $b_n$ the number of irreducible components of type 2) where $ n \not\in 3\mathbb{N}$, let $c \in 3\mathbb{N}$ denote the number of irreducible components of type 3) and let $d_k \in 3 \mathbb{N}$ denote the number of irreducible components of type 4). We summarize $e=a+c$.
A lower bound for $\delta_p$ is given by $$\begin{aligned}
\delta_p &\geq \sum_n b_n \frac{n^2+n-2}{2} + \sum_k d_k \frac{k^2-k}{2}\\
& + \frac{1}{2}e(e-1) + c +\sum_n e b_n n + \sum_k e d_k k\\
& + \frac{1}{2}\sum_n b_n(b_n-1)n^2 +\sum_{n_1 > n_2} b_{n_1}b_{n_2} n_1n_2 + \sum_{n,k} b_nd_k nk\\
& + \frac{1}{2}\sum_k d_k(d_k-1)k^2 + \sum_k d_k k+ \sum_{k_1 > k_2} d_{k_1}d_{k_2} k_1k_2.\end{aligned}$$ For simplicity, we first consider only one curve $\hat E$ intersecting $\hat B$. The formula for $\hat B^2$ becomes $$\begin{aligned}
\hat{B}^2 &= 2g(N) + 2\delta(b_X(\hat{B})) -2 - (\hat{E} \cdot \hat{B})^2 \notag\\
&=2g(N) + 2\delta(b_X(\hat{B})) -2 - \underset{(\hat{E} \cdot \hat{B})^2}{\underbrace{(e+ \sum_n b_n n + \sum_k d_k k)^2}} \notag\\
&=2g(N)-2- e +2c+ \sum_k d_k k + \sum_n b_n(n-2) \notag \\
&\geq 2g(N)-2- a. \label{ineq}\end{aligned}$$ The same formula also holds if we consider the general case of curves $\bigcup_i \hat E_i$ intersecting $\hat B$ since both the calculation of $\delta$ and the intersection number $\sum_i(\hat B, \hat E_i)^2$ can be obtained from the above by addition. The number $a$ now represents the number of points of type 1) in the union of curves $\hat E_i$.
The map $n \circ \hat \pi: N \to \hat B \to B$ is a degree three cover of the smooth curve $B$ branched at $V \subset B$. The genus of $B$ is three, the topological Euler characteristic is $e(B)= -4$. Let $\tilde V := B \cap (\bigcup E_i \cup \bigcup F_j)$ denote the branch locus of $\hat \pi: \hat B \to B$. Then $V \subset \tilde V$ and $V$ must contain those points in $\tilde V$ which correspond to smooth points on $\hat B$. In partcular, $|V| \geq a$.
The Euler characteristic of $N$ is given by $e(N) = 3e(B) - 2|V| = -12 -2|V| = 2- 2g(N)$. and inequality becomes $$\hat B ^2 \geq 12 + 2|V|-a \geq 12 + |V| \geq 0$$ contradicting the fact that $\hat B$ is exceptional.
### Conclusion {#conclusion-1 .unnumbered}
The above contradiction shows the non-existence of a K3-surface with an action of $G \times C_3$. This completes the prove of Theorem \[nonexist\].
The alternating group of degree six {#chapterA6}
===================================
In the previous chapters we have considered symplectic automorphisms groups of K3-surfaces centralized by an antisymplectic involution, i.e., the groups under consideration were of the form $\tilde G = G \times \langle \sigma \rangle$ where $\tilde G_\mathrm{symp} = G$. In this chapter we wish to discuss more general automorphims groups $\tilde G$ of mixed type: if $\tilde G $ contains an antisymplectic involution $\sigma$ with fixed points we consider the quotient by $\sigma$. In general, if $\sigma$ does not centralize the group $\tilde G_\mathrm{symp}$ inside $\tilde G$, the action of $\tilde G_\mathrm{symp}$ does *not* descend to the quotient surface. We therefore restrict our consideration to the centralizer $Z_{\tilde G}(\sigma)$ of $\sigma$ inside $\tilde G$ (or $\tilde G_\mathrm{symp}$) and study its action on the quotient surface.
If we are able to describe the family of K3-surfaces with $Z_{\tilde G} (\sigma)$-symmetry, it remains to detect the surfaces with $\tilde G$-symmetry inside this family. This chapter is devoted to a situation where the group $\tilde G$ contains the alternating group of degree six. Although, a precise classification cannot be obtained at present, we achieve an improved understanding of the equivariant geometry of K3-surfaces with $\tilde G$-symmetry and classify families of K3-surfaces with $Z_{\tilde G}(\sigma)$-symmetry (cf. Theorem \[classiA6\]). In this sense, this closing chapter serves as an outlook on how the method of equivariant Mori reduction allows generalization to more advanced classification problems.
The group $\tilde A_6$
----------------------
We let $\tilde G$ be any finite group which fits into the exact sequence $$\{\mathrm{id}\} \to A_6 \to \tilde G \overset {\alpha}{\to} C_n \to \{\mathrm{id}\}.$$ and in the following consider a K3-surface $X$ with an effective action of $\tilde G$. The group of symplectic automorphisms $(\tilde{G})_{\text{symp}}$ in $\tilde G$ coincides with $A_6$.
This particular situation is considered by Keum, Oguiso, and Zhang in [@KOZLeech] and [@KOZExten]. They lay special emphasis on the maximal possible choice of $\tilde G$ and therefore consider a group $\tilde G = \tilde A_6$ characterized by the exact sequence $$\label{tilde A6}
\{\mathrm{id}\} \to A_6 \to \tilde A_6 \overset {\alpha}{\to} C_4 \to \{\mathrm{id}\}.$$
Let $N := \mathrm{Inn}(\tilde{A_6}) \subset
\mathrm{Aut}(A_6)$ denote the group of inner automorphisms of $\tilde A_6$ and let $\mathrm{int} : \tilde A_6 \to N$ be the homomorphisms mapping an element $g \in \tilde A_6$ to conjugation with $g$. It can be shown that the group $\tilde{A_6}$ is a semidirect product $A_6
\rtimes C_4$ embedded in $N\times C_4$ by the map $(\mathrm{int}, \alpha)$ (Theorem 2.3 in [@KOZExten]). By Theorem 4.1 in [@KOZExten] the group $N$ is isomorphic to $M_{10}$ and the isomorphism class of $\tilde A_6$ is uniquely determined by and the condition that it acts on a K3-surface.
In [@KOZLeech] a lattice-theoretic proof of the following classification result (Theorem 5.1, Theorem 3.1, Proposition 3.5) is given.
A K3 surface $X$ with an effective action of $\tilde A_6$ is isomorphic to the minimal desingularization of the surface in $\mathbb P_1 \times \mathbb P_2$ given by $$S^2(X^3+Y^3 + Z^3) -3 (S^2 + T^2) XYZ =0.$$
Although this realization is very concrete, the action of $\tilde A_6$ on this surface is hidden. The existence of an isomorphism from a K3-surface with $\tilde A_6$-symmetry to the surface defined by the equation above follows abstractly since both surfaces are shown to have the same transcendental lattice. It is therefore desirable to achieve a more geometric understanding of K3-surfaces with $\tilde A_6$-symmetry in general and in particular to obtain an explicit realization of $X$ where the action of $\tilde A_6$ is visible.
We let the generator of the factor $C_4$ in the semidirect product $\tilde{A_6} = A_6
\rtimes C_4$ be denoted by $\tau$. The order four automorphism $\tau$ is nonsymplectic and has fixed points. It follows that the antisymplectic involution $\sigma := \tau^2$ fulfils $$\mathrm{Fix }_X(\sigma) \neq \emptyset.$$ Since $\sigma$ is mapped to the trivial automorphism in $\mathrm{Out}(A_6) = \mathrm{Aut}(A_6)/\mathrm{int}(A_6) \cong C_2 \times C_2$ there exists $h \in A_6$ such that $
\mathrm{int}(h) = \mathrm{int}(\sigma) \in \mathrm{Aut}(A_6)$. The antisymplectic involution $h \sigma$ centralizes $A_6$ in $\tilde{A_6}$.
If $\mathrm{Fix}_X(h \sigma) \neq \emptyset$, we are in the situation dealt with in Section \[A6Valentiner\], i.e., the K3-surface $X$ is an $A_6$-equivariant double cover of $\mathbb{P}_2$ where $A_6$ acts as Valentiner’s group and the branch locus is given by $F_{A_6}(x_1,x_2,x_3) = 10 x_1^3x_2^3+ 9 x_1^5x_3 + 9 x_2^3x_3^3-45 x_1^2 x_2^2 x_3^2-135 x_1 x_2 x_3^4 + 27 x_3^6$. By construction, there is an evident action of $A_6 \times C_2$ on the Valentiner surface, it is however not clear whether this surface admits the larger symmetry group $\tilde A_6$.
In the following we assume that $h \sigma$ acts without fixed points on $X$ as otherwise the remark above yields an $A_6$-equivariant classification of $X$.
### The centralizer $G$ of $\sigma$ in $\tilde{A_6}$ {#centralizer}
We study the quotient $ \pi : X \to X/\sigma = Y$. As mentioned above, the action of the centralizer of $\sigma$ descends to an action on $Y$. We therefore start by identifying the centralizer $G :=Z_{\tilde{A_6}}(\sigma)$ of $\sigma$ in $\tilde{A_6}$.
The group $G$ equals $Z_{A_6}(\sigma) \rtimes C_4$ and $Z_{A_6}(\sigma) = Z_{A_6}(h)$
The lemma follows from direct computations: we write an element of $\tilde A_6$ as $a\tau^k$ with $a \in A_6$. Then $a \tau^k$ is in $Z_{\tilde{A_6}}(\sigma)$ if and only if $a \tau^k \tau^2 = \tau^2 a \tau^k$. This is the case if and only if $a \tau^2 = \tau^2 a$, i.e., if $a \in Z_{A_6}(\sigma)$. Now $\langle \tau \rangle < Z_{\tilde{A_6}}(\sigma)$ implies the first part of the lemma. The second part follows from the equality $\mathrm{int}(\sigma) = \mathrm{int}(h)$.
$Z_{A_6}(h) = D_8$
Since $\mathrm{int}(\sigma) = \mathrm{int}(h)$ and $\sigma^2 = \mathrm{id}$, it follows that $h^2$ commutes with any element in $A_6$. As $Z(A_6)= \{ \mathrm{id} \}$, it follows that $h$ is of order two. There is only one conjugacy class of elements of order two in $A_6$. We calculate $Z_{A_6}(h) = D_8$ for one particular choice of $h$. Let $$h=
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
3 & 4 & 1 & 2 & 5 & 6
\end{pmatrix}.$$ Any element in the centralizer of $h$ must be of the form $$\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
* & * & * & * & 5 & 6
\end{pmatrix}
\quad \text{or} \quad
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
* & * & * & * & 6 & 5
\end{pmatrix}$$ It is therefore sufficient to perform all calculations in $S_4$. If an element of $S_4$ is a composition of an even (odd) number of transpositions, the corresponding element of $Z_{A_6}(h)$ is given by completing it with the identity map (transposition map) on the fifth and sixth letter.
Let $$g_1=
\begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 1 & 4 & 3
\end{pmatrix},
g_2=
\begin{pmatrix}
1 & 2 & 3 & 4 \\
3 & 2 & 1 & 4
\end{pmatrix},
g_3=
\begin{pmatrix}
1 & 2 & 3 & 4 \\
1 & 4 & 3 & 2
\end{pmatrix}.$$ and check that $g_1,g_2,g_3 \in Z_{A_6}(h)$. Define $g_1 g_2 =:c$ and check $$c=
\begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1
\end{pmatrix}, \quad
c^2
= h.$$ Now $g_3 c g_3 = c^3 $ and the subgroup of $S_4$ ($A_6$, respectively) generated by $c$ and $g_3$ is seen to be a dihedral group of order eight; $\langle g_3 \rangle \ltimes \langle c \rangle = D_8 < Z_{A_6}(h)$. In order to show equality, assume that $Z_{A_6}(h)$ is bigger. It then follows that the centralizer of $h$ in $S_4$ is a subgroup of order 12, in particular, it has a subgroup of order three. Going through the list of elements of order three in $S_4$ one checks that none commutes with $h$ and obtains a contradiction.
Let $D_8=C_2\ltimes C_4$ where $C_2$ is generated by $g= g_3$ and $C_4$ by $c$ and note that $c^2 =h$. We study the action of $\tau $ on $D_8$ by conjugation. Since $C_4$ is the only cyclic subgroup of order four in $D_8$, it is $\tau $-invariant. If $c$ is $\tau $-fixed, i.e. $\tau c = c \tau$, then $$(\tau c)^2= c \tau \tau c = c \sigma c \overset{c \in Z(\sigma)}{=} \sigma c^2 =
\sigma h.$$ In this case $\tau c$ generates a cyclic group of order four acting freely on $X$, a contradiction. So $\tau $ acts on $\langle c\rangle$ by $c \mapsto c^3$ and $c^2 \mapsto c^2$. Now $\tau g\tau^{-1}=c^kg$ for some $k \in \{0,1,2,3\}$. If $k=2$, then $$(\tau g)^2= \tau g \tau g = \tau g \tau^{-1} \tau^2 g = c^2 g \sigma g
\overset{g \in Z(\sigma)}{=} c^2 \sigma = \sigma h$$ and we obtain the same contradiction as above. So $k\in \{1,3\}$ and by choosing the appropriate generator of $\langle c\rangle$ we may assume that $k=3$. The action of $\tau $ on $Z_{A_6}(h)=D_8$ given by $g\mapsto c^3 g$ and $c\mapsto c^3$.
$G'=\langle c\rangle$.
The commutator subgroup $G'$ is the smallest normal subgroup $N$ of $G$ such that $G/N$ is Abelian. We use the above considerations about the action of $\tau$ on $D_8$ by conjugation. The subgroup $\langle c \rangle$ is normal in $G = D_8 \rtimes \langle \tau \rangle$ and $G/ \langle c \rangle$ is seen to be Abelian. Since $G / \langle c^2 \rangle$ is not Abelian, $ G' \neq \langle c^2 \rangle$ and the lemma follows.
### The group $H = G / \langle \sigma \rangle$
We consider the quotient $Y=X/\sigma $ equipped with the action of $G/\sigma =:H=Z_{\tilde A_6}(\sigma)/\langle \sigma \rangle=D_8\rtimes C_2$. The group $C_2$ is generated by $[\tau]_\sigma$. For simplicity, we transfer the above notation from $G$ to $H$ by writing e.g. $ \tau$ for $[\tau]_\sigma$. etc.. Since $\tau g\tau ^{-1}=c^3 g= g c$, it follows as above that $H'=\langle c\rangle$.
Let $K < G$ be the cyclic group of order eight generated by $g \tau $. $$K = \{ \mathrm{id}, g \tau , c \sigma, g \tau^3 c, c^2, g \tau c^2, \sigma
c^3, g c \tau^3 \}.$$ We denote the image of $K$ in $G/\sigma $ by the same symbol. Since $[\sigma c ]_\sigma = [c]_\sigma \in K$ it contains $H'=\langle c\rangle$ and we can write $H=\langle \tau \rangle \ltimes K=D_{16}$.
\[normal groups\] There is no nontrivial normal subgroup $N$ in $H$ with $N\cap H'=\{\mathrm{id}\}$.
If such a group exists, first consider the case $N \cap K =
\{\mathrm{id}\}$. Then $N \cong C_2$ and $H= K \times N$ would be Abelian, a contradiction. If $N \cap K \neq
\{\mathrm{id}\}$ then $N \cap K = \langle (g \tau) ^k\rangle $ for some $k \in
\{1,2,4\}$. This implies $(g \tau )^4 =c^2 \in N$ and contradicts $N \cap H' = N
\cap \langle c \rangle = \emptyset$.
The following observations strongly rely the assumption that $\sigma h$ acts freely on $X$.
\[free on B\] The subgroup $H'$ acts freely on the branch set $B = \pi(\mathrm{Fix}_X(\sigma))$ in $Y$.
If for some $b \in B$ the isotropy group $H'_b$ is nontrivial, then $c^2(b) = h(b)=b$ and $\sigma h$ fixes the corresponding point $\tilde b\in X$.
The subgroup $H'$ acts freely on the set $\mathcal R$ of rational branch curves. In particular, the number of rational branch curves $n$ is a multiple of four.
\[tau-fixed\] The subgroup $H'$ acts freely on the set of $\tau $-fixed points in $Y$.
We show $\mathrm{Fix}_Y(\tau) \subset B$. Since $\sigma = \tau ^2$ on $X$, a $\langle \tau \rangle$-orbit $\{x, \tau x,
\sigma x, \tau^3 x \}$ in $X$ gives rise to a $\tau$-fixed point $y$ in the quotient $Y = X / \sigma$ if and only if $\sigma x = \tau x $. Therefore, $\tau$-fixed points in $Y$ correspond to $\tau$-fixed points in $X$. By definition $\mathrm{Fix}_X(\tau) \subset \mathrm{Fix}_X(\sigma)$ and the claim follows.
$H$-minimal models of $Y$ {#reduction to del Pezzo}
-------------------------
Since $\mathrm{Fix}_X(\sigma) \neq \emptyset$, the quotient surface $Y$ is a smooth rational $H$-surface to which we apply the equivariant minimal model program. We denote by $Y_\mathrm{min}$ an $H$-minimal model of $Y$. It is known that $Y_\mathrm{min}$ is either a Del Pezzo surface or an $H$-equivariant conic bundle over $\mathbb P_1$.
\[no equivariant fibration\] An $H$-minimal model $Y_{\mathrm {min}}$ does not admit an $H$-equivariant $\mathbb P_1$-fibration. In particular, $Y_{\mathrm {min}}$ is a Del Pezzo surface.
In order to prove this statement we begin with the following general fact (cf. Proof of Lemma \[fixed points on mori fibers\]).
\[no increase\] If $Y\to Y_{\mathrm {min}}$ is an $H$-equivariant Mori reduction and $A$ a cyclic subgroup of $H$, then $$\vert \mathrm {Fix}_Y(A)\vert \geq
\vert \mathrm {Fix}_{Y_{\mathrm {min}}}(A)\vert \,.$$
Each step of a Mori reduction is known to contract a disjoint union of (-1)-curves. It is sufficient to prove the statement for one step in a Mori reduction. If such a step changes the number of fixed points, then some Mori fiber $E$ of the reduction is contracted to an $A$-fixed point. The rational curve $E$ is $A$-invariant and therefore contains two $A$-fixed points. The number of fixed points drops.
Suppose that some $Y_\mathrm{min}$ is an $H$-equivariant conic bundle, i.e., there is an $H$-equivariant fibration $p :Y_{\mathrm {min}}\to \mathbb P_1$ with generic fiber $\mathbb P _1$. We let $p _*:H\to \mathrm {Aut}(\mathbb P_1)$ be the associated homomorphism.
\[ker p\*\] $\mathrm {Ker}(p_*)\cap H'=\{\mathrm {id}\}\,.$
The elements of $\mathrm {Ker}(p_*)$ fix two points in every generic $p $-fiber. If $h = c^2 \in H' = \langle c \rangle$ fixes points in every generic $p$-fiber, then $h$ acts trivially on a one-dimensional subset $C \subset Y$. Since $h=c^2$ acts symplectically on $X$ it has only isolated fixed points in $X$. Therefore, on the preimage $\tilde C = \pi^{-1}(C) \subset X$, the action of $h$ coincides with the action of $\sigma$. But then $\sigma h | _{\tilde C} = \mathrm{id}| _{\tilde C}$ contradicts the assumption that $\sigma h$ acts freely on $X$.
Since there are no nontrivial normal subgroups in $H$ which have trivial intersection with $H'$ (Lemma \[normal groups\]), it follows from Lemma \[ker p\*\] that $\mathrm {Ker}(p_*)=
\{ \mathrm{id} \}$, i.e., the group $H$ acts effectively on the base.
We regard $H$ as the semidirect product $H=\langle \tau \rangle \ltimes K$, where $K=C_8$ is described above. The group $H$ acts on the base as a dihedral group and therefore $\tau$ exchanges the $K$-fixed points. We will obtain a contraction by analyzing the $K$-actions on the fibers over its two fixed points. Since $\tau $ exchanges these fibers, it is enough to study the $K$-action on one of them which we denote by $F$.
By Lemma \[singular fibers of conic bundle\] there are two situations which we must consider:
1. $F$ is a regular fiber of $Y_{\mathrm {min}}\to \mathbb P_1$.
2. $F=F_1\cup F_2$ is the union of two (-1)-curves intersecting transversally in one point.
We study the fixed points of $c$, $h=c^2$ and $g \tau $ in $Y_{\mathrm {min}}$. Recall that in $X$ the symplectic transformation $c$ has precisely four fixed points and $h$ has precisely eight fixed points. This set of eight points is stabilized by the full centralizer of $h$, in particular by $K = \langle g \tau \rangle \cong C_8$.
Since $h \sigma$ acts by assumption freely on $X$, it follows that $\sigma$ acts freely on the set of $h$-fixed points in $X$. If $hy=y$ for some $y \in Y$, then the preimage of $y$ in $X$ consists of two elements $x_1,\sigma x_1=x_2$. If these form an $\langle h \rangle $-orbit, then both are $ \sigma h $-fixed, a contradiction. It follows that $\{x_1,x_2 \} \subset \mathrm{Fix}_X(h)$ and the number of $h$-fixed points in $Y$ is precisely four. In particular, $h$ acts effectively on any curve in $Y$.
Let us first consider Case 2 where $F= F_1 \cup F_2$ is reducible. Since $\langle c\rangle $ is the only subgroup of index two in $K$, it follows that $\langle c \rangle $ stabilizes $F_i$ and both $c$ and $h$ have three fixed points in $F$ (two on each irreducible component, one is the point of intersection $F_1 \cap F_2$), i.e., six fixed points on $F \cup \tau F \subset Y_\mathrm{min}$. This is contrary to Lemma \[no increase\] because $h$ has at most four fixed points in $Y_{\mathrm {min}}$.
If $F$ is regular (Case 1), then the cyclic group $K$ has two fixed points on the rational curve $F$. Since $h \in K$, the four $K$-fixed points on $F \cup \tau F$ are contained in the set of $h$-fixed points on $Y_\mathrm{min}$. As $|\mathrm{Fix}_{Y_\mathrm{min}}(h)| \leq 4$, the $K$-fixed points coincide with the four $h$-fixed points in $Y_\mathrm{min}$; $$\mathrm{Fix}_{Y_\mathrm{min}}(h)= \mathrm{Fix}_{Y_\mathrm{min}}(K).$$ In particular, the Mori reduction does not affect the four $h$-fixed points $\{y_1, \dots y_4\}$ in $Y$. By equivariance of the reduction, the group $K$ acts trivially on this set of four points. Passing to the double cover $X$, we conclude that the action of $g \tau \in K$ on a preimage $\{x_i , \sigma x_i\}$ of $y_i$ is either trivial or coincides with the action of $\sigma$. In both cases it follows that $(g \tau )^2 = c\sigma$ acts trivially on the set of $h$-fixed points in $X$. As $\mathrm{Fix}_X(c) \subset \mathrm{Fix}_X(h)$, this is contrary to the fact that $\sigma$ acts freely on $\mathrm{Fix}_X(h)$.
In the following we wish to identify the Del Pezzo surface $Y_\mathrm{min}$. For thus, we use the Euler characteristic formulas, $$24= e(X)= 2 e(Y)-2n + \underset{\text{if $D_g$ is present}}{\underbrace{2g-2}},$$ where $D_g \subset B$ is of general type, $g = g(D_g) \geq 2$, and $$e(Y)= e(Y_{\mathrm{min}}) + m,$$ where $m = | \mathcal E|$ denotes the total number of Mori fibers. For convenience we introduce the difference $\delta =m -n$. If a branch curve $D_g$ of general type is present, then $
13-g-\delta =e(Y_{\mathrm {min}})
$ and if it is not present $
12-\delta =e(Y_{\mathrm {min}})
$.
For every Mori fiber $E$ the orbit $H.E$ consists of at least four Mori fibers.
We need to distinguish three cases: $$\text{1.)}\,\, E\cap B \neq \emptyset\text{\ and\ } E \not\subset B; \quad\quad
\text{2.)}\,\, E \subset B; \quad\quad
\text{3.)}\,\, E \cap B = \emptyset$$ [**Case 1** ]{} Since $H'$ acts freely on the branch curves and $E$ meets $B$ in at most two points, we know $\vert H'.E\vert \ge 2$. If $\vert H.E\vert =2$, then the isotropy group $H_E$ is a normal subgroup of index two which necessarily contains the commutator group $H'$, a contradiction.
[**Case 2** ]{} We show that the $H'$-orbit of $E$ consists of four Mori fibers. If it consisted of less than four Mori fibers, the stabilizer $H'_E \neq \{\mathrm{id}\}$ of $E$ in $H'$ would fix two points in $E \subset B$. This contradicts Lemma \[free on B\].
[**Case 3** ]{} All Mori fibers disjoint from $B$ have self-intersection (-2) and meet exactly one Mori fiber of the previous steps of the reduction in exactly one point. If $E \cap B = \emptyset$ there is a chain of Mori fibers $E_1, \dots, E_k =E$ connecting $E$ and $B$ with the following properties: The Mori fiber $E_1$ is the only one to have nonempty intersection with $B$ and is the first curve of this configuration to be blown down in the reduction process. The curves fulfil $(E_i, E_{i+1})=1$ for all $i \in \{1, \dots, k-1\}$ and $(E_i,E_j)=0$ for all $j\neq i+1$. The curves are blown down subsequently and meet no Mori fibers outside this chain.
The $H$-orbit of this union of Mori fibers consists of at least four copies of this chain. This is due to that fact that the $H$-orbit of $E_1$ consists of at least four Mori fibers by Case 1. In particular, the $H$-orbit of $E$ consists of at least four copies of $E$.
The difference $\delta $ is a non-negative multiple $4k$ of four. If $\delta =0$, then $X$ is a double cover of $Y = Y_\mathrm{min} = \mathbb P_1 \times \mathbb P_1$ branched along a curve of genus nine.
Above we have shown that $m$ and and $n$ are multiples of four. Therefore $\delta =4k$.
If $\delta$ was negative, i.e., $m < n$, there is no configuration of Mori fibers meeting the rational branch curves such that the corresponding contractions transform the (-4)-curves in $Y$ to curves on a Del Pezzo surface $Y_\mathrm{min}$. It follows that $\delta$ is non-negative.
If $\delta= 0$, then $n= m=0$ and $Y$ is $H$-minimal. The commutator subgroup $H' \cong C_4$ acts freely on the branch locus $B$ implying $e(B)\in \{0,-8,-16, \dots \}$. Since the Euler characteristic of the Del Pezzo surface $Y$ is at least 3 and at most 11, $$6 \leq 2e(Y)= 24 + e(B) \leq 22,$$ we only need to consider the case $e(Y)\in \{4,8\}$ and $B=D_{g}$ for $g \in \{9,5\}$.
The automorphism group of a Del Pezzo surface of degree 4 is $C_2^4 \rtimes \Gamma$ for $\Gamma \in \{C_2,C_4, S_3, D_{10} \}$. If $D_{16} < C_2^4 \rtimes \Gamma$ then $ A := D_{16} \cap C_2^4 \lhd D_{16}$ and $A$ is either trivial or isomorphic to $C_2$. In both case $D_{16} / A$ is not a subgroup of $\Gamma$ in any of the cases listed above. Therefore, $e(Y) \neq 8$.
A Del Pezzo surface of degree 8 is either the blow-up of $ \mathbb P_2$ in one point or $\mathbb P_1 \times \mathbb P_1$. Since the first is never equivariantly minimal, it follows that $Y \cong \mathbb P_1 \times \mathbb P_1$ and $g(B)=9$.
Any $H$-minimal model $Y_\mathrm{min}$ of $Y$ is $\mathbb P_1\times \mathbb P_1$ .
Suppose $\delta \neq 0$. Since $\delta \ge 4$, it follows that $e(Y_\mathrm{min})=13-g-\delta \le 7$ if a branch curve $D_g$ of general type is present, and $e(Y_\mathrm{min})=12-\delta \le 8$ if not. We go through the list of of Del Pezzo surfaces with $e(Y_\mathrm{min}) \leq 8$.
- If $e(Y_{\mathrm {min}})=8$, i.e., $\mathrm{deg}(Y_\mathrm{min}) = 4$, then the possible automorphism groups are very limited and we have alredy noted above that $D_{16}$ does not occur.
- If $e(Y_{\mathrm {min}})=7$, then $\mathrm {Aut}(Y_{\mathrm {min}})=S_5$. Since $120$ is not divisible by $16$, we see that a Del Pezzo surface of degree five does not admit an effective action of the group $H$.
- If $e(Y_{\mathrm {min}})=6$, then $A:=\mathrm {Aut}(Y_{\mathrm {min}})=
(\mathbb C^*)^2\rtimes (S_3\times C_2)$. We denote by $A^\circ \cong (\mathbb C^*)^2$ the connected component of $A$. If $q :A\to A/A^\circ$ is the canonical quotient homomorphism then $q (H') < q(A)'\cong C_3$. Consequently $H'=C_4 < A^\circ$. We may realize $Y_{\mathrm {min}}$ as $\mathbb P_2$ blown up at the three corner points and $A^\circ$ as the space of diagonal matrices in $\mathrm {SL}_3(\mathbb C)$. Every possible representation of $C_4$ in this group has ineffectivity along one of the lines joining corner points. But, as we have seen before, the elements of $H'$, in particular $c^2 = h$, have only isolated fixed points in $Y_{\mathrm {min}}$.
- A Del Pezzo surface obtained by blowing up one or two points in $\mathbb P_2$ is never $H$-minimal and therefore does not occur
- Finally, $Y_{\mathrm {min}}\not=\mathbb P_2$: If $e(Y_\mathrm{min}) =3$ then either $\delta =9$ (if $D_g$ is not present), a contradiction to $\delta = 4k$, or $g + \delta =10$. In the later case, $\delta =4,8$ forces $g= 6,2$. In both cases, the Euler characteristic $2-2g$ of $D_g$ is not divisible by 4. This contradicts the fact that $H'$ acts freely on $D_g$.
We have hereby excluded all possible Del Pezzo surfaces except $\mathbb P_1\times \mathbb P_1$ and the proposition follows.
Branch curves and Mori fibers {#branch-curves-and-mori-fibers-1}
-----------------------------
We let $M : Y \to Y_\mathrm{min} = \mathbb P_1 \times \mathbb P_1$ denote an $H$-equivariant Mori reduction of $Y$.
The length of an orbit of Mori fibers is at least eight.
Consider the action of $H$ on $\mathbb P_1 \times \mathbb P_1$. Both canonical projections are equivariant with respect to the commutator subgroup $H'= \langle c \rangle \cong C_4$. Since $c^2 \in H'$ does not act trivially on any curve in $Y$ or $Y_\mathrm{min}$, it follows that $H'$ has precisely four fixed points in $Y_\mathrm{min} =\mathbb P_1 \times \mathbb P_1$. Since $h = c^2$ has precisely four fixed points in $Y$ and $\mathrm{Fix}_Y (H') = \mathrm{Fix}_Y (c) \subset \mathrm{Fix}_Y (c^2)$, we conclude that $H'$ has precisely four fixed points in $Y$ and it follows that the Mori fibers do not pass through $H'$-fixed points. Note that the $H'$-fixed points in $Y$ coincide with the $h$-fixed points.
Suppose there is an $H$-orbit $H.E$ of Mori fibers of length strictly less then eight and let $p = M(E)$. We obtain an $H$-orbit $H.p$ in $\mathbb P_1 \times \mathbb P_1$ with $|H.p| \leq 4$. Now $| K.p| \leq 4$ implies that $K_p \neq \{\mathrm{id}\}$, in particular, $h= c^2 \in K_p$. It follows that $p$ is a $h$-fixed point. This contradicts the fact that the Mori fibers do not pass through fixed points of $h$.
The total number $m$ of Mori fibers equals 0, 8, or 16..
A total number of 24 or more Mori fibers would require 16 rational curves in $B$. This contradicts the bound for the number of connected components of the fixed point set of an antisymplectic involution on a K3-surface (cf. Corollary \[atmostten\])
Recalling that the number of rational branch curves is a multiple of four, i.e., $n \in \{0,4,8\}$ and using the fact $m \in \{0,8,16\}$ along with $m \leq n+9$, we conclude that the surface $Y$ is of one of the following types.
1. $m=0$\
The quotient surface $Y$ is $H$-minimal. The map $X \to Y \cong \mathbb P_1 \times \mathbb P_1$ is branched along a single curve $B$. This curve $B$ is a smooth $H$-invariant curve of bidegree $(4,4)$.
2. $m=8$ and $e(Y) = 12$\
The surface $Y$ is the blow-up of $\mathbb P_1 \times \mathbb P_1$ in an $H$-orbit consisting of eight points.
1. If the branch locus $B$ of $X \to Y$ contains no rational curves, then $e(B)=0$ and $B$ is either an elliptic curve or the union of two elliptic curves defining an elliptic fibration on $X$.
2. If the branch locus $B$ of $X \to Y$ contains rational curves, their number is exactly four (Observe that eight or more rational branch curves of self-intersection (-4) cannot be modified sufficiently and mapped to curves on a Del Pezzo surface by contracting eight Mori fibers). It follows that the branch locus is the disjoint union of an invariant curve of higher genus and four rational curves.
3. $m=16$ and $e(Y) =20$\
The map $X \to Y$ is branched along eight disjoint rational curves.
We can simplify the above situation by studying rational curves in $B$, their intersection with Mori fibers and their images in $\mathbb P_1 \times \mathbb P_1$.
If $e(Y)=12$, then $n=0$.
Suppose $n \neq 0$ and let $C_i \subset Y$ be a rational branch curve. Since $C_i^2 =-4$ and $M(C_i) \subset \mathbb P_1 \times \mathbb P_1$ has self-intersection $\geq 0$ it must meet the union of Mori fibers $\bigcup E_j$. All possible configurations of Mori fibers yield image curves $M(C_i)$ of self-intersection $\leq 4$. If $M(C_i)$ is a curve a bidegree $(a,b)$, then, by adjunction. $$2g(M(C_i)) -2 = (M(C_i))^2 + (M(C_i) \cdot K_{\mathbb P_1 \times \mathbb P_1} )= 2ab -2a-2b,$$ and $(M(C_i))^2 = 2ab \leq 4$ implies that $g(M(C_i))=0$. In particular, $M(C_i)$ must be nonsingular. Hence each Mori fiber meets $C_i$ in at most one point. It follows that $C_i$ meets four Mori fibers, each in one point, and $(M(C_i))^2 =0$. Curves of self-intersection zero in $\mathbb P_1 \times \mathbb P_1$ are fibers of the canonical projections $\mathbb P_1 \times \mathbb P_1 \to \mathbb P_1$. The curve $C_1$ meets four Mori fibers $E_1, \dots E_4$ and each of these Mori fibers meets some $C_i$ for $i \neq 1$. After renumbering, we may assume that $E_1$ and $E_2$ meet $C_2$ and therefore $M(C_1)$ and $M(C_2)$ meet in more than one point, a contradiction. It follows that $e(Y) = 12$ implies $n=0$
If $e(Y)=20$, then $Y$ is the blow-up of $\mathbb P_1 \times \mathbb P_1$ in sixteen points $$\{p_1, \dots p_{16}\} = (F_1 \cup F_2 \cup F_3 \cup F_4) \cap (F_5 \cup F_6 \cup F_7 \cup F_8),$$ where $F_1, \dots F_4$ are fibers of the canonical projection $\pi_1$ and $F_5, \dots F_8$ are fibers of $\pi_2$. The branch locus is given by the proper transform of $\bigcup F_i$ in $Y$.
We denote the eight rational branch curves by $C_1, \dots C_8$. The Mori reduction can have two steps. A slightly more involved study of possible configurations of Mori fibers shows that $0 \leq (M(C_i))^2 \leq 4$. As above $M(C_i)$ is seen to be nonsingular and each Mori fiber can meet $C_i$ in at most one point. Any configuration of curves with this property yields $(M(C_i))^2=0$ and $F_i = M(C_i)$ is a fiber of a canonical projection $\mathbb P_1 \times \mathbb P_1 \to \mathbb P_1$.
If there are Mori fibers disjoint from $B$ these are blown down in the second step of the Mori reduction. Let $E_1, \dots, E_8$ denote the Mori fibers of the first step and $\tilde E_1, \dots, \tilde E_8$ those of the second step. We label them such that $\tilde E_i$ meets $E_i$. Each curve $E_i$ meets two rational branch curves $C_i$ and $C_{i+4}$ and their images $F_i = M(C_i)$ and $F_{i+4}=M(C_{i+4})$ meet with multiplicity $\geq 2$. This is contrary to the fact that they are fibers of the canonical projections. It follows that there are no Mori fibers disjoint from $B$ and all 16 Mori fibers are contrancted simultaniously. There is precisely one possible configuration of Mori fibers on $Y$ such that all rational brach curves are mapped to fibers of the canonical projections of $\mathbb P_1 \times \mathbb P_1$: The curves $C_1, \dots C_4$ are mapped to fibers of $\pi_1$ and $C_5, \dots, C_8$ are mapped to fibers of $\pi_2$. The Mori reduction contracts 16 curves to the 16 points of intersection $\{p_1, \dots p_{16} \} = (\bigcup_{i=1}^4 F_i) \cap(\bigcup_{i=5}^8 F_i) \subset \mathbb P_1 \times \mathbb P_1$.
Let us now restrict our attention to the case where the branch locus $B$ is the union of two linearly equivalent elliptic curves and exclude this case.
### Two elliptic branch curves
In this section we prove:
\[two elliptic branch curves\] $\mathrm{Fix}_X(\sigma)$ is not the union of two elliptic curves.
We assume the contrary, let $\mathrm{Fix}_X(\sigma) = D_1 \cup D_2$ with $D_i$ elliptic and let $f :X\to \mathbb P_1$ denote the elliptic fibration defined by the curves $D_1$ and $D_2$. Recall that $\sigma $ acts effectively on the base $\mathbb P_1$ as otherwise $\sigma$ would act trivially in a neighbourhood of $D_i$ by a linearization argument (cf. Theorem \[FixSigma\]). It follows that the group of order four generated by $\tau $ acts effectively on $\mathbb P_1$.
Let $I$ be the ineffectivity of the induced $G$-action on the base $\mathbb P_1$. We regard $G=C_4\ltimes D_8$ where $C_4=\langle \tau \rangle$ and $D_8$ is the centralizer of $\sigma$ in $A_6$ (cf. Section \[centralizer\]) and define $J:=I\cap D_8$. First, note that $I$ is nontrivial:
The group $G$ does not act effectively on $\mathbb P_1$, i.e., $I \neq \{ \mathrm{id}\}$.
If $G$ acts effectively on $\mathbb P_1$, then $G$ is among the groups specified in Remark \[autP\_1\]. In our special case $|G| = 32$ and $G$ would have to be cyclic or dihedral. Since the group $G$ does not contain a cyclic group of order 16, this is a contradiction.
The intersection $J=I\cap D_8$ is nontrivial.
Assume the contrary and let $J = I \cap D_8 = \{e\}$. We consider the quotient $G \to G/D_8 \cong C_4$ and see that either $I \cong C_2$ or $I \cong C_4$.
- If $I\cap D_8=\{e\}$ and $I\cong C_2$, we write $I=\langle \sigma \xi\rangle $ with $\xi \in D_8$ an element of order two. Now $I$ is normal if and only if $\xi =h$, i.e., $I=\langle \sigma h\rangle$. In this case, since $\sigma h \notin K$, the image of $K$ in $G/I$ is a normal subgroup of index two and one checks that $G/I\cong D_{16}$ . The group $K$ is mapped injectively into $G/I$. The equivalence relation defining this quotient identifies $\sigma $ and $h$ and both are in the image of $K$. So $h$-fixed points in $X$ must lie in the fibers over the $\sigma$-fixed points in $\mathbb P_1$, i.e., the $\sigma $-fixed points sets $D_1, D_2$. Since $\sigma h$ acts freely on $X$, this is a contradiction.
- If $I\cap D_8=\{e\}$ and $I\cong C_4$ we write $I =\langle \tau \xi\rangle$ and show that for no choice of $\xi$ the group $I =\langle \tau \xi\rangle$ is normal in $G$: If $\xi =c^kg$, then $\langle \tau \xi\rangle =K$ is of order eight. If $\xi =c^k$, then $\langle \tau \xi\rangle$ is of order four and has trivial intersection with $D_8$. It is however not normalized by $g$.
As we obtain contradictions in all cases, we see that the intersection $J=I\cap D_8$ is nontrivial.
In the following, we consider the different possibilities for the order of $J$ and show that in fact none of these occur.
If $|J|=8$ then $D_8 \subset I$. Recall that any automorphism group of an elliptic curve splits into an Abelian part acting freely and a cyclic part fixing a point. Since $D_8$ is not Abelian, any $D_8$-action on the fibers of $f$ must have points with nontrivial isotropy and gives rise to a positive-dimensional fixed point set of some subgroup of $D_8$ on $X$ contradicting the fact that $D_8$ acts symplectically on $X$. It follows that the maximal possible order of $J$ is four.
\[I does not contain c\] The ineffectivity $I$ does not contain $\langle c\rangle$.
Assume the contrary and consider the fixed points of $c^2$. If a $c^2$-fixed point lies at a smooth point of a fiber of $f$, then the linearization of the $c^2$-action at this fixed point gives rise to a positive-dimensional fixed point set in $X$ and yields a contradiction. It follows that the fixed points of $c^2$ are contained in the singular $f $-fibers. Since $\langle \tau \rangle $ normalizes $\langle c\rangle$ and the $\langle \tau \rangle $-orbit of a singular fiber consists of four such fibers, we must only consider two cases:
1. The eight $c^2$-fixed points are contained in four singular fibers (one $\langle \tau \rangle $-orbit of fibers), each of these fibers contains two $c^2$-fixed points.
2. The eight $c^2$-fixed points are contained in eight singular fibers (two $\langle \tau \rangle$-orbits).
Note that $\langle c^2 \rangle$ is normal in $I$ and therefore $I$ acts on the set of $\langle c^2\rangle $-fixed points. In the second case, all eight $c^2$-fixed points are also $c$-fixed. This is contrary to $c$ having only four fixed points and therefore the second case does not occur.
The first case does not occur for similar reasons: If $c^2$ has exactly two fixed points $x_1$ and $x_2$ in some fiber $F$, then $\langle c\rangle $ either acts transitively on $\{x_1,x_2\}$ or fixes both points. Since $\mathrm{Fix}_X(c) \subset \mathrm{Fix}_X(c^2)$ and $\langle c\rangle $ must have exactly one fixed point on $F$, this is impossible.
$|J| \neq 4$.
Assume $|J| = 4$. Using $\tau$ we check that no subgroup of $D_8$ isomorphic to $C_2 \times C_2$ is normal in $G$. It follows that the group $\langle c\rangle$ is the only order four subgroup of $D_8$ which is normal in $G$ and therefore $J = \langle c\rangle$. By the lemma above this is however impossible.
It remains to consider the case where $|J|=2$. The only normal subgroup of order two in $D_8$ is $J=\langle h\rangle$.
If $|J|=2$, then $I=\langle \sigma c\rangle$.
We first show that $|J|=2$ implies $|I|=4$: If $|I|=2$, then $ I = \langle h \rangle$ and $G / I = C_4 \ltimes (C_2\times C_2)$. Since this group does not act effectively on $\mathbb P_1$, this is a contradiction. If $|I| \geq 8$, then $G/I$ is Abelian and therefore $I$ contains the commutator subgroup $G' = \langle c \rangle$. This contradicts Lemma \[I does not contain c\]. It follows that $|I|=4$ and either $I \cong C_4$ or $I \cong C_2 \times C_2$. In the later case, the only possible choice is $I = \langle \sigma \rangle \times \langle h \rangle$ which contradicts the fact that $ \sigma$ acts effectively on the base. It follows that $I=\langle \sigma \xi \rangle$, where $\xi ^2=h$ and therefore $\xi =c$.
Let us now consider the action of $G$ on $X$ with $I=\langle \sigma c\rangle $. Recall that the cyclic group $\langle \tau \rangle$ acts effectively on the base and has two fixed points there. Since $\sigma =\tau ^2$, these are precisely the two $\sigma $-fixed points. In particular, $\langle \tau \rangle$ stabilizes both $\sigma $-fixed point curves $D_1$ and $D_2$ in $X$. Furthermore, the transformations $\sigma c$ and $c$ stabilize $D_i$ for $i =1,2$. Since the only fixed points of $c$ in $\mathbb P_1$ are the images of $D_1$ and $D_2$, $$\mathrm {Fix}_X(c)\subset D_1\cup D_2=\mathrm {Fix}_X(\sigma).$$ On the other hand, we know that $\mathrm {Fix}_X(c)\cap \mathrm {Fix}_X(\sigma )=\emptyset$. Thus $I=\langle \sigma c\rangle $ is not possible and the case $|J|=2$ does not occur.
We have hereby eleminated all possibilities for $|J|$ and completed the proof of Theorem \[two elliptic branch curves\].
Rough classification of $X$
---------------------------
We summerize the observations of the previous section in the following classification result.
\[roughclassiA6\] Let $X$ be a K3-surface with an effective action of the group $G$ such that $\mathrm{Fix}_X(h\sigma) = \emptyset$. Then $X$ is one of the following types:
1. a double cover of $\mathbb P_1 \times \mathbb P_1$ branched along a smooth $H$-invariant curve of bidegree (4,4).
2. a double cover of a blow-up of $\mathbb P_1 \times \mathbb P_1$ in eight points and branched along a smooth elliptic curve $B$. The image of $B$ in $\mathbb P_1 \times \mathbb P_1$ has bidegree (4,4) and eight singular points.
3. a double cover of a blow-up $Y$ of $\mathbb P_1 \times \mathbb P_1$ in sixteen points $\{p_1, \dots p_{16}\} = (\bigcup_{i=1}^4 F_i) \cap(\bigcup_{i=5}^8 F_i)$, where $F_1, \dots F_4$ are fibers of the canonical projection $\pi_1$ and $F_5, \dots F_8$ are fibers of $\pi_2$. The branch locus ist given by the proper transform of $\bigcup F_i$ in $Y$. The set $\bigcup F_i$ is an invariant reducible subvariety of bidegree (4,4).
It remains to consider case 2. and show that the image of $B$ in $\mathbb P_1 \times \mathbb P_1$ has bidegree (4,4) and eight singular points. We prove that each Mori fiber $E$ meets the branch locus $B$ either in two points or once with multiplicity two, i.e., we need to check that $E$ may not meet $B$ transversally in exactly one point. If this was the case, the image $M(B)$ of the branch curve is a smooth $H$-invariant curve of bidegree $(2,2)$. The double cover $X'$ of $\mathbb P_1 \times \mathbb P_1$ branched along the smooth curve $M(B)= C_{(2,2)}$ is a smooth surface. Since $X$ is K3 and therefore minimal the induced birational map $X \to X'$ is an isomorphism. This is a contradiction since $X'$ is not a K3-surface.
As each Mori fiber meets $B$ with multiplicity two, the self-intersection number of $M(B)$ is 32 and $M(B)$ is a curve of bidegree (4,4) with eight singular points. These singularities are either nodes or cusps depending on the kind of intersection of $E$ and $B$. We obtain a diagram $$\begin{xymatrix}{
X_\mathrm {sing}\ar[d]^{2:1} & X \ar[d]^{2:1} \ar[l]^{\text{desing.}}\\
C_{(4,4)} \subset \mathbb P_1 \times \mathbb P_1 & \ar[l]_>>>>>>{M} Y \supset B}
\end{xymatrix}$$
In order to obtain a description of possible branch curves, we study the action of $H$ on $\mathbb P_1 \times \mathbb P_1$ and its invariants.
### The action of $H$ on $\mathbb P_1 \times \mathbb P_1$
Recall that we consider the dihedral group $H \cong D_{16}$ generated by $\tau g$ of order eight and $\tau$. For convenience, we recall the group structure of $H$: $$\begin{aligned}
c= ( g \tau )^2, & \quad \tau g \tau = gc,\\
g^2 = \mathrm{id}, & \quad \tau c \tau = c^3,\\
c^4 = \mathrm{id}, & \quad \tau ^2 = \mathrm{id}. \end{aligned}$$ In this section, we prove:
In appropriately chosen coordinates the action of $H$ on $\mathbb P_1\times \mathbb P_1$ given by
- $
c([z_0:z_1],[w_0:w_1])=
([iz_0:z_1],[-iw_0:w_1])
$
- $
\tau ([z_0:z_1],[w_0:w_1])=
([z_1:z_0],[iw_1:w_0])
$
- $
g([z_0:z_1],[w_0:w_1])=
([w_0:w_1],[z_0:z_1])\,.
$
First note that the index two subgroup $H_1$ of $H$ preserving the canonical projections is generated by $\tau$ and $c$, i.e, $H_1 = \langle \tau \rangle \ltimes \langle c \rangle \cong D_8$. We begin by choosing coordinates such that $$c([z_0:z_1],[w_0:w_1])=
([\chi _1(c)z_0:z_1],[\chi _2(c)w_0:w_1])$$ where $\chi _i : H' \to S^1$ are faithful characters. Since $\tau$ acts transitively on the set of $H'$-fixed points, we conclude that after an appropriate change of coordinates not affecting the $H'$-action $$\tau ([z_0:z_1],[w_0:w_1])=
([z_1:z_0],[w_1:w_0]).$$ The automorphism $g$ permutes the factors of $\mathbb P_1 \times \mathbb P_1$, stabilizes the fixed point set of $H'$ and fulfills $gcg^{-1}= c^3$ and $g \tau g^{-1} = c\tau$. Therefore, one finds
- $
c([z_0:z_1],[w_0:w_1])=
([iz_0:z_1],[-iw_0:w_1])
$
- $
\tau ([z_0:z_1],[w_0:w_1])=
([z_1:z_0],[w_1:w_0])
$
- $
g([z_0:z_1],[w_0:w_1])=
([\lambda w_0:w_1],[\lambda ^{-1}z_0:z_1])\,,
$ where $\lambda ^2=i$.
We introduce a change of coordinates such that $g$ is of the simple form $$g([z_0:z_1],[w_0:w_1]) = ([w_0:w_1],[z_0:z_1]).$$ This does affect the shape of the $\tau$-action and yields the action of $H$ described in the propostion.
### Invariant curves of bidegree $(4,4)$
Given the action of $H$ on $\mathbb P_1 \times \mathbb P_1$ discussed above, we wish to study the invariants and semi-invariants of bidegree $(4,4)$. The space of $(a,b)$- bihomogeneous polynomials in $[z_0 : z_1][w_0 : w_1]$ is denoted by $\mathbb C_{(a,b)} ([z_0 : z_1][w_0 : w_1])$.
An invariant curve $C$ is given by a $D_{16}$-eigenvector $f \in \mathbb C_{(4,4)} ([z_0 : z_1][w_0 : w_1])$. The kernel of the $D_{16}$-representation on the line $\mathbb C f$ spanned $f$ contains the commutator subgroup $H' = \langle c \rangle $. It follows that $f$ is a linear combination of $c$-invariant monomials of bidegree $(4,4)$. These are $$\begin{aligned}
z_0^4w_0^4,\,
z_0^4w_1^4, \,
z_1^4w_0^4, \,
z_1^4w_1^4, \,
z_0^2z_1^2w_0^2w_1^2, \,
z_0^3z_1w_0^3w_1, \,
z_0z_1^3w_0w_1^3.\end{aligned}$$ The polynomials $$\begin{aligned}
f_1 =z_0^4w_0^4 + z_1^4w_1^4, \quad
f_2 =z_0^4w_1^4 + z_1^4w_0^4, \quad
f_3 =z_0^3z_1w_0^3w_1 -i z_0z_1^3w_0w_1^3\end{aligned}$$ span the space of $D_{16}$-invariants. Semi-invariants are appropiate linear combinations of $$\begin{aligned}
g_1 =z_0^4w_0^4 - z_1^4w_1^4, \quad
g_2 =z_0^4w_1^4 - z_1^4w_0^4,\quad
g_3 =z_0^3z_1w_0^3w_1 +i z_0z_1^3w_0w_1^3,\quad
g_4 =z_0^2z_1^2w_0^2w_1^2.\end{aligned}$$ Note [$$\begin{array}{llll}
\tau (g_1) = -g_1, & \tau (g_2) = -g_2, & \tau (g_3) = -g_3, & \tau (g_4) = -g_4, \\
g(g_1) = g_1, & g(g_2) = -g_2, & g(g_3) = g_3, & g(g_4) = g_4.
\end{array}$$ ]{} It follows that a $D_{16}$-invariant curve of bidegree $(4,4)$ in $\mathbb P_1 \times \mathbb P_1$ is of the following three types $$\begin{aligned}
C_a &= \{a_1 f_1 + a_2 f_2 + a_3 f_3 = 0\}, \\
C_b &= \{b_1 g_1 + b_3 g_3 + b_4 g_4 =0\}, \\
C_0 &= \{g_2 =0\}.\end{aligned}$$
### Refining the classification of $X$
Using the above description of invariant curves of bidegree (4,4) we may refine Theorem \[roughclassiA6\].
#### Reducible curves of bidegree $(4,4)$
Let $X$ be a K3-surface with an effective action of the group $G$ such that $\mathrm{Fix}_X(h\sigma) = \emptyset$. If $e(X/\sigma) = 20$, then $X/\sigma$ is equivariantly isomorphic to the blow up of $\mathbb P_1 \times \mathbb P_1$ in the singular points of the curve $C = \{f_1-f_2=0\}$ and $X \to Y$ is branched along the proper transform of $C$ in $Y$.
It follows from Theorem \[roughclassiA6\] that $X$ is the double cover of $\mathbb P_1 \times \mathbb P_1$ blown up in sixteen points. These sixteen points are the points of intersection of eight fibers of $\mathbb P_1 \times \mathbb P_1$, four for each of fibration.
By invariance these fibers lie over the base points $[1:1], [1:-1], [1: i], [1:-1]$ and the configurations of eight fibres is defined by the invariant polynomial $f_1-f_2$.
The double cover $X \to Y$ is branched along the proper transform of this configuration of eight rational curves. This proper transform is a disjoint union of eight rational curves in $Y$, each with self-intersection (-4).
#### Smooth curves of bidegree $(4,4)$
Let $X$ be a K3-surface with an effective action of the group $G$ such that $\mathrm{Fix}_X(h\sigma) = \emptyset$. If $X/\sigma \cong \mathbb P_1 \times \mathbb P_1$, then after a change of coordinates the branch locus is $C_a$ for some $a_1,a_2,a_3 \in \mathbb C$.
The surface $X$ is a double cover of $\mathbb P_1 \times \mathbb P_1$ branched along a smooth $H$-invariant curve of bidegree (4,4). The invariant (4,4)-curves $C_b$ and $C_0$ discussed above are seen to be singular at $([1:0],[1:0])$ or $([1:0],[0:1])$.
Note that the general curve $C_a$ is smooth. We obtain a 2-dimensional family $\{C_a\}$ of smooth branch curves and a corresponding family of K3-surfaces $\{X_{C_a}\}$.
#### Curves of bidegree $(4,4)$ with eight singular points
It remains to consider the case 2. of the classification. Our aim is to find an example of a K3-surface $X$ such that $X/\sigma = Y $ has a nontrivial Mori reduction $M: Y \to \mathbb P_1 \times \mathbb P_1= Z$ contracting a single $H$-orbit of Mori fibers consisting of precisely 8 curves. In this case the branch locus $B \subset Y$ is mapped to a singular $(4,4)$-curve $C= M(B)$ in $Z$. The curve $C$ is irreducible and has precisely 8 singular points along a single $H$-orbit in $Z$.
As we have noted above, many of the curves $C_a,C_b,C_0$ are seen to be singular at $([1:0],[1:0])$ or $([1:0],[0:1])$. Since both points lie in $H$-orbits of length two, these curves are not candidates for our construction. This argument excludes the curves $C_b, C_0$ and $C_a$ if $a_1 = 0$ or $a_2 = 0$.
For $C_a$ with $a_3=0$ one checks that $C_a$ has singular points if and only if $a_1 = -a_2$, i.e., if $C_a$ is reducible. It therefore remains to consider curves $C_a$ where all coefficients $a_i \neq 0$. We choose $a_3=1$.
If $a_i\neq 0$ for $i=1,2,3$, then $C_a$ is irreducible.
First note that $C_a$ does not pass through $([1:0],[1:0])$ or $([1:0],[0:1])$. Therefore, possible singularities or points of intersection of irreducible components come in orbits of length eight. Assume that $C_a$ is reducible, consider the decomposition into irreducible components and the $H$-action on it. A curve of type $(n,0)$ is always reducible for $n>1$ and therefore does not occur in the decomposition.
If $C_a$ contains a (2,2)-curve $C_a^{(2,2)}$, then the $H$-orbit of $C_a^{(2,2)}$ has length $\leq2 $ and $C_a^{(2,2)}$ is stable with respect to the subgroup $H' = \langle c \rangle $ of $H$. All $c$-semi-invariants of bidegree (2,2) are, however, reducible. Similary, all $c$-semi-invariants of bidegree (1,2) or (2,1) are reducible an therefore $C$ does not have a curve of this type as an irreducible component.
The curve $C_a$ is not the union of a (1,3)- and a (3,1)-curve, since their intersection number is 10 and contradicts invariance. Similarly one excludes the union of a (1,1) and a (3,3)-curve.
If $C_a$ is a union of (1,1) or (1,0) and (0,1)-curves, one checks by direct computation that the requirement that $C_a$ is $H$-invariant gives strong restrictions and finds that in all cases at least one coefficient $a_i$ has to be zero.
One possible choice of an orbit of length eight is given by the orbit through a $\tau$-fixed point $p_\tau = ([1:1],[\pm \sqrt{i}:1])$. One checks that $p_\tau \in C_a$ for any choice of $a_i$. However, if we want $C_a$ to be singular in $p_\tau$, then $a_2=0$. It then follows that $C_a$ is singular at points outside $H .p_\tau$. It has more than eight singular points and is therefore reducible.
All other orbits of length eight are given by orbits through $g$-fixed points $p_x= ([1:x],[1:x])$ for $x \neq 0$. One can choose coefficients $a_i(x)$ such that $C_{a(x)}$ is singular at $p_x$ if and only if $x^8 \neq 1$. If the curve $C_{a(x)}$ is irreducible, then it has precisely eight singular points $H.p_x$ of multiplicity 2 (cusps or nodes) and the double cover of $\mathbb P_1 \times \mathbb P_1$ branched along $C_{a(x)}$ is a singular K3-surface with precisely eight singular points. We obtain a diagram $$\begin{xymatrix}{
X_\mathrm {sing}\ar[d]^{2:1} & X \ar[d]^{2:1} \ar[l]^{\text{desing.}}\\
C_{(4,4)} \subset \mathbb P_1 \times \mathbb P_1 & \ar[l]^<<<<<{M} Y \supset B}
\end{xymatrix}$$ If $p_x$ is a node in $C_{a(x)}$, then the corresponding singularity of $X_\mathrm{sing}$ is resolved by a single blow-up. The (-2)-curve in $X$ obtained from this desingularization is a double cover of a (-1)-curve in $Y$ meeting $B$ in two points.
If $p_x$ is a cusp in $C_{a(x)}$, then the corresponding singularity of $X_\mathrm{sing}$ is resolved by two blow-ups. The union of the two intersecting (-2)-curves in $X$ obtained from this desingularization is a double cover of a (-1)-curve in $Y$ tangent to $B$ is one point.
The information determining whether $p_x$ is a cusp or a node is encoded in the rank of the Hessian of the equation of $C_{a(x)}$ at $p_x$. The condition that this rank is one gives a nontrivial polynomial condition. For a general irreducible member of the family $\{C_{a(x)} \, | \, x\neq 0, \, x^8 \neq 1 \}$ the singularities of $C_{a(x)}$ are nodes.
We let $q$ be the polynomial in $x$ that vanishes if and only if the rank of the Hessian of $C_{a(x)}$ at $p_x$ is one. It has degree 24, but 16 of its solutions give rise to reducible curves $C_{a(x)}$. The remaining eight solution give rise to four different irreducible curves. These are identified by the action of the normalizer of $H$ in $\mathrm{Aut}(\mathbb P_1 \times \mathbb P_1)$ and therefore define equivalent K3-surfaces.
We summarize the discussion in the following main classification theorem.
\[classiA6\] Let $X$ be a K3-surface with an effective action of the group $G$ such that $\mathrm{Fix}_X(h \sigma) = \emptyset$. Then $X$ is an element of one the following families of K3-surfaces:
1. the two-dimensional family $\{X_{C_a}\}$ for $C_a$ smooth,
2. the one-dimensional family of minimal desingularization of double covers of $\mathbb P_1 \times \mathbb P_1$ branched along curves in $\{C_{a(x)} \, | \, x\neq 0, \, x^8 \neq 1 \}$. The general curve $C_{a(x)}$ has precisely eight nodes along an $H$-orbit. Up to natural equivalence there is a unique curve $C_{a(x)}$ with eight cusps along an $H$-orbit.
3. the trivial family consisting only of the minimal desingularization of the double cover of $\mathbb P_1 \times \mathbb P_1$ branched along the curve $C_a = \{f_1-f_2=0\}$ where $a_1 =1, a_2 =-1, a_3=0$.
Let $X$ be a K3-surface with an effective action of the group $\tilde A_6$. If $\mathrm{Fix}_X(h \sigma) = \emptyset$, then $X$ is an element of one the families 1. -3. above. If $\mathrm{Fix}_X(h \sigma) \neq \emptyset$, then $X$ is $A_6$-equivariantly isomorphic to the Valentiner surface.
Summary and outlook
-------------------
Recall that our starting point was the description of K3-surfaces with $\tilde A_6$-symmetry. Using the group structure of $\tilde A_6$ we have divided the problem into two possible cases corresponding to the question whether $\mathrm{Fix}_X(h\sigma)$ is empty or not. If it is nonempty, the K3-surface with $\tilde A_6$-symmetry is the Valentiner surface discussed in Section \[A6Valentiner\]. If is is empty, our discussion in the previous sections has reduced the problem to finding the $\tilde A_6$-surface in the families of surfaces $X_{C_a}$ with $D_{16}$-symmetry.
It is known that a K3-surface with $\tilde A_6$-symmetry has maximal Picard rank 20. This follows from a criterion due to Mukai (cf. [@mukai]) and is explicitely shown in [@KOZLeech].
All surfaces $X_{C_a}$ for $C_a \subset \mathbb P_1 \times \mathbb P_1$ a (4,4)-curve are elliptic since the natural fibration of $\mathbb P_1 \times \mathbb P_1$ induces an elliptic fibration on the double cover (or is desingularization).
A possible approach for finding the $\tilde A_6$-example inside our families is to find those surfaces with maximal Picard number by studying the elliptic fibration. It would be desirable to apply the following formula for the Picard rank of an elliptic surface $f: X \to \mathbb P_1$ with a section (cf. [@shiodainose]): $$\rho(X) = 2 + \mathrm{rank}(MW_f) + \sum_i (m_i-1)$$ where the sum is taken over all singular fibers, $m_i$ denotes the number of irreducible components of the singular fiber and $\mathrm{rank}(MW_f)$ is the rank of the Mordell-Weil group of sections of $f$. The number two in the formula is the dimension of the hyperbolic lattice spanned by a general fiber and the section.
First, one has to ensure that the fibration under consideration has a section. One approach to find sections is to consider the quotient $q:\mathbb P_1 \times \mathbb P_1 \to \mathbb P_2$ and the image of the curve $C_a$ inside $\mathbb P_2$. If we find an appropiate bitangent to $q(C_a)$ such that its preimage in $\mathbb P_1 \times \mathbb P_1$ is everywhere tangent to $C_a$, then its preimage in the double cover of $\mathbb P_1 \times \mathbb P_1$ is reducible and both its components define sections of the elliptic fibration. For $C_a$ the curve with eight nodes the existence of a section (two sections) follows from an application of the Plücker formula to the curve $q(C_a)$ with 3 cusps and its dual curve.
As a next step, one wishes to understand the singular fibers of the elliptic fibrations. Singular fibers occur whenever the branch curve $C_a$ intersects a fiber $F$ of the $\mathbb P_1 \times \mathbb P_1$ in less than four points. Depending on the nature of intersection $F \cap C_a$ one can describe the corresponding singular fiber of the elliptic fibration. For $C_a$ the curve with eight cusps one finds precisely eight singular fibres of type $I_3$, i.e., three rational curves forming a closed cycle. In particular, the contribution of all singular fibres $\sum_i (m_i-1)$ in the formula above is 16. In the case where $C_a$ is smooth or has eight nodes, this contribution is less.
In order to determine the number $ \rho(X_{C_a})$ it is neccesary to either understand the Mordell-Weil group and its $\mathrm{rank}(MW_f)$ or to find curves which give additional contribution to $\mathrm{Pic}(X_{C_a}$ not included in $2 + \sum_i (m_i-1)$.
In conclusion, the method of equivariant Mori reduction applied to quotients $X/\sigma$ yields an explicit description of a families of K3-surfaces with $D_{16} \times \langle \sigma \rangle$-symmetry and by construction, the K3-surface with $\tilde A_6$-symmetry is contained in one of these families. It remains to find criteria to characterize this particular surface inside this family. The possible approach by understanding the function $$a \mapsto \rho( X_{C_a})$$ using the elliptic structure of $X_{C_a}$ requires a detailed analysis of the Mordell-Weil group.
Actions of certain Mukai groups on projective space
===================================================
In this appendix, we derive the unique action of the group $N_{72}$ on $\mathbb P_3$ and the unique action of $M_9$ on $\mathbb P_2$ in the context of Sections \[N72\] and \[M9\]. We consider the homomorphism $\mathrm{SL}_n(\mathbb C) \to \mathrm{PSL}_n(\mathbb C)$ and determine preimages $\tilde g \in \mathrm{SL}_n(\mathbb C)$ of the generators $g \in G \subset \mathrm{PSL}_n(\mathbb C)$. Our considerations benefit from fact that both actions are induced by symplectic actions of the corresponding group on a K3-surface $X$.
The action of $N_{72}$ on $\mathbb P_3$ {#N72appendix}
---------------------------------------
One can calculate explicitly the realization of the $N_{72}$-action on $\mathbb P_3$ by using the decomposition $C_3^2 \rtimes D_8$ where $D_8 = C_2\ltimes (C_2 \times C_2) = \mathrm{Aut}(C_3^2)$. For each generator of $N_{72}$ we will specify the corresponding element in $\mathrm{SL}_4(\mathbb C)$. We denote the center of $\mathrm{SL}_4(\mathbb C)$ by $Z$. Recall that the action of $D_8 = C_2\ltimes (C_2 \times C_2) =
\langle \alpha \rangle \ltimes ( \langle \beta \rangle \times \langle \gamma \rangle)$ on $C_3 \times C_3$ is given by $$\alpha(a,b) = (b,a), \quad \beta(a,b)=(a^2,b), \quad \gamma(a,b) = (a,b^2).$$
In suitably chosen coordinates the generator $a$ of $C_3^2$ can be represented as $$\begin{aligned}
\tilde a=
\begin{pmatrix}
\xi & 0 & 0 & 0 \\
0 & \xi ^2& 0 & 0\\
0&0& 1 & 0\\
0&0&0&1
\end{pmatrix}\end{aligned}$$ where $\xi$ is a third root of unity. Next we wish to specify $\gamma$ in $\mathrm{SL}_4(\mathbb C)$. We know that $a \gamma = \gamma a$, i.e., $\tilde a \tilde \gamma \tilde a^{-1} \tilde \gamma^{-1} \in Z$, and $\gamma$ is seen to be of the form $$\begin{aligned}
\tilde \gamma =
\begin{pmatrix}
* & 0 & 0 & 0 \\
0 & * & 0 & 0\\
0&0& * & *\\
0&0& * & *
\end{pmatrix}\end{aligned}$$ where $*$ denotes a nonzero matrix entry. Since $a$ and $b$ commute in $N_{72}$, we know that $\tilde a \tilde b \tilde a^{-1} \tilde b^{-1} \in Z$ and $$\begin{aligned}
\tilde b =
\begin{pmatrix}
* & 0 & 0 & 0 \\
0 & * & 0 & 0\\
0&0& * & *\\
0&0& * & *
\end{pmatrix}\end{aligned}$$ Since $\gamma$ acts on $ b$ by $\gamma b \gamma = b^{-1} = b^2$, it follows that $$\begin{aligned}
\tilde b =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0&0& * & *\\
0&0& * & *
\end{pmatrix}.\end{aligned}$$ We apply a change of coordinates affecting only the lower $(2 \times 2)$-block of $b$ and therefore not affecting the shape of $a$ auch that $$\begin{aligned}
\tilde b=
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & \xi & 0 \\
0 & 0 & 0 & \xi^2
\end{pmatrix}.\end{aligned}$$ It follows that $\alpha$ interchanges the two $(2 \times 2)$-blocks of the matrices $a$ and $b$ and $$\begin{aligned}
\tilde \alpha=
\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}.\end{aligned}$$ Finally, $\gamma$ and $\beta$ can be put into the form $$\begin{aligned}
\tilde \gamma =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0&0& 0 & 1\\
0&0& 1 & 0
\end{pmatrix}, \quad
\tilde \beta =
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0\\
0&0& 1 & 0\\
0&0& 0 & 1
\end{pmatrix}.\end{aligned}$$
### Invariant quadrics and cubics
Let $f \in \mathbb C_2[x_1:x_2:x_3:x_4]$ be a semi-invariant homogeneous polynomial of degree two, $$f = \sum_i a_i x_1^2 + \sum _{i\neq j} b_{ij} x_ix_j.$$ If $a_i \neq 0$ for some $1 \leq i \leq 4$, then semi-invariance with respect to the transformations $\alpha, \beta, \gamma$ yields $a_1 = a_2 = a_3 = a_4$. It follows that $f$ is not semi-invariant with respect to $a$.
If $b_{13} \neq 0$, then semi-invariance with respect to the transformations $\alpha, \beta, \gamma$ yields $b_{13} = b_{23} = b_{24} = b_{14}$. As above, the polynomial $f$ is not semi-invariant with respect to $a$.
Therefore, if $f$ is semi-invariant, then $a_i = b_{13} = b_{23} = b_{24} = b_{14} =0$ and $b_{12} = b_{34}$. In particular, all degree two semi-invariants are in fact invariant. There is a unique $N_{72}$-invariant quadric hypersurface in $\mathbb P_3$ given by the equation $x_1 x_2 + x_3 x_4$ .
Analogous considerations show that a semi-invariant polynomial of degree three is a multiple of $f_\mathrm{Fermat} = x_1^3 +x_2^3 + x_3^3 +x_4^3$ and the Fermat cubic $\{f_\mathrm{Fermat} =0\}$ is seen to be the unique $N_{72}$-invariant cubic hypersurface in $\mathbb P_3$.
The action of $M_9$ on $\mathbb P_2$ {#M9 on P2}
------------------------------------
We consider the decompostion of $M_9 = (C_3 \times C_3) \rtimes Q_8$. The generators of $ C_3 \times C_3$ are denoted $a$ and $b$ and the generators of $Q_8$ are denoted by $I, J, K$. Recall $ I^2 = J^2 = K^2 = IJK = -1$. We choose the factorization of $C_3 \times C_3$ such that $-1$ acts as $$(-1)a(-1) = a^2, \quad (-1)b(-1) = b^2.$$ Furthermore, $I a (-I) = b$ and $Ja(-J) = b^2 a$.
We repeatedly use the fact that the action of $M_9$ is induced by a symplectic action of $M_9$ on a K3-surface $X$ which is a double cover of $\mathbb P_2$.
We begin by fixing a representation of $a$. Since $a$ may not have a positive dimensional set of fixed points in $\mathbb P_2$, it follows that in appropriately chosen coordinates $$\tilde a=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \xi & 0\\
0&0& \xi^2
\end{pmatrix},$$ where $\xi$ is third root of unity.
As a next step, we want to specify a representation of $b$ inside $\mathrm{SL}_3(\mathbb C)$. Since $a$ and $b$ commute in $\mathrm{PSL}_3(\mathbb C)$, we know that $$\tilde a \tilde b \tilde a^{-1} \tilde b^{-1} = \xi^k \mathrm{id}_{\mathbb C^3}$$ for $k \in \{0, 1,2\}$. Note that $\tilde b$ is not diagonal in the coordinates chosen above since this would give rise to $C_3^2 $-fixed points in $\mathbb P_2$. As these correspond to $C_3^2$-fixed points on the double cover $X \to Y$ and a symplectic action of $C_3^2 \nless \mathrm{SL}_2(\mathbb C)$ on a K3-surface does not admit fixed points, this is a contradiction. An explicit calculation yields $$\begin{aligned}
\tilde b = \tilde b_1=
\begin{pmatrix}
0 & 0 & * \\
* & 0 & 0\\
0 & * & 0
\end{pmatrix} \quad \text{or} \quad
\tilde b = \tilde b_2=
\begin{pmatrix}
0 & * & 0 \\
0 & 0 & *\\
* & 0 & 0
\end{pmatrix}.\end{aligned}$$ We can introduce a change of coordinates commuting with $\tilde a$ such that $$\begin{aligned}
\tilde b = \tilde b_1 =
\begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix} \quad \text{or} \quad
\tilde b = \tilde b_2=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1\\
1&0&0
\end{pmatrix}.\end{aligned}$$ Since $\tilde b_1 = \tilde b_2^2$, the two choices above correspond to choices of generators $b$ and $b^2$ of $\langle b \rangle$ and are therefore equivalent. In the following we fix the second choice of $b$. A direct computation yields that the element $-1$ must be represented in the form $$\begin{aligned}
\begin{pmatrix}
* & 0 & 0 \\
0 & 0 & *\\
0 & * & 0
\end{pmatrix} \quad \text{or} \quad
\begin{pmatrix}
0 & * & 0 \\
* & 0 & 0\\
0 & 0 & *
\end{pmatrix} \quad \text{or} \quad
\begin{pmatrix}
0 & 0 & * \\
0 & * & 0\\
* & 0 & 0
\end{pmatrix}.\end{aligned}$$ After reordering the coordinates, we can assume that $$\begin{aligned}
\widetilde{-1} =
\begin{pmatrix}
* & 0 & 0 \\
0 & 0 & *\\
0 & * & 0
\end{pmatrix}.\end{aligned}$$ The relation $(-1)b(-1) = b^2$ yields $$\begin{aligned}
\widetilde{-1} =
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 0 & \eta\\
0 & \eta^2 & 0
\end{pmatrix}.\end{aligned}$$ for some third root of unity $\eta$. The element $I$ fulfills $I a (-I) = b$ and, using the representation of $a$ and $b$ given above, we conclude $$\begin{aligned}
\widetilde{I} = \frac{1}{\xi -\xi ^2}
\begin{pmatrix}
1 & 1 & 1\\
\zeta^2 & \zeta^2 \xi & \zeta^2 \xi^2 \\
\zeta & \zeta \xi^2 & \zeta \xi
\end{pmatrix}\end{aligned}$$ for some third root of unity $\zeta$. Now $I^2 = -1$ implies $\zeta = 1$ and $\eta =1$. Analogous considerations yield the following shape of $J$: $$\begin{aligned}
\widetilde{J} = \frac{1}{\xi -\xi ^2}
\begin{pmatrix}
1 & \xi & \xi\\
\xi^2 & \xi & \xi^2 \\
\xi^2 & \xi^2 & \xi
\end{pmatrix}.\end{aligned}$$ In appropiately chosen coordinates the action on $M_9$ is precisely of the type claimed in Section \[M9\].
[BHPVdV04]{}
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|
---
abstract: 'A compact complex manifold $X$ is called [*elliptically connected*]{} if any pair of points in $X$ can be connected by a chain of elliptic or rational curves. We prove that the fundamental group of an elliptically connected compact complex surface is almost abelian. This confirms a conjecture which states that the fundamental group of an elliptically connected Kähler manifold must be almost abelian.'
author:
- 'K. Oguiso, M. Zaidenberg'
title: On fundamental groups of elliptically connected surfaces
---
ø ø v §[[S]{}]{} ß
Introduction {#introduction .unnumbered}
============
We use below the following
Let $X$ be a compact complex space. We say that $X$ is [*elliptically*]{} (resp. [*torically*]{}) [*connected*]{} if any two points $x',\,x'' \in X$ can be joined by a finite chain of (possibly, singular) rational or elliptic curves (resp. of holomorphic images of complex tori). Here we discuss the following
\[Z\]. [*Let $X$ be a compact Kähler manifold. If $X$ is torically connected, then the fundamental group $\pi_1 (X)$ is almost abelian (or, for a weaker form, almost nilpotent).*]{}
A group $G$ is called [*almost abelian*]{}[^1] (resp. [*almost nilpotent, almost solvable, etc.*]{}) if it contains an abelian (resp. nilpotent, solvable, etc.) subgroup of finite index. Obviously, each of these properties is stable under finite extensions.
More generally, we may ask whether a Kähler variety connected by means of chains of subvarieties with almost abelian (resp. almost nilpotent, almost solvable, etc.) fundamental groups has itself such a fundamental group.
ß
It is known that [a rationally connected]{} (i.e. connected by means of chains of rational curves) compact Kähler manifold is simply connected \[KMM, Cam1 (3.5), Cam2 (5.7), Cam3 ($2.4'$), Cam4 (5.2.3)\] (see also \[Se\]). Moreover, it follows from \[Cam1 (2.2), Cam2 (5.2), (5.4); Cam4 (5.2.4.1)\] that
ß
ß
This gives a motivation for the above conjecture. Another kind of motivation is provided by the following function–theoretic consideration. We introduce the next
We say that a complex space $X$ is [*sub–Liouville*]{} if its universal covering space $U_X$ is [*Liouville*]{}, i.e. if any bounded holomorphic function on $U_X$ is constant.
The complex tori yield examples of sub–Liouville compact manifolds. By a theorem of Lin \[Li\], any quasi–compact complex variety $X$ with an almost nilpotent fundamental group $\pi_1 (X)$ is a sub–Liouville one. It is easily seen that any complex space with countable topology, connected by means of chains of sub–Liouville subspaces, is itself sub–Liouville \[DZ, (2.3)\]. In particular, any torically connected variety is sub–Liouville. Thus, the question arises whether such a variety should also satisfy the assumption of Lin’s Theorem, which is just Conjecture 0.2 in its weaker form.
Note that even in its weaker form the conjecture fails for non–Kählerian compact complex manifolds. An example (communicated by J. Winkelmann[^2]) is a complex 3-fold which is a quotient of $SL_2 (\cz)$ by a discrete cocompact subgroup (for details see Appendix below).
In this note we consider the simplest case of complex surfaces. We prove the following
The above conjecture can be also formulated for non–compact Kähler manifolds, in particular, for smooth quasi–projective varieties. To this point, in Definition 0.1 one should consider, instead of chains of rational or elliptic curves (resp. compact complex tori), the chains of non–hyperbolic quasi–projective curves[^3] (resp. products of compact tori and factors $(\cz^* )^m,\,m \in \Bbb N$). L. Haddak[^4] has checked that Theorem 0.5 holds true for smooth quasi–projective surfaces. The proof is based on the Fujita classification results for open surfaces \[Fu\].
Proof of Theorem 0.5 {#proof-of-theorem-0.5 .unnumbered}
====================
In the proof we use the following two lemmas.
\[Fu, Thm. 2.12; No, Lemma 1.5.C\]. [*Let $X$ and $Y$ be connected compact complex manifolds, and let $f:\,X \to Y$ be a dominant holomorphic mapping. Then $f_* \pi_1 (X) \s \pi_1 (Y)$ is a subgroup of finite index. In particular, if the group $\pi_1 (X)$ is almost abelian (resp. almost nilpotent, almost solvable), then so is $\pi_1 (Y)$.*]{}
If the algebraic dimension $a(S)$ were zero, then $S$ would have only a finite number of irreducible curves \[BPV, IV.6.2\] and hence, it would not be elliptically connected. In the case when $a(S) = 1$, $S$ is not elliptically connected, either. Indeed, such an $S$ is an elliptic surface \[BPV, VI.4.1\], and any irreducible curve on it is contained in a fibre of the elliptic fibration $\pi\,:\,S \to B$, where $B$ is a smooth curve (because, if an irreducible curve $E \subset S$ were not contained in a fibre of $\pi$, then one would have $E\cdot F > 0$, where $F$ is a generic fibre of $\pi$, and hence $(E + nF)^2 > 0$ for $n$ large enough, which would imply that $S$ is projective \[BPV, IV.5.2\], a contradiction). Thus, $a(S) = 2$, and therefore, $S$ is projective (see e.g. \[BPV, IV.5.7\]).
Let $S$ be a smooth compact complex surface. Suppose that $S$ is torically connected. Then either $S$ itself is dominated by a complex torus, and then, by Lemma 1.1.$c)$, the group $\pi_1 (S)$ is almost abelian, or $S$ is elliptically connected. Consider the latter case. Due to the bimeromorphic invariance of the fundamental group, we may assume $S$ being minimal. $S$ being elliptically connected, by Lemma 1.2 it is a projective surface with a rational or elliptic curve passing through each point of $S$. Certainly, the Kodaira dimension $k(S) \le 1$. According to the possible values of $k(S)$, consider the following cases.
a\) Let $k(S) = -\infty$. Then $S$ is either a rational surface or a non–rational ruled surface over a curve $E$. In the first case, $S$ is simply connected. In the second one, $E$ should be an elliptic curve. Indeed, since $S$ is elliptically connected, $E$ is dominated by a rational or elliptic curve $C \se S$, and therefore it is itself rational or elliptic. The surface $S$ being non–rational, $E$ must be elliptic. Thus, we have a relatively minimal ruling $\pi :\,S \to E$, which is a smooth fibre bundle with a fibre $\pr^1$. From the exact sequence $${\bf 1} = \pi_2 (E) \to {\bf 1} = \pi_1 (\pr^1 ) \to \pi_1 (S) \to \pi_1 (E)
\to {\bf 1}$$ we obtain $\pi_1 (S) \cong \pi_1 (E) \cong \Gaz^2$.
b\) Let $k(S) = 0$. By the Enriques–Kodaira classification (see e.g. \[GH, p.590\] or \[BPV, Ch. VI\]), there are the following four possibilities: ß
$S$ is a K3–surface, and then $\pi_1 (S) = $[**1**]{}. ß
$S$ is an Enriques surface, and then $\pi_1 (S) \cong \Gaz / 2\Gaz$. ß
$S$ is an abelian surface, and then $\pi_1 (S) \cong \Gaz^4$. ß
$S$ is a hyperelliptic surface, and then, being a finite non–abelian extension of $\Gaz^4$, the group $\pi_1 (S)$ is almost abelian. ß
Note that in the last two cases $S$ is dominated by a torus.
c\) Suppose further that $k(S) = 1$. Then $S$ is an elliptic surface \[GH, p. 574\]; let $\pi_S\, :\,S \to B$ be an elliptic fibration. Since $S$ is elliptically connected, the base $B$ is dominated by a rational or elliptic curve $C \s S$. Hence, $B$ itself is rational or elliptic.
Fix a dominant morphism $g :\,E \to C$ from a smooth elliptic curve $E$. Set $f = \pi_S \circ g \,:\,E \to B$, and consider the product $X = S \times_B E$. The elliptic fibration $\pi_X \,:\,X \to E$ obtained from $\pi_S\, :\,S \to B$ by the base change $f \,:\,E \to B$ has a regular section $\sigma \,:\, E \ni e \longmapsto (e, g(e)) \in X = E \times_B S$. Passing to a normalization and a minimal resolution of singularities $X' \to X$ we obtain a smooth surface $X'$ with an elliptic fibration $\pi_{X'}\, :\,X' \to E$ and a section $\sigma' : \,E \to X'$. Thus, $\pi_{X'}$ has no multiple fibre. Replacing $X'$ by a birationally equivalent model we may also assume this fibration to be relatively minimal.
If it were no singular fibre, then $\pi_{X'}$ would be a smooth morphism, and so $\chi (X') = \chi(F) \chi (E) = 0$, where $F$ denotes the generic fibre of $\pi_{X'}$. The formula for the canonical class of a relatively minimal elliptic surface \[GH, p.572\] implies that $K_{X'} = \pi_{X'}^* (L)$, where $L$ is a line bundle over $E$. Hence, $c_2(X') = K_{X'}^2 = 0$, and by the Noether formula, $$\chi ({\cal O}_{X'}) = {c_1(X')^2 + c_2(X') \over 12} = {K_{X'}^2 + \chi (X')
\over 12} = 0\,.$$ Thus, ${\rm deg}\, L = 2g(E) - 2 + \chi (O_{X'}) = 0$. Therefore, the line bundle $K_{X'}$ is trivial, and so $k(X') = 0$, in contradiction with our assumption (indeed, since $X'$ rationally dominates $S$ and $k(S) =1$, we have $k(X') \ge 1$).
Hence, $\pi_{X'}\, :\,X' \to E$ is a minimal elliptic fibration with a singular fibre. By Proposition 2.1 in \[FM, Ch. II\], we have $\pi_1 (X') \cong
\pi_1 (E) \cong \Gaz^2$. Since $S$ is dominated by a surface birationally equivalent to $X'$, whose fundamental group is isomorphic to those of $X'$, by Lemma 1.1.$c)$, the group $\pi_1(S)$ is almost abelian. This completes the proof of Theorem 0.5.
For explicit examples of smooth elliptic surfaces $\pi_S\,:\,S \to \pr^1$ with a section $\si\,:\,
\pr^1 \to S\,\,(\pi_S \circ \si = {\rm id}_{\pr^1})$ of Kodaira dimension $1$, one may consider a (crepant) resolution of a surface in the projective bundle $\pr({\cal O} \bigoplus {\cal O}(-2m) \bigoplus {\cal O}(-3m))$ over $\pr^1$, where $m \ge 3$, defined by a general Weierstrass equation (see e.g. \[Ka, Mi\]). In the same way, replacing $\pr^1$ by $\pr^n$ and taking $m \ge n + 2$, one can construct examples of elliptically connected smooth projective varieties $X$ of Kodaira dimension $k(X) = $ dim$\,X - 1$.
APPENDIX: Winkelmann’s example {#appendix-winkelmanns-example .unnumbered}
==============================
We present here the example mentioned in (0.4) above, of an elliptically connected smooth compact non-Kählerian 3-fold $X$ such that the group $\pi_1(X)$ contains a non–abelian free subgroup, and hence, is not even almost solvable. We are grateful to D. Akhiezer for the detailed exposition reproduced below.
Let $\Ga \s SL_2(\cz)$ be a discrete cocompact subgroup. Due to Selberg’s Lemma, there exists a torsion free subgroup of $\Ga$ of finite index. Replacing $\Ga$ by this subgroup we may assume $\Ga$ being torsion free. By the Borel Density Theorem (see \[Ra, 5.16\]), $\Ga$ is Zariski dense in $SL_2(\cz)$.
Set $X = SL_2(\cz)/\Ga$. Thus, $X$ is a (non–Kählerian) compact homogeneous 3-fold with the fundamental group $\pi_1(X) \cong \Ga$. Suppose that $\Ga$ has a solvable subgroup $\Ga' \s \Ga$ of finite index. Then $\Ga'$ being Zariski dense in $SL_2(\cz)$, we would have that $SL_2(\cz)$ is solvable, too. $SL_2(\cz)$ being simple, $\Ga$ cannot be almost solvable. By Tits’ alternative \[Ti\], $\Ga$ must contain a non–abelian free subgroup.
Let $x \sim y$ mean that the points $x$ and $y$ in $X$ can be connected in $X$ by a chain of rational or elliptic curves. To show that $X$ is elliptically connected, it is enough to check this locally. That is to say, to show the existence of a neighborhood $U_0$ of the point $x_0 := \,$[**e**]{}$\,\cdot \Ga$ in $X= SL_2(\cz)/\Ga$ such that $x \sim x_0$ for any point $x \in U_0$.
Suppose we can find three one–dimensional algebraic tori (i.e. one–parametric subgroups isomorphic to $G_m \cong \cz^* $) $A_0,\,A_1,\,A_2 \s SL_2(\cz)$ such that
ß
\(i) $ A_i/ (A_i \cap \Ga)$ is compact, and therefore, the image $E_i$ of $A_i$ in $X$ is a smooth elliptic curve, $i = 0,\,1,\,2$;
ß
\(ii) the Lie subalgebras ${\goth a}_i \s {\goth{sl}}_2(\cz),\,\,i =
0,\,1,\,2$, span ${\goth{sl}}_2(\cz)$.
ß
Then, by (ii), any point $x$ in a small enough neighborhood $U_0$ of the point $x_0 \in X$ can be presented as $a_0 a_1 a_2 \cdot x_0$ with some $a_i \in A_i,\,\,i =
0,\,1,\,2$. Hence, by (i), $x$ and $x_0$ are joined in $X$ by the chain of elliptic curves $E_0,\,\,E_1' := a_0E_1,\,\,E_2' := a_0a_1E_2$. Indeed, we have $$x_0,\,a_0 \cdot x_0 \in A_0 \cdot x_0 = E_0\,,$$ $$a_0\cdot x_0,\,\,a_0a_1\cdot x_0 \in a_0A_1\cdot x_0 = E_1'\,,$$ $$a_0a_1 \cdot x_0,\,\,x = a_0a_1a_2\cdot x_0 \in a_0a_1A_2 \cdot x_0 = E_2'
\,.$$ This proves that $X$ is elliptically connected.
To find three tori $A_0,\,A_1,\,A_2$ in $SL_2(\cz)$ with properties (i) and (ii) note that the Zariski dense torsion free subgroup $\Ga \s SL_2(\cz)$ must contain at least one semisimple element $\g \neq \,$[**e**]{}. Indeed, there exists a Zariski open subset $\Omega \subset G$ such that all elements of $\Omega$ are semisimple. We may assume that [**e**]{} is not in $\Omega$. Since $\Gamma$ is Zariski dense in $G$, $\Gamma$ can not be contained in $G \setminus \Omega$. Thus, there is a semisimple element $\gamma \ne$ [**e**]{} in $\Gamma$.
Let $A_0 \s SL_2(\cz)$ be a torus which contains $\g$, and let $v \in {\goth
a}_0,\,\,v \neq 0$. In view of the Zariski density of $\Ga$, the orbit of $v$ by the adjoint action of $\Ga$ on ${\goth{sl}}_2(\cz)$ generates ${\goth{sl}}_2(\cz)$. Hence, we can find $\g_1,\,\g_2 \in \Ga$ such that $v,\,\,$Ad$\,(\g_1) \cdot v$ and Ad$\,\g_2 \cdot v$ form a basis of ${\goth{sl}}_2(\cz)$. Then for $A_i$ we can take the torus $\g_iA_0\g_i^{-1}$ through $\g_i\g_0\g_i^{-1},\,\,i=1,\,2$. Clearly, (ii) is fulfilled and (i) follows from the fact that $\Gamma$ has no torsion.
Finally, $X = G/\Gamma$ is non-Kähler. Indeed, by a theorem of Borel and Remmert (see \[Ak, 3.9\]), a complex compact homogeneous Kähler manifold is a product of a simply connected projective variety and a torus. Thus, it has an abelian fundamental group. Here $\Gamma$ is certainly non–abelian.
References {#references .unnumbered}
==========
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[*Added in proofs.*]{} Recently F. Campana has proved the above Conjecture 0.2 in its stronger form, and obtained interesting generalizations.
Keiji Oguiso:
Department of Mathematical Sciences, University of Tokyo, Komaba Megro, Tokyo, Japan
ß
e-mail: oguiso@ms.u-tokyo.ac.jp
Mikhail Zaidenberg:
Université Grenoble I, Institut Fourier des Mathématiques, BP 74, 38402 St. Martin d’Héres–cédex, France
ß
e-mail: zaidenbe@puccini.ujf-grenoble.fr
[^1]: or [*virtually abelian*]{}, or [*abelian-by-finite*]{}.
[^2]: we are thankful to J. Winkelmann for a kind permission to mention it here.
[^3]: i.e. those with abelian fundamental groups
[^4]: unpublished
|
---
abstract: 'Consider the stochastic differential equation $\mathrm dX_t = -A X_t \,\mathrm dt + f(t, X_t) \,\mathrm dt + \mathrm dB_t$ in a (possibly infinite-dimensional) separable Hilbert space, where $B$ is a cylindrical Brownian motion and $f$ is a just measurable, bounded function. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this stochastic differential equation. This extends A. M. Davie’s result from ${\mathbb{R}}^d$ to Hilbert space-valued stochastic differential equations.'
author:
- '[Lukas Wresch]{}\'
title: '[Path-by-path uniqueness of infinite-dimensional stochastic differential equations]{}'
---
Preliminaries
=============
Framework & Main result
-----------------------
Let us consider the following stochastic differential equation (SDE)
$$\begin{cases} {\,\mathrm{d}}\!\!\!\!\!\!& x(t) = -A x(t) \,{\mathrm}dt + f(t, x(t)) \,{\mathrm}dt + {\mathrm}dB_t \\ \!\!\!\!\!\!& x(0)\! = x_0 . \end{cases} \tag{SDE}$$
on a separable Hilbert space $H$ in mild form i.e. a solution $x$ satisfies
$$x(t) = e^{-tA} x_0 + \int\limits_0^t e^{-(t-s)A} f(s, x(s)) {\,\mathrm{d}}s + \int\limits_0^t e^{- (t-s)A} {\,\mathrm{d}}B_s \qquad {\mathbb{P}}\text{-a.s.},\ \forall t\in [0,T] . \tag{1.1.1}$$
Let, as in the previous article [@Wre16], $H$ be a separable Hilbert space over ${\mathbb{R}}$. Let\
$(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[}, {\mathbb{P}}, (B_t)_{t\in[0,\infty[} )$ be a filtered stochastic basis with sigma-algebra ${\mathcal}F$, a right-continuous, normal filtration ${\mathcal}F_t \subseteq {\mathcal}F$, a probability measure ${\mathbb{P}}$ and $(B_t)_{t\in[0,\infty[}$ an ${\mathcal}F_t$-Brownian motion on $(\Omega, {\mathcal}F, {\mathbb{P}})$ taking values in ${\mathbb{R}}^{\mathbb{N}}$. Let $A \colon D(A) \longrightarrow H$ be a positive definite, self-adjoint, linear operator with trivial kernel such that $A^{-1}$ is trace-class. Hence, there exists an orthonormal basis $(e_n)_{n\in{\mathbb{N}}}$ of $H$ such that
$$\hspace{28mm} A e_n = \lambda_n e_n, \qquad\qquad \lambda_n > 0, \ \forall n \in{\mathbb{N}}$$
with
$$\hspace{31.5mm} \lambda_n \leq \lambda_{n+1} , \qquad\qquad \hspace{14.8mm} \forall n \in{\mathbb{N}}.$$
By fixing this basis $(e_n)_{n\in{\mathbb{N}}}$ we identify $H$ with $\ell^2$, so that $H \cong \ell^2 \subseteq {\mathbb{R}}^{\mathbb{N}}$. Let $f \colon [0,1] \times H \longrightarrow H$ be a bounded, Borel measurable map.
\[RE-EXISTENCE\]
Using Girsanov’s Theorem (see e.g. [@LR15 Theorem I.0.2]) we can construct a filtered stochastic basis as above and an $({\mathcal}F_t)_{t\in[0,\infty[}$-adapted stochastic process $(X_t)_{t\in[0,T[}$ with ${\mathbb{P}}$-a.s. continuous sample paths in $H$ which solves (SDE). I.e. we have
$$\begin{cases} {\,\mathrm{d}}\!\!\!\!\!\!& X_t = -A X_t \,{\mathrm}dt + f(t, X_t) \,{\mathrm}dt + {\mathrm}dB_t \\ \!\!\!\!\!\!& X_0\! = x_0 . \end{cases}$$
On an *arbitrary* filtered stochastic basis $(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[}, {\mathbb{P}}, (B_t)_{t\in[0,\infty[} )$, as above, for which a priori it is not clear whether it carries a solution $(X_t)_{t\in[0,T[}$ as in Remark \[RE-EXISTENCE\], we study the equation (SDE)
$$\begin{cases} {\,\mathrm{d}}\!\!\!\!\!\!& x(t) = -A x(t) \,{\mathrm}dt + f(t, x(t)) \,{\mathrm}dt + {\mathrm}dB_t \\ \!\!\!\!\!\!& x(0)\! = x_0 . \end{cases} \tag{SDE}$$
for bounded, measurable $f \colon [0,T] \times H \longrightarrow H$ with $T>0$ and $x_0\in H$. We consider the so-called path-by-path approach where equation (SDE) is not considered as a stochastic differential equation, but as a random integral equation in the mild sense. More precisely, in the path-by-path picture we first plug in an $\omega\in\Omega$ into the corresponding integral equation (IE) of the mild form of equation (SDE)
[ @write auxout ]{} $$x_t = e^{-tA} x_0 + \int\limits_0^t e^{-(t-s)A} f(s, x_s) {\,\mathrm{d}}s + \left(\int\limits_0^t e^{- (t-s)A} {\,\mathrm{d}}B_s\right) (\omega) \tag*{$\text{(IE)}_\omega$}$$
and aim to find a (unique) continuous function $x \colon [0,T] \longrightarrow H$ satisfying this equation, which can now be considered as an ordinary integral equation (IE), that is perturbed by an Ornstein–Uhlenbeck path $Z^A(\omega)$. If such a (unique) function can be found for almost all $\omega\in\Omega$, the map $\omega \longmapsto x$ is called a (unique) path-by-path solution to the equation (SDE). Naturally, this notion of uniqueness is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.\
The main result of this article states that on every filtered stochastic basis as above there exists a *unique* mild solution to the equation (SDE) in the path-by-path sense. Although, in the finite dimensional setting many papers have been written about path-by-path uniqueness (see for example [@Dav07], [@Sha14], [@BFGM14], [@Pri15]) to the best of our knowledge this is the first result in a general infinite-dimensional Hilbert space setting. However, for the special case where $H = L^2([0,1], {\mathbb{R}})$ and $A=\Delta$ path-by-path uniqueness has been shown recently in [@BM16] for space-time white noise.\
Let us now state the assumptions on the drift $f$ and the main result.
\[ASS\]
From now on let $f \colon [0,1] \times H \longrightarrow H$ be a Borel measurable map with components $f = f^{(n)}$ w.r.t. our fixed basis $(e_n)_{n\in{\mathbb{N}}}$ satisfying the following conditions
$$\begin{aligned}
\|f\|_H &= \sup\limits_{t\in[0,1], x\in H} |f(t,x)|_H \leq 1 , \\
\|f\|_{\infty,A} :&= \sup\limits_{t\in[0,1], x\in H} \sum\limits_{n\in{\mathbb{N}}} \lambda_n e^{2\lambda_n} |f^{(n)}(t, x)|^2 \leq 1
\intertext{and}
\|f^{(n)}\|_\infty &= \sup\limits_{t\in[0,1], x\in H} |f^{(n)}(t,x)| \leq \exp\left( - e^{n^\gamma} \right)\end{aligned}$$
for some $\gamma > 6$.
[ \[THM-MAIN\] ]{}
Let $A$ and $f$ be as above and assume that $f$ fulfills Assumption \[ASS\]. Given *any* filtered stochastic basis $(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[}, {\mathbb{P}}, (B_t)_{t\in[0,\infty[})$ there exists $\Omega_0\in {\mathcal}F$ with ${\mathbb{P}}[\Omega_0]=1$ such that for every $\omega\in\Omega_0$ we have
$$\# \{ g \in {\mathcal}C( [0,T], H) | g \text{ solves } \text{(IE)}_\omega \} = 1 ,$$
i.e. (SDE) has a path-by-path unique mild solution.
Theorem \[THM-MAIN\] follows from the following
[ \[PRO-MAIN\] ]{}
Let $A$ and $f$ be as in Theorem \[THM-MAIN\]. Let $(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[}, {\mathbb{P}}, (B_t)_{t\in[0,\infty[})$ be a filtered stochastic basis and $(X_t)_{t\in[0,\infty[}$ a solution of (SDE) (as in Remark \[RE-EXISTENCE\]). Then path-by-path uniqueness holds, i.e. there exists $\Omega_0\in{\mathcal}F$ with ${\mathbb{P}}[\Omega_0]=1$ such that
$$\# \{ g \in {\mathcal}C( [0,T], H) | g \text{ solves } \text{(IE)}_\omega \} = 1$$
holds for every $\omega\in\Omega_0$.
Take an arbitrary filtered probability space and let $( (X^{1}_t)_{t\in[0,\infty[}, (B_t)_{t\in[0,\infty[} )$ and\
$( (X^{2}_t)_{t\in[0,\infty[}, (B_t)_{t\in[0,\infty[} )$ be two weak solutions driven by the same cylindrical $({\mathcal}F_t)_{t\in[0,\infty[}$-Brownian motion. Then by Proposition \[PRO-MAIN\] it follows that path-by-path uniqueness, and hence pathwise uniqueness, holds i.e. $X^1=X^2$ ${\mathbb{P}}$-a.s. Hence the Yamada–Watanabe Theorem (see [@RSZ08 Theorem 2.1]) implies that there exists even a *strong* solution for equation (SDE). In conclusion, by invoking Proposition \[PRO-MAIN\] again, this proves the existence *and* path-by-path uniqueness of solutions on *every* filtered stochastic basis $(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[}, {\mathbb{P}}, (B_t)_{t\in[0,\infty[})$.
Set $\Omega := L^2( [0,T], H)$ and ${\mathbb{P}}$ such that the projection $\pi_t(\omega) := \omega(t)$ is a cylindrical Brownian motion. As in the introduction consider the map
$$Z^A\colon L^2( [0,T], H) \longrightarrow {\mathcal}C([0,T], H), \qquad \omega \longmapsto \left( t \mapsto \int\limits_0^t e^{-(t-s)A} {\,\mathrm{d}}\omega(s) \right) .$$
Note that due to [@DZ92 Theorem 5.2] $\left({\mathbb{P}}\circ Z^A\right)^{-1}$ equals $N(0,K)$, the Gaussian measure on $L^2( [0,T], H)$ with covariance operator $K$ defined by
$$(K \varphi)(t) = \int\limits_0^T k(t,s) \varphi(s) {\,\mathrm{d}}s ,$$ where $$k(t,s) = \int\limits_0^{t \wedge s} e^{-(t-r)A} \left(e^{-(s-r)A}\right)^\star {\,\mathrm{d}}r$$
and $N(0,K)[ Z^A(\Omega) ] = 1$. Note that, since $Z^A$ is injective, Kuratowski’s Theorem (see [@Kal97 Theorem A1.7]) implies that $Z^A(\Omega)$ is a Borel set.
Let $f$ be as in Assumption \[ASS\] then path-by-path uniqueness holds for the SDE
$$\mathrm d x_t = -Ax_t \mathrm dt + f(t, x_t) \mathrm dt + \mathrm \omega(t) .$$
I.e. there exists $\Omega_0\subseteq {\mathcal}C([0,T], H)$ with ${\mathbb{P}}[\Omega_0]=1$ such that for every $\omega\in\Omega_0$ there exists a unique function $g \in {\mathcal}C( [0,T], H)$ solving the above equation.
If the function $f$ is independent of time and $A^{-1+\delta}$ is trace class for some $\delta \in ]0,1[$ is has already been proven that a strong mild solution to (SDE) exists for $\mu$-a.a. initial condition $x_0\in H$, where $\mu$ is the invariant measure of the Ornstein–Uhlenbeck process $Z^A$ (see [@DFPR13]). If in addition $f$ satisfies Assumption \[ASS\], our results improves this result to all initial conditions $x_0\in H$.
Structure of the article & Roadmap for the proof
------------------------------------------------
The structure of this article is the following: In the following section 2 we introduce approximation lattices and the notion of the effective dimension of an (infinite-dimensional) set. This is reminiscent to the Kolmogorov ${\varepsilon}$-entropy, which was used in the proof of A. V. Shaposhniko (see [@Sha14]) for the finite-dimensional case. In the third section we prove two regularization by noise estimates of the map
$$\varphi_{n,k} \colon (x,y) \longmapsto \int\limits_{k2^{-n}}^{(k+1)2^{-n}} \left[ f(s, Z^A_s + x) - f(s, Z^A_s + y) \right] \,\mathrm ds ,$$
which are based on the estimates previously obtained by the author in [@Wre16]. We show that for every $\delta > 0$
$$|\varphi_{n,k}(x,y)|_H \leq C_\delta \left( \sqrt n 2^{-n/6} |x-y|_H + {\varepsilon}_n \right)$$
with ${\varepsilon}_n \overset{n\rightarrow\infty}\longrightarrow 0$ for all $\omega\in\Omega$ outside a set of mass $\delta$. Here, $x$ and $y$ are in an approximation lattice of a suitable subset $Q$ of $H$ which includes the image of $f$. For fixed $\omega\in\Omega$ the map $\varphi_{n,k}$ is therefore “close to” being Lipschitz continuous. This estimate acts as a replacement for the lack of regularity for the non-linearity $f$ in equation (SDE).\
In the fourth section we extend these estimates: For sequences of functions $(h_m)_{m\in{\mathbb{N}}}$ converging to $h$ we prove, despite the lack of continuity in $f$, that
$$\int\limits_0^1 f(s, Z^A_s + h_m(s)) {\,\mathrm{d}}s \overset{m\rightarrow\infty}\longrightarrow \int\limits_0^1 f(s, Z^A_s + h(s)) {\,\mathrm{d}}s . \qquad {\mathbb{P}}\text{-a.s.} \tag{1.2.1}$$
This approximation theorem (Theorem \[THM-APPROX\]) implies that the above map $\varphi_{n,k}$ is continuous and therefore enables us to extend the estimates of the previous section from an approximation lattice to all $x,y$ of $Q$ (Corollary \[COR-SIGMA-RHO\]). The result obtained in this section is also necessary to justify the limiting argument in the proof of Theorem \[THM-FINAL\].\
It turns out that in the proof of the main result (Theorem \[THM-MAIN\]) we have to consider terms of type
$$\sum\limits_{q=1}^N |\varphi_{n,k+q}(x_{q+1}, x_q)|_H \tag{1.2.2}$$
for a sequence of points $\{x_q \in Q | q=1, ..., N\}$. Using just the estimates of Section 3 for each term under the sum of (1.2.2) is, unfortunately, insufficient to prove the main result (Theorem \[THM-MAIN\]) as this would merely give us an estimate of order ${\mathcal}O(2^{-(1/2-{\varepsilon})n} N)$.\
To overcome this, in Section 5 we use the fact that the above points $x_q\in Q$ are values of a solution of an integral equation and hence can be well approximated by a one-step Euler approximation. This enables us to prove much stronger estimates for expression (1.2.2). Namely, bounds of order ${\mathcal}O(2^{-n} N)$ (Theorem \[THM-EULER\]).\
Section 6 contains the proof of the main result (Theorem \[THM-MAIN\]). As a first step of the proof of the main theorem we reduce the problem via Girsanov’s Theorem to the following Proposition.
[ \[PRO-GIRSANOV\] ]{}
For every $f \colon [0,T] \times H \longrightarrow H$ be a Borel measurable function fulfilling Assumption \[ASS\]. Assume that for every process $(\tilde Z^A_t)_{t\in[0,\infty[}$ on $(\Omega, {\mathcal}F, ({\mathcal}F_t)_{t\in[0,\infty[})$ with $\tilde Z^A_0=0$, which is an Ornstein–Uhlenbeck process with drift term $A$ w.r.t. some measure $\tilde{\mathbb{P}}$ on $(\Omega, {\mathcal}F)$, there exists a set $\Omega'_{\tilde Z^A} \subseteq \Omega$ with $\tilde{\mathbb{P}}[\Omega'_{\tilde Z^A}] = 1$ such that for all fixed $\omega\in \Omega'_{\tilde Z^A}$ the only function $u \in {\mathcal}C( [0,T], H)$ solving
[ @write auxout ]{} $$u(t) = \int\limits_0^t e^{-(t-s)A} \left( f(s, \tilde Z^A_s(\omega) + u(s)) - f(s, \tilde Z^A_s(\omega)) \right) {\,\mathrm{d}}s \tag{\ref{DE-U}}$$
for all $t\in[0,T]$ is the trivial solution $u \equiv 0$, then the assertion of Proposition \[PRO-MAIN\] holds with $\Omega_0 := \Omega'_{\tilde Z^A}$, where $\tilde Z^A_t := X_t - e^{-tA} x_0$ with $X$ being a solution of (SDE). Recall that $X$ is an Ornstein–Uhlenbeck process under a measure $\tilde{\mathbb{P}}$ obtained via Girsanov transformation.
The set of “good omegas” $\Omega_0$ of the main result \[THM-MAIN\] therefore depends solely on the strong solution $X$, the initial condition $x_0$ and the drift $f$.
A proof of this proposition will be given in this section below. Now, let $u$ be a function solving equation and let us write $\varphi_{n,k}(x) := \varphi_{n,k}(x,0)$. To show that every solution to is trivial we use a discrete logarithmic Gronwall inequality of the form
$$|u((k+1)2^{-n})|_H \leq |u(k2^{-n})|_H \left(1+C 2^{-n} \log( 1 / |u(k2^{-n})|_H ) \right) .$$
In Section 6 we first show that
$$| u((k+1)2^{-n}) - u(k2^{-n}) |_H \approx |\varphi_{n,k}(u(\cdot))|_H .$$
Subsequently, we construct functions $u_\ell \overset{\ell\rightarrow\infty}\longrightarrow u$, which are constant on the dyadic intervals $[k2^{-\ell}, (k+1)2^{-\ell}[$. Using the equation (1.2.1) mentioned above this can be rewritten as
$$\lim\limits_{\ell\rightarrow\infty} |\varphi_{n,k}(u_\ell(\cdot))|_H \leq |\varphi_{n,k}(u_n(\cdot))|_H + \sum\limits_{\ell=n}^\infty |\varphi_{n,k}( u_{\ell+1}(\cdot)) , u_\ell(\cdot) )|_H .$$
Splitting the integrals and using that $u_\ell$ is constant on dyadic intervals of size $2^{-\ell}$ we can bring this is in the somewhat more complicated form
$$|\varphi_{n,k}(u(k2^{-n}))|_H + \sum\limits_{\ell=n}^\infty \sum\limits_{r=k2^{\ell+1-n}}^{(k+1)2^{\ell+1-n}} |\varphi_{\ell,r}( u((r+1)2^{-\ell-1}) , u(r2^{-\ell-1}) )|_H .$$
Using the estimates for $\varphi_{n,k}$ and expression (1.2.2) developed in the previous section we ultimately obtain an estimate of order
$$| u((k+1)2^{-n}) - u(k2^{-n}) |_H \leq C 2^{-n} |u(k2^{-n})|_H \log( 1 / |u(k2^{-n})|_H ) ,$$
where we have to impose the somewhat technical condition that $0< |u(k2^{-n})|_H<1$. We therefore obtain a discrete $\log$-Type Gronwall inequality of the form
$$|u((k+1)2^{-n})|_H \leq |u(k2^{-n})|_H \left(1+C 2^{-n} \log( 1 / |u(k2^{-n})|_H ) \right) ,$$
which, similar to the standard Grownall Inequality, implies that $u$ has to be trivial (Corollary \[COR-FINAL\]), so that the condition of Proposition \[PRO-GIRSANOV\] is fulfilled completing the proof.
Let $(X_t)_{t\in[0,T]}$ be a solution to (SDE). We set $\tilde Z^A_t := X_t - e^{-tA} x_0$ so that $\tilde Z^A$ is an Ornstein–Uhlenbeck process with drift term $A$ starting in $0$ under a measure $\tilde{\mathbb{P}}\approx{\mathbb{P}}$ obtained by Girsanov’s Theorem as mentioned in Remark \[RE-EXISTENCE\].
Then, by assumption there is a set $\Omega'_{\tilde Z^A}$ with ${\mathbb{P}}[\Omega'_{\tilde Z^A}] = \tilde{\mathbb{P}}[\Omega'_{\tilde Z^A}] = 1$ such that for all $\omega \in \Omega'_{\tilde Z^A}$ every solution $u$ to equation is trivial.
Let $\omega\in\Omega'_{\tilde Z^A}$ and $x \in {\mathcal}C( [0,T], H )$ be a solution to $\text{(IE)}_\omega$. We then have
$$x_t = e^{-tA} x_0 + \int\limits_0^t e^{-(t-s)A} f(s, x_s) {\,\mathrm{d}}s + \left(\int\limits_0^t e^{- (t-s)A} {\,\mathrm{d}}B_s\right) (\omega) .$$
Setting $u_t := x_t - X_t(\omega)$ yields that
$$\begin{aligned}
u_t &= \int\limits_0^t e^{-(t-s)A} f(s, x_s) {\,\mathrm{d}}s - \int\limits_0^t e^{-(t-s)A} f(s, X_s(\omega)) {\,\mathrm{d}}s \\
&= \int\limits_0^t e^{-(t-s)A} ( f(s, u_s + X_s(\omega)) - f(s, X_s(\omega)) ) {\,\mathrm{d}}s .\end{aligned}$$
By plugging in the definition of $\tilde Z^A$ and by setting
$$\tilde f_{x_0}(t, z) := f(t, z + e^{-tA} x_0)$$
we rewrite the above equation to
$$u_t = \int\limits_0^t e^{-(t-s)A} ( \tilde f_{x_0}(s, u_s + \tilde Z^A_s(\omega)) - \tilde f_{x_0}(s, \tilde Z^A_s(\omega) ) ) {\,\mathrm{d}}s$$
Since $\tilde Z^A$ is an Ornstein–Uhlenbeck process under $\tilde {\mathbb{P}}$ starting at zero and $\omega \in \Omega'_{\tilde Z^A}$ we conclude that $u \equiv 0$ and henceforth $x_t = X_t(\omega)$. Analogously, we obtain for any other solution $x'$ that $x'_t = X_t(\omega) = x_t$ so that all solutions of $\text{(IE)}_\omega$ coincide on $\Omega'_{\tilde Z^A}$ and are therefore unique.
Approximation Lattices
======================
In this section we define the set $Q$, where the function $u$ (see equation of Proposition \[PRO-GIRSANOV\]) takes values in. Additionally, we define the so-called *effective dimension* of a set, which is a variant of the Kolmogorov ${\varepsilon}$-entropy for lattices. At the end of this section we estimate the effective dimension of our set $Q$.
[ \[DE-Q\] ]{}
We define
$$Q := \{ x \in {\mathbb{R}}^{\mathbb{N}}\colon |x|_\infty \leq 2, \ |x_n| \leq 2 \exp\left( - e^{n^\gamma} \right) , \ x = (x_n)_{n \in {\mathbb{N}}} \} ,$$
where $\gamma$ is the constant from Assumption \[ASS\]. Additionally, for $r \in{\mathbb{N}}$ we set
$$Q_r := \{ x \in Q \colon |x|_\infty \leq 2\cdot2^{-r} \} ,$$
so that $Q_0=Q$. Note that for $m\in{\mathbb{N}}$ the lattice $Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ is the set of all points $x \in Q$, where the components $x_n$ of $x$ can be written as
$$x_n = k_n 2^{-m}$$
with certain $k_n \in {\mathbb{Z}}$ for every $n\in{\mathbb{N}}$.
[ \[DE-EFFDIM\] ]{}
Let $B \subseteq {\mathbb{R}}^{\mathbb{N}}$ with $0\in B$. For points $x\in B$ we write $(x_n)_{n\in{\mathbb{N}}} = x$ for the components of $x$. For every $m\in{\mathbb{N}}$ we set
$$d_m(B) := \sup\limits_{x \in B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}} \inf\limits \left\{ \left. n \right| x_{n'}=0 \ \forall n' \geq n \right\} \in \bar{\mathbb{N}}:= {\mathbb{N}}\cup \{ \infty \} .$$
I.e. given any point $(x_n)_{n\in{\mathbb{N}}}$ in the set $B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$, all components $x_n$ are zero for $n \geq d_m$ and $d_m$ is the smallest integer with this property.
We define the *effective dimension* of a set $B \subseteq {\mathbb{R}}^{\mathbb{N}}$ by $${\operatorname{ed}}\colon \{ B \subseteq {\mathbb{R}}^{\mathbb{N}}| 0 \in B \} \longrightarrow {\bar {\mathbb{N}}}^{\mathbb{N}}$$ $$B \longmapsto {\operatorname{ed}}(B) := (d_m(B))_{m\in{\mathbb{N}}} .$$
$B$ is called *effectively finite-dimensional* if $${\operatorname{ed}}(B)_m < \infty , \qquad \forall m\in{\mathbb{N}}.$$
Let $|\cdot|_1$ and $|\cdot|_2$ be two norm on $B$. $|\cdot|_1$ and $|\cdot|_2$ are called *effectively equivalent* if for every $m\in{\mathbb{N}}$ they are equivalent on the restricted domain $B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$. I.e. for every $m\in{\mathbb{N}}$ there exists constants $c_m,C_m\in {\mathbb{R}}$ such that
$$c_m|x|_1 \leq |x|_2 \leq C_m |x|_1 , \qquad \forall x \in B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}.$$
[ \[EQUIVALENCE\] ]{}
Let $B \subseteq {\mathbb{R}}^{\mathbb{N}}$ with $0\in B$ be an effectively finite-dimensional set then the norm $|\cdot|_2$ and the maximum norm $|\cdot|_\infty$ are effectively equivalent. More precisely, we have
$$|x|_2 \leq \sqrt{{\operatorname{ed}}(B)_m} |x|_\infty , \qquad m \in {\mathbb{N}}, \ x \in B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$$ and $$|x|_\infty \leq |x|_2 , \qquad\qquad\qquad\quad\!\! m \in {\mathbb{N}}, \ x \in B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}.$$
Let $m\in{\mathbb{N}}$. For every $x\in B \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ we have
$$|x|_2^2 = \sum\limits_{n=1}^\infty |x_n|^2 = \sum\limits_{n=1}^{{\operatorname{ed}}(B)_m} |x_n|^2 \leq {\operatorname{ed}}(H)_m |x|_\infty^2$$
and
$$|x|_\infty^2 \leq \sum\limits_{n=1}^\infty |x_n|^2 = |x|_2 .$$
[ \[EFFDIM\] ]{}
For $r,m\in{\mathbb{N}}$ with $m\geq r$ we have
$${\operatorname{ed}}(Q_r)_m \leq (\ln( m + 1 ))^{1/\gamma} .$$
Note that this implies that $Q_r$ is effectively finite-dimensional for every $r\in{\mathbb{N}}$.
Let $x \in Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$. Observe that every component $x_n$ is of the form $x_n = k_n 2^{-m}$ with $$k_n \in \{ - 2 \cdot 2^{m-r} , ..., 2 \cdot 2^{m-r} \} .$$
Set
$$d_m := (\ln( m + 1 ))^{1/\gamma} .$$
We are going to show that $k_n = 0$ holds for every $n \geq d_m$.
$$|k_n| 2^{-m} = |x_n| \leq 2 \exp\left(- e^{n^\gamma} \right) \Rightarrow |k_n| \leq 2^{m+1} \exp\left(- e^{n^\gamma} \right) ,$$
which implies that
$$|k_n| \leq 2^{m+1} \exp\left(- e^{n^\gamma} \right) \leq e^{\ln(2) (m+1)} \exp\left(- \exp\left( (d_m)^\gamma \right) \right) = e^{\ln(2) (m+1) - \exp\left( (d_m)^\gamma \right)}$$
$$\hspace{33mm} = e^{\ln(2) (m+1) - (m+1)} = e^{(\ln(2)-1) (m+1) } \leq e^{\ln(2)-1} < 1 .$$
In conclusion, $|k_n| = 0$ for all $n \geq d_m$ and hence we have
$${\operatorname{ed}}(Q_r)_m \leq d_m = (\ln( m + 1 ))^{1/\gamma} .$$
[ \[KOLTIK\] ]{}
Let $r\in{\mathbb{N}}$ and $m\in{\mathbb{N}}$. The number of points in the $m$-lattice of $Q_r$ can be estimated as follows
$$\# (Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}) \leq (4 \cdot 2^{m-r} + 1)^{{\operatorname{ed}}(Q_r)_m}$$
and
$$\# (2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}) \leq (8 \cdot 2^{m-r} + 1)^{{\operatorname{ed}}(2Q_r)_m} .$$
Let $m\in{\mathbb{N}}$ and $x \in Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ and note that, as in the last proof, every component $x_n$ is of the form $x_n = k_n 2^{-m}$ with $$k_n \in \{ - 2 \cdot 2^{m-r} , ..., 2 \cdot 2^{m-r} \} .$$
$k_n$ can take at most $4 \cdot 2^{m-r} + 1$ different values in the dimensions $1 \leq n < {\operatorname{ed}}(Q_r)_m$, so that the total number of points $x \in Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ can be estimated by
$$(4 \cdot 2^{m-r} + 1)^{{\operatorname{ed}}(Q_r)_m} .$$
Note that $k_n=0$ for $n \geq {\operatorname{ed}}(Q_r)_m$. The second part of the assertion follows analogously.
[ \[RE-PI\] ]{}
Let $r\in{\mathbb{N}}$. For every $m\in{\mathbb{N}}$ there exists a map
$$\pi_m^{(r)} \colon Q_r \longrightarrow Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$$
with the property that
$$| x - \pi_m^{(r)}(x) |_\infty \leq 2^{-m}$$
and
$$\not{\exists} y \in Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}\colon | x - y |_\infty < | x - \pi_m^{(r)}(x)|_\infty$$
holds for all $x\in Q_r$, $m\in{\mathbb{N}}$ and $r\in{\mathbb{N}}$.
Let $r, m\in{\mathbb{N}}$. By Theorem \[KOLTIK\] and Lemma \[EFFDIM\] $Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ is a finite set, hence we can write
$$Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}= \{ y_1, ..., y_N \} ,$$
where $N\in{\mathbb{N}}$ is some number depending on both $r$ and $m$. For every $x\in Q_r$ we set
$${\mathcal}I(x) := \left\{ i \in \{1 , ... , N \} : |x-y_i|_\infty = \min\limits_{1 \leq j \leq N} |x-y_j|_\infty \right\} .$$
Furthermore, we define
$$\pi_m^{(r)}(x) := y_{\min {\mathcal}I(x)} .$$
Observe that the map $\pi_m^{(r)}$ fulfills all the required properties.
[ \[DE-DYADIC\] ]{}
We set
$${\mathbb{D}}:= \left\{ \left. (x_n)_{n\in{\mathbb{N}}} \in {\mathbb{R}}^{\mathbb{N}}\right| \forall n\in{\mathbb{N}}, \ \exists m_n \in {\mathbb{N}}, \ x_n \in 2^{-m_n} {\mathbb{Z}}^{\mathbb{N}}\right\} .$$
We say that $x\in{\mathbb{R}}^{\mathbb{N}}$ is a dyadic point if $x \in {\mathbb{D}}$.
Regularization by Noise
=======================
In this section we are going to prove various estimates regarding the map $\varphi_{n,k}$ defined below. Surprisingly, although we do not assume any regularity on $b$, $\varphi_{n,k}$ is “close to” being Lipschitz continuous in space. This is due to the noise, which improves the situation significantly. From this point onwards let $(Z^A_t)_{t\in[0,\infty[}$ be an Ornstein–Uhlenbeck process on a probability space $(\Omega, {\mathcal}F, ({\mathcal}G_t)_{t\in[0,\infty[} , {\mathbb{P}})$ with drift term $A$ and initial sigma-algebra $({\mathcal}G_t)_{t\in[0,\infty[}$ as defined in the introduction.
[ \[DE-VARPHI\] ]{}
Let $b \colon [0,1] \times H \longrightarrow H$ be a Borel measurable function. For $n\in{\mathbb{N}}$, $k\in\{0, ..., 2^n-1\}$ and $x\in H$ we define $$\varphi_{n,k} \colon H \times \Omega \longrightarrow H$$ by $$\varphi_{n,k}(b ; x, \omega) := \int\limits_{k2^{-n}}^{(k+1)2^{-n}} b(s, Z^A_s(\omega) + x) - b(s, Z^A_s(\omega)) {\,\mathrm{d}}s .$$
Usually we drop the $b$ and $\omega$ and just write $\varphi_{n,k}(x)$ instead of $\varphi_{n,k}(b ; x, \omega)$. Additionally, we set
$$\varphi_{n,k}(x,y) := \int\limits_{k2^{-n}}^{(k+1)2^{-n}} b(s, Z^A_s + x) - b(s, Z^A_s + y) {\,\mathrm{d}}s .$$
[ \[RE-METRIC\] ]{}
Note that for fixed $n\in{\mathbb{N}}$, $k\in\{0, ..., 2^n-1\}$ and $\omega\in\Omega$ the map
$$|\varphi_{n,k}(\cdot,\cdot)|_H \colon H \times H \longrightarrow {\mathbb{R}}_+, \qquad (x,y) \longmapsto |\varphi_{n,k}(x,y)|_H$$
is a pseudometric on $H$.
[ \[LOG-CONVEX\] ]{}
For $r,m \in {\mathbb{N}}$ we have
$$\ln(r+m+1)^{1/\gamma} \leq \ln(r+1)^{1/\gamma} + \ln(m+1)^{1/\gamma} ,$$
where $\gamma$ is the constant from Assumption \[ASS\].
Let $r,m \in {\mathbb{N}}$. We have
$$r + m +1 \leq rm + r + m + 1 = (r+1)\cdot (m+1) ,$$
which implies that
$$\ln(r + m+1) \leq \ln((r+1)\cdot(m+1)) = \ln(r+1) + \ln(m+1) .$$
Since $\frac1\gamma \leq 1$ we immediately obtain
$$\ln(r + m+1)^{1/\gamma} \leq \ln(r+1)^{1/\gamma} + \ln(m+1)^{1/\gamma}$$
due to the fact that $x \longmapsto x^{1/\gamma}$ is concave which completes the proof.
[ \[THM-SIGMA\] ]{}
For every $0 < {\varepsilon}< \frac16$ there exists $C_{\varepsilon}\in {\mathbb{R}}$ such that for every Borel measurable function $b\colon [0,1] \times H \longrightarrow H$ satisfying Assumption \[ASS\], $n\in{\mathbb{N}}\setminus\{0\}$ and $k \in \{ 0 , ..., 2^n - 1 \}$ there exists a measurable set $A_{{\varepsilon},b,n,k} \in {\mathcal}G_{(k+1)2^{-n}} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},b,n,k}] \leq \frac{\varepsilon}3 e^{-n}$ such that on $A_{{\varepsilon},b,n,k}^c$
$$|\varphi_{n,k}(x)|_H \leq C_{\varepsilon}n^{\frac12 + \frac1\gamma} 2^{-n/2} \left( |x|_\infty + 2^{-2^n} \right)$$
holds for all points $x \in 2Q \cap {\mathbb{D}}$.
Note that the constant $C_{\varepsilon}$ depends on ${\varepsilon}$ and $\gamma$ from Assumption \[ASS\], but not on $b$. Conversely, the set of “good omegas” $A_{{\varepsilon},b,n,k}^c$ depends on ${\varepsilon}$, $b$, $n$ and $k$.
**Step 1:**\
Let $0 < {\varepsilon}< \frac1{6}$. For $r\geq 0$ recall that $Q_r := \{ x\in Q : |x|_\infty \leq 2 \cdot 2^{-r} \}$. Let $m$ be an integer with $m \geq r$ and $x,y \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$. We are going to estimate the probability of the event $\{|\varphi_{n,k}(x,y) |_H > \eta\}$ for a suitable $\eta \geq 0$. To this end let $\beta_A > 0$ be the constant from [@Wre16 Corollary 3.1] and for $0 < {\varepsilon}< \frac1{6}$ we set
[ @write auxout ]{} $$\eta_{\varepsilon}:= \sqrt{ \ln\left( \frac{6}{\varepsilon}\right) } \geq 1 . \tag{\ref{SIGMA-ETA-EPS}}$$
Let us consider the following probability.
$${\mathbb{P}}\left[ |\varphi_{n,k}(x,y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{ 9 (1+m-r) } ) {\operatorname{ed}}(2Q_r)_m |x-y|_\infty 2^{-n/2} \right] .$$
Since $x,y \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ and $|\cdot|_\infty$, $|\cdot|_2$ are effectively equivalent norms i.e. $|\cdot|_2 \leq \sqrt{{\operatorname{ed}}(2Q_r)_m} |\cdot|_\infty$ (see Proposition \[EQUIVALENCE\]) the above expression is smaller than
$${\mathbb{P}}\left[ |\varphi_{n,k}(x,y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{ 9 (1+m-r) } ) \sqrt{{\operatorname{ed}}(2Q_r)_m} |x-y|_2 2^{-n/2} \right] .$$
Due to Corollary [@Wre16 Corollary 3.1] this probability is smaller than
$$e^{- \eta_{\varepsilon}^2 {\operatorname{ed}}(2Q_r)_m} e^{- \eta_{\varepsilon}\left( \sqrt{2n} + \sqrt{9 (1+m-r)} \right)^2 {\operatorname{ed}}(2Q_r)_m} .$$
Using that $\eta_{\varepsilon}\geq 1$ and ${\operatorname{ed}}(2Q_r)_m\geq 1$ the above is bounded from above by
$$e^{- \eta_{\varepsilon}^2} e^{- \left( \sqrt{2n} + \sqrt{9 (1+m-r)} \right)^2 {\operatorname{ed}}(2Q_r)_m} \leq e^{- \eta_{\varepsilon}^2} e^{- (2n + 9 (1+m-r)) {\operatorname{ed}}(2Q_r)_m } = e^{- \eta_{\varepsilon}^2} e^{-2n} e^{- 9 (1+m-r) {\operatorname{ed}}(2Q_r)_m } .$$
In order to get a uniform bound we calculate
$${\mathbb{P}}\left[ \bigcup\limits_{r=0}^{2^n} \bigcup\limits_{m=r}^\infty \!\!\! \bigcup\limits_{{\genfrac{}{}{0pt}{}{x,y \in}{2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}}}} \!\!\!\!\!\! \left\{ |\varphi_{n,k}(x,y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{9 (1+m-r) } ) {\operatorname{ed}}(2Q_r)_m |x-y|_\infty 2^{-n/2} \right\} \right]$$
$$\hspace{-27mm} \leq \sum\limits_{r=0}^{2^n} \sum\limits_{m=r}^\infty \sum\limits_{{\genfrac{}{}{0pt}{}{x,y \in}{2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}}}} e^{- \eta_{\varepsilon}^2} e^{- 2n} e^{- 9 (1+m-r) {\operatorname{ed}}(2Q_r)_m}$$
$$\hspace{10mm} = e^{- \eta_{\varepsilon}^2} \sum\limits_{r=0}^{2^n} \sum\limits_{m=r}^\infty \# \{ (x,y) | x,y \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}\} e^{- 2n} e^{ - 9 (1+m-r) {\operatorname{ed}}(2Q_r)_m } .$$
Invoking Theorem \[KOLTIK\] results in $$\# \{ x | x \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}\} \leq \exp \left( 8 ( 1 + m - r ) {\operatorname{ed}}(2Q_r)_m \right) .$$
Hence, we can bound the above probability by
$$e^{- \eta_{\varepsilon}^2} \sum\limits_{r=0}^{2^n} \sum\limits_{m=r}^\infty \exp \left( 8 ( 1 + m - r ) {\operatorname{ed}}(2Q_r)_m \right) e^{- 2n} e^{- 9(1+m-r) {\operatorname{ed}}(Q_r)_m }$$
$$\hspace{-10mm} = e^{- \eta_{\varepsilon}^2} e^{- 2n } \sum\limits_{r=0}^{2^n} \sum\limits_{m=r}^\infty \exp \left( - ( 1 + m - r ) {\operatorname{ed}}(2Q_r)_m \right) .$$
Note that the last sum converges since ${\operatorname{ed}}(2Q_r)_m \geq 1$ and because of
$$\sum\limits_{m=r}^\infty \exp \left( - ( 1 + m - r ) {\operatorname{ed}}(2Q_r)_m \right) \leq \sum\limits_{m=0}^\infty \exp \left( - ( 1 + m ) \right) \leq 1$$
the above is smaller than
$$e^{- \eta_{\varepsilon}^2} \sum\limits_{r=0}^{2^n} e^{-2n} = e^{- \eta_{\varepsilon}^2} (2^n+1) e^{-2n} \leq 2 e^{- \eta_{\varepsilon}^2} e^{-n} .$$
Plugging in Definition of $\eta_{\varepsilon}$ the above is smaller than $\frac{\varepsilon}3 e^{-n}$. In conclusion there exists a measurable set $A_{{\varepsilon},b,n,k} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},b,n,k}] \leq \frac{\varepsilon}3 e^{-n}$ such that on $A_{{\varepsilon},b,n,k}^c$ we have
[ @write auxout ]{} $$\begin{aligned}
\tag{\ref{SIGMA-STEP-1}}
\begin{split}
|\varphi_{n,k}(x,y)|_H &\leq \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{ 9(1+m-r) } ) {\operatorname{ed}}(2Q_r)_m |x-y|_\infty 2^{-n/2} \\
&\leq 6 \beta_A^{-1/2} \eta_{\varepsilon}( \sqrt{n} + \sqrt{ 1+m-r } ) {\operatorname{ed}}(2Q_r)_m |x-y|_\infty 2^{-n/2}
\end{split}\end{aligned}$$
for $n \geq 1$, $k \in \{ 0, ..., 2^n-1 \}$, $r \in \{ 0, ..., 2^n$}, $m\geq r$ and $x,y \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$.\
**Step 2:**\
**Claim:** For every dyadic number $x \in 2Q_r$ with $r \in \{ 0 , ..., 2^n \}$ and $n \geq 1$, $k \in \{ 0, ..., 2^n-1 \}$ we have
[ @write auxout ]{} $$|\varphi_{n,k}(x)|_H \leq 384 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} 2^{-r} \sqrt{n} (\ln (r+3))^{1/\gamma} . \tag{\ref{SIGMA-STEP-2}}$$
on $A_{{\varepsilon},b,n,k}^c$. Indeed, let $x$ be a dyadic number such that $x \in 2Q_r$ with $r \in \{ 0 , ..., 2^n \}$. Recall Corollary \[RE-PI\]. For every $m\in{\mathbb{N}}$ with $m\geq r$ we set
$$x_m := 2\pi_{m+1}^{(r)}\left( \frac x2 \right) \in 2Q_r \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}},$$
where $\pi_m^{(r)}$ is the map from Corollary \[RE-PI\]. I.e. $|x - x_m|_\infty \leq 2^{-m}$. By the triangle inequality (see Remark \[RE-METRIC\]) and $\varphi_{n,k}(x)=\varphi_{n,k}(x,0)$ we immediately get
$$|\varphi_{n,k}(x)|_H \leq |\varphi_{n,k}(x_r, 0)|_H + \sum\limits_{m=r}^\infty |\varphi_{n,k}(x_{m+1}, x_m)|_H .$$
Note that the sum on the right-hand side is actually a finite sum, because $x$ is dyadic, so that $x_m = x$ for $m$ sufficiently large. Note that $x_m, x_{m+1} \in 2^{-(m+1)} {\mathbb{Z}}^{\mathbb{N}}$ hence, by using inequality , the above expression is bounded from above by
$$6 \beta_A^{-1/2} \eta_{\varepsilon}\left( \sqrt{n} + \sqrt{ 1+r-r } \right) {\operatorname{ed}}(2Q_r)_r |x_r|_\infty 2^{-n/2}$$ $$+ 6 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{m=r}^\infty ( \sqrt{n} + \sqrt{ 1+(m+1)-r } ) {\operatorname{ed}}(2Q_r)_{m+1} |x_{m+1} - x_m |_\infty 2^{-n/2} .$$
Using the definition of $x_m$ and $|x_{m+1} - x_m |_\infty \leq |x_{m+1} - x |_\infty + |x_m - x|_\infty \leq 2^{-m+1}$ this can be estimated from above by
$$\hspace{-23mm} 6 \beta_A^{-1/2} \eta_{\varepsilon}\left( \sqrt{n} + 1 \right) {\operatorname{ed}}(2Q_r)_r 2^{-r} 2^{-n/2}$$ $$\hspace{19mm} + 24 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{m=r}^\infty ( \sqrt{n} + \sqrt{ 1+(m+1)-r } ) 2^{-(m+1)} {\operatorname{ed}}(2Q_r)_{m+1} 2^{-n/2}$$
$$\leq 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} \sum\limits_{m=r}^\infty \left( \sqrt{n} + \sqrt{ 1+m-r } \right) {\operatorname{ed}}(2Q_r)_m 2^{-m} .$$
Invoking Lemma \[EFFDIM\] yields that ${\operatorname{ed}}(Q_r)_m \leq (\ln(m+1))^{1/\gamma}$, where $\gamma>1$ is the constant from Assumption \[ASS\]. Using this we can further estimate the above expression by
$$\hspace{11mm} 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} \sum\limits_{m=r}^\infty \left( \sqrt{n} + \sqrt{1+m-r} \right) (\ln(m+1))^{1/\gamma} 2^{-m}$$
$$\hspace{12mm} \leq 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} \sum\limits_{m=0}^\infty \left( \sqrt{n} + \sqrt{1+m} \right) (\ln(r+m+1))^{1/\gamma} 2^{-m-r} .$$
Using Lemma \[LOG-CONVEX\] the above is smaller than
$$\hspace{-7mm} 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} 2^{-r} \sum\limits_{m=0}^\infty \left( \sqrt{n} + \sqrt{1+m} \right) \left( (\ln (r+1) )^{1/\gamma} + \ln(m+1)^{1/\gamma} \right) 2^{-m}$$
$$\hspace{-15mm} \leq 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} 2^{-r} \left[ \sqrt{n} (\ln (r+1))^{1/\gamma} \sum\limits_{m=0}^\infty 2^{-m} + \sqrt{n} \sum\limits_{m=0}^\infty (\ln (m+1))^{1/\gamma} 2^{-m} \right.$$
$$\hspace{24mm} \qquad\qquad \left. + (\ln r)^{1/\gamma} \sum\limits_{m=0}^\infty \sqrt{1+m} 2^{-m} + \sum\limits_{m=0}^\infty \sqrt{1+m} (\ln (m+1))^{1/\gamma} 2^{-m} \right] .$$
Since $\gamma \geq 1$ we can estimate $(\ln (m+1))^{1/\gamma} \leq 2^{m/2}$. The above expression is therefore bounded by
$$24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} 2^{-r} \left[ 2 \sqrt{n} (\ln (r+1))^{1/\gamma} + \sqrt{n} \sum\limits_{m=0}^\infty 2^{-m/2} + 4 (\ln r)^{1/\gamma} + \sum\limits_{m=0}^\infty \sqrt{1+m} 2^{-m/2} \right]$$
$$\leq 24 \beta_A^{-1/2} \eta_{\varepsilon}2^{-n/2} 2^{-r} \left[ 2 \sqrt{n} (\ln (r+1))^{1/\gamma} + 4 \sqrt{n} + 4 (\ln r)^{1/\gamma} + 6 \right] .$$
And since we have $1 \leq (\ln(r+3))^{1/\gamma}$, we obtain
$$|\varphi_{n,k}(x)|_H \leq \underbrace{384 \beta_A^{-1/2} \eta_{\varepsilon}}_{=: C_{\varepsilon}} 2^{-n/2} 2^{-r} \sqrt{n} (\ln (r+3))^{1/\gamma} ,$$
which proves Claim .\
**Step 3:**\
For a fixed $n\in{\mathbb{N}}$ let $x\in 2Q \cap {\mathbb{D}}$ such that $|x|_\infty > 2^{-2^{n}}$. We set $$r := \lfloor \log_2 |x|_\infty^{-1} \rfloor \leq \lfloor 2^{n} \rfloor \leq 2^{n} .$$
And hence we have $$2^{-r} = 2^{- \log_2 \lfloor |x|_\infty^{-1} \rfloor} \leq 2^{- \log_2 |x|_\infty^{-1} + 1} = 2 |x|_\infty .$$
Additionally, we have $r\in\{-2, ..., 2^n\}$ and $x \in 2Q_r$, because of the fact that $$|x|_\infty = 2^{- \log_2 |x|_\infty^{-1}} \leq 2^{-r} .$$
Hence, we can apply Claim of Step 2 to obtain
$$\hspace{-53mm} |\varphi_{n,k}(x)|_H \leq C_{\varepsilon}2^{-r} \sqrt{n} 2^{-n/2} (\ln (r+3))^{1/\gamma}$$
$$\hspace{18mm} \leq C_{\varepsilon}\sqrt{n} 2^{-n/2} |x|_\infty \left(\log_2\left(2^{3n}\right)\right)^{1/\gamma} \leq C_{\varepsilon}\sqrt{n} (3n)^{1/\gamma} 2^{-n/2} |x|_\infty .$$
**Step 4:**\
Conversely to Step 3, for fixed $n\in{\mathbb{N}}$ let $x \in 2Q \cap {\mathbb{D}}$ such that $|x|_\infty \leq 2^{-2^{n}}$. Then $x \in Q_r$ with $r = 2^{n}$ so that by Invoking Step 2 (i.e. inequality ) we have
$$\hspace{-10mm} |\varphi_{n,k}(x)|_H \leq C_{\varepsilon}2^{-r} \sqrt{n} 2^{-n/2} (\ln (r+3))^{1/\gamma} \leq C_{\varepsilon}2^{-2^n} \sqrt{n} 2^{-n/2} \left(\log_2 \left(2^{3n}\right)\right)^{1/\gamma}$$
$$\hspace{51.5mm} \leq C_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-2^{n}} (3n)^{1/\gamma} .$$
This concludes the proof.
[ \[THM-RHO\] ]{}
For every $0 < {\varepsilon}< 1$ there exists $C_{\varepsilon}\in {\mathbb{R}}$ such that for every Borel measurable function $b\colon [0,1] \times H \longrightarrow H$ satisfying Assumption \[ASS\] there exists a measurable set $A_{{\varepsilon},b} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},b}] \leq {\varepsilon}$ such that on $A_{{\varepsilon},b}^c$
$$|\varphi_{n,k}(x, y)|_H \leq C_{\varepsilon}\left[ \sqrt{n} 2^{-n/6} |x - y|_\infty + 2^{- 2^{\theta n}} \right]$$
holds for all points $x,y \in Q \cap {\mathbb{D}}$ with $|x-y|_\infty \leq 1$, $n \geq 1$, $k \in \{ 0 , ..., 2^n - 1 \}$ and $\theta := \frac23 \frac\gamma{\gamma+2}$.
Note that the constant $C_{\varepsilon}$ depends on ${\varepsilon}$ and $\gamma$, but not on $b$. Conversely, the set of “good omegas” $A_{{\varepsilon},b}^c$ depends on both, ${\varepsilon}$ and $b$.
**Step 1:**\
Let $m\in{\mathbb{N}}$ and $x,y\in Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$. For $0 < {\varepsilon}< 1$ we set
$$\eta_{\varepsilon}:= \sqrt{ \ln\left( \frac1{\varepsilon}\right) } \leq 1 .$$
Analogously to the previous proof we estimate
$${\mathbb{P}}\left[ |\varphi_{n,k}(x, y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{ 5(1+m) } ) {\operatorname{ed}}(Q)_m |x-y|_\infty 2^{-n/2} \right] ,$$
where $\beta_A>0$ is the constant from [@Wre16 Corollary 3.1]. Since $x,y \in Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ and $|\cdot|_\infty$, $|\cdot|_2$ are effectively equivalent norms i.e. (see Proposition \[EQUIVALENCE\]) the above expression is smaller than
$${\mathbb{P}}\left[ |\varphi_{n,k}(x, y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}( 1 + \sqrt{2n} + \sqrt{ 5(1+m) } ) \sqrt{{\operatorname{ed}}(Q)_m} |x-y|_2 2^{-n/2} \right] .$$
Due to [@Wre16 Corollary 3.1] this expression is bounded by
$$e^{- \eta_{\varepsilon}^2 {\operatorname{ed}}(Q)_m} e^{- \eta_{\varepsilon}^2 \left( \sqrt{2n} + \sqrt{5(1+m)} \right)^2 {\operatorname{ed}}(Q)_m}$$
ans since $\eta_{\varepsilon}\geq 1$ as well as ${\operatorname{ed}}(Q)_m \geq 1$ the above expression can be estimated from above by
$$e^{- \eta_{\varepsilon}^2} e^{- (2n + 5(1+m)) {\operatorname{ed}}(Q)_m } \leq e^{- \eta_{\varepsilon}^2} e^{-2n} e^{- 5(1+m) {\operatorname{ed}}(Q)_m } .$$
Using this, we estimate the following probability
$${\mathbb{P}}\left[ \bigcup\limits_{n=1}^\infty \bigcup\limits_{m=0}^\infty \!\! \bigcup\limits_{{\genfrac{}{}{0pt}{}{x,y\in}{Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}}}} \!\! \bigcup\limits_{k=0}^{2^n-1} |\varphi_{n,k}(x, y)|_H > \beta_A^{-1/2} \eta_{\varepsilon}\left( 1 + \sqrt{2n} + \sqrt{5(1+m)} \right) {\operatorname{ed}}(Q)_m |x-y|_\infty 2^{-n/2} \right]$$
$$\hspace{-29mm} \leq \sum\limits_{n=1}^\infty \sum\limits_{m=0}^\infty \sum\limits_{{\genfrac{}{}{0pt}{}{x,y\in}{Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}}}} \sum\limits_{k=0}^{2^n-1} e^{- \eta_{\varepsilon}^2} e^{- 2n} e^{- 5(1+m) {\operatorname{ed}}(Q)_m}$$
$$\hspace{5mm} \leq e^{- \eta_{\varepsilon}^2} \sum\limits_{n=1}^\infty \sum\limits_{m=0}^\infty \# \{ (x,y) | x,y \in Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}\} 2^n e^{- 2n} e^{ - 5(1+m) {\operatorname{ed}}(Q)_m } .$$
By invoking Theorem \[KOLTIK\] for $r=0$ we have $$\# \{ (x,y) | x,y \in Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}\} \leq \exp \left( 4 ( 1 + m ) {\operatorname{ed}}(Q)_m \right) .$$
So that we can bound the above probability by
$$\hspace{5mm} e^{- \eta_{\varepsilon}^2} \sum\limits_{n=1}^\infty \sum\limits_{m=0}^\infty \exp \left( 4 ( 1 + m ) {\operatorname{ed}}(Q)_m \right) 2^n e^{- 2n} e^{- 5(1+m) {\operatorname{ed}}(Q)_m }$$
$$\hspace{-21.5mm} \leq e^{- \eta_{\varepsilon}^2} \sum\limits_{n=1}^\infty \sum\limits_{m=0}^\infty 2^n e^{- 2n } \exp \left( - ( 1 + m ) {\operatorname{ed}}(Q)_m \right) .$$
Note that the last sum converges since ${\operatorname{ed}}(Q)_m \geq 1$. Hence, the above is bounded from above by
$$e^{- \eta_{\varepsilon}^2} \sum\limits_{n=1}^\infty 2^n e^{- 2n } \underbrace{\sum\limits_{m=0}^\infty \exp \left( - ( 1 + m ) \right)}_{\leq 1}$$
so that, in conclusion, we have estimated the above probability by
$$e^{- \eta_{\varepsilon}^2} \sum\limits_{n=1}^\infty 2^n e^{- 2n } \leq e^{- \eta_{\varepsilon}^2} = {\varepsilon}.$$
Therefore, we obtain
[ @write auxout ]{} $$\begin{aligned}
\tag{\ref{RHO-STEP-1}}
\begin{split}
|\varphi_{n,k}(x, y)|_H &\leq \beta_A^{-1/2} \eta_{\varepsilon}\left( 1 + \sqrt{2n} + \sqrt{5(1+m)} \right) {\operatorname{ed}}(Q)_m |x-y|_\infty 2^{-n/2} \\
&\leq 5 \beta_A^{-1/2} \eta_{\varepsilon}\left( \sqrt{n} + \sqrt{1+m} \right) {\operatorname{ed}}(Q)_m |x-y|_\infty 2^{-n/2}
\end{split}\end{aligned}$$
for $n \geq 1$, $k \in \{ 0, ..., 2^n-1 \}$, $m \in {\mathbb{N}}$ and for all $x,y \in Q \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ on a set $A_{{\varepsilon},b}^c \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},b}] \leq {\varepsilon}$.\
**Step 2:**\
**Claim:** For all points $x,y \in Q\cap{\mathbb{D}}$, with $|x-y|_\infty \leq 1$, $n \geq 1$ and $k \in \{ 0, ..., 2^n-1 \}$ we have
[ @write auxout ]{} $$|\varphi_{n,k}(x,y)|_H \leq 7200 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt n \left[ 2^{-n/6} |x - y|_\infty + 2^{- 2^{\theta n}} \right] . \tag{\ref{RHO-STEP-2}}$$
on $A_{{\varepsilon},b}^c$. Indeed, let $x,y \in Q$ be two dyadic points in $Q$ with $|x-y|_\infty \leq 1$. W.l.o.g. we assume $x \neq y$. Fix $m\in{\mathbb{N}}$ be so that $2^{-m-1} \leq |x - y|_\infty \leq 2^{-m}$. Note that this implies that $m \geq 0$. Using Corollary \[RE-PI\] for every $r\in{\mathbb{N}}$ with $r\geq m$ we set
$$x_r := \pi_r^{(0)}(x) \in Q \cap 2^{-r} {\mathbb{Z}}^{\mathbb{N}},$$ $$y_r := \pi_r^{(0)}(y) \in Q \cap 2^{-r} {\mathbb{Z}}^{\mathbb{N}}.$$
By the triangle inequality (see Remark \[RE-METRIC\]) we immediately get
$$|\varphi_{n,k}(x,y)|_H \leq |\varphi_{n,k}(x_m, y_m)|_H + \sum\limits_{r=m}^\infty |\varphi_{n,k}(x_{r+1}, x_r)|_H + \sum\limits_{r=m}^\infty |\varphi_{n,k}(y_{r+1}, y_r)|_H .$$
Note that both sums on the right-hand side are actually a finite sums, because $x$ and $y$ are dyadic points. Also note that $x_r, x_{r+1}, y_r, y_{r+1} \in 2^{-(r+1)} {\mathbb{Z}}^{\mathbb{N}}$, so that by using inequality the above expression is bounded from above by
$$5 \beta_A^{-1/2} \eta_{\varepsilon}\left( \sqrt{n} + \sqrt{1+m} \right) {\operatorname{ed}}(Q)_{m} |x_m - y_m|_\infty 2^{-n/2}$$ $$\hspace{-7.5mm} + 10 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{r=m}^\infty ( \sqrt{n} + \sqrt{ r+2 } ) {\operatorname{ed}}(Q)_{r+1} 2^{-(r-1)} 2^{-n/2} ,$$
where we have used that by the definition of $x_r$ we have $|x_{r+1} - x_r |_\infty \leq |x_{r+1} - x|_\infty + |x_r - x|_\infty \leq 2^{-(r-1)}$ and an analogous calculation for $|y_{r+1} - y_r |_\infty$. Since $|x_m-y_m|_\infty \leq |x_m-x|_\infty + |x-y|_\infty + |y-y_m|_\infty \leq 2^{-(m-2)}$ this can be further estimated from above by
$$40 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{r=m}^\infty ( \sqrt{n} + \sqrt{r+1} ) {\operatorname{ed}}(Q)_{r} 2^{-r} 2^{-n/2} .$$
Invoking Lemma \[EFFDIM\] yields that ${\operatorname{ed}}(Q)_r \leq (\ln(r+1))^{1/\gamma}$, where $\gamma>0$ is the constant from Assumption \[ASS\]. Using this we can further estimate the above expression by
$$\hspace{8mm} 40 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{r=m}^\infty ( \sqrt{n} + \sqrt{r+1} ) (\ln (r+1))^{1/\gamma} 2^{-r} 2^{-n/2}$$
and since $\sqrt{n} + \sqrt{r+1} \leq 2 \sqrt{n(r+1)}$ this is bounded by
$$80 \beta_A^{-1/2} \eta_{\varepsilon}\sum\limits_{r=m}^\infty \sqrt{n} \sqrt{r+1} (\ln (r+1))^{1/\gamma} 2^{-r} 2^{-n/2} .$$
By performing an index shift this can be written as
$$80 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-m} \sum\limits_{r=0}^\infty \sqrt{r+m+1} (\ln (r+m+1))^{1/\gamma} 2^{-r} .$$
We use $\sqrt{r+m+1} \leq \sqrt{r+1} + \sqrt m$ and invoke Lemma \[LOG-CONVEX\] to estimate this further from above by
$$80 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-m} \sum\limits_{r=0}^\infty (\sqrt{r+1} + \sqrt{m}) \left( (\ln(r+1))^{1/\gamma} + (\ln (m+1))^{1/\gamma} \right) 2^{-r} .$$
Expanding the terms yields
$$\hspace{-10mm} 80 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-m} \sum\limits_{r=0}^\infty \left[ \sqrt{r+1} (\ln (r+1))^{1/\gamma} + \sqrt{r+1} (\ln (m+1))^{1/\gamma} \right.$$ $$\hspace{40mm} \left. + \sqrt{m} (\ln (r+1))^{1/\gamma} + \sqrt{m} (\ln (m+1))^{1/\gamma} \right] 2^{-r} .$$
Plugging in $(\ln(m+1))^{1/\gamma} \leq 2^{m/2}$ and evaluating the sum term by term leads us to the following upper bound
$$80 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-m} \left[ \sum\limits_{r=0}^\infty \sqrt{r+1} 2^{-r/2} + (\ln (m+1))^{1/\gamma} \sum\limits_{r=0}^\infty \sqrt{r+1} 2^{-r} \right.$$ $$\hspace{38mm} \left. + \sqrt{m} \sum\limits_{r=0}^\infty 2^{-r/2} + \sqrt{m} (\ln (m+1))^{1/\gamma} \sum\limits_{r=0}^\infty 2^{-r} \right]$$
$$\leq 80 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} 2^{-m} \left[ 6 + 3 (\ln (m+1))^{1/\gamma} + 4 \sqrt{m} + 2 \sqrt{m} (\ln (m+1))^{1/\gamma} \right]$$
$$\leq 1200 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} (\ln (m+1))^{1/\gamma} \sqrt{m+1} 2^{-m} .$$
$$\leq 2400 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} (m+1)^{1/\gamma} \sqrt{m+1} 2^{-(m+1)} .$$
In conclusion we finally obtained
[ @write auxout ]{} $$|\varphi_{n,k}(x, y)|_H \leq 2400 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt{n} 2^{-n/2} (m+1)^{1/\gamma} \sqrt{m+1} 2^{-(m+1)} . \tag{\ref{RHO-STEP-2-PRE}}$$
We are going to estimate this further using the following claim:\
Set $\theta := \frac23 \frac\gamma{\gamma+2} > 0$.\
**Claim:**\
For every $n,m \in N$ we have that
[ @write auxout ]{} $$\sqrt{n} m^{\frac12 + \frac1\gamma} 2^{-m} 2^{-n/2} \leq 3 \sqrt{n} 2^{-n/6} 2^{-m} + 3\cdot 2^{-2^{\theta n}} \tag{\ref{RHO-CLAIM}}$$
holds.
**Proof of Claim :**\
**Case 1:** $m \leq 2^{1 + \theta n}$
$$\sqrt{n} m^{\frac12 + \frac1\gamma} 2^{-n/2} \leq \sqrt{n} 2^{(1 + \theta n) (\frac12 + \frac1\gamma)} 2^{-n/2} = \sqrt{n} \underbrace{2^{\frac12 + \frac1\gamma}}_{\leq3} \underbrace{2^{\theta (\frac12 + \frac1\gamma) n}}_{= 2^{n/3}} 2^{-n/2} \leq 3 \sqrt{n} 2^{-n/6} .$$
**Case 2:** $2^{1 + \theta n} < m$
$$\underbrace{\sqrt{n} 2^{-n/2}}_{\leq1} m^{\frac12 + \frac1\gamma} 2^{-m} \leq \underbrace{m^{\frac12 + \frac1\gamma} 2^{-m/2}}_{\leq 3} 2^{-m/2} \leq 3 \cdot 2^{-2^{\theta n}} .$$
This ends the proof of Claim . Using inequality and we conclude that
$$|\varphi_{n,k}(x,y)|_H \leq 7200 \beta_A^{-1/2} \eta_{\varepsilon}\sqrt n \left[ 2^{-n/6} 2^{-m} + 2^{- 2^{\theta n}} \right] .$$
Recall that $2^{-m-1} \leq |x - y|_\infty$ so that the above is smaller than
$$14400 \beta_A^{-1/2} \eta_{\varepsilon}\left[ \sqrt{n} 2^{-n/6} |x - y|_\infty + 2^{- 2^{\theta n}} \right] ,$$
which finishes the proof of Claim and hence the assertion.
Continuity of the map $\boldsymbol{\varphi_{n,k}}$
==================================================
In this section we will prove that for almost all $\omega\in\Omega$ the map
$$\varphi_{n,k}(\omega) \colon H \longrightarrow H$$
is continuous. Furthermore, we will show that on a suitable class of Lipschitz functions $h$ and their dyadic piecewise approximations that the map
$$h \longmapsto \int\limits_0^1 b(t, Z_t(\omega) + h(t)) {\,\mathrm{d}}t$$
is continuous w.r.t. the maximum norm.
[ \[DE-QA\] ]{}
We set
$$Q^A := \{ x = (x_n)_{n\in{\mathbb{N}}} \in{\mathbb{R}}^{\mathbb{N}}| \lambda_n e^{2\lambda_n} x_n^2 \leq 1 \} ,$$
where $(\lambda_n)_{n\in{\mathbb{N}}}$ are the eigenvalues of the operator $A$ of our Ornstein–Uhlenbeck process $Z^A$.
[ \[DE-PHI\] ]{}
We define
$$\begin{aligned}
\Phi &:= \{ h \colon [0,1] \longrightarrow Q \cap Q^A \colon |h(s) - h(t)|_\infty \leq 2|s-t|,\ \forall s,t\in[0,1] \} , \\
\Phi_n &:= \left\{ h \colon [0,1] \longrightarrow Q \cap Q^A \cap {\mathbb{D}}\left| {\genfrac{}{}{0pt}{}{\forall 0\leq k < 2^n \colon \forall s,t\in [k2^{-n}, (k+1)2^{-n}[ \colon h(s)=h(t) \text{ and}}{\forall m,\ell\in{\mathbb{Z}}\cap [0,2^n] \colon |h(m2^{-n}) - h(\ell 2^{-n})|_\infty \leq 2|m-\ell|2^{-n}}} \right.\right\} , \\
\Phi^* &:= \Phi \cup \bigcup\limits_{n=1}^\infty \Phi_n .\end{aligned}$$
Note that elements in $\Phi$ are continuous, since functions in $\Phi$ are Lipschitz continuous (with Lipschitz constant at most $2$). $\Phi_n$ will be used to approximate elements in $\Phi$. Also note that $\Phi$ and $\Phi_n$ are separable w.r.t. the maximum norm and hence $\Phi^*$ is separable.
Observe that the above spaces are constructed in such a way that the assumptions we impose on $b$ (see Assumption \[ASS\]) implie that the function $u$ from Proposition \[PRO-GIRSANOV\] is in the space $\Phi$. I.e. the difference of two solutions of always lives in the space $\Phi$ due to Assumption \[ASS\].
[ \[APPROX-LEM\] ]{}
Let $h\in\Phi^*$ and $n\in{\mathbb{N}}$. We then have
$$\sum\limits_{k=0}^{2^n-1} \left| h((2k+1)2^{-(n+1)}) - h(2k2^{-(n+1)}) \right|_\infty \leq 1 .$$
Let $h\in\Phi^*$ and $n\in{\mathbb{N}}$ be as in the assertion. If $h\in\Phi$ the inequality follows immediately from the Lipschitz continuity of $h$. Let $h\in\Phi_m$ for some $m\in{\mathbb{N}}$.\
**Case 1:** $m\geq n+1$
We have
$$\sum\limits_{k=0}^{2^n-1} \left| h((2k+1) 2^{-(n+1)}) - h(2k 2^{-(n+1)}) \right|_\infty$$
$$= \sum\limits_{k=0}^{2^n-1} \left| h((2k+1) 2^{m-(n+1)}2^{-m}) - h(2k 2^{m-(n+1)}2^{-m}) \right|_\infty .$$
Using the assumption that $h\in\Phi_m$ by definition of $\Phi_m$ the above expression us bounded from above by
$$\sum\limits_{k=0}^{2^n-1} 2 \cdot 2^{m-(n+1)}2^{-m} = 1 .$$
**Case 2:** $m < n+1$
Since $h\in\Phi_m$ is constant on all intervals of the form $[k2^{-m} , (k+1)2^{-m}[$ the sum simplifies to
$$\sum\limits_{k=0}^{2^n-1} \left| h((2k+1) 2^{-(n+1)}) - h(2k 2^{-(n+1)}) \right|_\infty = \sum\limits_{k=0}^{2^{m-1}-1} \left| h((2k+1) 2^{-m}) - h(2k 2^{-m}) \right|_\infty .$$
And using the definition of $\Phi_m$ the above sum is bounded by
$$\sum\limits_{k=0}^{2^{m-1}-1} 2\cdot2^{-m} = 1 .$$
[ \[APPROX\] ]{}
For every $0 < {\varepsilon}< 1$ there exist $\delta > 0$ such that for every open set $U \subseteq [0,1] \times H$ with mass $\mu[U] \leq \delta$, where $\mu = \mathrm dt \otimes Z^A_t({\mathbb{P}})$ there is a measurable set $\Omega_{{\varepsilon},U} \subseteq \Omega$ with
$${\mathbb{P}}[\Omega \setminus \Omega_{{\varepsilon},U}] \leq {\varepsilon}$$
such that the inequality
$$\int\limits_0^1 {\mathbbm}1_U( s, Z^A_s + h(s)) {\,\mathrm{d}}s \leq {\varepsilon}$$
holds on $\Omega_{{\varepsilon},U}$ uniformly for any $h\in \Phi^*$.
Let $0 < {\varepsilon}< 1$ and let $C_{{\varepsilon}/2}$ be the constant from Theorem \[THM-RHO\]. Recall that $\theta := \frac23 \frac\gamma{\gamma+2}$. Choose $m\in{\mathbb{N}}$ sufficiently large, so that
[ @write auxout ]{} $$6C_{{\varepsilon}/2} \sum\limits_{n=m}^\infty \sqrt n 2^{-n/6} \leq \frac{\varepsilon}2 \qquad \text{and} \qquad m \geq \frac4{\theta^2 \ln(2)^2} . \tag{\ref{APPROX-LEMMA-DEF-M}}$$
Set ${\mathcal}N_m := Q \cap Q^A \cap 2^{-m} {\mathbb{Z}}^{\mathbb{N}}$ and note that ${\mathcal}N_m$ is a finite $2^{-m}$-net of $Q \cap Q^A$ w.r.t. the maximum norm.
Also, observe that $Z^A_t({\mathbb{P}})$ is equivalent to the invariant measure $N(0, \frac12 A^{-1})$ due to [@DZ92 Theorem 11.13] and analogously $(Z_t^{A} + h(t))({\mathbb{P}})$ to $N(h(t), \frac1{2} A^{-1})$. Let $z\in{\mathcal}N_m$ then $z$ is in the domain of $A$ because
$$\sum\limits_{n\in{\mathbb{N}}} {\langle}z,e_n{\rangle}^2 \lambda_n^2 = \sum\limits_{n\in{\mathbb{N}}} |z_n|^2 \lambda_n^2 < \infty .$$
Set $y := 2 A z$ then $y\in H$ hence, [@Bog98 Corollary 2.4.3] is applicable which implies that the measures $N(0, \frac12 A^{-1})$ and $(Z^A+z)({\mathbb{P}})$ are equivalent. We set
$$\mu := \mathrm dt \otimes Z^A_t({\mathbb{P}}), \qquad \mu_z := \mathrm dt \otimes (Z^A_t + z)({\mathbb{P}})$$
for all $z\in D(A)$. By the Radon–Nikodyn Theorem there exist densities $\rho_z$ so that
$$\frac{{\mathrm}d\mu_z}{{\mathrm}d\mu} = \rho_z .$$
Furthermore, the family $\{ \rho_z | z\in{\mathcal}N_m \}$ is uniformly integrable, since ${\mathcal}N_m$ is finite. Hence, there exists $\delta > 0$ such that
[ @write auxout ]{} $$\int\limits_A \rho_z(t,x) {\,\mathrm{d}}\mu(t,x) \leq \frac{{\varepsilon}^2}{4 \cdot 2^m \#({\mathcal}N_m)}, \qquad \forall z\in{\mathcal}N_m \tag{\ref{APPROX-LEMMA-DENSITIES}}$$
for every measurable set $A \subseteq \Omega$ with $\mu[A] \leq \delta$. Let $U \subseteq [0,1]\times H$ be open with mass $\mu[U] \leq \delta$. Then, by invoking Theorem \[THM-RHO\] for the function ${\mathbbm}1_U$ with the constant $C_{{\varepsilon}/2}$, there exists a measurable set $A_{{\varepsilon},U} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},U}] \leq \frac{\varepsilon}2$ such that
$$\left| \int\limits_{k 2^{-n}}^{(k+1) 2^{-n}} \!\! {\mathbbm}1_U(t, Z^A_t + x) - {\mathbbm}1_U(t, Z^A_t + y) {\,\mathrm{d}}t \right| \leq C_{{\varepsilon}/2} \left( \sqrt n 2^{-n/6} |x-y|_\infty + 2^{-2^{\theta n}} \right) .$$
holds for every $n\geq 1$, $k\in \{0, ..., 2^n -1 \}$ and $x,y \in Q \cap {\mathbb{D}}$ on $A_{{\varepsilon},U}^c$. Furthermore, we define the events $B_{{\varepsilon},U}$ by
$$B_{{\varepsilon},U} := \bigcup\limits_{z\in{\mathcal}N_m} \left\{ \int\limits_0^1 {\mathbbm}1_U (s, Z^A_s + z) {\,\mathrm{d}}s > \frac{\varepsilon}{2\cdot 2^m} \right\} .$$
We then have
$$\hspace{-60mm} {\mathbb{P}}[B_{{\varepsilon},U}] = {\mathbb{P}}\left[ \bigcup\limits_{z\in{\mathcal}N_m} \left\{ \int\limits_0^1 {\mathbbm}1_U (s, Z_s + z) {\,\mathrm{d}}s > \frac{\varepsilon}{2\cdot 2^m} \right\} \right]$$
$$\hspace{12mm} \leq \sum\limits_{z\in{\mathcal}N_m} {\mathbb{P}}\left[ \int\limits_0^1 {\mathbbm}1_U (s, Z^A_s + z) {\,\mathrm{d}}s > \frac{\varepsilon}{2\cdot2^m} \right] \leq \frac{2\cdot2^m}{\varepsilon}\sum\limits_{z\in{\mathcal}N_m} {\mathbb E}\int\limits_0^1 {\mathbbm}1_U (s, Z^A_s + z) {\,\mathrm{d}}s$$
$$\hspace{2mm} = \frac{2\cdot2^m}{\varepsilon}\sum\limits_{z\in{\mathcal}N_m} \int\limits_{[0,1] \times H} {\mathbbm}1_U (s, x) {\,\mathrm{d}}\mu_z(s, x) = \frac{2\cdot2^m}{\varepsilon}\sum\limits_{z\in{\mathcal}N_m} \int\limits_U \rho_z(s,x) {\,\mathrm{d}}\mu(s, x) .$$
Since $\mu[U] \leq \delta$ using inequality the above is bounded from above by
$$\frac{2\cdot2^m}{\varepsilon}\#({\mathcal}N_m) \frac{{\varepsilon}^2}{4\cdot 2^m \#({\mathcal}N_m)} = \frac{\varepsilon}2 .$$
In conclusion we proved that we have ${\mathbb{P}}[ B_{{\varepsilon},U} ] \leq \frac{\varepsilon}2$ and therefore obtained that
$${\mathbb{P}}[ A_{{\varepsilon},U}^c \cap B_{{\varepsilon},U}^c] \geq 1 - {\varepsilon}.$$
For every $h\in\Phi$ and $n\in{\mathbb{N}}$ we define
[ @write auxout ]{} $$h_n(t) := \sum\limits_{k=0}^{2^n-1} {\mathbbm}1_{[k2^{-n}, (k+1)2^{-n}[}(t) \frac{\lfloor 2^n h(k2^{-n}) \rfloor}{2^n} \in \underbrace{Q \cap Q^A \cap 2^{-n} {\mathbb{Z}}^{\mathbb{N}}}_{={\mathcal}N_n} , \qquad \forall t \in [0,1] , \tag{\ref{APPROX-LEMMA-H_N}}$$
where $\lfloor \cdot \rfloor$ denotes the componentwise floor function. Note that $h_n$ is $Q \cap Q^A$-valued since $h$ is $Q \cap Q^A$-valued. Furthermore, $h_n(t)$ is a dyadic number for all $t\in [0,1]$. Also note that $h_n$ converges to $h$ for $n\rightarrow\infty$.
Now, let
$$E_{{\varepsilon},U} := \bigcap\limits_{h \in \Phi^*} \left\{ \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t + h(t)) {\,\mathrm{d}}t \leq {\varepsilon}\right\} .$$
We are going to prove that $A_{{\varepsilon},U}^c \cap B_{{\varepsilon},U}^c \subseteq E_{{\varepsilon},U}$ holds. To this end let $\omega \in A_{{\varepsilon},U}^c \cap B_{{\varepsilon},U}^c$. Using that $\omega \in B_{{\varepsilon},U}^c$ we have
$$\left| \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_m(t)) {\,\mathrm{d}}t \right| \leq \sum\limits_{k=0}^{2^m-1} \left| \int\limits_{k2^{-m}}^{(k+1)2^{-m}} \right. \!\!\! {\mathbbm}1_U(t, Z^A_t(\omega) + \underbrace{h_m(t)}_{\in{\mathcal}N_m}) {\,\mathrm{d}}t \left. \vphantom{\int\limits_{k2^{-m}}^{(k+1)2^{-m}}} \right| \leq \sum\limits_{k=0}^{2^m-1} \frac{{\varepsilon}}{2 \cdot 2^m} = \frac{\varepsilon}2 .$$
And since $\omega\in A_{{\varepsilon},U}^c$ we obtain for $n\geq m$
$$\hspace{-35mm} \left| \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_{n+1}(t)) - {\mathbbm}1_U(t, Z^A_t(\omega) + h_n(t)) {\,\mathrm{d}}t \right|$$
$$\hspace{7mm} \leq \sum\limits_{k=0}^{2^{n+1}-1} \left| \int\limits_{k2^{-(n+1)}}^{(k+1)2^{-(n+1)}} \!\! \right. {\mathbbm}1_U(t, Z^A_t(\omega) + \underbrace{h_{n+1}(t)}_{\in Q \cap {\mathbb{D}}}) - {\mathbbm}1_U(t, Z^A_t(\omega) + \underbrace{h_n(t)}_{\in Q \cap {\mathbb{D}}}) {\,\mathrm{d}}t \left. \vphantom{\int\limits_{k2^{-(n+1)}}^{(k+1)2^{-(n+1)}}} \right|$$
$$\hspace{-5mm} \leq \sum\limits_{k=0}^{2^{n+1}-1} C_{{\varepsilon}/2} \left( \sqrt{n} 2^{-n/6} |h_{n+1}(k2^{-n-1}) - h_n(k2^{-n-1})|_\infty + 2^{-2^{\theta n}} \right)$$
$$\hspace{8mm} \leq C_{{\varepsilon}/2} \left[ 2^{n+1} 2^{-2^{\theta n}} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} |h_{n+1}(k2^{-n-1}) - h_n((k/2)2^{-n})|_\infty \right] .$$
Note that since $h_n$ is constant on intervals of the form $[k2^{-n}, (k+1)2^{-n}[$ we have $h_n((k/2)2^{-n}) = h_n( \lfloor k/2 \rfloor 2^{-n})$, so that the above equals
$$C_{{\varepsilon}/2} \left[ 2^{n+1} 2^{-2^{\theta n}} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} |h_{n+1}(k2^{-n-1}) - h_n(\lfloor k/2 \rfloor 2^{-n})|_\infty \right] .$$
Plugging in Definition yields that the above expression can be written as
$$C_{{\varepsilon}/2} \left[ 2^{n+1-2^{\theta n}} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} 2^{-n-1} \left| \lfloor 2^{n+1}h(k2^{-n-1}) \rfloor - 2\left\lfloor 2^n h\left( \left\lfloor k/2 \right\rfloor 2^{-n} \right) \right\rfloor \right|_\infty \right]$$
$$\leq C_{{\varepsilon}/2} \left[ 2^{n - \frac12 \theta^2 \ln(2)^2 n^2} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} \right. 2^{-n-1} \underbrace{\left| \lfloor 2^{n+1}h(k2^{-n-1}) \rfloor - 2^{n+1}h(k2^{-n-1}) \right|_\infty}_{\leq 1}$$
$$\hspace{13.5mm} \left. + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} \left| h(k2^{-n-1}) - h\left( \left\lfloor k/2 \right\rfloor 2^{-n} \right) \right|_\infty \right.$$
$$\hspace{43.4mm} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} 2^{-n} \underbrace{\left| 2^n h\left( \left\lfloor k/2 \right\rfloor 2^{-n} \right) - \left\lfloor 2^n h\left( \left\lfloor k/2 \right\rfloor 2^{-n} \right) \right\rfloor \right|_\infty}_{\leq1} \left. \vphantom{\sum\limits_{k=0}^{2^{n+1}-1}} \right]$$
$$\leq C_{{\varepsilon}/2} \left[ 2^{n - \frac12 \theta^2 \ln(2)^2 m n} + 3 \sqrt n 2^{-n/6} + \sqrt n 2^{-n/6} \sum\limits_{k=0}^{2^{n+1}-1} \left| h\left(k2^{-(n+1)}\right) - h\left( 2\left\lfloor k/2 \right\rfloor 2^{-(n+1)} \right) \right|_\infty \right] .$$
Since $k = 2 \lfloor k/2 \rfloor$ in case $k$ is even the sum can be restricted to $k$ of the form $k = 2k' + 1$ for $k' \in \{ 0, ..., 2^n -1 \}$. with the help of the above is bounded by
$$C_{{\varepsilon}/2} \left[ 2^{n - 2n} + 3 \sqrt n 2^{-n/6} + \sqrt n 2^{-n/6} \sum\limits_{k'=0}^{2^n-1} \left| h\left((2k'+1)2^{-(n+1)}\right) - h\left( 2k' 2^{-(n+1)} \right) \right|_\infty \right] .$$
Using Lemma \[APPROX-LEM\] we can further estimate the above sum by $1$ so that in conclusion we obtain
$$\left| \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_{n+1}(t)) - {\mathbbm}1_U(t, Z^A_t(\omega) + h_n(t)) {\,\mathrm{d}}t \right| \leq 6 C_{{\varepsilon}/2} \sqrt{n} 2^{-n/6} .$$
Therefore as long as $\omega \in A_{{\varepsilon},U}^c \cap B_{{\varepsilon},U}^c$ we have by Lebesgue’s dominated convergence Theorem, the lower semi-continuity of ${\mathbbm}1_U$ and by the above calculation
$$\int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h(t)) {\,\mathrm{d}}t \leq \lim\limits_{n\rightarrow\infty} \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_n(t)) {\,\mathrm{d}}t$$
$$= \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_m(t)) {\,\mathrm{d}}t + \sum\limits_{n=m}^\infty \int\limits_0^1 {\mathbbm}1_U(t, Z^A_t(\omega) + h_{n+1}(t)) - {\mathbbm}1_U(t, Z^A_t(\omega) + h_n(t)) {\,\mathrm{d}}t$$
$$\leq \frac{\varepsilon}2 + 6 C_{{\varepsilon}/2} \sum\limits_{n=m}^\infty \sqrt n 2^{-n/6} \overset{\eqref{APPROX-LEMMA-DEF-M}}\leq \frac{\varepsilon}2 + \frac{\varepsilon}2 = {\varepsilon}.$$
In conclusion we have proven that $A_{{\varepsilon},U}^c \cap B_{{\varepsilon},U}^c \subseteq E_{{\varepsilon},U}$ and hence ${\mathbb{P}}[E_{{\varepsilon},U}] \geq 1 - {\varepsilon}$ which completes the proof.
[ \[THM-APPROX\] ]{}
Let $b \colon [0,1] \times H \longrightarrow H$ be a bounded, Borel measurable function satisfying Assumption \[ASS\]. There exists a measurable set $\Omega'\subseteq \Omega$ with ${\mathbb{P}}[\Omega'] = 1$ such that for every sequence $(h_m)_{m\in{\mathbb{N}}} \subseteq \Phi^*$ which converges pointwise to a function $h\in\Phi^*$ i.e. $\lim\limits_{m\rightarrow\infty} |h(t)-h_m(t)|_H=0$ we have
$$\lim\limits_{m\rightarrow\infty} \int\limits_0^1 b(s, Z^A_s + h_m(s)) {\,\mathrm{d}}s = \int\limits_0^1 b(s, Z^A_s + h(s)) {\,\mathrm{d}}s$$
on $\Omega'$.
Let $b$ be as in the assertion. For $\ell\in{\mathbb{N}}$ let ${\varepsilon}_\ell := 2^{-\ell}$. By Lemma \[APPROX\] for every ${\varepsilon}_\ell$ there exists for every $\ell\in{\mathbb{N}}$ a $\delta_\ell$ such that for every pair $({\varepsilon}_\ell, \delta_\ell)$ the conclusions of Lemma \[APPROX\] holds. Applying Lusin’s Theorem to the pair $(b, \delta_\ell)$ yields for every $\ell\in{\mathbb{N}}$ a closed set $K_\ell \subseteq [0,1]\times H$ with $\mu[K_\ell^c] \leq \delta_\ell$, where $\mu := \mathrm dt \otimes Z^A_t({\mathbb{P}})$, so that
$$b\mid_{K_\ell} \colon K_\ell \longrightarrow H, \qquad (t,x) \longmapsto b(t,x)$$
is continuous. By Dugundji’s Extension Theorem (see [@Dug51 Theorem 4.1]) (applied to the above maps) there exist functions $\bar b_\ell \colon [0,1] \times H \longrightarrow H$ such that
$$\hspace{25mm} b (t, x) = \bar b_\ell (t, x ), \qquad \forall (t,x) \in K_\ell ,$$ $$\hspace{-10mm} \|\bar b_\ell\|_\infty \leq 1$$
and
$$\bar b_\ell \text{ is continuous.}$$
Then, by invoking Lemma \[APPROX\] for $({\varepsilon}_\ell, \delta_\ell, K_\ell^c)$ we obtain for every $\ell\in{\mathbb{N}}$ a measurable set $\Omega'_\ell$ with ${\mathbb{P}}[\Omega'_\ell] \geq 1 - {\varepsilon}_\ell$ such that for any $\omega \in \Omega'_\ell$ and $h\in \Phi^*$
$$\int\limits_0^1 {\mathbbm}1_{K_\ell^c}(s, Z^A_s(\omega) + h(s)) {\,\mathrm{d}}s \leq {\varepsilon}_\ell$$
holds. Let
$$\Omega' := \liminf\limits_{\ell\rightarrow\infty} \Omega'_\ell .$$
Since we have $$\sum\limits_{\ell\in{\mathbb{N}}} {\mathbb{P}}[ \Omega'^c_\ell ] \leq \sum\limits_{\ell\in{\mathbb{N}}} {\varepsilon}_\ell = \sum\limits_{\ell\in{\mathbb{N}}} 2^{-\ell} < \infty$$
the Borel–Canteli Lemma implies that
$${\mathbb{P}}[ \limsup\limits_{\ell\rightarrow\infty} \Omega'^c_\ell ] = 0 \quad \Rightarrow \quad {\mathbb{P}}[\Omega'] = 1 .$$
Let $\omega \in \Omega'$ be fixed. Then, there is an $N(\omega) \in{\mathbb{N}}$ such that for all $\ell>N(\omega)$ we have $\omega \in \Omega_\ell$ and therefore for all $m\in{\mathbb{N}}$ we obtain
[ @write auxout ]{} $$\left| \int\limits_0^1 {\mathbbm}1_{K_\ell^c} (s, Z^A_s(\omega) + h_m(s)) {\,\mathrm{d}}s \right| \leq {\varepsilon}_\ell \tag{\ref{APPROX-THM-INVOC}} .$$
Note that inequality also holds if we replace $h_m$ by $h$, since $h\in\Phi^*$ by assumption.
The assertion now follows easily by the following calculation
$$\left| \int\limits_0^1 b(s, Z^A_s(\omega) + h_m(s)) - \bar b_\ell(s, Z^A_s(\omega) + h_m(s)) {\,\mathrm{d}}s \right|_H$$
$$\leq \int\limits_0^1 {\mathbbm}1_{K_\ell^c} (s, Z^A_s(\omega) + h_m(s)) \underbrace{\left| b(s, Z^A_s(\omega) + h_m(s)) - \bar b_\ell(s, Z^A_s(\omega) + h_m(s)) \right|_H}_{\leq 2} {\,\mathrm{d}}s$$
$$\leq 2 \underbrace{\int\limits_0^1 {\mathbbm}1_{K_\ell^c} (s, Z^A_s(\omega) + h_m(s)) {\,\mathrm{d}}s}_{\leq {\varepsilon}_\ell \text{ by \eqref{APPROX-THM-INVOC}}} .$$
In conclusion we have
$$\lim\limits_{m\rightarrow\infty} \left| \int\limits_0^1 b(s, Z^A_s(\omega) + h_m(s)) - b(s, Z^A_s + h(s)) {\,\mathrm{d}}s \right|_H$$
$$\leq \lim\limits_{m\rightarrow\infty} \left| \int\limits_0^1 b(s, Z^A_s(\omega) + h_m(s)) - \bar b_\ell(s, Z^A_s + h_m(s)) {\,\mathrm{d}}s \right.$$
$$\hspace{13mm} + \left. \int\limits_0^1 \bar b_\ell(s, Z^A_s(\omega) + h_m(s)) - b(s, Z^A_s(\omega) + h(s)) {\,\mathrm{d}}s \right|_H .$$
Using the above calculation this is bounded from above by
$$2 {\varepsilon}_\ell + \lim\limits_{m\rightarrow\infty} \left| \int\limits_0^1 \bar b_\ell(s, Z^A_s(\omega) + h_m(s)) - b(s, Z^A_s(\omega) + h(s)) {\,\mathrm{d}}s \right|_H .$$
Since $\bar b_\ell$ is continuous and $h_m$ converges pointwise to $h$ this is the same as
$$2 {\varepsilon}_\ell + \left| \int\limits_0^1 \bar b_\ell(s, Z^A_s(\omega) + h(s)) {\,\mathrm{d}}s - b(s, Z^A_s(\omega) + h(s)) {\,\mathrm{d}}s \right|_H$$
$$\leq 2 {\varepsilon}_\ell + \int\limits_0^1 {\mathbbm}1_{K_\ell^c} (s, Z^A_s(\omega) + h(s)) \underbrace{\left| \bar b_\ell(s, Z^A_s(\omega) + h(s)) - b(s, Z^A_s(\omega) + h(s)) \right|_H}_{\leq 2} \! {\,\mathrm{d}}s \leq 4 {\varepsilon}_\ell ,$$
where the last inequality follows by invoking inequality for $h_m$ replaced by $h$. Taking the limit $\ell\rightarrow\infty$ completes the proof of the assertion, since the left-hand side is independent of $\ell$.
Using the above Approximation Theorem we can now extend the estimates obtained in Section 2 to the whole space $Q$ as the following Corollary shows.
\[COR-SIGMA-RHO\]
For every $0 < {\varepsilon}< \frac16$ there exists $C_{\varepsilon}\in {\mathbb{R}}$ such that for every function $b\colon [0,1] \times H \longrightarrow H$ satisfying Assumption \[ASS\], $n\in{\mathbb{N}}$ and $k \in \{ 0 , ..., 2^n - 1 \}$ there exists a measurable set $A_{{\varepsilon},b,n,k} \in {\mathcal}G_{(k+1)2^{-n}} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},b,n,k}] \leq \frac{\varepsilon}3 e^{-n}$ such that
$${\mathbbm}1_{A_{{\varepsilon},b,n,k}^c} |\varphi_{n,k}(x)|_H \leq C_{\varepsilon}n^{\frac12 + \frac1\gamma} 2^{-n/2} \left( |x|_\infty + 2^{-2^n} \right)$$
holds for every $x\in 2Q$ and by setting
$$A_{{\varepsilon},b} := \bigcup\limits_{n=1}^\infty \bigcup\limits_{k=0}^{2^n-1} A_{{\varepsilon},b,n,k}$$
we have ${\mathbb{P}}[A_{{\varepsilon},b}] \leq {\varepsilon}$ with the property that
$${\mathbbm}1_{A_{\varepsilon}^c} |\varphi_{n,k}(x, y)|_H \leq C_{\varepsilon}\left[ \sqrt{n} 2^{-n/6} |x - y|_\infty + 2^{- 2^{\theta n}} \right]$$
holds for all $x,y \in 2Q$, $n \geq 1$ and $k \in \{ 0 , ..., 2^n - 1 \}$, where $\theta := \frac23 \frac{\gamma}{\gamma+2}$.
The first inequality follows from Theorem \[THM-SIGMA\] for all points $x \in 2Q \cap {\mathbb{D}}$. For general points $x \in 2Q$ this follows by approximating $2Q \cap {\mathbb{D}}\ni x_n \longrightarrow x$ and using Theorem \[THM-APPROX\].
The second inequality follows in the same way by combining Theorem \[THM-RHO\] and Theorem \[THM-APPROX\]. Note that the estimate can be trivially extended from points $x,y \in Q$ with $|x-y|_\infty \leq 1$ to $x,y \in 2Q$ by changing the constant $C_{\varepsilon}$ and using that $\varphi_{n,k}$ is a seminorm.
Observe that one can choose ($C_{\varepsilon}$ / $A_{{\varepsilon},b}$), so that the conclusion of Theorem \[THM-SIGMA\] and \[THM-RHO\] hold (with the same constant / one the same set).
Long-time Regularization by Noise via Euler Approximation
=========================================================
In this section we will prove estimates for terms of the type
$$\sum\limits_{q=1}^N |\varphi_{n,k+q}(x_{q+1}, x_q)|_H .$$
We will first prove a concentration of measure result for the above term in Lemma \[LEM-EULER\]. Using this we prove a ${\mathbb{P}}$-a.s. sure version of this estimate in Theorem \[THM-EULER\]. However, this estimate only holds for medium-sized $N$. By splitting the sum and using Theorem \[THM-EULER\] repetitively we conclude the full estimate in Corollary \[COR-GLUING\].
Note that applying Corollary \[COR-SIGMA-RHO\] to every term under the sum would result in an estimate of order ${\mathcal}O(\sqrt n 2^{-n/6} N)$. Since $N$ will later be chosen to be of order $2^n$ this is of no use. The technique to overcome this is two-fold:
On the one hand the $\varphi_{n,k+q}$ terms have to “work together” to achieve an expression of order ${\mathcal}O(N)$. However, since $\{ \varphi_{n,k+q}(x_q) | q=1, ..., N\}$ are “sufficiently uncorrelated” the law of large numbers tells us to expect on average an estimate of order ${\mathcal}O(\sqrt N)$.
On the other hand in later applications $x_q$ will be values from the solution of the integral equation $\text{(IE)}_{\omega}$, so that it is reasonable to assume that $|x_{q+1} - x_q|_H \approx |\varphi_{n,k+q}(x_q)|_H$. Exploiting this enables to use *both* of our previous established estimates for every $|\varphi_{n,k+q}(x_{q+1},x_q)|_H$ term.
Using both techniques we end up with an estimate of order ${\mathcal}O(2^{-n} N)$ (see Corollary \[COR-GLUING\]).
Let $(M_n, {\mathcal}F_n)_{n\in{\mathbb{N}}}$ be a real-valued martingale. For $2 \leq p < \infty$ we have
[ @write auxout ]{} $$({\mathbb E}|M_n|^p)^{1/p} \leq p ({\mathbb E}|{\langle}M {\rangle}_n^{p/2} )^{1/p} . \tag{\ref{BDG-1}}$$
In the celebrated paper [@Dav76 Section 3] it is shown that the optimal constant in our case is the largest positive root of the Hermite polynomial of order $2k$. We refer to the appendix of [@Ose12] for a discussion of the asymptotic of the largest positive root. See also [@Kho14 Appendix B], where a self-contained proof of the Burkholder–Davis–Gundy Inequality with asymptotically optimal constant can be found for the one-dimensional case.
[ \[LEMMA-MARTINGALE-2\] ]{}
Let $(M_n)_{n\in{\mathbb{N}}}$ be a martingale of the form
$$M_r := \sum\limits_{k=1}^r X_k$$
with ${\mathbb E}[X_k^{p}] \leq C^p p^p$ for all $k\in{\mathbb{N}}$ and $p\in[1,\infty[$ then
$${\mathbb E}\left[ \exp\left( \frac18 \left(\frac{M_r}{C \sqrt r}\right)^{1/2} \right) \right] \leq 2$$
holds for all $r\in{\mathbb{N}}$.
Let $(M_n)_{n\in{\mathbb{N}}}$ be a martingale. Using the Burkholder–Davis–Gundy Inequality for every $r,p\in{\mathbb{N}}$ with $p\geq 2$ we have
$$\begin{aligned}
{\mathbb E}[ M_r^p ] &\leq p^{p} {\mathbb E}[ {\langle}M {\rangle}_r^{p/2} ] = p^{p} {\mathbb E}\left[ \left( \sum\limits_{k=1}^r X_k^2 \right)^{p/2} \right] \\
&\leq p^{p} r^{p/2-1} {\mathbb E}\left[ \sum\limits_{k=1}^r X_k^{p} \right] \leq p^{p} r^{p/2-1} r C^p p^p = C^p r^{p/2} p^{2p} .\end{aligned}$$
In conclusion we obtain
[ @write auxout ]{} $${\mathbb E}[ M_r^p ] \leq C^p r^{p/2} p^{2p} \tag{\ref{MART2-3}}$$
for every $p\geq 2$. Furthermore, using inequality for $p=2$, we trivially have by Jensen’s Inequality
[ @write auxout ]{} $${\mathbb E}[ M_r^{1/2} ] \leq {\mathbb E}[ M_r^2 ]^{1/4} \leq C^{1/2} r^{1/4} , \tag{\ref{MART2-1}}$$
[ @write auxout ]{} $${\mathbb E}[ M_r^{1} ] \leq {\mathbb E}[ M_r^2 ]^{1/2} \leq C r^{1/2} 2^2 \tag{\ref{MART2-2}}$$
and
[ @write auxout ]{} $$\hspace{2mm} {\mathbb E}[ M_r^{3/2} ] \leq {\mathbb E}[ M_r^2 ]^{3/4} \leq C^{3/2} r^{3/4} 2^{3} . \tag{\ref{MART2-4}}$$
Hence, we obtain
$${\mathbb E}\left[ \exp\left( \frac18 \left(\frac{M_r}{C \sqrt r}\right)^{1/2} \right) \right] = \sum\limits_{p=0}^\infty 8^{-p} \frac{{\mathbb E}[ M_r^{p/2} ]}{p! C^{p/2} r^{p/4}} .$$
We split the sum for different $p$ and use the above inequalities , , and to bound the above expression by
$$1 + \underbrace{8^{-1} \frac{C^{1/2} r^{1/4}}{C^{1/2} r^{1/4}}}_{\leq 4^{-1}} + \underbrace{8^{-2} \frac{C r^{1/2} 2^2}{C r^{1/2}}}_{=4^{-2}} + \underbrace{8^{-3} \frac{C^{3/2} r^{3/4} 2^3}{C^{3/2} r^{3/4}}}_{= 4^{-3}} + \sum\limits_{p=4}^\infty \underbrace{8^{-p} \frac{(p/2)^p}{p!}}_{\leq 4^{-p} \frac{p^p}{p!}} \leq 1 + \sum\limits_{p=1}^\infty 4^{-p} \frac{p^p}{p!} .$$
Using Stirling’s Formula for $p\geq 1$
$$3 p^p e^{-p} \leq e^{\frac1{12p+1}} \sqrt{2\pi p} p^p e^{-p} \leq p!$$
and the above calculation we finally obtain
$${\mathbb E}\left[ \exp\left( \frac18 \left(\frac{M_r}{C \sqrt r}\right)^{1/2} \right) \right] \leq 1 + \frac13 \sum\limits_{p=1}^\infty 4^{-p} e^{p} \leq 2 .$$
[ \[LEM-EULER\] ]{}
Let $0 < {\varepsilon}< \frac16$, $(b_q)_{q\in{\mathbb{N}}}$ be a sequence of functions $b_q \colon [0,1] \times H \longrightarrow H$ each satisfying Assumption \[ASS\] then there exists a measurable set $A_{{\varepsilon},b} := A_{{\varepsilon},(b_q)_{q\in{\mathbb{N}}}} \subseteq \Omega$, an absolute constant $C \in {\mathbb{R}}$ and $N_{\varepsilon}\in {\mathbb{N}}$ such that for all $x_0 \in Q$, all $n\in{\mathbb{N}}$ with $n \geq N_{\varepsilon}$, all $r\in{\mathbb{N}}$ with $r \leq 2^{n/4}$, $k \in \{ 0 , ..., 2^n - r - 1 \}$ and for every $\eta > 0$ we have
$${\mathbb{P}}\left[ {\mathbbm}1_{A_{{\varepsilon},b}^c} \sum\limits_{q=1}^{r} |\varphi_{n,k+q}(b_q ; x_{q-1}, x_q)|_H > \eta C \left( 2^{-n} \sqrt r |x_0|_H + \sqrt r 2^{-2^{n}} \right) + C 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H \right] \leq 4 e^{-\eta^{1/2}} ,$$
where $x_{q+1} := x_q + \varphi_{n,k+q}(b_q ; x_q)$ for $q\in\{0,...,r-1\}$ is the Euler approximation.
Let $0 < {\varepsilon}< \frac16$, $n\in{\mathbb{N}}$ and $b_q\colon [0,1] \times H \longrightarrow H$ be as in the assertion. Using Corollary \[COR-SIGMA-RHO\] there exists $C_{\varepsilon}\in{\mathbb{R}}$ and $A_{{\varepsilon},b_q,n,k+q} \in {\mathcal}G_{(k+1)2^{-n}}$ with ${\mathbb{P}}[A_{{\varepsilon},b_q,n,k+q}] \leq 2^{-n/24}{\varepsilon}$ such that for all $x\in 2Q$ we have
[ @write auxout ]{} $$|\varphi_{n,k+q}(b_q ; x)|_H \leq C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( |x|_H + 2^{-2^n} \right) . \tag{\ref{THM-EULER-SIGMA}}$$
on $A_{{\varepsilon},b_q,n,k+q}^c$. Note that $x$ is allowed to be a random variable and we have used that $|\cdot|_\infty \leq |\cdot|_H$. We now set
$$N_{\varepsilon}:= \min \left\{ n\in{\mathbb{N}}\setminus \{0\} | C_{\varepsilon}n^{\frac12+\frac1\gamma} \leq 2^{n/4} \right\} .$$
Let, as in the assertion, be $n\in{\mathbb{N}}$ with $n \geq N_{\varepsilon}$, $r\leq 2^{n/4}$, $k \in \{ 0 , ..., 2^n - r - 1 \}$ and $x_0 \in Q$. Additionally, let $x_{q+1} := x_q + \varphi_{n,k+q}(b_q ; x_q)$ be the Euler approximation defined for $q \in \{0, ..., r-1\}$. We write $x_q = (x_q^{(i)})_{i\in{\mathbb{N}}}$ for the components of $x_q$ and for $q\in\{1,...,r\}$ we calculate
$$| x_q^{(i)} | \leq | x_{q-1}^{(i)} | + \left| \int\limits_{(k+q)2^{-n}}^{(k+q+1)2^{-n}} \!\!\!\! b_q^{(i)}(s , Z_s^A+x_q) - b^{(i)}_q(s , Z_s^A) {\,\mathrm{d}}s \right| \leq | x_{q-1}^{(i)} | + 2 \| b_q^{(i)} \|_\infty 2^{-n} .$$
Via induction on $q$ we deduce
$$| x_q^{(i)} | \leq | x_0^{(i)} | + 2q 2^{-n} \| b_q^{(i)} \|_\infty \leq
|x_0^{(i)}| + \underbrace{2r 2^{-n}}_{\leq 1} \| b_q^{(i)} \|_\infty .$$
and since both $x_q\in Q$ and by Assumption \[ASS\] $b$ takes values in $Q$ we conclude that $x_q \in 2Q$ for all $q \in \{1,...,r\}$. Note that $x_q$ is ${\mathcal}G_{(k+q)2^{-n}}$-measurable. Due to the fact that Inequality only holds on $A_{{\varepsilon},b_q,n,k}^c \subseteq \Omega$ we modify $x_q$ in the following way $$\begin{aligned}
{\hat}x_0 &:= x_0 , \\
{\hat}x_{q+1} &:= {\hat}x_q + {\mathbbm}1_{A_{{\varepsilon},b_q,n,k+q}^c} \varphi_{n,k+q}({\hat}x_q) .\end{aligned}$$
Observe that we lose the property that $x_{q+1} - x_q = \varphi_{n,k+q}(b_{n,k,q} ; x_q)$, but we still have ${\hat}x_q \in 2Q$ and
[ @write auxout ]{} $$|{\hat}x_{q+1} - {\hat}x_q|_H \leq |\varphi_{n,k+q}(b_q ; {\hat}x_q)|_H . \tag{\ref{THM-EULER-MODIFIED}}$$
Furthermore, the modified Euler approximation ${\hat}x_q$ is still ${\mathcal}G_{(k+q)2^{-n}}$-measurable. We set
$$A_{{\varepsilon},b} := A_{{\varepsilon},(b_q)_{q\in{\mathbb{N}}}} := \bigcup\limits_{n\in{\mathbb{N}}} \bigcup\limits_{k=0}^{2^n-1} \bigcup\limits_{q\in{\mathbb{N}}} A_{{\varepsilon},b_q,n,k}$$
in a similar way as in Corollary \[COR-SIGMA-RHO\]. We obviously have ${\mathbb{P}}[A_{{\varepsilon},b}] \leq {\varepsilon}$ and for the modified Euler approximation we obtain for every $q \in \{0, ..., r-1\}$
$$|{\hat}x_{q+1}|_H = |{\hat}x_q + {\mathbbm}1_{A_{{\varepsilon},b_q,n,k+q}^c} \varphi_{n,k+q}(b_{n,k,q} ; {\hat}x_q)|_H \leq |{\hat}x_q|_H + {\mathbbm}1_{A_{{\varepsilon},b_q,n,k+q}^c} |\varphi_{n,k+q}({\hat}x_q)|_H$$
and using Inequality for $x$ replaced by ${\hat}x_q$ and $C_{\varepsilon}n^{\frac12+\frac1\gamma} \leq 2^{n/4}$ this is bounded from above by
$$|{\hat}x_q|_H + C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( |{\hat}x_q|_H + 2^{-2^n} \right) \leq (1 + 2^{-n/4}) |{\hat}x_q|_H + 2^{-n/4} 2^{-2^n} .$$
By induction over $q \in \{ 0 , ..., r\}$ we have
$$|{\hat}x_q|_H \leq (1 + 2^{-n/4})^q |{\hat}x_0|_H + \sum\limits_{\ell=0}^{q-1} (1 + 2^{-n/4})^\ell 2^{-n/4} 2^{-2^n} .$$
Using that $q \leq r \leq 2^{n/4}$ this can be further estimated by
$$(1 + 2^{-n/4})^r |x_0|_H + r (1 + 2^{-n/4})^{r} 2^{-n/4} 2^{-2^n} \leq \underbrace{(1 + 2^{-n/4})^{2^{n/4}}}_{\leq e} |x_0|_H + \underbrace{r 2^{-n/4}}_{\leq 1} \underbrace{(1 + 2^{-n/4})^{2^{n/4}}}_{\leq e} 2^{-2^n} .$$
In conclusion we obtain
[ @write auxout ]{} $$|{\hat}x_q|_H \leq e \left( |x_0|_H + 2^{-2^n} \right) . \tag{\ref{THM-EULER-x_q}}$$
for all $q \in \{ 0 , ..., r\}$.\
For the next step we define
$$\hspace{-53mm} Y_q := |\varphi_{n,k+q}(b_q ; {\hat}x_{q-1}, {\hat}x_q)|_H ,$$ $$Z_q := {\mathbb E}[ Y_q | {\mathcal}G_{(k+q)2^{-n}} ] = {\mathbb E}[ |\varphi_{n,k+q}(b_q ; {\hat}x_{q-1}, {\hat}x_q)|_H | {\mathcal}G_{(k+q)2^{-n}} ] ,$$ $$\hspace{-78.5mm} X_q := Y_q - Z_q ,$$
and
$$\hspace{-81mm} M_\tau := \sum\limits_{q=1}^{r \wedge \tau} X_q .$$
with $\tau\in{\mathbb{N}}$. Note that $M_\tau$ is a ${\mathcal}G_{(k+\tau+1)2^{-n}}$-Martingale with $M_0 = 0$. Furthermore, for every $p\in{\mathbb{N}}$ we have the following bound of the increments of $M$
$${\mathbb E}[ |X_q|^p ] \leq 2^{p-1} {\mathbb E}[ |Y_q|^p + |Z_q|^p ] \leq 2^p {\mathbb E}[ |\varphi_{n,k+q}(b_q ; {\hat}x_{q-1}, {\hat}x_q)|_H^p ] .$$
Using Corollary [@Wre16 Corollary 3.2] and inequality this is bounded by
$$2^p 3 \beta_A^{p/2} p^{p/2} 2^{-pn/2} {\mathbb E}[ |{\hat}x_{q-1} - {\hat}x_q|_H^p ] \leq 2^p 3 \beta_A^{p/2} p^{p/2} 2^{-pn/2} {\mathbb E}[ |\varphi_{n,k+q-1}(b_{q-1} ; {\hat}x_{q-1})|_H^p ] .$$
Using Corollary [@Wre16 Corollary 3.2] again this is bounded by
$$2^p 9 \beta_A^p p^p 2^{-pn} {\mathbb E}[ |{\hat}x_{q-1} |_H^p ] \leq 18^p \beta_A^p p^p 2^{-pn} {\mathbb E}[ |{\hat}x_{q-1} |_H^p ] .$$
Applying inequality yields
$${\mathbb E}[ |X_q|_H^p ] \leq 18^p \beta_A^p p^p 2^{-pn} e^{p} \left( |x_0|_H + 2^{-2^{n}} \right)^p .$$
Note that $x_0$ is deterministic. Using this bound we invoke Lemma \[LEMMA-MARTINGALE-2\] with
$$C := 18 \beta_A 2^{-n} \left( |x_0|_H + 2^{-2^n} \right)$$
and hence we obtain the following bound for the Martingale $(M_\tau)_{\tau\in{\mathbb{N}}}$
[ @write auxout ]{} $${\mathbb E}\left[ \exp \left( \frac18 \left( \frac{r^{-1/2} 2^{n} M_r}{18 \beta_A \left(|x_0|_H + 2^{-2^n}\right)} \right)^{1/2} \right) \right] \leq 2 . \tag{\ref{THM-EULER-M}}$$
In a similar way as $(X_q, Y_q, Z_q, M_\tau)$ we define
$$V_q := {\mathbb E}[ Z_q | {\mathcal}G_{(k+q-1)2^{-n}} ] ,$$ $$\hspace{-18mm} W_q := Z_q - V_q ,$$
and
$$\hspace{-19mm} M'_\tau := \sum\limits_{\tau=1}^{r \wedge \tau} W_q .$$
Observe that $M'_\tau$ is a ${\mathcal}G_{(k+\tau)2^{-n}}$-Martingale and in a completely analogous way as above we obtain
[ @write auxout ]{} $${\mathbb E}\left[ \exp \left( \frac18 \left( \frac{r^{-1/2} 2^{n} M'_r}{18 \left(|x_0|_H + 2^{-2^n}\right)} \right)^{1/2} \right) \right] \leq 2 . \tag{\ref{THM-EULER-M'}}$$
Let us now consider the term $V_q$
$$V_q = {\mathbb E}[ Z_q | {\mathcal}G_{(k+q-1)2^{-n}} ] = {\mathbb E}[ {\mathbb E}[ |\varphi_{n,k+q}(b_q ; {\hat}x_{q-1}, {\hat}x_{q})|_H | {\mathcal}G_{(k+q)2^{-n}} ] | {\mathcal}G_{(k+q-1)2^{-n}} ]$$
Using Corollary [@Wre16 Corollary 3.2] for $p=1$ and inequality this is bounded by
$$3 \beta_A^{1/2} 2^{-n/2} {\mathbb E}[ |{\hat}x_{q-1} - {\hat}x_{q}|_H | {\mathcal}G_{(k+q-1)2^{-n}} ] \leq 3 \beta_A^{1/2} 2^{-n/2} {\mathbb E}[ |\varphi_{n,k+q-1}(b_{q-1} ; {\hat}x_{q-1})|_H | {\mathcal}G_{(k+q-1)2^{-n}} ] .$$
Invoking Corollary [@Wre16 Corollary 3.2] again this can be further bounded from above by
$$9 \beta_A 2^{-n} {\mathbb E}[ |{\hat}x_{q-1}|_H | {\mathcal}G_{(k+q-1)2^{-n}} ] = 9 \beta_A 2^{-n} |{\hat}x_{q-1}|_H .$$
This leads us to
[ @write auxout ]{} $$\sum\limits_{q=1}^r V_q \leq 9 \beta_A 2^{-n} \sum\limits_{q=0}^{r-1} |{\hat}x_{q}|_H . \tag{\ref{THM-EULER-V}}$$
For notational ease we set $C' := 18 \beta_A$. Finally, starting from the left-hand side of the assertion and using $Y_q = X_q + W_q + V_q$ we get for every $\eta > 0$
$$\hspace{4mm} {\mathbb{P}}\left[ {\mathbbm}1_{A_{{\varepsilon},b}^c} \sum\limits_{q=1}^r \right. |\varphi_{n,k+q}(b_q ; x_{q-1}, x_q)|_H \left. > \eta C' \left( 2^{-n} \sqrt r |x_0|_H + \sqrt r 2^{-2^{n}} \right) + C' 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H \right]$$
$$\hspace{-0.5mm} \leq {\mathbb{P}}\left[ \sum\limits_{q=1}^r \right. {\mathbbm}1_{A_{{\varepsilon},b}^c} \underbrace{|\varphi_{n,k+q}(b_q ; {\hat}x_{q-1}, {\hat}x_q)|_H}_{=Y_q = X_q + W_q + V_q} \left. > \eta C' \left( 2^{-n} \sqrt r |x_0|_H + \sqrt r 2^{-2^{n}} \right) + C' 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H \right]$$
$$\hspace{-14mm} \leq \underbrace{{\mathbb{P}}\left[ \sum\limits_{q=1}^r V_q > C' 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H \right] }_{=0 \text{ by } \eqref{THM-EULER-V}} + {\mathbb{P}}\left[ \sum\limits_{q=1}^r X_q+W_q > \eta C' \sqrt r \left( 2^{-n} |x_0|_H + 2^{-2^{n}} \right) \right]$$
$$\leq {\mathbb{P}}\left[ \vphantom{\sum\limits_{q=1}^r} \right. \underbrace{\sum\limits_{q=1}^r X_q}_{=M_r} > C' \eta \sqrt r \left( 2^{-n} |x_0|_H + 2^{-2^{n}} \right) \left. \vphantom{\sum\limits_{q=1}^r} \right] + {\mathbb{P}}\left[ \vphantom{\sum\limits_{q=1}^r} \right. \underbrace{\sum\limits_{q=1}^r W_q}_{=M'_r} > C' \eta \sqrt r \left( 2^{-n} |x_0|_H + 2^{-2^{n}} \right) \left. \vphantom{\sum\limits_{q=1}^r} \right]$$
$$= {\mathbb{P}}\left[ \frac{r^{-1/2} 2^{n}}{C' \left(|x_0|_H + 2^{-2^{n}}\right)} M_r > \eta \right] + {\mathbb{P}}\left[ \frac{r^{-1/2} 2^{n}}{C' \left(|x_0|_H + 2^{-2^{n}} \right)} M'_r > \eta \right]$$
By applying the increasing function $x\mapsto \exp( x^{1/2})$ to both sides and using Chebyshev’s Inequality this can be bounded from above by
$$\exp(- \eta^{1/2} ) \left( {\mathbb E}\left[ \exp\left( \frac{r^{-1/2} 2^{n}}{C' \left(|x_0|_H + 2^{-2^{n}}\right)} M_r \right)^{1/2} + \exp \left( \frac{r^{-1/2} 2^{n}}{C' \left( |x_0|_H + 2^{-2^{n}} \right)} M'_r \right)^{1/2} \right] \right) .$$
Using inequality and we can conclude that
$${\mathbb{P}}\left[ {\mathbbm}1_{A_{{\varepsilon},b}^c} \sum\limits_{q=1}^r |\varphi_{n,k+q}(b_q ; x_{q-1}, x_q)|_H > \eta C' \left( 2^{-n} \sqrt r |x_0|_H + \sqrt r 2^{-2^{n}} \right) + C' 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H \right] \leq 4 e^{- \eta^{1/2} } ,$$
which completes the proof.
\[THM-EULER\]
For every $0 < {\varepsilon}< \frac1{40}$ there exist $C_{\varepsilon}\in {\mathbb{R}}$, $\Omega_{{\varepsilon},b} \subseteq \Omega$ with ${\mathbb{P}}[\Omega_{{\varepsilon},b}^c] \leq {\varepsilon}$ and $N_{\varepsilon}\in{\mathbb{N}}$ such that for all sequences $(b_q)_{q\in{\mathbb{N}}}$ of functions $b_q \colon [0,1] \times H \longrightarrow H$ with $b_q$ fulfilling Assumption \[ASS\] for all $q\in{\mathbb{N}}$, all $n\in{\mathbb{N}}$ with $n\geq N_{\varepsilon}$, $k\in \{0, ..., 2^n - r - 1\}$ and for all $y_0, ..., y_r \in Q$ we have
$$\sum\limits_{q=1}^r |\varphi_{n,k+q}(b_q ; y_{q-1}, y_q)|_H \leq C_{\varepsilon}\left[ 2^{-n} \max \left(r, n^{2+\frac2\gamma} \sqrt r\right) |y_0|_H + 2^{-n/24} \sum\limits_{q=0}^{r-1} |\gamma_{n,k,q}|_H + r 2^{-2^{\theta n}} \right] ,$$
on $\Omega_{{\varepsilon},b}$ for $1 \leq r \leq 2^{n/24}$, where $\gamma_{n,k,q} := y_{q+1} - y_q - \varphi_{n,k+q}(b_q ; y_q)$ for $q\in\{0,...,r-1\}$ is the error between $y_q$ and the Euler approximation .
**Step 1:**
Let $0 < {\varepsilon}< \frac1{40}$ and $C_{{\varepsilon}/2}$ the constant from Corollary \[COR-SIGMA-RHO\]. Similar to the proof of Lemma \[LEM-EULER\] we set
[ @write auxout ]{} $$N_{\varepsilon}:= \min \left\{ n\in{\mathbb{N}}\setminus\{0\} | C_{{\varepsilon}/2} n^{\frac12+\frac1\gamma} \leq 2^{13n/24} \right\} . \tag{\ref{THM-EULER-DE-N}}$$
For the sake of readability we write $b = (b_q)_{q\in{\mathbb{N}}}$. By Lemma \[LEM-EULER\] there is $A_{{\varepsilon}/2,b} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon}/2,b}] \leq \frac{\varepsilon}2$ and a constant $C\in{\mathbb{R}}$ such that for $x_{q+1} := x_q + \varphi_{n,k+q}(b_q ; x_q)$ and $x_0\in Q$ we have
[ @write auxout ]{} $${\mathbb{P}}\left[ \vphantom{\sum\limits_{q=1}^r} \right. \underbrace{{\mathbbm}1_{A_{{\varepsilon}/2,b}^c} \sum\limits_{q=1}^{r} |\varphi_{n,k+q}(b_q ; x_{q-1}, x_q)|_H > \eta C \sqrt r \left( 2^{-n} |x_0|_H + 2^{-2^{n}} \right) + C 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H}_{=: B_{{\varepsilon}/2,b,n,r,k} } \left. \vphantom{\sum\limits_{q=1}^r} \right] \leq 4 e^{-\eta^{1/2}} \tag{\ref{THM-EULER-LEMMA}}$$
for all $\eta > 0$. In order to obtain an almost sure bound we define
$$B_{{\varepsilon}/2,b} := \bigcup\limits_{n=N_{\varepsilon}}^\infty \bigcup\limits_{r=0}^{2^{n/24}} \bigcup\limits_{k=0}^{2^n-r-1} \bigcup\limits_{s=0}^{2^{2n}} \bigcup\limits_{x_0 \in Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}} B_{{\varepsilon}/2,b,n,r,k} .$$
Setting
$$\tilde \eta_{\varepsilon}:= \log \frac{40}{\varepsilon}\geq 1$$
and applying Lemma \[LEM-EULER\] in the form of inequality with $\eta := (1 + 2(3n)^{1+\frac1\gamma})^2 \tilde\eta_{\varepsilon}^2$ yields
$$\hspace{-28mm} {\mathbb{P}}\left[ B_{{\varepsilon}/2,b} \right] \leq 4 \sum\limits_{n=N_{\varepsilon}}^\infty \sum\limits_{r=0}^{2^{n/24}} \sum\limits_{k=0}^{2^n-r-1} \sum\limits_{s=0}^{2^{2n}} \sum\limits_{x_0 \in Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}} e^{-\eta^{1/2}}$$
$$\hspace{10mm} \leq 4 \sum\limits_{n=N_{\varepsilon}}^\infty 2^{n/24} 2^n \sum\limits_{s=0}^{2^{2n}} \#(Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}) e^{-2(3n)^{1+\frac1\gamma}} e^{-\tilde\eta_{\varepsilon}} .$$
Using Theorem \[KOLTIK\] this is smaller than
$$4 e^{-\tilde\eta_{\varepsilon}} \sum\limits_{n=N_{\varepsilon}}^\infty 2^{2n} \sum\limits_{s=0}^{2^{2n}} (2\cdot2^n + 1)^{{\operatorname{ed}}(Q_s)_{s+n}} e^{-2(3n)^{1+\frac1\gamma}}$$
and by invoking Lemma \[EFFDIM\] this can be again bounded by
$$\hspace{3mm} 4 e^{-\tilde \eta_{\varepsilon}} \sum\limits_{n=N_{\varepsilon}}^\infty 2^{2n} \sum\limits_{s=0}^{2^{2n}} (2\cdot2^n + 1)^{\ln (s+n+1)^{1/\gamma}} e^{-2(3n)^{1+\frac1\gamma}}$$
$$\leq 4 e^{-\tilde \eta_{\varepsilon}} \sum\limits_{n=N_{\varepsilon}}^\infty 2^{4n} (2\cdot 2^n + 1)^{\ln (1 + 2^{2n} + n)^{1/\gamma}} e^{-2(3n)^{1+\frac1\gamma}} \leq 4 e^{-\tilde \eta_{\varepsilon}} \sum\limits_{n=N_{\varepsilon}}^\infty 2^{4n} (3^n)^{ (3n)^{1/\gamma} } e^{-2(3n)^{1+\frac1\gamma}}$$
$$\hspace{10mm} = 4 e^{-\tilde \eta_{\varepsilon}} \sum\limits_{n=N_{\varepsilon}}^\infty 2^{4n} \underbrace{3^{(3 n)^{\frac1\gamma}} e^{-(3n)^{1+\frac1\gamma}}}_{\leq1} e^{-(3n)^{1+\frac1\gamma}} \leq 4 e^{-\tilde \eta_{\varepsilon}} \underbrace{\sum\limits_{n=N_{\varepsilon}}^\infty 2^{4n} e^{-3n}}_{\leq 5} \leq 20 e^{-\tilde \eta_{\varepsilon}} = \frac{\varepsilon}2 .$$
Henceforth, ${\mathbb{P}}[B_{{\varepsilon}/2,b}] \leq \frac{\varepsilon}2$. We set $\Omega_{{\varepsilon},b} := A_{{\varepsilon}/2,b}^c \cap B_{{\varepsilon}/2,b}^c$. Note that ${\mathbb{P}}[\Omega_{{\varepsilon},b}^c] \leq {\varepsilon}$.
In conclusion there exists $C_{\varepsilon}\in{\mathbb{R}}$ such that for all $n\geq N_{\varepsilon}$, $r \leq 2^{n/24}$, $k \in \{0,...,2^n-r-1\}$ and $x_0 \in Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}$ with $s \in \{ 0, ..., 2^{2n} \}$
[ @write auxout ]{} $$\sum\limits_{q=1}^r |\varphi_{n,k+q}(b_q ; x_{q-1}, x_q)|_H \leq C_{\varepsilon}\left[ n^{2+\frac2\gamma} 2^{-n} \sqrt r |x_0|_H + 2^{-n} \sum\limits_{q=0}^{r-1} |x_q|_H + r 2^{-2^{\theta n}} \right] \tag{\ref{THM-EULER-STEP-1}}$$
holds on $\Omega_{{\varepsilon},b}$ with $x_q := x_q + \varphi_{n,k+q}(x_q)$. Recall that $\theta := \frac23 \frac{\gamma}{\gamma+2}$ and note that we have $\theta \leq \frac23$.\
**Step 2:**
Let $n$, $k$, $r\in{\mathbb{N}}$ and $y_0$, ..., $y_r \in Q$ be as in the statement of this theorem. From now on fix an $\omega\in\Omega_{{\varepsilon},b}$. Let $s$ be the largest integer in $\{ 0, ..., 2^{2n} \}$ such that
$$|y_0|_H \leq 2^{-s}$$
holds. This implies that $y_0 \in Q_s$. Since $s$ is maximal with the above property we have
$$2^{-(s+1)} < |y_0|_H \qquad \text{or} \qquad |y_0|_H \leq 2^{-s} = 2^{-2^{2n}}$$
and hence
[ @write auxout ]{} $$2^{-s} \leq \max( 2|y_0|_H, 2^{-2^{2n}} ) \leq 2|y_0|_H + 2^{-2^{2n}} . \tag{\ref{THM-EULER-2S}}$$
Since $y_0\in Q_s$ we can construct $z_0 \in Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}$, which is close to $y_0$, in the following way: Set $d:=\ln(2s+2n)^{1/\gamma}$. For the components $i < d$ we choose $z_0$ so that
[ @write auxout ]{} $$|y_0^{(i)} - z_0^{(i)}| \leq 2^{-s-n} \tag{\ref{THM-EULER-FIRST-COMP}}$$
and $z_0^{(i)} := 0$ for $i \geq d$. The distance between $y_0$ and $z_0$ can now be estimated by
$$|y_0 - z_0|_H^2 = \sum\limits_{0\leq i<d} |y_0^{(i)} - z_0^{(i)}|^2 + \sum\limits_{d \leq i<\infty} |y_0^{(i)}|^2 .$$
Using the above inequality and the fact that $y_0\in Q$ this can be estimated by
$$d 2^{-2s-2n} + \sum\limits_{i=d}^{\infty} 4 \exp\left(-2 e^{i^\gamma} \right) \leq d 2^{-2s-2n} + 4 \underbrace{\exp\left(-e^{d^\gamma} \right)}_{=e^{-2s-2n}} \underbrace{\sum\limits_{i=0}^\infty \exp\left(-e^{i^\gamma} \right)}_{=: C_\gamma^2 < \infty} ,$$
where we have used $\exp(-2e^{i^\gamma}) \leq \exp(-e^{d^\gamma}) \exp(-e^{i^\gamma})$ in the last step. Therefore, we get
$$|y_0 - z_0|_H \leq 2C_\gamma \sqrt{\ln(2s+2n)^{1/\gamma}} 2^{-s-n}$$
and hence by inequality we obtain
$$|y_0 - z_0|_H \leq 2C_\gamma \sqrt{\ln(2s+2n)^{1/\gamma}} \left( 2^{1-n} |y_0|_H + 2^{-n} 2^{-2^{2n}} \right)$$
$$\leq 4C_\gamma \sqrt{\ln(2^{2n+1}+2n)^{1/\gamma}} \left( 2^{-n} |y_0|_H + 2^{-n} 2^{-2^{2n}} \right) \leq 4C_\gamma \sqrt{\ln(2^{4n})^{1/\gamma}} \left( 2^{-n} |y_0|_H + 2^{-n} 2^{-2^{2n}} \right)$$
$$\hspace{7.5mm} = 4C_\gamma \sqrt{ \frac{(\log_2(2^{4n}))^{1/\gamma}}{\ln(2)^{1/\gamma}} } \left( 2^{-n} |y_0|_H + 2^{-n} 2^{-2^n} \right) = 4C_\gamma \sqrt{ \frac{(4n)^{1/\gamma}}{\ln(2)^{1/\gamma}} } \left( 2^{-n} |y_0|_H + 2^{-n} 2^{-2^n} \right) .$$
In conclusion we have
[ @write auxout ]{} $$|y_0 - z_0|_H \leq \tilde C_\gamma \left( n^{\frac1{2\gamma}} 2^{-n} |y_0|_H + 2^{-2^n} \right) . \tag{\ref{THM-EULER-Y_0-Z_0}}$$
We define $z_1, ..., z_r$ recursively by
$$z_{q+1} := z_q + \varphi_{n,k+q}(b_q ; z_q) .$$
Note that $z_0, ..., z_q$ are deterministic since we have fixed $\omega$. Using the definition of $z_q$ we have
$$|z_{q+1}|_H \leq |z_q|_H + |\varphi_{n,k+q}(b_q ; z_q)|_H .$$
Recall that $\omega\in\Omega_{{\varepsilon},b} \subseteq A_{{\varepsilon}/2,b}^c$ and hence we can invoke the conclusion of Corollary \[COR-SIGMA-RHO\], so that the above expression is bounded from above by
$$|z_q|_H + C_{{\varepsilon}/2} n^{\frac12+\frac1\gamma} 2^{-n/2} ( |z_q|_H + 2^{-2^n}) \leq (1 + 2^{-n/24}) |z_q|_H + 2^{-n/24} 2^{-2^n} ,$$
where we have used Definition to conclude that $C_{{\varepsilon}/2} n^{\frac12+\frac1\gamma} 2^{-n/2} \leq 2^{n/24}$. By induction on $q \in \{ 1, ..., r-1 \}$ and using $r \leq 2^{n/24}$ we obtain
$$|z_q|_H \leq (1 + 2^{-n/24})^q |z_0|_H + \sum\limits_{\ell=0}^{q-1} (1 + 2^{-n/24})^\ell 2^{-n/24} 2^{-2^{n}}$$
$$\leq \underbrace{(1 + 2^{-n/24})^r}_{\leq e} |z_0|_H + \underbrace{(1 + 2^{-n/24})^r}_{\leq e} \underbrace{r 2^{-n/24}}_{\leq 1} 2^{-2^{n}} \leq e \left( |z_0|_H + 2^{-2^{n}} \right) .$$
Since $z_0$, ..., $z_r$ is by definition an Euler approximation and $z_0 \in Q_s \cap 2^{-(s+n)} {\mathbb{Z}}^{\mathbb{N}}$ the conclusion of Step 1 (inequality ) with $x_q$ replaced by $z_q$ holds and we obtain that
$$\hspace{-23mm} \sum\limits_{q=1}^{r} |\varphi_{n,k+q}(z_{q-1}, z_q)|_H \leq C_{\varepsilon}\left[ n^{2+\frac2\gamma} 2^{-n} \sqrt r |z_0|_H + r 2^{-2^{\theta n}} + 2^{-n} \sum\limits_{q=0}^{r-1} |z_q|_H \right]$$
$$\hspace{35mm} \leq C_{\varepsilon}\left[ n^{2+\frac2\gamma} 2^{-n} \sqrt r |z_0|_H + r 2^{-2^{\theta n}} + 2^{-n} \sum\limits_{q=0}^{r-1} e ( |z_0|_H + 2^{-2^n} ) \right]$$
$$\hspace{30mm} \leq C_{\varepsilon}\left[ n^{2+\frac2\gamma} 2^{-n} \sqrt r |z_0|_H + r 2^{-2^{\theta n}} + 2^{-n} r e \left( |z_0|_H + 2^{-2^{n}} \right) \right]$$
$$\hspace{8mm} \leq eC_{\varepsilon}\left[ \max\left(n^{2+\frac2\gamma} \sqrt r, r\right) 2^{-n} |z_0|_H + r 2^{-2^{\theta n}} \right]$$
$$\hspace{35mm} \leq eC_{\varepsilon}\left[ \max\left(n^{2+\frac2\gamma} \sqrt r, r\right) 2^{-n} ( |y_0|_H + |y_0-z_0|_H ) + r 2^{-2^{\theta n}} \right] .$$
Applying inequality yields that the above expression is bounded from above by
$$eC_{\varepsilon}\left[ \max\left(n^{2+\frac2\gamma} \sqrt r, r\right) 2^{-n} \left( |y_0|_H + \tilde C_\gamma \left( \underbrace{n^{\frac1{2\gamma}} 2^{-n}}_{\leq1} |y_0|_H + 2^{-2^{n}} \right) \right) + r 2^{-2^{\theta n}} \right]$$
$$\hspace{36mm} \leq C_{{\varepsilon},\gamma} \left[ \max\left(n^{2+\frac2\gamma} \sqrt r, r\right) 2^{-n} |y_0|_H + r 2^{-2^{\theta n}} \right] .$$
Therefore we obtain
[ @write auxout ]{} $$\sum\limits_{q=1}^{r} |\varphi_{n,k+q}(z_{q-1}, z_q)|_H \leq C_{{\varepsilon},\gamma} \left[ \max\left(n^{2+\frac2\gamma} \sqrt r, r\right) 2^{-n} |y_0|_H + r 2^{-2^{\theta n}} \right] . \tag{\ref{THM-EULER-PHI-Z-Z}}$$
**Step 3:**\
**Claim:**
[ @write auxout ]{} $$\sum\limits_{q=1}^r |\varphi_{n,k+q}(z_q, y_q)|_H \leq C_{\varepsilon}' \left[ r 2^{-n} |y_0|_H + r 2^{-2^{\theta n}} + 2^{-n/24} \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right] . \tag{\ref{THM-EULER-PHI-Z-Y}}$$
**Proof of** :
We set $u_q := z_q - y_q$ for $q \in \{0, ..., r \}$ and bound the increments of $u_q$ in the following way.
$$|u_{q+1} - u_q|_H = |z_{q+1} - y_{q+1} - z_q + y_q|_H = | \varphi_{n,k+q}(b_q ; z_q) - y_{q+1} + y_q |_H$$
$$\hspace{8mm} \leq | \varphi_{n,k+q}(b_q ; z_q) - y_{q+1} + y_q + \gamma_{n,k,q} |_H + | \gamma_{n,k,q} |_H$$
$$\hspace{2mm} = | \varphi_{n,k+q}(b_q ; z_q) - \varphi_{n,k+q}(b_q ; y_q) |_H + | \gamma_{n,k,q} |_H .$$
$$\hspace{-22mm} = |\varphi_{n,k+q}(b_q ; z_q, y_q)|_H + | \gamma_{n,k,q} |_H$$
We therefore deduce that
$$|u_{q+1}|_H \leq |u_{q+1} - u_q|_H + |u_q|_H \leq | \varphi_{n,k+q}(b_q ; z_q, y_q) |_H + | \gamma_{n,k,q} |_H + |u_q|_H .$$
By the conclusion of Corollary \[COR-SIGMA-RHO\] and Definition this is bounded by
$$C_{{\varepsilon}/2} \left( \vphantom{2^{-2^{\theta n}}} \right. \sqrt{n} 2^{-n/6} \underbrace{|z_q-y_q|_H}_{=|u_q|_H} + 2^{-2^{\theta n}} \left. \vphantom{2^{-2^{\theta n}}} \right) + | \gamma_q |_H + |u_q|_H \leq ( 1 + 2^{-n/24} ) |u_q|_H + C_{{\varepsilon}/2} 2^{-2^{\theta n}} + | \gamma_{n,k,q} |_H .$$
Induction on $q \in \{0,...,r\}$ yields
$$|u_q|_H \leq C_{{\varepsilon}/2} ( 1 + 2^{-n/24} )^r \left( |u_0|_H + r 2^{-2^{\theta n}} + \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right)$$
$$\hspace{-9mm} \leq e C_{{\varepsilon}/2} \left( |u_0|_H + r 2^{-2^{\theta n}} + \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right) .$$
Using inequality together with the above calculation yields
$$|u_q|_H \leq e C_{{\varepsilon}/2} \left( \tilde C_\gamma n^{\frac1{2\gamma}} 2^{-n} |y_0|_H + 2r 2^{-2^{\theta n}} + \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right)$$
and hence by combining this estimate with Corollary \[COR-SIGMA-RHO\] we have
$$\hspace{-10mm} |\varphi_{n,k+q}(z_q, y_q)|_H \leq C_{{\varepsilon}/2} \left( \sqrt n 2^{-n/6} |z_q-y_q|_H + 2^{-2^{\theta n}} \right) \leq C_{{\varepsilon}/2} \left( 2^{-n/12} |u_q|_H + 2^{-2^{\theta n}} \right)$$
$$\hspace{27mm} \leq eC_{{\varepsilon}/2}^2 2^{-n/12} \left( \tilde C_\gamma n^{\frac1{2\gamma}} 2^{-n} |y_0|_H + 2r 2^{-2^{\theta n}} + \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right) + C_{{\varepsilon}/2} 2^{-2^{\theta n}} .$$
In conclusion since $r\leq 2^{n/24}$ we obtain
$$|\varphi_{n,k+q}(z_q, y_q)|_H \leq C_{\varepsilon}' \left[ 2^{-n} |y_0|_H + 2^{-2^{\theta n}} + 2^{-n/12} \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right]$$
and hence summing over $q=1,...,r$ and using again that $r\leq 2^{n/24}$ complete the proof of Claim .
**Step 4:**
Finally, using the identity $y_{q-1} - y_q = y_{q-1} - z_{q-1} + z_{q-1} - z_q + z_q - y_q$ the left-hand side of the assertion can be bounded as follows
$$\!\! \sum\limits_{q=1}^r \! |\varphi_{n,k+q}( b_q ; y_{q-1}, y_q )|_H \leq \sum\limits_{q=1}^r \! |\varphi_{n,k+q}( b_q ; y_{q-1}, z_{q-1} )|_H + |\varphi_{n,k+q}( b_q ; z_{q-1}, z_q )|_H + |\varphi_{n,k+q}( b_q ; z_q, y_q )|_H .$$
Applying inequalities , and with $z_q$, $y_q$ replaced by $z_{q-1}$, $y_{q-1}$ respectively yields that this is bounded by
$$C_{\varepsilon}'' \left[ 2^{-n} \max \left(r, n^{2+\frac2\gamma} \sqrt r\right) |y_0|_H + r 2^{-2^{\theta n}} + 2^{-n/24} \sum\limits_{q=0}^{r-1} | \gamma_{n,k,q} |_H \right] .$$
[ \[COR-GLUING\] ]{}
For every $0 < {\varepsilon}< \frac1{40}$ there exists $C_{\varepsilon}\in {\mathbb{R}}$ such that for every sequence $(b_q)_{q\in{\mathbb{N}}}$ of Borel measurable functions $b_q \colon [0,1] \times H \longrightarrow H$ satisfying Assumption \[ASS\] there exists a measurable set $\Omega_{{\varepsilon},(b_q)_{q\in{\mathbb{N}}}} \subseteq \Omega$ with ${\mathbb{P}}[\Omega_{{\varepsilon},(b_q)_{q\in{\mathbb{N}}}}^c] \leq {\varepsilon}$ such that for all sufficiently large $n\in{\mathbb{N}}$, $N \in{\mathbb{N}}$ with $N \leq 2^n$, $k\in \{0, ..., 2^n - N \}$ and for all $x_q \in Q$ for $q \in \{ 0 ,..., N \}$, we have
$$\sum\limits_{q=0}^{N-1} |\varphi_{n,k+q}(b_q ; x_{q+1}, x_q)|_H \leq C_{\varepsilon}\left[ 2^{-n} \sum\limits_{q=0}^{N} |x_q|_H + 2^{-3n/4} |x_0|_H + 2^{-n/24} \sum\limits_{q=0}^{N-1} |\gamma_{n,k,q}|_H + N 2^{-2^{\theta n}} \right] ,$$
on $\Omega_{{\varepsilon},b}$, where $\gamma_{n,k,q} := x_{q+1} - x_q - \varphi_{n,k+q}(b_q ; x_q)$ is the error between $x_{q+1}$ and the Euler approximation for $x_{q+1}$ given $x_q$.
We set $r := \lfloor 2^{n/24} \rfloor$. For the sake of notional ease we set $x_{q'} = 0$ whenever $q'>N$. In order to estimate the left-hand side of the assertion we will use Theorem \[THM-EULER\]. To this end we split the sum into $s$ pieces of size $r$. Choose $i\in\{ 0, ..., r-1 \}$ such that
$$\sum\limits_{t=0}^{\lfloor r^{-1} N \rfloor} |x_{i + tr}|_H \leq \frac1r \sum\limits_{q=0}^{r-1} \sum\limits_{t=0}^{\lfloor r^{-1} N \rfloor} |x_{q +tr}|_H$$
holds. Since we calculate the mean of $\sum\limits_{t=0}^{\lfloor r^{-1} N \rfloor} |x_{q + tr}|_H$ on the right-hand side, it is clear that such an $i$ always exists. Set $s := \lfloor r^{-1} (N - i) \rfloor$ and note that $s\leq \lfloor r^{-1} N \rfloor$. Using this we have
$$\sum\limits_{t=0}^s |x_{i + tr}|_H \leq \frac1r \sum\limits_{q=0}^{r-1} \sum\limits_{t=0}^{\lfloor r^{-1} N \rfloor} |x_{q +tr}|_H .$$
Hence, we obtain
[ @write auxout ]{} $$\sum\limits_{t=0}^s |x_{i + tr}|_H \leq r^{-1} \sum\limits_{q=0}^{N-1} |x_q|_H . \tag{\ref{COR-GLUING-PSI}}$$
Starting with the left-hand side of the assertion we split the sum into three parts. The first part contains the terms $x_q$ for $q=0$ to $q=i$. Since $i \leq r \leq 2^{n/24}$ this can be handled by applying Theorem \[THM-EULER\] directly. The second part contains $s$ sums of size $r$. Here, Theorem \[THM-EULER\] is applicable for every term of the outer sum running over $t$. The last part can be handled, in the same way as the first part, by directly applying Theorem \[THM-EULER\]. This strategy yields
$$\hspace{-50mm} \sum\limits_{q=0}^{N-1} |\varphi_{n,k+q}( b_q ; x_{q+1}, x_q )|_H = \sum\limits_{q=0}^{i-1} |\varphi_{n,k+q}( b_q ; x_{q+1}, x_q )|_H$$ $$\hspace{10mm} \qquad\qquad\qquad + \sum\limits_{t=0}^{s-1} \sum\limits_{q=0}^{r-1} |\varphi_{n,k + i + tr + q}( b_q ; x_{q+1 + i + tr}, x_{q + i + tr} )|_H$$ $$\hspace{13mm} \qquad\qquad\qquad + \sum\limits_{q=0}^{N-i-rs-1} |\varphi_{n,k + i + sr +q}( b_q ; x_{q+1 + i + sr}, x_{q + i + sr} )|_H$$
$$\hspace{-25mm} \leq C_{\varepsilon}\left[ 2^{-n} \max \left(r, n^{2+\frac2\gamma} \sqrt r\right) |x_0|_H + 2^{-n/24} \sum\limits_{q=0}^{i-1} |\gamma_{n,k,q}|_H + r 2^{-2^{\theta n}} \right]$$
$$\hspace{-24mm} + C_{\varepsilon}\sum\limits_{t=0}^{s-1} \left[ 2^{-n} r |x_{i + tr}|_H + 2^{-n/24} \sum\limits_{q=0}^{r-1} |\gamma_{n,k,i + tr + q}|_H + r 2^{-2^{\theta n}} \right]$$
$$\hspace{-1mm} \quad\qquad + C_{\varepsilon}\left[ 2^{-n} \max \left(r, n^{2+\frac2\gamma} \sqrt r\right) |x_{i + sr}|_H + 2^{-n/24} \sum\limits_{q=0}^{N-i-rs-1} |\gamma_{n,k,i + sr + q}|_H + r 2^{-2^{\theta n}} \right] .$$
$$\hspace{-7mm} \leq C_{\varepsilon}\left[ 2^{-n} r |x_0|_H + 2^{-n} r \sum\limits_{t=0}^s |x_{i + tr}|_H + 2^{-n/24} \sum\limits_{q=0}^{N-1} |\gamma_{n,k,q}|_H + (s+2) r 2^{-2^{\theta n}} \right] .$$
Estimating this further by using inequality and $r \leq 2^{n/24}$ yields the following bound
$$2 C_{\varepsilon}\left[ 2^{-3n/4} |x_0|_H + 2^{-n} \sum\limits_{q=0}^{N-1} |x_q|_H + 2^{-n/24} \sum\limits_{q=0}^{N-1} |\gamma_{n,k,q}|_H + N 2^{-2^{\theta n}} \right] ,$$
which completes the proof.
Proof of the main result
========================
In this section we are going to formulate a $\log$-type Gronwall inequality of the form
$$f((j+1)2^{-n}) - f(j2^{-n}) \leq C 2^{-n} f(j2^{-n}) \log(1/f(j2^{-n}))$$ $$\Rightarrow f(j2^{-n}) \leq C f(0)$$
for $j \in \{0, ..., 2^n\}$.
In Lemma \[LEM-GRONWALL\] we prove this implication in an abstract setting. Using all our previous considerations we show in Theorem \[THM-FINAL\] that our function $u$ from Proposition \[PRO-GIRSANOV\] satisfies such a Gronwall inequality and hence has to coincide with the zero function (Corollary \[COR-FINAL\]).
[ \[LEM-GRONWALL\] ]{}
Let $K>0$, $m\in{\mathbb{N}}$ “sufficiently big” i.e. $K \leq \ln(2) 2^{m}$ and $0 < \beta_0, ..., \beta_{2^m} < 1$ and assume that
$$\Delta \beta_j \leq K 2^{-m} \beta_j \log_2(1/\beta_j), \qquad \forall j \in \{ 0, ..., 2^m-1 \}$$
holds, where $\Delta \beta_j := \beta_{j+1} - \beta_j$. Then, we have
$$\beta_j \leq \exp\left( \log_2(\beta_0) e^{-2K-1} \right), \qquad \forall j \in \{ 0, ..., 2^m \} .$$
For every $j \in \{ 0, ..., 2^m \}$ we define
$$\gamma_j := \log_2(1/\beta_j) .$$
By assumption we have
$$\hspace{-34mm} \gamma_{j+1} = - \log_2(\beta_{j+1}) \geq - \log_2( \beta_j + K 2^{-m} \beta_j \gamma_j)$$
$$\hspace{12mm} = - \log_2(\beta_j) - \log_2( 1 + K 2^{-m} \gamma_j)
= \gamma_j - \frac1{\ln 2} \ln( 1 + K 2^{-m} \gamma_j) .$$
Using the inequality $\ln(1+x) \leq x$ the above, and hence $\gamma_{j+1}$, is smaller than
$$\gamma_j \left( 1 - \frac{K}{\ln 2} 2^{-m} \right) .$$
By induction on $j\in \{0, ..., 2^m\}$ we obtain
$$\gamma_j \geq \gamma_0 \left( 1 - \frac{K}{\ln 2} 2^{-m} \right)^j .$$
Since $m$ is “sufficiently big” the term inside the brackets is in the interval $[0,1]$ so that $\gamma_j$ is bounded from below by
$$\gamma_0 \left( 1 - \frac{K}{\ln 2} 2^{-m} \right)^{2^m} \geq \gamma_0 e^{- K/\ln(2) - 1} \geq \gamma_0 e^{- 2K - 1} .$$
Plugging in the definition of $\gamma_j$ implies that
$$\log_2(1/\beta_j) \geq \log_2(1/\beta_0) e^{- 2K - 1} .$$
Isolating $\beta_j$ yields
$$\beta_j \leq \exp\left( \log_2(\beta_0) e^{- 2K - 1} \right) .$$
[ \[THM-FINAL\] ]{}
Let $0 < {\varepsilon}< \frac1{40}$ and $f$ be as in Assumption \[ASS\] then there exist $A_{{\varepsilon},f} \subseteq \Omega$, $K=K({\varepsilon})>0$ and $m_0 = m_0({\varepsilon})\in{\mathbb{N}}$ with ${\mathbb{P}}[A_{{\varepsilon},f}^c ]\leq {\varepsilon}$ such that for any function $u\in\Phi$ being a solution of equation for a fixed $\omega\in A_{{\varepsilon},f}$, for all integers $m$ with $m \geq m_0$, $j\in\{0,...,2^m-1\}$ and $\beta$ satisfying
$$2^{m-2^{\theta m}} \leq \beta \leq 2^{-2^{(\frac\theta2+\frac14)m}} \quad \text{and} \quad |u(j2^{-m})|_H \leq \beta$$
we have
$$|u((j+1)2^{-m})|_H \leq \beta \left( 1 + K2^{-m} \log_2(1/\beta) \right) .$$
Let $0 < {\varepsilon}< \frac1{40}$ and $f$ be as in the assertion. For all $n\in{\mathbb{N}}$ and $k\in\{0, ..., 2^n-1\}$ we set
$$b_{n,k}(t, x) := e^{-((k+1)2^{-n}-t)A} f(t,x) , \qquad \forall t \in [0,1] , \ x \in H .$$
Note that $b_{n,k}$ fulfills Assumption \[ASS\] since $|b_{n,k}(t,x)|_H \leq |f(t,x)|_H$. Choose $A_{{\varepsilon},f} \subseteq \Omega$ with ${\mathbb{P}}[A_{{\varepsilon},f}] \leq {\varepsilon}$ such that the conclusions of Corollary \[COR-SIGMA-RHO\], Theorem \[THM-APPROX\] and Corollary \[COR-GLUING\] hold with the same constant $C_{\varepsilon}\geq 1$ for all functions $b_{n,k}$ on $A_{{\varepsilon},f}^c$. We set
$$m_0 := \max\left( 3\log_2(584C_{\varepsilon}'), 24\log_2(72C_{\varepsilon}) , \frac{1}{2\theta-1} \right) ,$$
where $C_{\varepsilon}'$ will be defined later. Recall that we defined $\theta := \frac23 \frac\gamma{\gamma+2}$. Fix an $\omega\in A_{{\varepsilon},f}^c$, $m\geq m_0$, $j$, $u$ and $\beta$ as in the statement and suppose $|u(j2^{-m})|_H \leq \beta$. We set $N := 7 \lfloor \log_2(1/\beta) \rfloor$. Observe that
[ @write auxout ]{} $$m^{\frac12+\frac1\gamma} 2^{m/2} \leq 7 \cdot 2^{(\frac\theta2+\frac14)m} - 7 \leq N \leq 7 \cdot 2^{\theta m} \leq 7 \cdot 2^{2m/3} , \tag{\ref{THM-FINAL-DE-N}}$$
where we have used that $\frac12 < \theta \leq \frac23$ (due to $\gamma>6$ in Assumption \[ASS\] and $\theta := \frac23 \frac{\gamma}{\gamma+2}$). Suppose $u\in\Phi$ satisfies equation as stated in the assertion. We define for every $n\in{\mathbb{N}}$ and $t\in[0,1]$
$$u_n(t) := \sum\limits_{k=0}^{2^n-1} {\mathbbm}1_{[k2^{-n}, (k+1)2^{-n}[}(t) u(k2^{-n}) .$$
Note that $u_n$ converges pointwise to $u$ on $[0,1[$ and $u_n\in\Phi^*$ by construction and since $u\in\Phi$. Let $\alpha$ be the smallest real number such that
[ @write auxout ]{} $$\sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\! |u((k+1)2^{-n}) - u(k2^{-n})|_H \leq \alpha 2^{-m} \left[ N + n^{\frac12+\frac1\gamma} 2^{n/2} \right], \quad \forall n\in\{ m, ..., N\} \tag{\ref{THM-FINAL-DE-ALPHA}}$$
holds. I.e.
$$\alpha := \max\limits_{m \leq n \leq N} \frac{2^m}{N + n^{\frac12+\frac1\gamma} 2^{n/2}} \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\! |u((k+1)2^{-n}) - u(k2^{-n})|_H .$$
For $n\geq m$ we define
[ @write auxout ]{} $$\psi_n := \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\! |u(k2^{-n})|_H . \tag{\ref{THM-FINAL-DE-PSI}}$$
By splitting the sum in in two sums, one where $k$ is even and one where $k$ is odd, we can estimate $\psi_n$ by $\psi_{n-1}$. To this end let $n\in\{m+1,...,N\}$. We then have
$$\psi_n = \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\mid k}}}^{(j+1)2^{n-m}-1} |u(k2^{-n})|_H + \!\!\! \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\nmid k}}}^{(j+1)2^{n-m}-1} \!\!\!\! |u(k2^{-n})|_H$$
$$\leq \!\!\!\!\!\! \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\mid k}}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\!\! |u(k2^{-n})|_H + \!\!\!\!\!\!\! \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\nmid k}}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\!\! |u(k2^{-n}) - u((k-1)2^{-n})|_H + |u((k-1)2^{-n})|_H + |u((k+1)2^{-n}) - u(k2^{-n})|_H .$$
Since $k-1$ is even whenever $k$ is odd, rewriting the term $|u((k-1)2^{-n})|_H$ yields that the above equals
$$\hspace{-84mm} \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\mid k}}}^{(j+1)2^{n-m}-1} |u( k 2^{-n})|_H + |u( k 2^{-n})|_H$$
$$\hspace{10mm} + \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\nmid k}}}^{(j+1)2^{n-m}-1} |u(k2^{-n}) - u((k-1)2^{-n})|_H + |u((k+1)2^{-n}) - u(k2^{-n})|_H$$
$$= 2 \!\!\! \sum\limits_{k=j2^{n-m-1}}^{(j+1)2^{n-m-1}-1} \!\!\!\! |u(k 2^{-n+1})|_H + \! \sum\limits_{{\genfrac{}{}{0pt}{}{k=j2^{n-m}}{2\nmid k}}}^{(j+1)2^{n-m}-1} \!\!\! |u(k2^{-n}) - u((k-1)2^{-n})|_H + |u((k+1)2^{-n}) - u(k2^{-n})|_H$$
$$\hspace{-32mm} = 2 \! \sum\limits_{k=j2^{n-1-m}}^{(j+1)2^{n-1-m}-1} \! |u(k 2^{-(n-1)})|_H + \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\! |u((k+1)2^{-n}) - u(k2^{-n})|_H .$$
And, henceforth, since $n\in \{m+1,...,N\}$ using inequality and the definition of $\psi_n$ (equation ) we have the following bound
$$\psi_n \leq 2 \psi_{n-1} + \alpha 2^{-m} N + \alpha 2^{-m} n^{\frac12+\frac1\gamma} 2^{n/2} .$$
By induction we deduce
$$\hspace{-42mm} \psi_n \leq 2^{n-m} \psi_m + \sum\limits_{\ell=m+1}^n \alpha 2^{n-\ell-m} N + \sum\limits_{\ell=m+1}^n \alpha 2^{n-\ell-m} \ell^{\frac12+\frac1\gamma} 2^{\ell/2}$$
$$\hspace{11mm} \leq 2^{n-m} |u(j2^{-m})|_H + \alpha 2^{n-m} N \sum\limits_{\ell=m+1}^n 2^{-\ell} + \alpha 2^{n-m} \sum\limits_{\ell=m+1}^n 2^{-\ell/3} , \qquad \forall n\in\{m,...,N\} .$$
We use $|u(j2^{-m})|_H \leq \beta$ to bound the above by
$$2^{n-m} \left[ \beta + \alpha N \sum\limits_{\ell=m+1}^n 2^{-\ell} + \alpha \sum\limits_{\ell=m+1}^n 2^{-\ell/3} \right] \leq 2^{n-m} \left[ \beta + \alpha 2^{-m} N + \alpha 2^{-m/3} \right] .$$
Furthermore, using inequality i.e. $N \leq 7\cdot 2^{2m/3}$ we bound the above by
$$2^{n-m} \left[ \beta + 7 \alpha 2^{-m/3} + \alpha 2^{-m/3} \right] \leq 8 \cdot 2^{n-m} \left[ \beta + \alpha 2^{-m/3} \right] .$$
In conclusion we obtain
[ @write auxout ]{} $$\psi_n \leq 8\cdot 2^{n-m} \left( \beta + \alpha 2^{-m/3} \right), \qquad\qquad \forall n\in\{m,...,N\} . \tag{\ref{THM-FINAL-PSI-ESTIMATE}}$$
Since $u$ solves equation we have
$$\left| u((k+1)2^{-n}) - u(k2^{-n}) - \varphi_{n,k}(b_{n,k} ; u(k2^{-n})) \right|_H$$
$$= \left| u((k+1)2^{-n}) - u(k2^{-n}) - \int\limits_{k2^{-n}}^{{(k+1)2^{-n}}} b_{n,k}(t,Z_t^A(\omega) + u(k2^{-n})) - b_{n,k}(t,Z_t^A) {\,\mathrm{d}}t \right|_H$$
$$\hspace{-33mm} \overset{\eqref{DE-U}}= \left| \int\limits_0^{(k+1)2^{-n}} e^{-((k+1)2^{-n}-t)A} ( f(t, Z^A_t(\omega) + u(t)) - f(t, Z^A_t(\omega)) ) {\,\mathrm{d}}t \right.$$
$$\hspace{-25mm} - \int\limits_0^{k2^{-n}} e^{-(k2^{-n}-t)A} (f(t, Z^A_t(\omega) + u(t)) - f(t, Z^A_t(\omega))) {\,\mathrm{d}}t$$
$$\hspace{-20mm} - \left. \int\limits_{k2^{-n}}^{{(k+1)2^{-n}}} b_{n,k}(t,Z_t^A(\omega) + u(k2^{-n})) - b_{n,k}(t,Z_t^A(\omega)) ) {\,\mathrm{d}}t \right|_H$$
$$\hspace{-32mm} = \left| \int\limits_{k2^{-n}}^{(k+1)2^{-n}} e^{-((k+1)2^{-n}-t)A} (f(t, Z^A_t(\omega) + u(t)) - f(t, Z^A_t(\omega))) {\,\mathrm{d}}t \right.$$
$$\hspace{-28mm} - \int\limits_{k2^{-n}}^{{(k+1)2^{-n}}} b_{n,k}(t,Z_t^A(\omega) + u(k2^{-n})) - b_{n,k}(t,Z_t^A(\omega)) ) {\,\mathrm{d}}t$$
$$\hspace{18mm} + \left. \int\limits_0^{k2^{-n}} \left( e^{-((k+1)2^{-n} - t)A} - e^{-(k2^{-n} - t)A} \right) \cdot ( f(t,Z_t^A(\omega) + u(t)) - f(t,Z_t^A(\omega)) ) {\,\mathrm{d}}t \right|_H .$$
Using the definition of $b_{n,k}$ this can be simplified and bounded by
$$\hspace{-32mm} \left| \int\limits_{k2^{-n}}^{(k+1)2^{-n}} b_{n,k}(t, Z^A_t(\omega) + u(t)) - b_{n,k}(t, Z^A_t(\omega) + u(k2^{-n})) {\,\mathrm{d}}t \right|_H$$
$$\hspace{5mm} + \underbrace{\left| e^{-2^{-n} A} - 1 \right|_{\text{op}}}_{\leq C2^{-n}} \cdot \left| \int\limits_0^{k2^{-n}} e^{-(k2^{-n} - t)A} ( f(t,Z_t^A(\omega) + u(t)) - f(t,Z_t^A(\omega)) ) {\,\mathrm{d}}t \right|_H .$$
Since $u_n$ is constant on $[k2^{-n}, (k+1)2^{-n}[$ and using again that $u$ solves equation we can estimate this by
$$\left| \int\limits_{k2^{-n}}^{(k+1)2^{-n}} b_{n,k}(t, Z^A_t(\omega) + u(t)) - b_{n,k}(t, Z^A_t(\omega) + u(k2^{-n})) {\,\mathrm{d}}t \right|_H + C 2^{-n} |u(k2^{-n})|_H .$$
By invoking Theorem \[THM-APPROX\] this can be rewritten as
$$\lim\limits_{\ell\rightarrow\infty} \left| \int\limits_{k2^{-n}}^{(k+1)2^{-n}} b_{n,k}(t, Z^A_t(\omega) + u_\ell(t)) - b_{n,k}(t, Z^A_t(\omega) + u_n(t)) {\,\mathrm{d}}t \right|_H + C 2^{-n} |u(k2^{-n})|_H$$
$$\!\!\!\! \!\! \!\!\!\!\!\! \leq C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \left| \int\limits_{k2^{-n}}^{(k+1)2^{-n}} \! b_{n,k}(t, Z^A_t(\omega) + u_{\ell+1}(t)) - b_{n,k}(t, Z^A_t(\omega) + u_\ell(t)) {\,\mathrm{d}}t \right|_H .$$
$$\!\!\!\! = C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \! \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \left| \!\int\limits_{2r2^{-\ell-1}}^{(2r+2)2^{-\ell-1}}\!\!\! b_{n,k}(t, Z^A_t(\omega) + u_{\ell+1}(t)) - b_{n,k}(t, Z^A_t(\omega) + u_\ell(t)) {\,\mathrm{d}}t \right|_H$$
$$\!\!\!\! = C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \!\! \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \left| \!\int\limits_{2r2^{-\ell-1}}^{(2r+1)2^{-\ell-1}}\!\!\!\!\!\!\!\!\! \right. \underbrace{b_{n,k}(t, Z^A_t(\omega) + u(2r2^{-\ell-1})) - b_{n,k}(t, Z^A_t(\omega) + u(r2^{-\ell}))}_{=0} \!{\,\mathrm{d}}t \left. \vphantom{\int\limits_{2r2^{-\ell-1}}^{(2r+1)2^{-\ell-1}}}\right|_H$$
$$+ \sum\limits_{\ell=n}^\infty \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \left| \ \int\limits_{(2r+1)2^{-\ell-1}}^{(2r+2)2^{-\ell-1}}\!\!\!\!\! b_{n,k}(t, Z^A_t(\omega) + u((2r+1)2^{-\ell-1})) - b_{n,k}(t, Z^A_t(\omega) + u(r2^{-\ell})) {\,\mathrm{d}}t \!\right|_H$$
$$= C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \! \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \!\!\! |\varphi_{\ell+1,2r+1}\left( b_{n,k} ; u\left((2r+1)2^{-\ell-1}\right), u\left(r2^{-\ell}\right) \right)|_H .$$
Summing over $k\in\{j 2^{n-m}, ..., (j+1)2^{n-m}-1 \}$ leads us to
$$\hspace{-49mm} \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\! |u((k+1)2^{-n}) - u(k2^{-n}) - \varphi_{n,k}(b_{n,k} ; u(k2^{-n}))|_H$$
$$\leq \!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \! \left( C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \! \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \!\!\!\! |\varphi_{\ell+1,2r+1}\left( b_{n,k} ; u\left((2r+1)2^{-\ell-1}\right), u\left(r2^{-\ell}\right) \right)|_H \right) .$$
$$= \!\!\!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\! C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \sum\limits_{r=k2^{\ell-n}}^{(k+1)2^{\ell-n}-1} \!\!\!\!\!\! |\varphi_{\ell+1,2r+1}\left(b_{n,k} ; u\left((2r+1)2^{-\ell-1}\right), u\left(r2^{-\ell}\right)\right)|_H$$
$$= \!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\! C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \sum\limits_{r=j2^{\ell-m}}^{(j+1)2^{\ell-m}-1} \!\! |\varphi_{\ell+1,2r+1}\left(b_{n,\lfloor r2^{n-\ell}\rfloor} ; u\left((2r+1)2^{-\ell-1}\right), u\left(r2^{-\ell}\right)\right)|_H$$
$$= \!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\! C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-2} \!\!\!\!\!\! |\varphi_{\ell+1,r+1} \left(b_{n,\lfloor r2^{n-\ell-1}\rfloor} ; u\left((r+1) 2^{-\ell-1}\right), u\left(r 2^{-\ell-1}\right)\right)|_H .$$
We set for $\ell \geq n$
$$\Lambda_\ell := \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-2} \!\! |\varphi_{\ell+1,r+1}\left(b_{n,\lfloor r2^{n-\ell-1}\rfloor} ; u\left((r+1)2^{-(\ell+1)}\right), u\left(r2^{-(\ell+1)}\right)\right)|_H$$
and obtain
$$\sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\! \left| u((k+1)2^{-n}) - u(k2^{-n}) - \varphi_{n,k}(b_{n,k} ; u(k2^{-n})) \right|_H \leq \!\!\!\!\! \sum\limits_{r=j2^{\ell-m}}^{(j+1)2^{\ell-m}-1} \!\!\!\!\!\! C 2^{-n} |u(k2^{-n})|_H + \sum\limits_{\ell=n}^\infty \Lambda_\ell . \label{T27E20}$$
From the reversed triangle inequality we deduce
[ @write auxout ]{} $$\begin{aligned}
\tag{\ref{THM-FINAL-MAIN}}
\begin{split} \hspace{-80mm} &\sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\! |u((k+1)2^{-n}) - u(k2^{-n})|_H \\ \hspace{20mm} \leq &\sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\! \!\!\! \left( C 2^{-n} |u(k2^{-n})|_H + |\varphi_{n,k}(b_{n,k} ; u(k2^{-n}))|_H \right) + \sum\limits_{\ell=n}^\infty \Lambda_\ell .
\end{split}\end{aligned}$$
The idea of the proof is the following: We will obtain estimates for the two sums on the right-hand side of the above inequality . For the first sum we simply use Theorem \[THM-SIGMA\] (in the form of Corollary \[COR-SIGMA-RHO\]) to obtain estimate . We will split the second sum in the cases $\ell < N$ and $N \leq \ell$. In the first case we use Corollary \[COR-GLUING\], which will lead us to inequality . For the second case we have to do a more direct computation, which heavily relies on the fact that $u$ is Lipschitz continuous (inequality ).\
Combining all of this will result the final bound . Using the knowledge of the already established estimate and the definition of $\alpha$ we will be able to estimate $\alpha$ in terms of $\beta$ (inequality \[THM-FINAL-ALPHA-ESTIMATE\]). Feeding this back into inequality for $n=m$ completes the proof.\
We will now estimate the two sums on the right-hand side starting with the $\varphi_{n,k}$ sum. We apply Corollary \[COR-SIGMA-RHO\] to obtain
$$\hspace{-19mm} \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \! \left( C 2^{-n} |u(k2^{-n})|_H + |\varphi_{n,k}(b_{n,k} ; u(k2^{-n}))|_H \right)$$
$$\leq \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\! \left(C 2^{-n} |u(k2^{-n})|_H + C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( \left| u(k2^{-n})\right|_H + 2^{-2^{n}} \right)\right)$$
$$\hspace{-37mm} \leq \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\! 2 \tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( \left| u(k2^{-n})\right|_H + 2^{-2^{n}} \right)$$
and since $n\geq m$ this is smaller than
$$\hspace{-9mm} 2\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \left( |u(k2^{-n})|_H + 2^{-2^{m}} \right)$$
$$= 2\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( 2^{n-m} 2^{-2^{m}} + \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} |u(k2^{-n})|_H \right) .$$
Again, using that $n\in\{m,...,N\}$ and the definition of $\psi_n$ (equation ) this can be written as
$$2\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( 2^{n-m} 2^{-2^{m}} + \psi_n \right) .$$
Using inequality this can be further estimated by
$$2\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{-n/2} \left( 2^{n-m} 2^{-2^{m}} + 2^{n-m} \left(\beta + \alpha 2^{-m/3} \right)\right)$$
$$= 2\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{n/2} 2^{-m} \left( 2^{-2^{m}} + \beta + \alpha 2^{-m/3} \right) \overset{2^{-2^{m}}\leq\beta}\leq 4\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{n/2} 2^{-m} \left( \beta + \alpha 2^{-m/3} \right)$$
and hence for the first sum we obtain for all $n\in\{m,...,N\}$
[ @write auxout ]{} $$\!\!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\!\! C 2^{-n} |u(k2^{-n})|_H + |\varphi_{n,k}(b_{n,k} ; u(k2^{-n}))|_H \leq 4\tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{n/2} 2^{-m} \left( \beta + \alpha 2^{-m/3} \right) . \tag{\ref{THM-FINAL-SIGMA}}$$
Now consider the term $\Lambda_\ell$ for $\ell \geq N$. Applying Corollary \[COR-SIGMA-RHO\] we obtain
$$\hspace{-6mm} \sum\limits_{\ell=N}^\infty \Lambda_\ell = \sum\limits_{\ell=N}^\infty \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-2} \left|\varphi_{\ell+1,r+1}\left(b_{n,\lfloor r2^{n-\ell-1}\rfloor} ; u\left((r+1)2^{-(\ell+1)}\right), u\left(r2^{-(\ell+1)}\right)\right)\right|_H .$$
$$\hspace{15mm} \leq \sum\limits_{\ell=N}^\infty \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-2} C_{\varepsilon}\left( \sqrt{\ell+1} 2^{-\ell/6} \left| u\left((r+1)2^{-(\ell+1)} \right) - u\left(r2^{-(\ell+1)}\right) \right|_\infty + 2^{-2\ell} \right) .$$
By the Lipschitz continuity of $u$ this is smaller than
$$C_{\varepsilon}\sum\limits_{\ell=N}^\infty \sqrt{\ell+1} 2^{-\ell/6} \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-1} \left( |(r+1)2^{-\ell-1} - r2^{-\ell-1}| + 2^{-\ell} \right)$$
$$\hspace{-37mm} = C_{\varepsilon}\sum\limits_{\ell=N}^\infty \sqrt{\ell+1} 2^{-\ell/6} \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-1} ( 2^{-\ell} + 2^{-\ell-1} )$$
$$= \frac32 C_{\varepsilon}\sum\limits_{\ell=N}^\infty 2^{\ell-m} \sqrt{\ell+1} 2^{-\ell/6} 2^{-\ell} = \frac32 C_{\varepsilon}2^{-m} \sum\limits_{\ell=N}^\infty \sqrt{\ell+1} 2^{-\ell/6} \leq 2 C_{\varepsilon}2^{-m} 2^{-N/7} .$$
And hence we obtain
[ @write auxout ]{} $$\sum\limits_{\ell=N}^\infty \Lambda_\ell \leq 2C_{\varepsilon}2^{-m} 2^{-N/7} . \tag{\ref{THM-FINAL-OMEGA-L}}$$
Now consider the case $n \leq \ell \leq N$. We define
$$\gamma_{\ell,r} := u((r+1)2^{-\ell}) - u(r2^{-\ell}) - \varphi_{\ell,r}(b_{n,\lfloor r2^{n-\ell-1}\rfloor} ; u(r2^{-\ell})), \qquad\qquad \forall r\in\{0,...,2^\ell-1\}$$
and note that due to inequality we have
[ @write auxout ]{} $$\sum\limits_{r=j2^{\ell-m}}^{(j+1)2^{\ell-m}-1} \!\! |\gamma_{\ell,r}|_H \leq \sum\limits_{\ell'=\ell}^\infty \Lambda_{\ell'} . \tag{\ref{THM-FINAL-GAMMA-SMALLER-OMEGA}}$$
Recall the definition of $\Lambda_\ell$:
$$\Lambda_\ell = \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-2} \!\! \left|\varphi_{\ell+1,r+1}\left(b_{n,\lfloor r2^{n-\ell-1}\rfloor} ; u\left((r+1)2^{-(\ell+1)}\right), u\left(r2^{-(\ell+1)}\right)\right)\right|_H .$$
Using Corollary \[COR-GLUING\] yields that this is bounded from above by
$$C_{\varepsilon}\left[ 2^{-\ell} \psi_\ell + 2^{-\ell/24} \!\! \sum\limits_{r=j2^{\ell-m}}^{(j+1)2^{\ell-m}-1} \!\!\!\! |\gamma_{\ell,r}|_H + 2^{-3\ell/4} |u(j2^{-m})|_H + 2^{\ell+1-m} 2^{-2^{\theta \ell}} \right] .$$
Since $\ell \leq N$ we can use inequality and the assumption $|u(j2^{-m})|_H \leq \beta$ to obtain the following estimate
$$\sum\limits_{\ell=n}^N \Lambda_\ell \leq C_{\varepsilon}\sum\limits_{\ell=n}^N \left[ 2^{-m} (\beta + \alpha 2^{-m/3}) + 2^{-\ell/24} \!\! \sum\limits_{r=j2^{\ell+1-m}}^{(j+1)2^{\ell+1-m}-1} \!\!\! |\gamma_{\ell,r}|_H + 2^{-3\ell/4} \beta + 2^{\ell+1-m} 2^{-2^{\theta \ell}} \right] .$$
Using inequality and $2^\ell 2^{-2^{\theta\ell}} \leq 2^{n-\ell} 2^n 2^{-2^{\theta n}}$ this can be further estimated by
$$\hspace{4mm} C_{\varepsilon}\left[ 2^{-m} N (\beta + \alpha 2^{-m/3}) + \sum\limits_{\ell=n}^N 2^{-\ell/24} \sum\limits_{\ell'=\ell}^\infty \Lambda_{\ell'} + 2^{-3n/4} \beta + 2^{-m} 2^{n+1} 2^{-2^{\theta n}} \right]$$
and since $m \leq n$ this is smaller than
$$36 C_{\varepsilon}\left[ 2^{-m} N (\beta + \alpha 2^{-m/3}) + 2^{-m/24} \sum\limits_{\ell=n}^\infty \Lambda_\ell + 2^{-3m/4} \beta + 2 \cdot 2^{-2^{\theta m}} \right] .$$
Recall that by we have $2^{-3m/4} = 2^{-m} 2^{m/4} \leq 2^{-m}N$ as well as $2^{-2^{\theta m}} \leq 2^{-m} \beta$, so that in conclusion we deduce
[ @write auxout ]{} $$\sum\limits_{\ell=n}^N \Lambda_\ell \leq 144 C_{\varepsilon}2^{-m} N (\beta + \alpha 2^{-m/3}) + \frac12 \sum\limits_{\ell=n}^\infty \Lambda_\ell , \tag{\ref{THM-FINAL-OMEGA-EULER}}$$
where we have used that $36 C_{\varepsilon}2^{-m/24} \leq 36 C_{\varepsilon}2^{-m_0/24} \leq \frac12$.
Putting together the both estimates and for $\Lambda_\ell$ we have
$$\sum\limits_{\ell=n}^\infty \Lambda_\ell = \sum\limits_{\ell=n}^N \Lambda_\ell + \sum\limits_{\ell=N+1}^\infty \Lambda_\ell \leq 144 C_{\varepsilon}2^{-m} N (\beta + \alpha 2^{-m/3}) + \frac12 \sum\limits_{\ell=n}^\infty \Lambda_\ell + 2 C_{\varepsilon}2^{-m} 2^{-N/7} .$$
Henceforth, we deduce
$$\sum\limits_{\ell=n}^\infty \Lambda_\ell \leq 288 C_{\varepsilon}2^{-m} N (\beta + \alpha 2^{-m/3}) + 4 C_{\varepsilon}2^{-m} 2^{-N/7}$$
and since $N$ reads $N = 7 \lfloor \log_2(1/\beta) \rfloor$ this expression is bounded by
$$292 C_{\varepsilon}2^{-m} N (\beta + \alpha 2^{-m/3}) .$$
Therefore, we have
[ @write auxout ]{} $$\sum\limits_{\ell=n}^\infty \Lambda_\ell \leq 292 C_{\varepsilon}2^{-m} N (\beta + \alpha 2^{-m/3}) . \tag{\ref{THM-FINAL-OMEGA-FULL-ESTIMATE}}$$
Looking back to inequality , with the help of and , we estimate the sum by
$$\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\! |u((k+1)2^{-n}) - u(k2^{-n})|_H \leq \!\!\!\!\!\!\! \sum\limits_{k=j2^{n-m}}^{(j+1)2^{n-m}-1} \!\!\!\!\!\!\!\! \left( C2^{-n} |u(k2^{-n})|_H + |\varphi_{n,k}(b_{n,k} ; u(k2^{-n}))|_H \right) + \sum\limits_{\ell=n}^\infty \Lambda_\ell$$
$$\leq 4 \tilde C_{\varepsilon}n^{\frac12+\frac1\gamma} 2^{n/2} 2^{-m} \left[ \beta + \alpha 2^{-m/3} \right] + 292 C_{\varepsilon}2^{-m} N \left[ \beta + \alpha 2^{-m/3} \right ]$$
$$\hspace{-30mm} \leq 292 C_{\varepsilon}' 2^{-m} \left[ n^{\frac12+\frac1\gamma} 2^{n/2} + N \right] \cdot \left[ \beta + \alpha 2^{-m/3} \right] .$$
Note that the above argument holds for all $n \in\{ m , ..., N\}$. Hence, by the minimality of $\alpha$ and inequality we have
$$\alpha 2^{-m} \left[ n^{\frac12+\frac1\gamma} 2^{n/2} + N \right] \leq 292 C_{\varepsilon}' 2^{-m} \left[ n^{\frac12+\frac1\gamma} 2^{n/2} + N \right] \cdot \left[ \beta + \alpha 2^{-m/3} \right]$$
for all $n\in\{m,...,N\}$. This implies that
$$\alpha \leq 292 C_{\varepsilon}' \left[ \beta + \alpha 2^{-m/3} \right] .$$
Since $292 C_{\varepsilon}' 2^{-m/3} \leq 292 C_{\varepsilon}' 2^{-m_0/3} \leq \frac12$ holds for all $m\geq m_0$. It now follows
$$\alpha \leq 292 C_{\varepsilon}' \beta + \alpha 292 C_{\varepsilon}' 2^{-m/3} \leq 292 C_{\varepsilon}' \beta + \frac\alpha2 .$$
From which we deduce that
[ @write auxout ]{} $$\alpha \leq 584 C_{\varepsilon}' \beta . \tag{\ref{THM-FINAL-ALPHA-ESTIMATE}}$$
Setting $n=m$ in reads
$$|u((j+1)2^{-m}) - u(j2^{-m})|_H \!\! \overset{\eqref{THM-FINAL-DE-ALPHA}}\leq \!\! \alpha 2^{-m} \left[ m^{\frac12+\frac1\gamma} 2^{m/2} + N \right] .$$
Putting $|u(j2^{-m})|_H$ to the right-hand side yields
$$|u((j+1)2^{-m})|_H \leq |u(j2^{-m})|_H + \alpha 2^{-m} \left[ m^{\frac12+\frac1\gamma} 2^{m/2} + N \right]$$
and since we have $\alpha \leq 584 C_{\varepsilon}' \beta$ as well as $m^{\frac12+\frac1\gamma} 2^{m/2} \leq N$ by definition of $N$ using our estimate yields that the above expression is smaller than
$$\beta + 584 C_{\varepsilon}' \beta 2^{-m} N = \beta \left( 1 + 584 C_{\varepsilon}' 2^{-m} \lfloor \log_2(1/\beta) \rfloor \right) \leq \beta \left( 1 + K 2^{-m} \log_2(1/\beta) \right) ,$$
where the constant is defined as $K := 584 C_{\varepsilon}'$ which completes the proof.
[ \[COR-FINAL\] ]{}
Let $f$ be a $H$-valued Borel function such that the Assumption \[ASS\] is fulfilled then there exists a set $N_f\subseteq\Omega$ with ${\mathbb{P}}[N_f]=0$ such that for all $\omega\in N_f^c$ if $u$ is a solution to
$$u(t) = \int\limits_0^t e^{-(t-s)A} ( f(s, u(s) + Z^A_s(\omega)) - f(s, Z^A_s(\omega)) ) {\,\mathrm{d}}s , \qquad \forall t\in [0,1] .$$
then $u \equiv 0$.
**Step 1:**\
\
Let $0 < {\varepsilon}< \frac1{40}$ and $\Omega_{{\varepsilon},f}$ be the of set of Theorem \[THM-FINAL\]. Fix $\omega \in \Omega_{{\varepsilon},b}$ and let $u$, as stated in the assertion, be a solution to the above equation. Since $\|f\|_\infty \leq 1$ the function $u$ is Lipschitz continuous with Lipschitz constant at most $2$. Furthermore, Assumption \[ASS\] on $f$ implies that $u$ is $Q$ as well as $Q^A$-valued.
Therefore $u\in\Phi$. Applying Theorem \[THM-FINAL\] gives us a $K>0$ and $m_0\in{\mathbb{N}}$. For sufficiently large $m\in{\mathbb{N}}$ (i.e. $K \leq \ln(2) 2^m$ and $m\geq m_0$) we define
$$\beta_0 := 2^{m-2^{\theta m}}$$
and
$$\beta_{j+1} := \beta_j(1+K2^{-m} \log_2(1/\beta_j))$$
for $j \in \{0, ..., 2^m -1 \}$. By the very definition we have
$$\beta_{j+1} - \beta_j = K2^{-m} \beta_j \log_2(1/\beta_j)$$
for every $j \in \{0, ..., 2^m -1 \}$. Hence, Lemma \[LEM-GRONWALL\] is applicable which implies that
$$\beta_j \leq \exp\left( \log_2(\beta_0) e^{-2K-1} \right) = \exp\left( \left(m-2^{\theta m}\right) e^{-2K-1} \right)$$
$$\hspace{-17mm} \leq \exp\left( - \ln(2) 2^{\left(\frac\theta2+\frac14\right) m} \right) = 2^{-2^{\left(\frac\theta2+\frac14\right) m}} .$$
Together with the fact that $\beta_j$ is increasing we have
$$2^{m-2^{\theta m}} \leq \beta_j \leq 2^{-2^{(\frac\theta2+\frac14) m}} .$$
Since $u$ is a solution to equation we know that $u(0) = 0 \leq \beta_0$, so that we are able to invoke Theorem \[THM-FINAL\] from which we deduce that $|u(2^{-m})|_H \leq \beta_1$. By induction on $j$ we obtain
$$|u(j2^{-m})|_H \leq \beta_j \leq \beta_{2^m} \leq 2^{-2^{\left(\frac\theta2+\frac14\right) m}} , \qquad \forall j\in\{0, ..., 2^m \}$$
for every $j \in \{0, ..., 2^m -1 \}$. By letting $m\rightarrow\infty$, we deduce that $u$ vanishes at all dyadic points. By continuity of $u$ it follows $u \equiv 0$.\
**Step 2:**\
\
Let $k\in{\mathbb{N}}$. By setting ${\varepsilon}:=1/k$ in Step $1$ we conclude that there is $\Omega_{k,f} \subseteq\Omega$ with ${\mathbb{P}}[\Omega_{k,f}^c] \leq 1/k$ such that $u \equiv 0$ for all $\omega\in \Omega_{k,b}$. By defining
$$N_f := \bigcap\limits_{k=41}^\infty \Omega_{k,f}$$
we have $u\equiv 0$ for all $\omega\in N_f^c$ which concludes the proof.
By Corollary \[COR-FINAL\] the assumption of Proposition \[PRO-GIRSANOV\] is fulfilled, so that by invoking Proposition \[PRO-GIRSANOV\] the conclusion of Proposition \[PRO-MAIN\] follows. Theorem \[THM-MAIN\] then follows from Proposition \[PRO-MAIN\] as explained in the introduction.
[BFGM14\]]{}
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|
---
abstract: 'I reexamine some elements of the theory of black holes, pointing out where the usual treatment seems to be adequate and where there is reason to want to improve it. Some of the problematic elements of the theory can be clarified by studying holonomies relating the neighborhood of the horizon to the regime occupied by distant observers.'
author:
- 'Adam D. Helfer'
title: Black Holes Reconsidered
---
[ address=[Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.]{} ]{}
Introduction {#introduction .unnumbered}
============
The current basis for the theoretical investigation of black holes largely grew out of the tremendously productive period 1963–1974, sometimes called the Golden Age of Black-Hole Theory, during which key concepts and results were obtained; from these, an extraordinarily rich and exciting field has developed.
There have remained, however, several awkward elements of the foundations: the [*problem of teleology*]{} (that the strict definition of a black hole requires the knowledge of the entire space–time), difficulties fleshing out the notion of black-hole thermodynamics, and problems surrounding Hawking’s prediction of black-hole radiation. For many investigations, these concerns are not directly relevant, but for others they are. The problem of teleology, for instance, underlies the difficulties in formulating an observational criterion for the identification of black holes, figures in many arguments involving black holes and causal processes, and is ultimately responsible for the attempts to find useful substitutes for the event horizon (trapping, isolated and dynamical horizons). And the “information paradox” and “holography” are evidently bound up with Hawking’s argument for black-hole radiation.
In this paper I take up these awkard elements: partly, my aim is to review the usual development of the theory, pointing out open questions and concerns; partly, it is to describe some new ideas which help either to resolve or to elucidate the difficulties. In particular, perhaps surprisingly, studying the holonomy around paths linking the neighborhood of the horizon to the regime distant observers occupy both brings out important features of the incipient black hole in a non-teleological way, and clarifies the trans-Planckian problem associated with Hawking’s prediction. It also helps approach the problem of delineating just where the classical treatment of black holes becomes inadequate.
While at present the quantum issues appear to be almost beyond hope of observation, the case for reviewing the structure of the neighborhood of the event horizon and casting it in terms which are at least in principle observable rather than teleological is more pressing. There is a real prospect that within the next decade or so we will begin to be able to resolve some of the structure of the (presumed) supermassive black hole at the center of the Milky Way with submillimeter VLBI networks [@Doeleman:2009te].
Section 1 reviews black holes and causal structure, emphasizing the conceptual issues.
Section 2 introduces the theory of the holonomy around paths relating the neighborhood of the horizon to the regime occupied by distant observers. It is shown that there are certain [*universal*]{} features, which should be observable, associated with the development of an incipient event horizon.
Section 3 is a sketch of the elements of quantum field theory in curved space–time, without taking up the question of what special effects might be due to black holes.
Section 4 reviews Hawking’s prediction of thermal radiation from black holes, with attention to the problematic elements of the derivation, and also to some indirect arguments which have been suggested for it.
Section 5 probes deeper into the relation between black holes and quantum theory. It is first shown that it is reasonable to expect that the close neighborhood of an event horizon will not have an operational interpretation as a classical portion of space–time. Then it is shown that the trans-Planckian problem is promoted from a virtual to a real one if we consider any of three common quantum effects: the results of sequences of measurements; the nonlinear interactions of quantum fields; and Casimir energies. These points lead me to suggest that the current theory of quantum fields in curved space–time, although so natural as to appear unexceptionable, may need to be reconsidered.
The remainder of this section is devoted to a brief discussion of the problem of teleology and associated terminological issues.
The conventions used here are those of @Penrose:1986ca; the metric has signature $+{}-{}-{}-$. In some places, factors of $c$, $G$, $\hbar$ are omitted.
Teleology and Terminology {#teleology-and-terminology .unnumbered}
-------------------------
Different workers mean different things by “black hole.” Astrophysicists generally have in mind very compact objects which would explain certain phenomena; they are often not directly concerned with whether an event horizon in a strict sense exists, although this question is increasingly coming into the astrophysical literature (usually as the question of whether an object “has a surface”). At the other end of the spectrum, for mathematical relativists, the existence of the event horizon is central. It will be clearest to begin by attempting to formalize the concept:
> A [*black hole*]{} is a region of space–time from which causal signals can never escape; the [*event horizon*]{} is the boundary of the black-hole.
These concepts are problematic, because of the words “escape” and “never.” Since there is no fixed sense of space in general relativity, the only invariant way of defining escape is a limiting one of “escape to infinity.” Even more severe are the consequences of “never.” To verify that one has a black hole, according to this standard formulation, one would have to know the entire future development of the space–time, including over cosmic scales. This is the [*problem of teleology*]{} [@Carter:1979; @Booth:2005qc].
One might hope to get around this by finding some local criterion which would [*imply*]{} that one had found an event horizon or a black hole. However, such hopes turn out to be misdirected. While one can indeed show that under certain assumptions event horizons will form, simple arguments also show that the locations of these cannot be fixed from local data, and indeed the location of the horizon could always potentially be altered by the [*later*]{} evolution of the space–time. The problem of teleology is central — at least if we keep the usual definition of “black hole.”
Because this issue is so severe, we should properly be open to reconsidering the definition. But the present usage is firmly embedded, and until such time as a compelling alternative appears, it is pointless to try to change it. Still, to carry the discussion forward, it is necessary to give some indication of what appears to be the problem and how we might resolve it, which involves contemplating some sort of change in what is to be called a black hole.
It seems clear that the root of the problem is an over-idealization, bound up with considering arbitrarily large times and distances. Any practical reference to black holes must come down to considering times which are “sufficiently long” but not actually infinite, and likewise “sufficiently large” regions of space. These in turn show us that [*a practical notion of black holes should be linked to systems which are complete enough to be considered isolated*]{}, in both time and space. For the moment in will not be necessary to be more specific; we will accept that in some circumstances it is possible to characterize such systems and to use this sense of isolation to define (perhaps only up to a well-understood ambiguity) black holes and event horizons.
We therefore need terminology which allows us to specify just what system is under discussion, and what assumptions are going to go into modeling it mathematically.
I shall reserve the term [*Universe*]{} for the physical world. By [*a space–time*]{} I mean the conventional mathematical concept of a manifold ${{\mathscr M}}$ equipped with a Lorentzian metric $g_{ab}$.[^1] We are usually interested in a space–time as [*modeling*]{} a portion of the Universe, most often a relatively small portion, such as an object we think is, in a suitable sense, a black hole. When we do this, we should be as precise as possible about the purposes for which the model is supposed to be valid (which physical quantities we intend to be calculable and how accurately), and its extent in time and space. Given the problem of teleology, it it especially important to consider how sensitive the predictions might be to any boundary conditions we assume, and how well justified those assumptions are.
It will be important to distinguish between [*complete*]{} and [*incomplete*]{} models.[^2] By a complete model, I mean a space–time which is considered to model the entire physical system in question the entire region of interest; an incomplete model is a space–time which only models a portion (in time and space) of the system. Suppose, for example, we have in mind a system with a central black hole which is surrounded by vacuum over some periods, but during others accretes matter. It may be a good approximation for many purposes to model the exterior vacuum region by Schwarzschild solutions (and use some other solutions during the accreting periods). In these cases the Schwarzschild solutions are incomplete models, but the entire patched-together space–time may be a complete model, if it covers the system over all times of interest.
In an example like the one just given, there may well be an event horizon associated with the complete model space–time. The location of such a horizon cannot be determined from the incomplete models alone. For instance, while a Schwarzschild solution of mass $M$ might form one of the incomplete models, the event horizon of the complete model will generally not lie at $r=2M$, even within the portion on which the incomplete model is valid. It will be helpful to have a term for the surface $r=2M$ in this case, or more generally for the surface where the event horizon would lie, were the black-hole space–time for the incomplete model valid; I will call this a [*stand-in horizon*]{}.
In much of the literature, the term “event horizon” is used for what I have called a stand-in horizon; also, typically Killing horizons are of interest as stand-in horizons.
Black Holes and Causal Structure
================================
In this section, I will review the ideas leading up to the definition of black holes, the relation with singularities, and some of the fundamental theoretical results.
Trapped Regions and Singularities: Weak Cosmic Censorship
---------------------------------------------------------
While the history of the concept of black holes is complex and nuanced [@Israel:1987ae], a decisive step towards the modern view took place with the work of @Oppenheimer:1939ue. These authors considered the indefinite gravitational collapse of a simplified (spherically symmetric, zero pressure) model of a star, and uncovered two key features: the development of singularities and of a trapped region. While the two features are known to be linked (via some of the singularity theorems), we have at present only an incomplete understanding of the relationship between the two. It will help to begin by discussing the concepts individually.
We start by distinguishing between [*local*]{} and [*global*]{} notions of trapping. The definitions of black holes and event horizons are global: they are given in terms of whether causal signals can escape “to infinity,” which formalizes the idea of a region of space–time closed off from the inspection of distant observers.
The notion of trapping which appears in the singularity theorems, however, is a local one, and it is an assumption, not a conclusion. A [*trapped surface*]{} is a spacelike two-surface, both of whose null normals are converging. In other words, a trapped surface is one from which a flash of light emitted “outwards” would (at least initially) decrease in area. Notice that this concept does not require us to make any assumptions about the existence of an asymptotic regime where very distant observers are considered to lie. However, if such an asymptotic regime does exist, then one can show under mild positivity-of-energy assumptions that the trapped surface must lie within an event horizon.
(That trapped surfaces lie within event horizons but can be studied locally has led to a great deal of work on them, and the related concepts of apparent, isolated and dynamical horizons [@Hawking:1973uf; @Ashtekar:2004cn]. These concepts, while of interest, do not in general situations capture the sense of an event horizon, as a barrier to distant observers’ examination. Precisely because trapped surfaces lie behind event horizons, we hope and expect never to find them observationally! And easy examples (e.g. [@Booth:2005qc]; see also [@Williams:2008] for more sophisticated work) show that in general trapped surfaces do not encode much of the geometry of the horizon, unless one makes teleological assumptions.)
Now let us turn to singularities.
What, precisely, do we mean by a space–time singularity? Even in cases where the metric is known explicitly, it can be quite difficult to analyze the structure of a singularity, or even to verify that there is a singularity. If one can show that freely falling observers would, in a finite interval of proper time, perceive divergent curvature tensors, for example, one has in a clear sense a curvature singularity, but it is not at all clear that these are the only possible sorts of singular behavior. (See Clarke’s book [@Clarke:1993] for general results.) And usually we only know the metric explicitly on an initial-data set, or in some region, and the explicit determination of the evolution is beyond our present technical capabilities.
For this reason, most of the strong results we have use an indirect characterization of singularities: their existence is signalled by the presence of causal geodesics whose affine parameters do not take all real values — incomplete causal geodesics. One must then make sure that the incompleteness really does indicate a singularity, and not simply that some points have been left out of an otherwise non-singular space–time. For this reason, the singularity theorems typically are phrased in the form, “If certain conditions hold, then space–time cannot be causally complete.”
For instance, the 1970 singularity theorem of @HP1970 asserts that space–time cannot be causally complete if the following conditions hold: (a) there is a trapped two-surface; (b) a positvity-of-energy condition holds ($R_{ab}t^at^b\leq 0$ for all timelike vectors $t^a$); (c) there are no closed timelike curves; and (d) a certain genericity condition holds (each causal geodesic contains at least one point along which the tangent is not an algebraically special vector for the Riemann curvature). The important points are that: we do expect that such situations can occur; and, given any set of data satisfying the hypotheses, all “sufficiently close” data will as well. Thus the occurrence of singularities is a stable phenomenon (but the theorem does not tell us that the character of the singularities is stable).
So if the hypotheses of the theorem hold certain singularities will develop; if additionally there is a “nice” asymptotic region in space–time, those singularities will lie behind an event horizon; we say the horizon [*clothes*]{} them. However, we do [*not*]{} know that those singularities are the only ones which may occur; in particular, we do not know that all singularities must lie behind an event horizon. One which did not, that is, which would be visible to distant observers, would be called [*naked*]{}. The question of whether naked singularities can exist is usually considered the most important open issue in classical general relativity: it is the
> [**Weak Cosmic Censorship Question.**]{}
>
> In classical general relativity, for physically realistic matter, in generic circumstances, must any singularities which develop lie behind an event horizon?
This is somewhat imprecise, as the notion of what counts as “physically reasonable” has not been specified, nor has the notion of genericity, and also implicit in the formulation is the notion that the space–time has a regular enough asymptotic region that it makes sense to define an event horizon. Nevertheless, it does capture the idea.
There is no definitive progress, one way or the other, on this problem. Penrose’s article [@Penrose:1999] is to be recommended as a prolegomenon. I will make only a few comments:
First, it is quite surprising to conjecture a link between singularities, which are local, and event horizons, which are global. For this reason, it has been thought that if weak cosmic censorship holds, it will do so as a special case of [*strong*]{} cosmic censorship, which should forbid the development of locally naked singularities.
Second, the genericity condition makes it very hard to approach this question via the study of exact solutions. No single solution could provide a negative answer to the question; one would need to show that naked singularities persisted for an open set of data sufficiently close to the solution. It should be emphasized that a really rigorous argument, and not (say) simply a linearized analysis would be required. On the other hand, a generic numerical space–time with physically reasonable matter with a naked singularity could be strong circumstantial evidence for the failure of cosmic censorship.
Similarly, proof that a specific class of space–times (whose initial data lie only on a thin set in the space of all physically acceptable data) can only develop clothed singularities does not settle the question in the affirmative. However, such results could give clues about approaching the problem; the recent work of @Ringstrom:2008 has generated excitement among those approaching the problem from the point of view of partial differential equations.
For the purposes of this paper, there are two points to keep in mind about naked singularities:
(The bad news)
: Because we have so little definitive progress on the question of weak cosmic censorship, we know little about what a naked singularity (were one to form) would look like.
(The good news)
: A naked singularity (were one to form) would, unlike an event horizon, be by definition discernible by distant observers at finite times. Thus naked singularities, whatever they might be, should be distinguishable from black holes.
Hypotheses on the Asymptotic Regime
-----------------------------------
In order to formalize the notion of a black hole, we must say what it means for causal signals to escape from a system. To do this we must define a regime which counts as “very far away” and is accessible by causal curves. There is in fact no single accepted notion. Mathematically, this can be thought of as an issue of the regularity one wants to hypothesize in the asymptotic regime; physically, one is deciding what exactly should qualify as a “system.” In general relativity, the questions of what the system is and what its asymptotics are must be addressed together.
For most purposes, one does not want to have to consider the whole Universe in order to analyze a system potentially containing a black hole; at the same time, one would hope that the precise boundaries of the system are not too important and one can substitute some sort of idealized boundary conditions for (what would be more accurate) a precise specification the relevant data and its uncertainties at the boundary. One expects this to be largely true, but there will be places where it will be important to recall that we are really working with idealizations which cannot be pushed too far. Again, we are here distinguishing the physical [*Universe*]{} from the mathematical [*space–time*]{}, which signifies a model of a portion of the Universe subject to some idealizations.
With few exceptions, one is interested in modeling black holes within systems which can be viewed as isolated. The relevant model of the asymptotic regime is then that first given by Bondi, Sachs and coworkers [@Bondi:1962; @Sachs:1962]. While it was originally developed by finding an appropriate class of asymptotic coordinate systems with which general radiating systems could be studied, @Penrose:1964 showed that it could be recast neatly and powerfully in terms of the space–time admitting a “conformal boundary.”
While the end result of the Bondi–Sachs treatment is easy to work with, the logic justifying it includes some fine points. I am going to quickly review it, with two aims: first, I want to emphasize that it is not at all clear how closely parallel a structure would exist in other dimensions, so one should be cautious of assuming that the theory of black holes (or isolated systems generally) in higher dimensions will follow the same pattern as in four;[^3] second, I want to raise a mild concern about one assumption common in some of the literature. I will not give the details of the calculations or the differentiability assumptions, for which see @Penrose:1986ca.
I will say a space–time $({{\mathscr M}},g_{ab})$ [*admits a future null conformal infinity*]{} ${{{\mathscr I}^{+}}}$ if: (a) ${{\mathscr M}}$ embeds as the interior of a manifold $\overline{{\mathscr M}}$ with boundary ${{{\mathscr I}^{+}}}$; (b) there exists a smooth function $\Omega$ on $\overline{{\mathscr M}}$ with $\Omega >0$ on ${{\mathscr M}}$ and $\Omega =0$ on ${{{\mathscr I}^{+}}}$ and ${\hat N}_a=-{\hat\nabla}_a\Omega$ is non-zero on ${{{\mathscr I}^{+}}}$; (c) the conformally rescaled metric ${\hat g}_{ab}=\Omega ^2g_{ab}$ extends smoothly to a non-degenerate metric on $\overline{{\mathscr M}}$; (d) each point of ${{{\mathscr I}^{+}}}$ contains the future, but not the past, end-points of null geodesics in ${{\mathscr M}}$.[^4]
That ${\hat g}_{ab}$ should be regular at ${{{\mathscr I}^{+}}}$ is in fact a powerful constraint. For instance, the Ricci scalars of the original and rescaled metrics are related by $$\label{Rsc}
R=\Omega ^2{\hat R} -6\Omega{\hat\nabla}_a{\hat\nabla}^a\Omega +12{\hat N}_a{\hat N}^a\, .$$ Einstein’s equation gives us $-R+4\lambda =-8\pi G T_a{}^a$, where $\lambda$ is the cosmological constant of the space–time (not the Universe). This cosmological constant would be relevant [*only*]{} if we wanted to treat the model as applicable on cosmological scales, and we usually do not: in the Bondi–Sachs scheme we assume $\lambda =0$. And because points on ${{{\mathscr I}^{+}}}$ are reached by going out to infinity along null geodesics, we generally expect that the only matter fields which can survive in this limit are null radiation, which has $T_a{}^a=0$. In this case, we have ${\hat N}_a{\hat N}^a=0$ at ${{{\mathscr I}^{+}}}$, and ${{{\mathscr I}^{+}}}$ is a null hypersurface. As such, it is foliated by null geodesics (of the rescaled metric), called its [*generators*]{}.
Each point on ${{{\mathscr I}^{+}}}$ can be thought of as the set of null geodesics in the physical space–time which terminate at the point. Since there is a five-parameter family of null geodesics (neglecting the two additional choices needed to fix affine parameterizations of them), and ${{{\mathscr I}^{+}}}$ is three-dimensional, we expect each point on ${{{\mathscr I}^{+}}}$ to correspond to a two-parameter family of null geodesics. Each such family can be identified with an asymptotically plane-fronted wave-front. Note that each null generator of ${{{\mathscr I}^{+}}}$ corresponds to a one-dimensional family of such wave-fronts, at a succession of later and later times. We may interpret this as saying that all of these wave-fronts have the same asymptotic direction. We then make the natural assumption that ${{{\mathscr I}^{+}}}$ is topologically $S^2\times{{\mathbb R}}$, the angular variables corresponding to the different possible directions for the asymptotic wave-fronts, and the ${{\mathbb R}}$ factors to the null generators, that is, the different possible retarded times. (If one is willing to assume asymptotic simplicity — see below — then one can prove that ${{{\mathscr I}^{+}}}$ must have this topology [@Penrose:1965; @Geroch:1971].)
It is often convenient to think of ${{{\mathscr I}^{+}}}$ as a bundle over $S^2$, projecting along the generators; in this context, a section of the bundle is called a [*cut*]{} of ${{{\mathscr I}^{+}}}$. The base space $S^2$ is then naturally the set of asymptotic directions in which null geodesics might escape. However, it is not at all obvious that this base space has structure beyond that of a smooth point-set. But in fact it turns out that this $S^2$ has well-defined conformal structure, and this is critical to some of the deepest results.
We know already that any cut of ${{{\mathscr I}^{+}}}$ has a conformal structure, simply by restricting the metric ${\hat g}_{ab}$ to the cut. What is not obvious is that flowing along the generators of ${{{\mathscr I}^{+}}}$ should preserve this conformal structure. The condition that this should be the case is that the null tangent ${\hat N}^a$ should be [*shear-free*]{}, that is ${\hat\nabla}_{(a}{\hat N}_{b)}$ should be pure trace. However, by a calculation like that leading to , this turns out to follow from the (very weak) assumption that $\Omega (T_{ab}-(1/4)g_{ab}T_c{}^c)$ vanishes at ${{{\mathscr I}^{+}}}$, and we henceforth assume this holds.
There is further important structure, which again is not obvious. In general, if one only knows the conformal class of a metric, its null geodesics are well-defined only up to reparameterization: in particular, they do not have a well-defined affine structure. This would appear, at first blush, to be the case for ${{{\mathscr I}^{+}}}$. However, it turns out that the null generators of ${{{\mathscr I}^{+}}}$ [*do*]{} have well-defined affine structures. This arises from the invariance of the quantity ${\hat g}_{ab}{\hat N}^c{\hat N}^d$, sometimes called the [*strong conformal geometry*]{} of ${{{\mathscr I}^{+}}}$. Picking any one metric structure for the base $S^2$, we may pull this back to ${{{\mathscr I}^{+}}}$, and this fixes a scale for the tangent ${\hat N}^a$ to the generators. While the scale may change with the metric chosen, it can only do so by a factor which is constant along each generator.
We may get still more, for we may restrict the metric on the base $S^2$ to be that of a unit sphere, compatible with its conformal structure. There is only a three-dimensional family of such choices, which is naturally identifiable with the set of unit timelike vectors. Therefore each such choice amounts to an asymptotic choice of timelike unit vector.
What this means is that, for each asymptotic direction, there is a well-defined sense of retarded time $u$ along the corresponding generator, transforming naturally with the choice of asymptotic unit timelike vector, and otherwise free only up to the addition of a (generator-dependent) term. (This freedom is the famous [*supertranslation ambiguity*]{}.) In particular, it is meaningful to say whether the generator extends infinitely far, in either the future or the past. To model black-hole space–times, we shall assume that the generators are infinitely long in both directions. (One might consider relaxing the requirement that they be infinitely long to the past.)
Much more can be deduced from this, including [*Sachs peeling,*]{} which describes the asymptotic fall-off of the curvature, and this formalism is also the basis for the Bondi–Sachs energy–momentum [@Penrose:1986ca] and twistorial angular momentum [@Helfer:2007].
[*A Comment on Weak Future Asymptotic Simplicity.* ]{} The treatment that I have given here is almost completely standard. However, there is one assumption which is commonly made.
A space–time $({{\mathscr M}},g_{ab})$ is [*future asymptotically simple*]{} if it admits a future conformal null infinity and if additionally [*every*]{} null geodesic in $(M,g_{ab})$ acquires a future end-point on ${{{\mathscr I}^{+}}}$.
Now, it is well-known that it is overly restrictive to impose future asymptotic simplicity; for example, in Schwarzschild there are null geodesics which orbit at $r=3M$ and do not escape. For this reason, it has become common to consider [*weakly future asymptotically simple*]{} space–times, which are those which admit an open set isometric to an open neighborhood of ${{{\mathscr I}^{+}}}$ in an auxiliary future-simple space–time. (More precisely, isometric to an open [*deleted*]{} neighborhood of ${{{\mathscr I}^{+}}}$, that is, deleting ${{{\mathscr I}^{+}}}$ itself.)
It seems to me that perhaps the term “simplicity” occurring in these definitions has given the impression that they correspond to expected situations. While it is certainly of interest to try to investigate the future limits of null geodesics, it is not [*a priori*]{} clear that the set of those future limits which correspond to the idea of escape from a given system should have the property of weak asymptotic future simplicity. So I would suggest that until this point is clarified, it would be prudent not to assume that all isolated systems of interest are weakly future asymptotically simple.
Black Holes, Event Horizons, and Causal Structure
-------------------------------------------------
In a space–time $({{\mathscr M}},g_{ab})$, we may define the [*causal past*]{} of any subset $S\subset {{\mathscr M}}$ as $$J^-(S)=\{ p\in {{\mathscr M}}\mid\mbox{there is a causal curve from }p\mbox{ to some point in }S\}\, .$$ (A causal curve, if smooth, is one whose tangent is everywhere a future-directed timelike or null vector. For the non-smooth case see [@Penrose:1972].) Note that this depends only on the conformal class of the metric. Thus if $({{\mathscr M}},g_{ab})$ admits a future null conformal infinity ${{{\mathscr I}^{+}}}$ as defined above, we may define $$J^-({{{\mathscr I}^{+}}})=\{ p\in {{\mathscr M}}\mid\mbox{there is a causal curve from }p\mbox{ to some point in }{{{\mathscr I}^{+}}}\}\, ;$$ this will be the set of events in ${{\mathscr M}}$ from which at least one causal curve escapes to the asymptotic regime ${{{\mathscr I}^{+}}}$. If ${{{\mathscr I}^{+}}}$ has been defined so that it includes all possible means of “causal escape” — usually, this is taken to mean that the space–time is weakly future asymptotically simple, but see the comment at the end of the last subsection — then we may interpret $${{\mathscr B}}={{\mathscr M}}-J^-({{{\mathscr I}^{+}}})\, ,$$ if it exists, as the [*black-hole*]{} region of the space–time. The [*(future) event horizon*]{} is the boundary of this: $${{{\mathscr H}^{+}}}=\partial{{\mathscr B}}\, .$$ (Most authors assume, as part of the definition of a black hole, that the space–time is [*globally hyperbolic*]{} (see, e.g. [@Penrose:1972]), which essentially means that it admits a good initial-data surface. I have omitted this because it is bound up with the question of cosmic censorship [@Penrose:1980].)
These points should be emphasized:
- We are only entitled to interpret ${{\mathscr B}}$ as a black hole if we are satisfied that ${{{\mathscr I}^{+}}}$ represents all possible means of causal escape.
- The definition of the black hole (and that of the event horizon) is [*highly time-asymmetric*]{}. The time-reverse concept would be a [*white hole*]{} (given by ${{\mathscr M}}-J^+({{{\mathscr I}^{-}}})$ in an obvious notation), a region of space–time which incoming causal signals could not penetrate — something quite different.
- The black hole and its event horizon are determined [*highly nonlocally*]{}, in terms of whether escape is ever possible. This has been one of the most difficult features to work with in the theory.
There is sometimes a tendency to think that because black holes are expected in many cases to settle down to stationary states this time asymmetry is not very important. However, it is crucial for some purposes in surprising ways — for instance, it was Hawking’s appreciation of this which led him to predict black-hole radiation, when others had intuitively ruled it out based on stationarity arguments. It is worthwhile, when dealing with black holes, to make one’s arguments about which parts of the space–time are expected to be (approximately) stationary carefully.
### Structure of the Event Horizon {#structure-of-the-event-horizon .unnumbered}
The causal differential topology of the event horizon ${{{\mathscr H}^{+}}}$ is elegant. For proofs of the following, see @Penrose:1972:
\(a) It is not hard to see that ${{{\mathscr H}^{+}}}$ must be [*achronal*]{}, that is, no timelike curve can join one of its points to another. (If $p,q\in{{{\mathscr H}^{+}}}$, if there were a timelike curve from $p$ to $q$, then, displacing $q$ slightly to a point $q'\in {{\mathscr M}}-{{\mathscr B}}$ to the past of $q$, we could get a causal curve from $p$ to $q'$ and then escaping to ${{{\mathscr I}^{+}}}$, a contradiction.) (b) The event horizon must also be a topological three-manifold which is locally the graph of a Lipschitz function (that is, has bounded difference quotients). (c) Any point $p\in{{{\mathscr H}^{+}}}$ lies on a null geodesic segment (not necessarily unique) on ${{{\mathscr H}^{+}}}$ extending to the future of $p$, and this segment lies on ${{{\mathscr H}^{+}}}$ as far as the segment extends. (d) No two null geodesic segments on ${{{\mathscr H}^{+}}}$ can end at the same point, unless they are the same segment in a neighborhood of the point.
These results again point up the highly time-asymmetric character of the horizon: it may acquire new generators as one moves to the future but, once a null geodesic joins the horizon it can never leave. New generators join at [*caustics,*]{} and it is of great interest to understand just how this happens and what observational properties the neighborhood a black hole where a caustic forms might have.
I have pointed out that the (standard) arguments so far give what is usually regarded as a rather low degree of regularity for event horizons: they are locally Lipschitz. Is this really as much as can be said?
As far as the purely mathematical question goes, not much more regularity can be expected in general. (See @Chrusciel:2001 [@Chrusciel:2002] for slightly better results.) The precise statements of the results are technical (for example, horizons are differentiable almost everywhere in the sense of measure theory but may well fail to be differentiable over all open sets [@Chrusciel:1998]). The cleanest general result is one of @Beem:1998, who show that differentiability fails precisely at places where new generators join.
But are such mathematical oddities really relevant physically? In order to investigate this, let us suppose we are modeling a system with Bondi–Sachs asymptotics. The non-black region is the interior of $J^-({{{\mathscr I}^{+}}})$, and we may think of forming this by a limiting process. Let $Z$ be any cut of ${{{\mathscr I}^{+}}}$, and let ${{{\mathscr I}^{+}}}_Z$ be the portion of ${{{\mathscr I}^{+}}}$ at or prior to $Z$. Then $J^-({{{\mathscr I}^{+}}})=\bigcup _{Z\uparrow +\infty} J^-({{{\mathscr I}^{+}}}_Z)$, where the union is over later and later cuts. We may take this union over a family of smooth cuts.
Each $\partial J^-({{{\mathscr I}^{+}}}_Z)$ will be, in the vicinity of $Z$, a smooth null hypersurface meeting $Z$ orthogonally. As we follow it into the past, it may develop singularities, associated with what is called the [*cut locus*]{} (this use of “cut” is different from $Z$; unfortunately, both uses are firmly embedded) — the family of points at which its generators begin to cross; at these cut points, some of the null geodesics, followed into the past, would move to the chronological past of $Z$ and hence leave $\partial J^-({{{\mathscr I}^{+}}}_Z)$. However, such cut points would arise by solving for the intersection of two null generators of $\partial J^-({{{\mathscr I}^{+}}}Z)$, and generically (that is, at points where neither generator is conjugate to $Z$) this variety will be a smooth two-surface.
Also one would expect that for generically positioned smooth $Z$ only finitely many such cut loci will occur as one follows $\partial J^-({{{\mathscr I}^{+}}}_Z)$ inward to a neighborhood of any point on ${{{\mathscr H}^{+}}}$. On the other hand, as we move $Z$ later and later, it is quite possible that the number of such cut loci increases, and that they accumulate, these $\partial J^-({{{\mathscr I}^{+}}}_Z)$ accumulating in an irregular way as the horizon is approached.
It thus seems to me likely that the question of how irregular practical approximations to the horizon are in realistic models is a reflection of how increasingly complex the system of cut loci associated to the cuts $Z$ become as $Z$ moves towards the future.
The Area Theorem
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The famous Area Theorem is a key conceptual result. It asserts that, if the weak energy condition holds ($T_{ab}l^al^b\geq 0$ for all null vectors $l^a$), then the areas of cross-sections of the event horizon increase as the sections move towards the future.
A “physicist’s” statement and proof of this theorem was first given by Hawking in 1972 [@Hawking:1971vc], and I shall outline this below. While this does provide the template for a rigorous treatment, the technicalities in achieving that are formidable. Given the low regularity of the event horizon, it is not even clear that there are many cross-sections whose areas can be defined; also the physicist’s argument is overly cavalier in its use of differential inequalities for surface area elements and in its treatment of caustics. A proof addressing the necessary regularity issues — and indeed clarifying some interpretational points — was only given in 2001, by Chrusćiel, Delay, Galloway and Howard [@Chrusciel:2001].
Here is the “physicist’s argument”: Let $l^a$ be the tangents to the affinely parameterized null generators of ${{{\mathscr H}^{+}}}$. (For now, we only consider points at which there is a unique tangent.) Then the [*Raychaudhuri equation*]{} (derived from the geodesic deviation equation) is $$l^a\nabla _a\rho =\rho ^2+|\sigma |^2 +4\pi G T_{ab}l^al^b\, ,$$ where $\rho$ (real) and $\sigma$ (complex) are the convergence and shear of $l^a$. Assuming the [*weak energy condition*]{} $T_{ab}l^al^b\geq 0$, this shows that $l^a\nabla _a\rho\geq \rho ^2$. Any function satisfying this inequality will diverge to infinity within a finite parameter range, if $\rho$ ever becomes positive. But if this were to happen either the generator would be incomplete (which would correspond to a singularity) or it would develop a conjugate point, which is not allowed because that would require a null generator to enter the black-hole region. Thus, excepting the possibility of an incomplete generator, we must have $\rho\leq 0$ everywhere.
Since $\rho$ is the convergence of $l^a$, it can be interpreted as $-(1/2)({{\mathcal L}}_l dA)/dA$, where $dA$ is the area element transverse to the generator. Thus $\rho \leq 0$ is a statement that the area element cannot decrease as we flow forward along generators. Since generators can never, as we move forward in time, leave ${{{\mathscr H}^{+}}}$, but new generators can join, we expect the appearance of caustics only to lead to further increases in the area. This completes the “physicist’s argument.”
[r]{}[0.4]{}
[{width=".35\textwidth"}]{}
\[fig:telarea\]
Note the teleological character of the argument: the convergence can never, [*in the future*]{}, become positive, and therefore the Raychaudhuri equation and the weak energy condition imply that it must be non-positive [*at earlier times.*]{} In fact, if we push this argument a little further, we see that what we have is a differential equation with a [*final*]{}, rather than an initial, condition. We have seen that $\rho$ is a monotonically increasing function, bounded above, on each generator, and so its limit must exist. But this limit cannot be negative, for that is not allowed by the Raychaudhuri equation. Thus $\rho$ must tend to zero, and we have the integral equation $$\label{rhoint}
\rho (s)=-\int _{s}^\infty ( (\rho ^2+|\sigma |^2 +4\pi G T_{ab}l^al^b)({{\acute s}}) )d{{\acute s}}\, ,$$ where the bizarre range of integration, from the affine parameter value $s$ in question [*forward*]{} in time, makes manifest the teleological character of the situtation.
Imagine, for example, a spherically symmetric black hole forming from an initial infall of matter to a vacuum region; say this matter has crossed the event horizon for affine parameters $s_1\leq s\leq s_2$. In general, the event horizon will have formed [*before*]{} the matter got there, in anticipation of its arrival. Similarly, if during a later interval $s_3\leq s\leq s_4$ more matter falls in, then during the no-infall period $s_2\leq s\leq s_3$ the event horizon will expand in anticipation of that infall. In fact, the growth of the square root of the area element can be linear in the no-infall periods but only sublinear in the infall periods; see Fig. 1.[^5] It is this behavior which the Area Theorem reflects.
Stationary Black Holes
----------------------
The Area Theorem is the strongest general, dynamical, result on black holes. The problem of developing further dynamical results in full general relativity is largely open, although there is a large body of perturbative results and also an increasing body of numerical work. There does exist, however, important work on stationary black holes.
This material may be roughly divided into a family of results and expectations which is usually regarded as adequate for most physical purposes, and the serious mathematical problems involved in justifying them. I am here going to sketch the results and the (sometimes rough) arguments given for them. There is much rigorous work in this area, and it is very active, but giving an account on it would involve many qualifications and technicalities. Among older works, Carter’s review article [@Carter:1979] is a good introduction to the material, and of course many important ideas are in @Hawking:1973uf. A recent sampler, with references, will be found in section 3 of the useful paper by Chrusćiel, Galloway and Pollack [@Chrusciel:2010fn].
There is a general expectation that in many cases a black hole will settle down to be well approximated, at least at large distances, by a stationary space–time, that is, one which possesses a Killing vector $\xi ^a$ which is timelike at large distances. (That $\xi ^a$ need not be timelike everywhere will be discussed in more detail shortly.) We assume that at large distances $\xi ^a$ approaches a unit future-directed vector.
One would expect that in order to be stationary, the space–time must have a significant degree of symmetry, for typically inhomogeneities lead to multipole moments which result in gravitational radiation, which is not compatible with stationarity. One possibility is that the space–time is not only stationary but in fact static, that is, possesses a time-reversal symmetry. Such a situation is usually regarded as very specialized; I shall not discuss it further. Another possibility is that the space–time is axisymmetric, that is, possesses a second Killing field $\phi ^a$, with closed spacelike orbits, commuting with $\xi ^a$. (If $\phi ^a$ did not commute with $\xi ^a$, their commutator would be another Killing field and one would be in a more specialized situation. Under mild assumptions one can prove that the two must commute [@Carter:1970ea; @Carter:1979].) One can in fact prove, subject to certain assumptions, that a non-static stationary black-hole space–time must be axisymmetric in this sense [@Hawking:1973uf; @Chrusciel:2010fn].
When suitable assumptions are made on the matter fields, one may hope to classify the stationary axisymmetric space–times. The “no-hair” principle asserts that the only Einstein–Maxwell such solutions are those of the Kerr–Newman family; in particular, a stationary, axisymmetric vacuum solution is expected to be a Kerr solution.
[*Ergoregions and Energy Extraction.*]{} The portion of space–time exterior to the black hole in which $\xi ^a$ is not timelike is the [*ergoregion;*]{} the boundary where $\xi ^a$ makes the transition to timelike is the [*stationary limit surface*]{}. It can be shown that at any event in the exterior the two-plane spanned by $\xi ^a$ and $\phi ^a$ is timelike (becoming null precisely at the event horizon), and so even a point in the ergoregion will have a timelike Killing field in its vicinity. However, the discrepancy between this field and $\xi ^a$ gives rise to the possibility of extracting significant amounts of energy from a highly rotating black hole.
If a particle’s four-momentum is $p_a$, its $\xi ^a$[*-energy*]{} is $E=\xi ^ap_a$. It is not hard to show that this is conserved for a freely falling particle, and coincides with the usual energy measured in a frame defined by $\xi ^a$ in the asymptotic region. But now suppose the particle falls into the ergoregion, and splits into two, locally conserving energy–momentum. Because the vector $\xi ^a$ is space-like there, it is possible for the daughter particles to have $\xi ^a$-energies $E_1>E$, $E_2<0$. Then the first of these may escape to infinity with an augmented energy (while the second, having $E_2<0$, cannot escape to the asymptotic regime). In this way [*energy may be extracted from a black hole*]{}; this is called the [*Penrose process*]{} [@Penrose:1965].
The real point of the argument is not, however, the specific process, but the possibility of energy extraction. Suppose, for example, one very gradually extracted energy from a Kerr solution, giving it time to equilibrate. Then, by the no-hair principle, it would pass through a family of Kerr solutions, and for slow enough extractions one would expect this could be done adiabatically, that is, without ever deviating substantially from the Kerr family. Now, only a more detailed analysis could define and track the event horizon; each Kerr solution has rather what I called at the end of the introductory section a stand-in horizon. Still, for the adiabatic process just described, it seems plausible that the event horizon should track the stand-in horizons. Then the Area Theorem would provide a limit on the energy extraction, but it is hard to see that that there should be any other fundamental limit. For a Kerr black hole with mass $m$ and specific angular momentum $a$, this limit is $$m(1-2^{-1/2}(1+(1-a^2/m^2)^{1/2})^{1/2})\, .$$ If $a$ is a significant fraction of $m$, this is a significant fraction of $m$ as well, approaching $m(1-2^{-1/2})\simeq .29 m$ as $a\uparrow m$. Thus if astrophysical black holes acquire large specific angular momenta, colossal energies could be extracted from them.
[*Angular Velocity and Surface Gravity.*]{} The Killing fields $\xi ^a$, $\phi ^a$ must be tangent to the event horizon ${{{\mathscr H}^{+}}}$ (for it is a geometric invariant). The [*rigidity theorem*]{} [@Hawking:1971vc; @Hawking:1973uf] asserts that there is a Killing field which is tangent to the generators of ${{{\mathscr H}^{+}}}$, and (assuming there are no symmetries beyond those generated by $\xi ^a$ and $\phi ^a$), this field can only be a linear combination with constant coefficients of $\xi ^a$ and $\phi$. We normalize it to be $\chi ^a=\xi ^a+\omega\phi ^a$. Then $\omega \phi ^a$ can be interpreted as the [*angular velocity*]{} of the event horizon relative to infinity.
On the horizon itself, since the generators are geodesics, we must have $\chi ^b\nabla _b\chi ^a=\kappa \chi ^a$ for some scalar $\kappa$, necessarily constant up the generators. It is a remarkable fact that $\kappa$ [*must in fact be constant over the horizon*]{}, which can be shown by differentiating Killing’s equation and using the [*dominant energy condition*]{} (the energy–momentum density $T_{ab}t^b$ measured by any observer with future-directed timelike tangent $t^a$ should itself be future-directed). The constant $\kappa$ is called the [*surface gravity*]{} of the black hole.
It is important to note that the definition of surface gravity is non-local, for it requires the vector $\xi ^a$ to be normalized at infinity. In fact, suppose a test particle rigidly co-rotates with a black hole, just outside the event horizon. Then the surface gravity turns out to be the particle’s acceleration divided by $u^a\xi _a$ (where $u^a$ is its unit future-directed timelike tangent) [@Bardeen:1973gs], so one might more properly call $\kappa$ the surface gravity, measured relative to infinity (for that is where $\xi ^a$ becomes a unit vector).
@Wald:1984 gives a nice interpretation of surface gravity, for spherically symmetric holes. Consider lowering a small mass slowly on a very light but inelastic string towards the event horizon from infinity. As one lowers it, one can recover potential energy. By working out the change in energy as the string is lowered, one can find the tension that must be applied by the holder of the string, and thus the applied acceleration to keep the string stationary. One finds that as the end of the string approaches the horizon, the acceleration approaches the surface gravity. (This argument works for strings lowered along the axis of spinning black holes, too.)
Another interpretation (valid for spherically symmetric holes, again) is as the logarithmic rate of change of the red-shift factor along radial null geodesics from the distant past to the distant future; this will be discussed in the next section.
The existence of a constant surface gravity is quite remarkable, and not fully understood. It plays an essential role in “classical black-hole thermodynamics,” as well as in Hawking’s quantum analysis of black holes.
[*Energy and Angular Momentum.*]{} In general we do not have good definitions of energy–momentum or angular momentum in general relativity. However, there are certain classes of situations for which we do have satisfactory results. In particular, for space–times admitting Bondi–Sachs asymptotics these quantities can be defined at arbitatrary cuts of ${{{\mathscr I}^{+}}}$, and, if the space–time is stationary, there is a well-define sense in which they are constant (independent of the cut). In the stationary axisymmetric case, the energy and angular momentum turn out to be given by the [*Komar integrals*]{} $$E=(8\pi G)^{-1}\oint \epsilon _{abcd}\nabla ^{[c}\xi ^{d]}
\, ,\quad
J=-(16\pi G)^{-1}\oint \epsilon _{abcd}\nabla ^{[c}\phi ^{d]}\, ,$$ where the integrals are taken over a shear-free cut of ${{{\mathscr I}^{+}}}$.
In the case of vacuum exteriors, the surfaces of integration can be continuously deformed without affecting the results, and so it has been common to define the energy and angular momentum “locally,” or “of the hole itself” by deforming the surface to the event horizon. However, while there is a good justification of this in the vacuum-exterior case, I would caution that in general the physical interpretation of the Komar integrals at the horizon is obscure. Some hint of this is seen in the famous factor of $2$ between the two, which signifies that the Komar integrals do [*not*]{} give the conserved quantities as linear functions of the associated Killing vectors, even at ${{{\mathscr I}^{+}}}$. (That is, the conserved quantity associated with $a\xi ^a +b\phi ^a$ is $aE-bJ$,[^6] but this is not the Komar integral of the Killing field $a\xi ^a+b\phi ^a$.) The logic justifying the physical interpretation of the Komar integrals as energy and angular momentum applies [*only*]{} at infinity.
Classical Black-Hole Thermodynamics
-----------------------------------
The Area Theorem is clearly suggestive of the second law of thermodynamics, and soon after it was discovered @Bardeen:1973gs pointed out that there were other parallels. These authors in fact cautioned against taking the correspondence more literally, although very shortly thereafter several lines of thought came together which suggested that it was much more than formal. Some of those later arguments, which depend on quantum effects, will be described in more detail in Section 3; here the treatment is classical. It will be clearest to take the laws out of order.
The [*second law*]{} of classical black-hole thermodynamics is, as noted above, the Area Theorem. There are, however, difficulties, some more serious than others, in making the correspondence with ordinary thermodynamics precise.
First, the area of a black hole is a dimensionful quantity, whereas thermodynamic entropies are dimensionless (in units where Boltzmann’s constant is one); Hawking’s quantum analysis suggests that the entropy is $A/(4l_{\rm Pl}^2)$, where $l_{\rm Pl}$ is the Planck length. Second, thermodynamic entropies are only defined when some sort of averaging process (perhaps an implicit coarse-graining) is used; by contrast, no such procedure is evident in the definition of a back hole’s area. But the most important difference is that the area of the black hole is defined teleologically and is neither directly measurable in principle nor calculable from theory without teleological assumptions — something which is very different from ordinary physical reasoning.
The [*zeroth law*]{} of thermodynamics (existence of temperature) corresponds, in black-hole theory, to the existence of surface gravity, that is, to the fact that the surface gravity of a stationary hole is not only defined but is constant on the horizon. Classically, however, it is hard to see how this can be interpreted as a thermodynamic temperature (and its units are not those of temperature). Hawking’s remarkable quantum analysis, however, suggests that the incipient hole really does radiate at a temperature $T_{\rm H}=\hbar \kappa /c$.
Note however that the temperature and entropy have in principle very different bases in that the temperature is defined in terms of the [*stand-in horizon*]{}, whereas the justification for interpreting the area as an entropy requires using the [*event horizon*]{}. Also, while Hawking’s quantum analysis does suggest that a black hole really has a temperature in a conventional sense, it does not provide an [*independent*]{} explanation of just what the link between area and entropy is; the formula $A/(4l_{\rm Pl}^2)$ for the entropy is inferred, given the temperature, only by analogy with ordinary thermodynamics. The problem of justifying the interpretation of black-hole area as an entropy is a major open one. (This is one of the areas of work in quantum gravity theories.)
For the [*first law*]{}, we want a statement of conservation of energy in terms of physically satisfactory heat and work terms. We do not really have that. One underlying difficulty is that, except in the simplest cases, we must have theory which allows relativistic changes of reference frames (for example, one could imagine a process which resulted in the direction of the black hole’s energy–momentum changing), and at present there is as yet no agreed fully [*special*]{}-relativistic thermodynamics (see e.g. [@DHH:2009]). There is a related difficulty with angular momentum, in that one must be able to accommodate supertranslations.
There are two distinct versions of the first law discussed in the literature. The first is the [*equilibrium-state*]{} version. It only compares invariant information about two neighboring stationary black-hole states [*individually*]{}; it does not attempt to keep track of the additional invariant information needed to specify one of these [*relative*]{} to the other (asymptotically, the boost, rotation and supertranslation). It is therefore really a differential relation on the invariant parameters (mass and total angular momentum) of the family of stationary black-hole space–times.
The equilibrium-state version is most straightforward in the stationary axisymmetric vacuum case (which is believed to comprise only the Kerr solutions), where it takes the form $$\delta M=(8\pi )^{-1}\kappa\delta A +\omega\delta J\, ,$$ which one may compare to $$\delta E=T\delta S -\delta W$$ for an ordinary thermodynamic system, with $\delta W$ being the work done on the environment. The analogy here is certainly very attractive.
An extension of this to the general axisymmetric stationary case has been given by @Carter:1979; it has the form $$\delta M=\omega\delta J_{\rm H}+(8\pi )^{-1}\kappa \delta A
+\delta\int T_{ab}\xi ^a d\Sigma ^b -(1/2)\int T_{bc} h^{bc} \xi _a d\Sigma ^a\, ,$$ where $\Sigma$ is a spacelike three-surface joining the horizon to infinity, and $h_{ab}=\delta g_{ab}$ is the associated metric perturbation. Two points are important here: First, the diffeomorphism freedom has been used to adjust $h_{ab}$ so that the Killing vectors are preserved by the perturbation, and hence the relative boost, rotation and supertranslation of the two states are not encoded. Second, the change in angular momentum $\delta J_{\rm H}$ which appears here is that defined by the Komar integral applied at the horizon. Since we do not have a good justification for this except in the vacuum case, one is left with interpretational questions.
The second version of the first law is the [*physical-process*]{} form, and this aims to treat the true dynamical problem of passage of a system from one equilibrium state to another [@Wald:1995yp; @Gao:2001ut]. However, so far, work on this has only treated the case where one need not consider perturbations of the background geometry, and in particular this means that one is restricted to situations where one does not have to consider changes in the Killing vectors (for example, to accommodate relative boosts, rotations or supertranslations).
Finally, the literature also contains discussion of a possible [*third law*]{}, which would hold that one cannot by a finite sequence of even idealized processes reduce the temperature (i.e., surface gravity) of a black hole to zero.
Holonomy and Black Holes
========================
Our present definition of black holes is teleological, which is counter to virtually all conventional physical considerations. It is worthwhile to try to distance ourselves from this aspect of the theory, and try to identify what seem to be the prominent features of black holes for which we need not invoke teleology.
In this section, I will sketch an approach to this problem which seems promising. I will show that useful and potentially observable information is encoded in the infinitesimal holonomies relating the neighborhood of the event horizon to the regime occupied by distant observers. More precisely, we imagine parallel propagation along an outgoing null geodesic from a point near the event horizon to a very distant point; the change in this propagation as we consider later and later null geodesics is (essentially) the infinitesimal holonomy.
We will see that this quantity encodes certain [*universal*]{} features of the incipient hole, in that it is directly recoverable from signals emitted by matter falling towards the horizon, independent of the peculiar velocity of the matter or the point at which it will cross the horizon. The verification of this universal behavior, and even more so, of its persistence for long times, provides strong circumstantial evidence that an event horizon will form — and since this behavior is observable and the event horizon is not, one is led to suggest that this is a candidate for the physical quantity of interest.
This infinitesimal holonomy has other important features: it can be shown, in the stationary case, to give the surface gravity; it also makes clear at a quantitative level the way the exponential attenuation and red-shift of signals from matter near the horizon, which was discovered in the early model of @Oppenheimer:1939ue, extends to more general situations.
These results do need further development. The situation is cleanest, and best worked out, in the case of radial null geodesics in a spherically symmetric space–time [@Helfer:2001]; there are also results for families of null geodesics approaching the generators of an event horizon, without any symmetry assumptions. However, for a full connection with potential observables, one also needs to know about families of null geodesics which emerge from near the event horizon, but do not quite approach generators. Too, one would like to know whether these ideas can help with the teleological problems in classical black-hole thermodynamics. Finally, precisely because the approach here is so far from teleological, it is not clear how the ideas here are linked to notions of trapping.
Spherically Symmetric Black Holes {#sphol}
---------------------------------
We consider here a spherically symmetric distribution of matter which collapses to form a black hole. The matter need not be sharply bounded (although it may help to think of this case), but we do require that it fall off, as one moves outwards towards either the future or past along radial null geodesics, that future and past null infinities ${{{\mathscr I}^{+}}}$ and ${{{\mathscr I}^{-}}}$ exist, with their usual Bondi structures.
We shall use $u$ as the Bondi retarded time parameter at ${{{\mathscr I}^{+}}}$, and similarly $v$ as the Bondi advanced time at ${{{\mathscr I}^{-}}}$. Each of these coordinates can be extended uniquely inwards as far as the center of symmetry by requiring them to be spherically symmetric and null. Note that the spherical symmetry gives us a preferred time direction $t^a$ (orthogonal to the spheres and to $\nabla _a r$). Both $u$ and $v$ are normalized relative to this, so that $t^a\nabla _au=t^a\nabla _av=1$ (at ${{{\mathscr I}^{+}}}$, ${{{\mathscr I}^{-}}}$).
We define a future-pointing null vector field $l^a$ as follows. We let $l^a$ be the tangent vector to affinely parameterized radial null geodesics, normalized by $l^at_a\Bigr| _{{{\mathscr I}^{+}}}=1$. (This will, strictly speaking, make $l^a$ ill-defined at the center of symmetry, since any point at the center has a sphere’s worth of radial null geodesics through it, but that will not matter.) Then define a null vector field $n^a$ by requiring that it be future-pointing and directed radially inwards, parallel transported along $l^a$ (so $l^a\nabla _a n^b=0$), and normalized so that $l^an_a=1$. (It is common to define $l^a$ and $n^a$ like this near ${{{\mathscr I}^{+}}}$, but to use a different choice near ${{{\mathscr I}^{-}}}$. However, the present system will be most convenient for us.) Note that $n^a=\partial /\partial u$ and $l_a=du$.
We now define a function which will be of central importance, both here and in the discussion of Hawking’s prediction of black-hole radiation. Imagine following a radial null geodesic backwards in time. We start from the geodesic’s future end-point at some retarded time $u$ on ${{{\mathscr I}^{+}}}$, and end with its past end-point at a retarded time $v={{v}}(u)$ on ${{{\mathscr I}^{-}}}$. The function ${{v}}(u)$ is, in the sense of geometric optics, the [*mapping of surfaces of constant phase*]{} for spherically symmetric waves. (One could also consider, of course, the inverse function $u={{u}}(v)$, but most often ${{v}}(u)$ is what comes up.)
Note that ${{v}}(u)$ is a strictly monotonically increasing function, and so as $u\to +\infty$ we must have either ${{v}}(u)\to +\infty$ or ${{v}}(u)\to v_*$, a finite limit. the latter case represents formation of a black hole, for then no radial geodesic starting from an advanced time $v\geq v_*$ can emerge to ${{{\mathscr I}^{+}}}$. The limiting value $v_*$ is [*the advanced time of formation of the black hole.*]{}
Now consider, in geometric optics, the propagation of a spherically symmetric wave. If two crests of the wave are separated by a period $\delta v$ at ${{{\mathscr I}^{-}}}$, they will emerge with separation $\delta u$ at ${{{\mathscr I}^{+}}}$, where $\delta v={{v}}'(u) \delta u$ (assuming the period is short enough that ${{v}}'$ is practically constant over the period). [*Thus ${{v}}'(u)$ is the factor by which frequencies are red-shifted,*]{} in the passage of spherically symmetric waves from the distant past to the distant future.
[r]{}[0.5]{}
[{width=".45\textwidth"}]{}
\[fig:spherhol\]
In fact, what we have found is really a holonomy, for we are comparing the parallel propagation of a wave along the radial null geodesic with, implicitly, parallel propagation along a very distant path from the point of emission on ${{{\mathscr I}^{-}}}$ to the point of detection on ${{{\mathscr I}^{+}}}$. Precisely, if we consider the closed path formed by starting at retarded time $u$ at ${{{\mathscr I}^{+}}}$, going backwards along a very distant path to advanced time ${{v}}(u)$ at ${{{\mathscr I}^{-}}}$, and then going forward along radial null geodesic to retarded time $u$ at ${{{\mathscr I}^{+}}}$ again, the holonomy acting on vectors orthogonal to the spheres of symmetry is $$\Lambda ^a{}_b(u)=({{v}}' (u))^{-1} l^an_b+ {{v}}'(u) n^al_b\, .$$ Since the curvature gives the infinitesimal parallel transport, we have $\partial _u\Lambda ^a{}_b=-(\int R_{pqc}{}^a l^pn^q\, ds)\Lambda ^c{}_b$, where we understand that the integrand is parallel-transported to the point at the end of the geodesic on ${{{\mathscr I}^{+}}}$. We thus find that $$\label{vac}
{{v}}''(u)={{v}}'(u) \int R_{pqrs}l^pn^ql^rn^s{}\, ds\, .$$
Now suppose that the integral $\int R_{abcd}l^an^bl^cn^d\, ds\leq -a<0$. Then eq. (\[vac\]) implies that ${{v}}'(u)$ will be driven to zero at least exponentially quickly, and we will have $\lim _{u\to +\infty} {{v}}(u)=v_*$, a finite value. In this case, an event horizon will form at advanced time $v_*$. This is in fact typical of what one expects in most spherically symmetric models. (Conceivably there could be models in which black holes form “softly,” and ${{v}}'(u)$ decreases to zero but not exponentially quickly.)
There is rather more to the story than this, however. Suppose that an event horizon will form. Then $v$ is a good coordinate along the horizon, but $u$ is not, as it diverges. On the other hand, we can take $U={{v}}(u)$ as a coordinate near the horizon; in fact, this coordinate is regular across the horizon (except at the origin), for it is also equal to the advanced time of emission of the radial geodesic which passes outwards through the point. Thus $U={{v}}(u)$ [*resolves the singularity*]{} $u=+\infty$ at the horizon. (In the Schwarzschild case, one finds $U$ is a null Kruskal coordinate.)
[r]{}[0.4]{}
[{width=".35\textwidth"}]{}
\[fig:spherhola\]
Now note that we have $n^a=\partial _u=-{{v}}'(u)\partial _U$. This the vector $n^a$ will tend to zero as rapidly as ${{v}}' (u)$ does as we approach any fixed point, or indeed any compact subset, of the horizon. On the other hand, the corresponding covector $n_a$ can be viewed as the normalized affine one-form $ds$ along the radial geodesics. This means that [*the affine measure $ds$ along the outgoing radial null geodesics, normalized relative to ${{{\mathscr I}^{+}}}$, tends to zero, as rapidly as ${{v}}'(u)$ does, along any compact interval of the geodesic approaching the horizon.*]{}
What this result means is that, not only does ${{v}}'(u)$ tend to zero because the integral on the right of eq. (\[vac\]) is tending to a negative value, but in fact [*the contributions to the integral from portions of the geodesic near the horizon or to the past of it are suppressed by $\sim {{v}}' (u)$ as well.*]{} Thus [*the ratio $-{{v}}''(u) /{{v}}' (u)$ rapidly becomes independent of all information along the geodesic near, or to the past of, the horizon.*]{} This ratio can be shown, in the case of a black hole which, once formed, is stationary, to be the surface gravity [@Helfer:2001]. We thus set $\kappa =-{{v}}''(u)/{{v}}' (u)$, the [*(running) surface gravity*]{} of the incipient black hole.
It is worthwhile emphasizing the limiting process involved here. We have a family of later and later radial null geodesics, and on these we have segments. We speak of these segments approaching the horizon in the natural sense of the topological structure of space–time. This is very different, however, from the coordinate-dependent senses of approach often used in the literature. (Many authors, for example, consider measuring closeness by a Schwarzschild coordinate $r$ or $r_*$, and neglecting $u$.)
This result has potentially important physical consequences. Consider some matter falling across the horizon at some event $p$, and emitting radiation radially outwards as it does so. That radiation emerges at ${{{\mathscr I}^{+}}}$ (in the geometric-optics limit) with a red-shift which is partly due to its climb out of the gravitational potential, but partly due to the peculiar velocity of its source. The climb out of the potential in turn depends partly on segments of the null geodesic from the point of emission outwards close to the event horizon, and partly on the climb outwards to the asymptotic regime — although there is no clean way of saying where one portion of this climb ends and the other begins. But what the result above shows is that [*the logarithmic derivative of the red-shift approaches $-{{v}}''(u)/{{v}}' (u)$ and so becomes independent of the peculiar velocity of the source, independent of the precise point of emission, and indeed independent of the geometry along any segment of the null geodesic close to any compact portion of the corresponding generator of the horizon*]{},[^7] as the horizon is approached. This is therefore a [*universal*]{} quantity associated with the horizon: all matter crossing any compact part of the horizon and emitting signals radially outwards will give rise to the same fractional rate of change of the red-shift factor.
I close this subsection with some comments: (a) It ought to be possible to test this universality observationally, and indeed the existence and persistence of it would be circumstantial evidence for a black hole. (b) It is of considerable interest to study non-radial null geodesics as well as radial ones. (c) Because of the exponential red-shifts (and associated exponential attentuations), one quickly reaches a point where any real quanta emitted in the vicinity of the horizon have been so severely red-shifted on their way out that they are below the limits of any detection. Thus we are confronted with quantum limitations on the measurement of the vicinity of the incipient black hole, a point which will be discussed in section 5.
The General Case
----------------
I will now drop the assumption of spherical symmetry, and take up the case of general black holes. The theory here is less well developed than in the previous case, and I shall indicate the approach, an attractive result, and some problematic issues.
In the previous case, we could use the symmetry to identify a candidate family of null geodesics which would have to approach the generators of a horizon, were one to form: the radial null geodesics. In the present, general, case, the problem of non-teleologically determining which escaping null geodesics count as approaching the generators of the horizon (should one form) is more difficult, and a solution to it which is adequate for interpreting terrestrial observations will probably require analyzing also geodesics which are slightly off-kilter with respect to the generators, so that we can say what special properties do distinguish those more directly approaching the generators. Such an analysis has not yet been developed. Here I will suppose that we are following a family of null geodesics which is approaching a generator of the horizon, and leave open the question of how the family is determined.
Any generator of ${{{\mathscr H}^{+}}}$ is a null geodesic segment; let us consider a one-parameter family $\gamma _u(s)$ of affinely parameterized null geodesics approaching such a segment (along compact subsets), escaping to points $\gamma _u(+\infty )$ at Bondi retarded time $u$ on ${{{\mathscr I}^{+}}}$, and with tangents $l^a$ normalized to the Bondi frame at ${{{\mathscr I}^{+}}}$. While $u$ and $s$ will not be good coordinates at the horizon, it is nevertheless meaningful to require that the family $\gamma _u(s)$ of geodesics approach the generator smoothly along compact subsets, for example by requiring that there are two curves $s=s_1(u)$, $s=s_2(u)$ for which $\gamma _u(s_j(u))$ tend smoothly to two distinct points on the generator. Note that there will be many one-parameter families of null geodesics approaching the same generator.
We will suppose that the vector field $l^a$ along this one-parameter family of geodesics has been completed to a parallel-propagated standard null basis ($l^a$, $m^a$, ${\overline m}^a$, $n^a$, the only non-zero inner products among them being $l^an_a=1$ and $m^a{\overline m}_a=-1$). Then $w^a=\partial _u\gamma$ is a connecting vector field, satisfying the Jacobi equation $$(l\cdot\nabla )^2w^a=l^pl^qR_{pbq}{}^aw^b\, .$$ Using the peeling of the curvature in the Bondi–Sachs asymptotics, one can show that $w^a\sim n^a +s(\omega m^a +\overline\omega {\overline m}^a)$ (plus a possible multiple of $l^a$) for some complex $\omega$ asymptotically, and we may identify $\omega m^a +\overline\omega {\overline m}^a$ as the (running, with respect to $u$) angular velocity of the family of geodesics approaching the particular generator of ${{{\mathscr H}^{+}}}$, relative to ${{{\mathscr I}^{+}}}$. (In the present, fully dynamical, situation, it is not clear that one can define a “running angular velocity” of a generator on ${{{\mathscr H}^{+}}}$, independent of the family used to approximate it.)
Let us now consider the construction of a holonomy operator. Without additional assumptions, there is no guarantee that the extensions of the geodesics $\gamma _u(s)$ to the past reach ${{{\mathscr I}^{-}}}$. On the other hand, we learned from the spherically symmetric case that for many purposes it is not necessary to consider the details of the pasts of these geodesics. Let us rather fix any point $p$ of interest on the generator of ${{{\mathscr H}^{+}}}$ that the geodesics are approaching, determined by $\lim _{u\to +\infty}\gamma _u(s(u))$ for some function $s(u)$. Consider the holonomy along a path from the end-point $\gamma _u(+\infty )$ of one of the geodesics, backwards along ${{{\mathscr I}^{+}}}$ to $\gamma _{u_0}(+\infty )$, then backwards along the $u_0$-geodesic to $\gamma _{u_0}(s(u_0))$, then along $\gamma _u(s(u))$, and finally outwards along the $u$-geodesic again.
In this non-symmetric case, the infinitesimal holonomy cannot be characterized simply by a scalar, but will be a generator of Lorentz motions. It is again given by integrating the curvature; the contribution from the geodesic[^8] is $$\lambda ^a{}_b=-\int R_{pqb}{}^al^p w^q\, ds\Bigr| _{{{\mathscr H}^{+}}}^{{{\mathscr I}^{+}}}\, ,$$ where we understand that the integrand has been parallel-propagated along $\gamma _u$ to $\gamma _u(+\infty )$, and the limits of integration stand for the point $\gamma _u(s(u))$ near ${{{\mathscr H}^{+}}}$ and the point $\gamma _u(+\infty )$ on ${{{\mathscr I}^{+}}}$. We note the identity $$\label{lambdident}
\lambda ^a{}_b l^b=-l\cdot\nabla w^a\Bigr| _{{{\mathscr H}^{+}}}^{{{\mathscr I}^{+}}}\, ,$$ where we understand that the vector $l\cdot\nabla w^a$ at the lower limit must be parallel-transported to the point at ${{{\mathscr I}^{+}}}$.
Let us now consider the case of a black hole which becomes stationary, and let $w^a=\chi ^a$ be the Killing field which becomes tangent to the generators of the horizon, as usual. If we contract both sides of eq. (\[lambdident\]) with $w^a({{{\mathscr H}^{+}}})$, we get $$\label{sugra}
\lambda ^a{}_bl^bw_a({{{\mathscr H}^{+}}})=-w_a({{{\mathscr H}^{+}}})(l\cdot \nabla w^a )({{{\mathscr I}^{+}}})
+w_a({{{\mathscr H}^{+}}})(l\cdot \nabla w^a)({{{\mathscr H}^{+}}})\, .$$ In the limit of later and later geodesics, the first term on the right will vanish in the stationary case, for $l\cdot\nabla w^a ({{{\mathscr I}^{+}}}) =\omega m^a+\overline\omega {\overline m}^a$, and $w^a({{{\mathscr H}^{+}}})$ will become proportional to $l^a$. In this limit, the second term (divided by $l\cdot w({{{\mathscr H}^{+}}})$) becomes a well-known expression for the surface gravity, and thus we conclude that, in the stationary case, the limiting value of $ \lambda ^a{}_bl^bw_a({{{\mathscr H}^{+}}}) /l^aw_a({{{\mathscr H}^{+}}})$ represents the surface gravity, where $w^a$ is chosen to be the usual Killing field. We have therefore $$\kappa =\lambda ^a{}_bl^bw_a({{{\mathscr H}^{+}}})/l^aw_a({{{\mathscr H}^{+}}})\, ,$$ where we understand the symbol ${{{\mathscr H}^{+}}}$ is a short-hand for taking the limit as the event horizon is approached. We therefore get a new interpretation of surface gravity, in case of stationary black holes, as this infinitesimal holonomy. The quantity (\[sugra\]) is well-defined even in the non-stationary case, however, and may be of interest to those searching for a general definition of surface gravity.
Quantum Fields in Curved Space–Time
===================================
While we are ultimately interested in those aspects of quantum field theory related to black holes, it will be helpful in this section to first review the general theory in curved space–time in the absence of extreme circumstances.
The present theory of quantum fields in curved space–time is so natural and attractive that one should be chary of trying to modify it. Still, it would be wise to keep the following points in mind:
- It is almost entirely untested experimentally. While we have evidence for many quantum-field-theoretic processes operating in the Universe, none of these so far probes deeply the distinctive elements of the curved-space theory.
- While the theory does give an extremely natural prescription for computing some important observables (for instance, field-strength operators and particle number density operators asymptotically far from sources), it runs into difficulties when we try to push it past a certain point, especially when we try to construct the stress–energy operator.
This latter point is not usually considered to be very important: a fly in the ointment only — but I shall explain in Section 5 why I believe it may be a hint of more serious foundational concerns: a cloud on the horizon.
A few remarks for students: The development of quantum field theory took decades, both to work out techniques of computation and to sort out interpretational issues, and there are still some foundational problems with it, while at the same time it is unarguably powerful and successful. Some parts of the field are understood in a practical sense better than they are in a foundational one. This is only to be expected in such a conceptually difficult area, and is a mark of how close the material is to the frontier of research. Expect to learn the material in stages and to refine your understanding as you go along — and always try to verify claims you encounter.
Contrast with Special-Relativistic Theory
-----------------------------------------
The development of quantum field theory in curved space–time forced physicists to reconsider the foundations of the usual theory. In doing so, it became apparent that much of the way the special-relativistic theory is usually explained — starting with Fock spaces and creation and annihilation operators — relies heavily on intuitions about how the theory should come out. These intuitions turn out to be correct in Minkowski space, but the difficulty is that appealing to them skips over the issues of just how they are justified — and in curved space–time, where the intuitions are not generally correct, we need to look at what the justifications were and how the conclusions must be modified.
The most fundamental point is that quantum field theory is, primitively, a theory of [*fields,*]{} and [*particles are a derived concept*]{} in it. This is of course almost backwards of the way the special-relativistic theory was developed, is usually taught, and is generally used by particle theorists. Nevertheless, to understand how the theory is generalized to curved space–time, this is the key point. We must think, for example, of quantum electromagnetism as being, at a primitive level, about the electric and magnetic field strength operators (or more properly their averages over small space–time volumes), rather than about photons. (Willis Lamb wanted to issue “photon licenses” only to those people who could use the term properly. He presumably did not have curved space–time in mind, but he had a deep sense of how less-than-straightforward the particle concept was.)
In fact, particles (and the related concepts of the vacuum — the no-particle state — and ladder operators) depend heavily on Poincaré invariance. They are defined by resolving the fields into positive- and negative-frequency parts, and it is a critical consequence of Poincaré invariance that the same positive-/negative-frequency decomposition is obtained no matter which inertial time axis is used for the Fourier transformation. However, in curved space–time, such a decomposition would indeed depend on the choice of timelike curves one Fourier-analyzed along.
One might think this is caviling, that in familiar cases the space–time curvature is such a tiny effect that that there is little substantive ambiguity in the definitions of positive and negative frequency. And indeed this is true for ordinary particle-physics purposes. But in crucial cases — in particular, for Hawking’s work on black holes — it is just these ambiguities, and their physically correct resolution, which are at the heart of things.
Linear Fields in Minkowski Space
--------------------------------
While the case of curved space–time differs from that of Minkowski space, it is worthwhile sketching the special-relativistic case here. In particular, I want to emphasize some points not often discussed in introductory treatments but important for us: the nonlocality of particle concepts, and the character of quantum field fluctuations.
In order to have a manifestly relativistically invariant theory, we work in the Heisenberg picture, so that the field operators will be functions on space–time (more properly, they will be operator-valued distributions). The state vector will be fixed, except when a quantum measurement is made, when it will be projected into a subspace according to the usual rules of quantum theory. However, those comments describe what we are aiming at; we must show how to achieve it: to construct the operators and the Hilbert space.
Let us consider a massless (there is little difference in the massive case) real scalar field satisfying the wave equation $$\nabla _a\nabla ^a\phi =0$$ and the canonical commutation relations $$\begin{aligned}
[\partial _t\phi (t,{\bf x}),\phi (t,{\bf y})] &=&i\delta ^{(3)}({\bf x}-{\bf y})\\
\left[\phi (t,{\bf x}),\phi (t,{\bf y})\right]&=&
[\partial _t\phi (t,{\bf x}),\partial _t\phi (t,{\bf y})]=0 \, .\end{aligned}$$ To quantize this, one typically proceeds as follows:
1. One Fourier-transforms the field, using the field equations to write $$\phi (t,{\bf x})=\int \left( e^{iEt-i{\bf k}\cdot {\bf x}} a({\bf k}) +e^{-iEt+i{\bf k}\cdot {\bf x}} a^*({\bf k})\right)
\frac{d^3{\bf k}}{2^{1/2}(2\pi )^{3/2}E^{1/2}}\, ,$$ where $E=E({\bf k})=\| {\bf k}\|$. At this point, the q-numbers $a({\bf k})$, $a^*({\bf k})$ have yet to be determined.
2. One then examines what the canonical commutation relations imply about $a({\bf k})$, $a^*({\bf k})$, recognizing in them the usual quantum-mechanical [*ladder algebras*]{}: $$\begin{aligned}
[a({\bf k}),a^*({\bf l})] &=& \delta ^{(3)}({\bf k}-{\bf l})\\
\left[a ({\bf k}),a({\bf l})\right] &=&
[a^* ({\bf k}),a^*({\bf l})]=0\, .\end{aligned}$$ Notice that at this point we have not yet constructed the Hilbert space, so it would not really be correct, at this stage, to refer to the fields or the q-numbers as [*operators*]{} (they do not yet operate on anything). What we have defined at this point is the [*field algebra*]{} ${{\mathcal A}}$, that is, the commutation relations the fields must satisfy, rather as if we had defined a group by a multiplication table but had not yet given a representation of it by matrices acting on a vector space.
3. One then constructs the [*Fock representation*]{}, by [*assuming that there exists a Poincaré-invariant ground state*]{} $|0\rangle$ which is normalized and annihilated by all the $a({\bf k})$. The rest of the construction follows directly by applying the canonical commutation relations, just as one shows in elementary quantum mechanics that one can deduce the spectrum of the harmonic oscillator from the ladder-operator algebra together with the assumption that there is a normalizable ground state.
4. Finally, one establishes the particulate interpretation of the theory by showing that the vacuum and one-particle states are the appropriate eigenstates of the mass and spin operators, and the $n$-particle states are tensor products of the one-particle states.
There are a number of important points to make about this:
\(a) The entire procedure uses Poincaré invariance, and most especially time-translation invariance, very strongly. It is this which gives us the split of the field operators into positive- and negative-frequency parts, and the vacuum state, from which all others are constructed, is characterized in terms of this split.
\(b) There are other representations of the field algebra, but they do not admit Poincaré-invariant vacuum states.
\(c) The particle states are non-local. If, for example, we work out the inner product on one-particle states, we find that it has the form $$\langle \phi |\psi\rangle =\int {\overline\phi}(0,{\bf x})K({\bf x},{\bf y})\psi (0,{\bf y})\, d^3{\bf x}\, d^3{\bf y}\, ,$$ where the kernel $K({\bf x},{\bf y})$ is not simply the delta function $\delta ^{(3)}({\bf x}-{\bf y})$, but is non-zero even for ${\bf x}\not={\bf y}$. This non-locality is not very important for modes of wavelengths above the Compton scale $\|{\bf k}\| ^{-1}$.[^9] However, for shorter-wavelength modes the non-local nature of the kernel must be taken into account. Note that this means [*it is impossible to say that a particle is within a volume of space of characteristic size $\lesssim \hbar c/E({\bf k})$.*]{}
\(d) Measurements of the field operators in arbitrarily small space–time volumes are defined, however. But the particle states, and in particular the vacuum state, are [*not*]{} eigenstates of these operators. For example, if $\Phi (a) =(4\pi a^3/3)^{-1}\int _{{\bf x}\leq a}\phi (0,{\bf x})\, d^3{\bf x}$ is the average of the field over a sphere of radius $a$, then one finds $\langle 0|\Phi (a)|0\rangle =0$ but $$\langle 0|(\Phi (a))^2|0\rangle\sim a^{-2}\mbox{ as }a\to 0\, ,$$ the larger and larger fluctuations as $a\to 0$ representing the deviation of the vacuum state from an eigenstate of the field operator. These are the famous [*vacuum fluctuations*]{}. It will be vacuum fluctuations which will ultimately be responsible for the production of quanta in Hawking’s proposal of black-hole radiation.
\(e) The two-point function $$\label{twopoint}
\langle 0|\phi (p)\phi (q)|0\rangle =
-\frac{1}{4\pi ^2}\cdot\frac{1}{(p-q-i\epsilon )^2}
\, ,$$ where I am now usuing $p$, $q$ to stand for points in Minkowski space and $\epsilon$ is an infinitesimal future-directed timelike vector, plays a key role.[^10] In fact, for any “reasonable” state $|\Psi\rangle$, the two-point function $\langle\Psi |\phi (p)\phi (q)|\Psi\rangle$ will have the same ultraviolet asymptotics (that is, as $p\to q$), since there will be an energy scale beyond which the state is unexcited and on correspondingly small space–time scales the state will “look like” the vacuum.
\(f) Note in particular that the two-point function does not vanish at spacelike separations. This means that [*the vacuum state contains field correlations at spacelike separations.*]{}
A consequence of this is that one cannot simply restrict the Fock quantization to a subvolume of Minkowksi space, for the fields in that volume are correlated with those elsewhere. For many problems, of course, the effects of such correlations are negligible; on the other had, in some cases they are important and one must look to the precise physics of the situation. But it is the contrapositive of this which is the most important for many of the arguments in this area: it is generally impossible to “quantize in a given volume” and have the results there accurately reflect the Fock quantization. In particular, arguments about “quantization in the Rindler wedge” should be examined carefully.
The [*origin*]{} of the spacelike correlations in the field is a profound, ultimately cosmological, problem.
Construction of a Quantum Field Theory
--------------------------------------
I shall here outline the construction of the theory of a linear, massless, minimally coupled real scalar field in curved space–time. This is really a stand-in for the physically more interesting case of the electromagnetic field, but the constructions depend little on the mass or spin. The main advantage of the scalar case is that we do not have to deal with technical issues relating to the quantum treatment of gauge invariance; such issues do not affect the points we will make. As above, we will work in the Heisenberg picture.
It is helpful to think of the construction in two stages. First, we construct the [*field algebra*]{} ${{\mathcal A}}$, that is, we specify the commutation relations which must hold. One should think of this as an abstract algebraic structure, much like specifying a group by giving its multiplication table. The second stage is to construct a [*representation*]{} of that algebra, that is, a physical Hilbert space ${{\mathcal H}}$, together with an interpretation of the fields as operators on that space, satisfying the correct commutation relations. In order to have a sensible theory, we require the space–time to be globally hyperbolic, that is, it admits Cauchy surfaces on which initial data for the field may be given resulting in a well-posed evolution problem.
Let us denote the field $\phi$. Then the field algebra ${{\mathcal A}}$ is characterized by two requirements: that the field equation $$\nabla ^a\nabla _a\phi =0$$ be satisfied; and that the canonical commutation relations $$[t^a\nabla _a\phi (p),\phi (q)] =i\delta _\Sigma (p,q)\, ,\quad
[\phi (p),\phi (q)]=[t^a\nabla _a\phi (p),t^b\nabla _b\phi (q)]=0$$ on any Cauchy surface $\Sigma$ with future-directed normal $t^a$, hold.[^11] In fact, the canonical commutation relations can be written in a manifestly invariant form first noted by Peierls: $$[\phi (p),\phi (q)] = \Delta _{\rm a}(p,q) -\Delta _{\rm r}(p,q)\, ,$$ where $\Delta _{\rm a}$, $\Delta _{\rm r}$ are the advanced and retarded Green’s functions for the field equation.
We now turn to the representation.
In quantum mechanics, the Stone–von Neumann Theorem asserts that all representations of the canonical commutation relations are unitarily equivalent. Thus in quantum mechanics there is no question of principle as to which representation to use: one may be more convenient than another for some purposes, but all are interconvertible. In quantum field theory, the situation is different. There, there are an infinite number of distinct equivalence classes of representations, and one must decide which are physically appropriate. In the special-relativistic theory, this was done by requiring the existence of a Poincaré-invariant state, the vacuum; here, we must find another criterion.
It turns out that the equivalence classes of the representations are essentially specified by the asymptotic forms of the two-point functions $$\langle\Psi | \phi (p)\phi (q) |\Psi\rangle$$ for $p$ and $q$ close to each other, and as one (or both) recede to infinity. These correspond, in standard parlance, to the ultraviolet and infrared asymptotics of the theory. We shall not worry about infrared issues, which typically bear on only cosmological questions. On the other hand, the ultraviolet asymptotics go to the question of how the theory, on short scales, compares with that in Minkowski space.[^12]
It turns out that there is a distinguished equivalence class of representations (modulo infrared issues) for which the leading ultraviolet asymptotics have the same form as in Minkowski space. These are the [*Hadamard*]{} representations, and they are the physically natural candidates for quantization. It is an important and non-trivial theorem that these are well-defined, that is, that the correct ultraviolet asymptotics are preserved under evolution.
We assume henceforth that we have constructed a Hadamard representation. We can do this in practice, for instance, by selecting a Cauchy surface and identifying the Cauchy data with those for the corresponding field equation on a Cauchy surface in Minkowski space in the usual Fock representation, and then defining the field elsewhere by its evolution.
We now have, in principle at least, enough data to define the quantum field theory. We have shown that there is a well-defined field algebra ${{\mathcal A}}$, and a preferred (up to infrared issues) representation of that algebra as operators on a Hilbert space ${{\mathcal H}}$. Of course, the description is so far fairly abstract, and we must develop it in more detail if we are to learn how to analyze particular physical problems. There are two main issues to take up: how to effectively compute the evolution of the fields; and how to specify (and analyze) the physical content of the states. The former depends on the particular space–time, but the latter is a more general issue.
It has sometimes been suggested that one can dispense with the construction of representations in the sense I have described, and work only with the algebra ${{\mathcal A}}$ and what can be thought of as generalized density matrices (and it is these which are called states in the algebraic approach). While this is adequate for some purposes, any density-matrix-like approach does not detect phase information which is generally important in quantum theory; also, one must somehow cut the freedom in the choice of class of admissible density matrices down to correspond to that in an irreducible representation of the field algebra, and in doing this one must either use the Hadamard condition or come up with a physically plausible substitute.
The Physical Content of a State
-------------------------------
I emphasized earlier that the primitive elements of our theory are the quantum fields. I have been discussing, for simplicity, a scalar field $\phi$, for which the field measurements would correspond to (weighted averages of) $\phi $ and its canonically conjugate momentum $t^a\nabla _a\phi$ near a hypersurface with normal $t^a$, but this is really a stand-in for the more physically realistic case of the electromagnetic field, where one measures (weighted averages of) the magnetic ${\bf B}$ and electric ${\bf E}$ field strength operators.[^13] These operators ($\phi$ and $t^a\nabla _a\phi$, or ${\bf B}$ and ${\bf E}$) are complementary, and so one cannot measure them simultaneously. Thus one could specify the state by giving a “wave functional” $\Psi _q[\varphi ]$ or $\Psi _q [{\mathcal B}]$ analogous to the position wave function in quantum mechanics, or a momentum-type functional $\Psi _p[t^a\nabla _a\varphi ]$ or $\Psi _p[{\mathcal E}]$, in each case referring to the field eigenvalues on a specific hypersurface. These wave functionals are just the coefficients of the corresponding eigenstates of the field measurements, just as the wave functions in quantum mechanics are the coefficients of the corresponding position or momentum eigenstates.[^14] Such expressions are said to concern the [*field aspect*]{} of the state.
At least in special-relativistic theory, however, we are more often concerned with a state’s [*particle aspect*]{} than its field aspect, that is, we wish to know its particle content. How is this defined?
In special relativity, the particles are supposed to be eigenstates of mass, spin, and other appropriate quantum numbers. The mass and spin (or, more precisely, their squares) are in fact the Casimir operators for the Poincaré group, and so analyzing the particle-content of a state involves decomposing it into irreducible representations of the Poincaré group.
In the case of linear quantum fields, the end-result of this is characterized by [*number density operators*]{} associated with the different modes of the field, analogous to the number operator for the harmonic oscillator. For the scalar field discussed earlier, these would be $$n({\bf k})=a^*({\bf k}) a({\bf k})\, ;$$ the number density of particles with momentum $\hbar{\bf k}$. (For the electromagnetic field, the operator depends on the polarization as well as the wave-number ${\bf k}$.) One sees quite explicitly that this depends on the Fourier transform of the field, which in turn depends heavily on the Poincaré invariance.
Poincaré invariance need not hold exactly for us to have a reasonably good, although in principle imperfect, definition of number density, however. Evidently, if we have a region of space–time which is well-approximated by a portion of Minkowski space of linear size $\sim l$, then the Fourier components of wave-packets within that region, of nominal wave-numbers $\gg l^{-1}$, will be well-defined, and we will have good approximate number-density operators for the corresponding modes. In fact, since all real particle detectors function only over finite volumes of space–time, they really respond to such approximate number density operators only. The discrepancy between these and the true operators is analogous to considering the corrections to an optical system due to finite-aperture and finite integration-time effects.
In many cases one can identify regions far from the gravitational sources in which the curvature is quite small and one thus has good definitions of number density operators for at least substantial regimes in wave-number space. In fact, often we can cover hypersurfaces in the distant past or future with regions which are approximately Minkowskian, and then we can analyze the particle-contents within these regions fairly well, and get overall characterizations of the states up to the problems associated with patching the regions together. On the other hand, when we attempt to link the vicinity of a black hole’s event horizon to the region near ${{{\mathscr I}^{+}}}$, we have seen that the holonomy typically involves exponentially growing an decaying terms, and thus curvature effects over this entire regime cannot be discounted. We will examine this in more detail in the next section.
The reader may have noticed that something curious has crept in in the localization of the particle content just discussed: while we may have good (approximate) definitions of particle number-density at different times, there is no guarantee that those at one time should be compatible with those at another. This is indeed the case, and corresponds to the possibility of particle creation, as will now be discussed.
Bogoliubov Transformations {#BTsec}
--------------------------
Suppose that we have a field $\phi$ propagating through space–time, and on two Cauchy hypersurfaces $\Sigma _{\rm p}$, $\Sigma _{\rm f}$ we have physically-justified resolutions of the field into positive- and negative-frequency modes and associated annihilation and creation operators $a_{\rm p,f}({\bf k})$, $a^*_{\rm p,f}({\bf k})$. Here, the subscripts “p,” “f” are meant to suggest past and future, but that is to fix ideas only. The relative positions of $\Sigma _{\rm p}$, $\Sigma _{\rm f}$ are immaterial, and they may overlap. Also, I have written the mode label ${\bf k}$ as if it were a wavenumber, but this interpretation is not necessary; one only needs the ladder algebra to be obeyed. A simple if overidealized example would occur in a space–time which was initially stationary and close to Minkowskian, then passed through a time-dependent phase, and returned finally to a stationary, approximately Minkowskian, state.
Because the field algebra can be expressed equally well on either Cauchy surface, we may express the annihilation operators with respect to $\Sigma _{\rm p}$ in terms of those at $\Sigma _{\rm f}$ (or vice versa): $$\label{bog}
a_{\rm p}({\bf k}) =\int \alpha ({\bf k},{\bf l}) a_{\rm f}({\bf l})d^3{\bf l}
+\int \beta ({\bf k},{\bf l}) a^*_{\rm f}({\bf l})d^3{\bf l}\, ,$$ or more briefly $$\label{bogb}
a_{\rm p} =\alpha a_{\rm f}
+\beta a^*_{\rm f}\, .$$ That is, because the field has propagated through a time-dependent region, its modes have mixed relative to the two different ways of resolving it. The quantities $\alpha$ and $\beta$ are called [*Bogolioubov coefficents*]{}, and eqs. (\[bog\]), (\[bogb\]) are called a [*Bogoliubov transformation*]{}. Note that the field modes, that is, the coefficients of the annihilation and creation operators in the expansion of the field operators, will also transform via a Bogoliubov transformation, contragredient to the q-numbers $a$, $a^*$.
We may construct Fock-like representations of the field relative to each of $\Sigma _{\rm p,f}$, beginning in each case with a vacuum state $|0_{\rm p,f}\rangle$ characterized by $a_{\rm p,f}|0_{\rm p,f}\rangle =0$, and applying the creation $a^*_{\rm p,f}$ and annihilation $a_{\rm p,f}$ operators. These will both be Hadamard representations (assuming that the positive-/negative-frequency decompositions at $\Sigma _{\rm p,f}$ were physically correct, that is, gave the right ultraviolet asymptotics), and so they will be equivalent, but there will be a non-trivial transformation between them.
We can see this explicitly. Suppose the state is the p-vacuum $|0_{\rm p}\rangle$, characterized by $a_{\rm p}|0_{\rm p}\rangle =0$. Using eq. (\[bogb\]), we can rewrite this condition as $$(\alpha a_{\rm f}
+\beta a^*_{\rm f})|0_{\rm p}\rangle =0\, .$$ We can regard this as an equation for $|0_{\rm p}\rangle$ in terms of the data at $\Sigma _{\rm f}$. In fact, simply using the ladder algebra, we find $$\begin{aligned}
\label{vactr}
|0_{\rm p}\rangle &=&\mbox{(normalization)}\cdot
\exp [-(\alpha ^{-1}\beta /2)a^*_{\rm f}a^*_{\rm f}]
\, |0_{\rm f}\rangle\\
&=&\mbox{(normalization)}\cdot
\sum _{n=0}^\infty \frac{(-(\alpha ^{-1}\beta /2)a^*_{\rm f}a^*_{\rm f})^n}{n!}\,
|0_{\rm f}\rangle \, ,\end{aligned}$$ showing that the p-vacuum appears as a superposition of states of different f-particle numbers, the first contribution being an f-vacuum one, then a two f-particle one, and so on. It is sometimes said that eq. (\[vactr\]) shows that the p-vacuum is a Gaussian distribution of f-particles, but one should bear in mind that this equation really applies at the level of probability [*amplitudes*]{} and not probability [*distributions*]{}.
Formulas like (\[vactr\]) show that [*the particle content of a state is a function of where it is observed*]{}, and is not absolute. There is a sort of limited observer-independence, in that it may well happen that in regions around $\Sigma _{\rm p}$ or $\Sigma _{\rm f}$ separately there may be agreement among relatively boosted, rotated or translated observers about the particle-content, but there will not be agreement in comparing $\Sigma _{\rm p}$ with $\Sigma _{\rm f}$. The physical interpretation of this is that the passage of the field through a time-dependent potential, or region of time-dependent curvature, has resulted in the creation or destruction of particles. Just this point was a main reason for developing quantum field theory in contradistinction to quantum mechanics: the need to accommodate changing particle numbers.
A well-known example of this — predicted, but not yet experimentally verified — is the [*Schwinger effect*]{}, where one considers the electron–positron field propagating through a time-dependent classical electromagnetic potential [@Schwinger:1951nm]. Even an initially vacuum state of the charged field is predicted to give rise to pairs of real charged particles as time passes.
It is important to note that one consequence of this observer-dependence of particle-content is that there will be regimes in which there is no very good definition of particle-content. For instance, in a region where the potential or the space–time geometry is changing on a timescale $\sim T$, it will not be possible to give much meaning to the notion of particles with energy $\sim\hbar /T$. This issue affects all of the arguments which attempt to explain black-hole radiation quantum-[*mechanically*]{}, that is, without confronting the [*field-theoretic*]{} issues bound up with the ambiguity in the definition of particles.
The Unruh Process
-----------------
@Unruh:1976db, in studying the foundations of Hawking’s black-hole radiation prediction, considered the response of a detector uniformly accelerating in Minkowski space, and predicted that it would respond as if it were in a thermal bath of temperature $T_{\rm U}=\hbar a/2\pi c$ (with $a$ the acceleration, Boltzmann’s constant being taken to be unity). This is not really a prediction of quantum field theory on curved space–time — the production of the quanta coming from the acceleration of the detector rather than any space–time curvature or even potential — but it is closely allied to those ideas. And as it is often linked with Hawking’s prediction, it is worth outlining here.
Consider a scalar field $\phi$ in Minkowski space, and suppose there is a measuring device which follows a world-line $\gamma (s)$ parameterized by proper time. We will be mostly interested in the case where the world-line is initially inertial, and then smoothly changes to uniformly accelerate for a period. Because we shall only be interested in causal considerations, the behavior of the world-line as $s\to +\infty$ will not enter.
I have so far not specified just how the detector responds, and indeed this will depend on just how it is constructed. But the main point is that the detector can only respond to the field $\phi (\gamma (s))$, since it is local to the world-line. Thus it cannot really be a particle detector, for that would require averaging over some finite spatial extent, a feature we are (so far) ignoring; it is a field-strength detector, or more precisely, it responds to weighted averages of field strengths along the world-line $\gamma (s)$. Thus the portion of the field algebra accessible to the detector is that generated by $\phi (\gamma (s))$. Whatever the internal physics of the detector is which determines what it measures, the assumption is that it will respond according to its internal, proper, time.
It should be evident that the physics of the operators $\phi (\gamma (s))$ will depend on the particular world-line $\gamma (s)$. We can see this already in considering the two-point function $\langle 0|\phi (\gamma (s_1))\phi (\gamma (s_2))|0\rangle$. This function can be interpreted as describing the spectrum of vacuum fluctuations of field measurements relative to the proper time along the world-line, and that spectrum will depend on the details of world-line. Thus the results of measurements by an accelerating detector will in general depend on the details of the acceleration.
In particular, if for some range of $s$-values the world-line corresponds to uniform acceleration, say $$\gamma (s) =(a^{-1}\sinh (as),a^{-1}\cosh (as),0,0)$$ in a Cartesian coordinate system, then one has $$\langle 0|\phi (\gamma (s_1))\phi (\gamma (s_2))|0\rangle =
-\frac{1}{4\pi^2}\frac{a^2}{(\sinh as_1-\sinh as_2 -i\epsilon )^2-(\cosh as_2-\cosh as_1)^2}\, .$$ Note that this is periodic in imaginary time with angular frequency $T_{\rm U}=\hbar a/(2\pi c)$. This sort of periodicity is characteristic of thermal behavior, and indeed one can check that the $n$-point functions are those of a thermal field at this Unruh temperature, over the interval in which the world-line accelerates uniformly.
A few comments about this are in order:
Unruh’s analysis is sometimes criticized on the grounds that particle states have finite spatial extent, and he worked only along a given world-line. However, the treatment I have given avoids this problem, by using not particle, but field-strength, detectors. It has also been criticized because he considered a world-line accelerating for all time. The treatment here avoids this problem too. I believe this analysis is secure, but there has been no experimental verification of Unruh radiation as yet.
Many authors have noted that a uniformly accelerated detector follows an integral curve of a boost Killing vector field in Minkowski space, and have considered the “Rindler wedge” formed by a connected family of such curves which are timelike. There is certainly much beautiful geometry in such analyses, but some cautions should be borne in mind. First, one cannot truly “quantize in the Rindler wedge” alone and have the theory really represent a restriction of a full Minkowski-space theory to the wedge, because, as we have seen, the field in the wedge is correlated with the field outside of it. Thus, if the relevant space–time is really Minkowski space, any work purely in the Rindler wedge must be justified in terms of the theory in the full Minkowski space. And finally, the mathematical structure of the Rindler wedge is a result of the very high degree of symmetry present, and it is hard to draw general lessons from such special situations.
Let me close by mentioning a very beautiful result which was obtained at about the same time Unruh’s was, in axiomatic field theory. The Bisognano–Wichmann theorem [@Bisognano:1976za] asserts that in a Poincaré-invariant theory with non-negative energy spectrum, even an interacting theory, the $n$-point functions of the fields along any uniformly accelerating world-line are the same as those of a non-accelerating observer but seeing the theory in a thermal (KMS) state at the Unruh temperature.
Hawking’s Prediction of Black-Hole Radiation
============================================
In 1974 and 1975, Hawking asserted that, taking quantum field theory into account, black holes are not in fact black, but emit thermal radiation [@Hawking:1974rv; @Hawking:1974sw]. Whether this prediction turns out to be correct or not, Hawking’s work contributed immensely to advancing our understanding of quantum fields in curved space–time. That there are problems with the analysis (and with others which have been offered in support of its conclusions) is because physical issues still deeper than those originally considered are implicated.
The issues are fundamental. The point not so much that there is a question about whether black holes radiate, as that Hawking’s work leads directly to the conclusion that the theory of quantum fields in curved space–time is [*inadequate*]{} for treating the system. The core of his analysis is a very beautiful and insightful understanding of the propagation of the field modes through the black-hole space–time, but just that analysis shows that the Hawking quanta have, as their precursors in the distant past, vacuum fluctuations of exponentially increasing frequencies. Those frequencies quickly pass the Planck scale (and indeed any scale, including the total estimated energy of the Universe), and quantum field theory surely becomes inadequate. This is the [*trans-Planckian problem*]{}.
The force of this problem has not always been appreciated, for two reasons. The first is that its invariance has not always been recognized; this has led a number of workers to suggest that it is an artifact of the way Hawking’s computations were done and might be avoided by suitable mathematical transformations. But the problem is invariant, and we shall see that the holonomy ideas discussed earlier clarify this issue.
The second reason that the severity of the trans-Planckian problem has not always been appreciated is that it is a [*virtual*]{} issue. It does not appear explicitly in the predicted thermal radiation, and thus the failure of the theory is not apparent. Indeed, at the time Hawking wrote his papers, the significance of a virtual trans-Planckian problem was clear only to a small number of workers. However, this point is more broadly understood now. And in Section \[moreqft\] I shall show that this problem is in any case promoted from virtual to real when interactions or measurement effects are taken into account.
Closely related to the trans-Planckian problem is another concern, which is that Hawking’s analysis has as a premise that explicitly quantum-gravitational effects can be neglected (at least until the decaying hole approaches Planck size). While this may seem plausible, we now know that it cannot be a foregone conclusion, for there are simple dimensional arguments showing that quantum-gravitational effects may very well be of a size to seriously alter Hawking’s predictions. These, together with the trans-Planckian problem, mean that we cannot have confidence in any treatment of the quantum effects of black holes without an understanding of at least some aspects of quantum gravity.
In this section, I will review the main elements of Hawking’s analysis, with an emphasis on these foundational concerns. I shall also go over some of the indirect arguments which have been offered in support of Hawking’s conclusions: intuitive pictures about pairs of virtual particles near event horizons; suggestions that they are necessary to avoid thermodynamic paradoxes; appeals to CPT invariance, etc. (For further treatment of such matters, see the review article [@Helfer:2003va].) It should be clear that no such arguments could really resolve the trans-Planckian problem, or the question of whether explicit quantum-gravitational effects should be included. In fact, some of the arguments are simply incorrect, some would require radical changes in physics, and the others have resisted attempts to make them precise. In objective terms, this can be read as evidence against Hawking radiation as easily as evidence for it: attempts to argue for the radiation seem to uniformly run into difficulties at key steps, and perhaps that is Nature trying to tell us something. Thus while it is certainly possible that black holes emit thermal radiation, it would be unwise to assume that they necessarily do; there could be some other link between black holes and thermodynamics which results from the physically correct solution of the trans-Planckian problem.
I have gone to some length to outline the inadequacies of the present treatment of quantum theory and black holes, but I would like to emphasize that Hawking’s work does strongly suggest that something very deep does link thermodynamics, gravity and quantum theory. The difficulty is that we know the present treatment is wrong and that we will be unable to be confident in any analysis until we know more about quantum gravity. While in one sense this is certainly a negative statement, in another it reveals opportunities far deeper than those which would apply had the treatment been adequate. Yet there is no disguising the fact that this means there is little hope of knowing what work is correct and what is not until we have a breakthrough on quantum gravity.
In these circumstances, it is essential that those making arguments about Hawking radiation (and related concepts, such as the “information paradox” and holography) examine which parts of their analyses depend on the problematic elements of the theory. If, for instance, an analysis only depends on the assertion that black holes radiate, but not on the details of the radiative mechanism, the results evidently have a degree of robustness. On the other hand, for the deepest progress one wants analyses which do [*not*]{} have this robustness, for one wants to be able to confront the trans-Planckian problem and to discriminate among the different possible resolutions to it.
The Predictions and Their Scale
-------------------------------
Before looking at the analysis in detail, it will be helpful to have a sense of the scale of the effects we are concerned with.
According to Hawking, a spherically symmetric black hole will radiate at the [*Hawking temperature*]{} $T_{\rm H}=1/8\pi M$ in natural units, where $M$ is its mass. For stellar-mass or greater holes this is very small: $$T_{\rm H}=6.2\cdot 10^{-8} \left(\frac{M_\odot}{M}\right) \, {\rm K}\, ,$$ where $M_\odot\simeq 2\cdot 10^{33}\, {\rm g}$ is the mass of the Sun. Thus, unless we find “mini” black holes, there is little prospect for verifying its presence experimentally.
The hole’s luminosity will be given by a Stefan–Boltzmann law $L=\sigma T^4_{\rm H}A_{\rm eff}$, where $\sigma$ is the Stefan–Boltzmann constant appropriate to the field species and $A_{\rm eff}$ is the effective radiating area. It is not a bad approximation to take $A_{\rm eff}=4\pi (2M)^2=16\pi M^2$. Notice that this means [*we have a black body whose linear dimension is of the same scale as the wavelength of the dominant quanta at the temperature. It is therefore in an invariant sense very dim.*]{} We may see this semiquantitatively by taking $T_{\rm H}/L$ to be a characteristic time between the emission of quanta, and $1/T_{\rm H}=8\pi M$ to be the period of a Hawking quantum. The ratio of these times is $$T_{\rm H}^2/L\simeq 4\pi/\sigma \, .$$ Since $\sigma$ is typically a small number, the prediction is that the hole occasionally emits quanta. It would be more accurate to say it [*flickers*]{} than that it glows.
The main point of the above computation however is to emphasize how strongly quantum the process is. The emission of radiation is not simply supposed to be due to quantum effects: it is quantum in its appearance as well.
Hawking’s Analysis
------------------
With the background already given, it is not hard to follow the key elements of Hawking’s analysis and to appreciate the trans-Planckian problem. As these points are present even in the simplest case (spherical symmetry and scalar field), we treat those.
So consider a spherically symmetric space–time containing a bounded distribution of matter which is initially dispersed (enough so that we need need not consider strong-field gravitational effects) and collapses to form a black hole, and a minimally coupled massless scalar field $\phi$ on this background. We specify the state $|\Psi\rangle$ of the field in terms of its content at early times; it will turn out that the results are largely independent of the particular state, and so one can think of $|\Psi\rangle$ as the state which appears to be vacuum $|0_{\rm p}\rangle$ (as far as the $\phi$ field goes) in the distant past.
We wish to analyze the content of the state as measured by observers far from the incipient hole at late retarded times. In this regime, the curvature is small and the particle-content of a state is physically well-defined (except for those of extremely low wavelengths, wavelengths of order the distance to the collapsing matter or more). We may therefore use the formalism of Bogoliubov transformations to compute the state’s appearance in the future. We saw in section \[BTsec\] that if the state appeared to be vacuum in the past, in the future it will be a Gaussian amplitude of particles: $$|\Psi \rangle =\mbox{(normalization)}\exp \left[-(\alpha ^{-1}\beta /2)a^*a^*\right]\, |0_{\rm f}\rangle\, .$$ Our task is thus to compute the Bogoliubov coefficients, and especially to focus on the modes for which $\beta$ is non-zero.
To work out the Bogoliubov coefficients, first resolve the field into spherical harmonics: $\phi =\sum _{l,m}\phi _{l,m} Y_{l,m}$, and then consider the reduced wave equation for each components $\phi _{l,m}$, in the two-dimensional space where the angles have been factored out. These reduced equations will, as usual, consist of a part independent of $l$ and $m$ together with a centrifugal potential $l(l+1)/r^2$. Note that the space–time geometry exterior to the matter is characterized by a single dimensionful quantity $M$ (the mass).
Now imagine starting with data for one of these for a wave-packet given near ${{{\mathscr I}^{+}}}$, with nominal angular frequency $\omega$ around a nominal retarded time $u$, and propagating this backwards in time towards the past. (This packet is just a c-number field mode.) As the packet moves inwards, it will be partially dispersed, and partially reflected by the potential. In fact, the centrifugal terms (measured at a few Schwarzschild radii) go like $l(l+1)/M^2$, whereas the kinetic terms go like $\omega /M$, and it turns out that most of the wave is reflected for $\omega \lesssim l(l+1)/M$. This reflected portion of the wave propagates through a static space–time only and so does not give rise to interesting effects. In fact it is the s-wave, $l=0$, sector, which is almost entirely responsible for Hawking radiation.
Let us continue following the transmitted portion of the wave backwards in time, or more precisely, let us consider a family of such wave packets at increasing nominal retarded times $u$. Relative to any fixed observer near the horizon, these packets appear to blue-shift by a factor $\sim\exp +u/4M$. Thus one quickly reaches a point where geometric optics applies. The propagation backwards through the remaining space–time, including through the region where matter is present, will be governed by geometric optics and the mapping of surfaces of constant phase ${{v}}(u)$ discussed in section \[sphol\].
It is precisely the computation of $\alpha$ and $\beta$ in the geometric-optics approximation, with ${{v}}(u)\sim -\exp -u/4M$, which is the central part of Hawking’s paper. While one can write the analytic formulas explicitly in terms of the Gamma function, it is not hard to understand their main properties without a detailed computation. For angular frequencies $\omega \gg 1/4M$ the main effect is to blue-shift the packets to angular frequencies $\omega /{{v}}'(u)$. On the other hand, for $\omega \lesssim 1/4M$, the oscillations in the packet change over the same scale as does ${{v}}'(u)$, and one gets a significant dispersion; in particular, there are contributions to the $\beta$’s as well as the $\alpha$’s. We can thus see that particle production is predicted to occur for s-wave modes of angular frequencies $\lesssim 1/4M$. Hawking computes the spectrum in detail, and shows that it is thermal with temperature $T_{\rm H}=1/8\pi M$. (See also @Wald:1975kc.) We shall not need this, but it is also not entirely a surprise, for there is only one dimensionful parameter in the system, the mass.
Thus what we find is that a field mode of finite angular frequency $\omega$ in the future (near ${{{\mathscr I}^{+}}}$) has, in the past (near ${{{\mathscr I}^{-}}}$), an exponentially blue-shifted distribution of frequencies, as measured by the coefficients $\alpha$ and $\beta$. If $\omega\gtrsim (4M)^{-1}$, then the distribution is fairly sharply peaked, and the main contribution to $\alpha _{\acute\omega \omega}$ is for $\acute\omega\simeq \omega /{{v}}'(u)$; one has $\beta _{\acute\omega \omega}\simeq 0$. On the other hand, for $\omega\sim (4M)^{-1}$ one has significant contributions to both $\alpha _{\acute\omega \omega}$ and $\beta _{\acute\omega \omega}$ for $\acute\omega$ within perhaps an order of magnitude of $\omega /{{v}}'(u)$.
We can now see why the initial state is not very important. Since if we examine field modes of finite angular frequency $\omega$ in the future, they appear terribly blue-shifted in the past, only detecting the deep ultraviolet content of the state in the past, whose structure is governed by the Hadamard asymptotics, independent of the particular state.
Where do the Hawking quanta come from? Their precursors, in the distant past, are evidently coded in field modes of angular frequencies $\sim (4M)^{-1}/{{v}}'(u)\sim (4 M)^{-1}\exp +u/4M$. These modes are unoccupied in the distant past; what is supposed to create the particles is the [*non-linear*]{} distortion of the retarded time $u$ relative to the advanced time $v$, a distortion which mixes positive- and negative-frequency modes and thus changes the sense of whether modes are occupied or not. It would be correct to say that the Hawking quanta have their origins in vacuum fluctuations in the distant past.
It is important to appreciate that this particle production is non-local. We can only confidently speak of identifying particles in regimes in which we have a clear enough sense of time to resolve the field into positive and negative frequencies, and, while these do exist in the future and in the past, there is no clear physically justified means of interpolating between these.
The Argument of Bekenstein and Mukhanov
---------------------------------------
@Bekenstein:1995ju gave a simple argument (whose importance was emphasized by @Ashtekar:1998) which shows that quantum-gravitational effects could easily be of such a magnitude as to completely alter Hawking’s predictions.
Consider a spherically symmetric black hole, and suppose that quantum gravity only allows the area to change in multiples of $\alpha l_{\rm Pl}^2$, where $\alpha$ is a constant. While there is no strong argument that this would be the case, it is the sort of effect which might very plausibly arise, and it is an attractive hypothesis if we wish to think of the black hole’s area as an information-theoretic entropy.
We have $A=4\pi (2M)^2=16\pi M^2$, and so $\Delta A=32\pi M\Delta M$, or $\Delta M=\Delta A/32\pi M=(n\alpha /4)T_{\rm H}$. In other words, the hole’s mass-energy can only change by discrete units, and so Hawking quanta can only be emitted if their energies are one of these units. If $\alpha\ll 1$, then the allowed lines are very dense, in effect quasi-continuous, and there is little change to Hawking’s analysis. But if $\alpha \sim 1$, only a fraction of the Hawking quanta will have the correct energy, and the radiation would be significantly curtailed; for $\alpha\gtrsim 10$ there would be almost no Hawking radiation.
Thus quantum-gravitational effects could, although they need not, completely alter Hawking’s analysis, and we cannot have confidence in the prediction of black-hole radiation without understanding some features of quantum gravity.[^15]
Some Arguments on Hawking Radiation
-----------------------------------
It should be clear that no argument within conventional quantum field theory in curved space–time could remove the trans-Planckian problem, or eliminate the possibility that quantum-gravitational effects might seriously alter the prediction of black-hole radiation. These issues are recognized in [*non-standard propagation models*]{}, which explicitly aim to save the predictions by altering the propagation rules for the quantum fields (see [@Helfer:2003va] for references and discussion).
One can also try to find indirect arguments in favor of thermal black-hole radiation. These all run into difficulties at essential points, and it is not clear whether those difficulties are signals that the aim is incorrect or just that the analysis is not deep enough. I have elsewhere [@Helfer:2003va] critiqued a number of these (claims that the radiation is necessary for the thermodynamic consistency of general relativity and that it follows from Unruh radiation via the equivalence principle), and I shall not repeat these here. I will briefly comment on some other arguments which have appeared.
[*Hawking radiation is implied by CPT invariance.*]{} There are several versions of this statement in the literature. As far as I know, they break down into: (a) interesting but radically speculative arguments to the effect that Hawking radiation and CPT invariance [*together*]{} form a picture the author finds attractive, but which cannot objectively be considered independent evidence for Hawking radiation; (b) incorrect arguments; (c) assertions without supporting arguments.
Examples of arguments of class (a) are those of @Hawking:1976de and @Hooft:1996tq. While it would be out of place to describe these works in detail here, as indices of their speculative character I’ll note that Hawking argues that black holes and white holes ought — quantum-theoretically — to be the same thing! (Since the classical causal structures of black and white holes are clearly distinct, this view implies gross modifications to causal structure via quantum effects, and thus really amounts to speculations about quantum-gravitational effects severely altering classical relativity.) And ’t Hooft’s argument [*starts*]{} from the premise that a black hole should have the same properties as a special-relativistic particle.
Arguments of type (b) assert that because black holes form due to processes $\mbox{(particles)}\to\mbox{(black hole)}$ there should be corresponding decay processes $\mbox{(black hole)}\to\mbox{(particles)}$. This is seriously incorrect, at two levels. First, it is hardly a foregone conclusion that the CPT theorem will hold — or even be formulatable — in a unified theory of quantum fields and gravity, and indeed experimental searches for CPT violation are of considerable interest as evidence for quantum-gravitational effects. And even if the CPT Theorem holds in this context, the (b)-type arguments do not apply it correctly. The theorem relates processes to their time-reversed (and CP-transformed) versions, so it would relate the black-hole formation process to a [*white*]{}-hole decay process $\mbox{(white hole)}\to\mbox{(particles)}$.
A correct application of the sorts of ideas underlying this actually provides a pair of alternatives, neither of them really arguments in favor of Hawking radiation (and one tending against it). If Hawking’s notion that black holes radiate completely away is correct, that is, the process $\mbox{(black hole)}\to\mbox{(particles)}$ exists, and CPT applies, then the process $\mbox{(particles)}\to\mbox{(white hole)}$ must be possible, that is, a white hold could be formed from a non-problematic initial state. This would be almost universally considered highly counterintuitive, and so tends to make Hawking radiation less plausible.
If, on the other hand, Hawking’s notion that black holes decay is correct but they leave remnants, and CPT applies, the forward process would be $\mbox{(black hole)}\to\mbox{(particles + remnant)}$, and so the reverse would be $\mbox{(particles + seed)}\to\mbox{(white hole)}$, where “seed” is the CPT-reversal of a remnant. This would say that white-hole formation via this mechanism would require a seed, and, if such objects are sufficiently exotic then the white-hole formation mechanism itself might be plausible.
[*Analytic-continuation arguments.*]{} These include appeals to the very beautiful structure of the analytically-continued Schwarzschild solution which arises when it is used to model a thermal state at the Hawking temperature, and Hawking’s Euclidean quantum gravity program.
First and foremost, these arguments do not address any of the difficulties associated with Hawking radiation. Instead, what they do is provide elegant formal mathematical structures which however it is hard to link in detail with the physical processes. (Even leaving aside the serious doubts about whether physically relevant solutions admit the requisite analytic continuations, in general analytic continuation is a highly nonlocal procedure.)
Hawking [@Hawking:1996jh] has argued that Euclidean quantum gravity is simply not meant to be anything more than a calculational tool for working out transition probabilities, and thus that it is wrong to fault its internal mathematics for not connecting clearly to the physical world. If one holds this position, however, one cannot argue that at this level Euclidean quantum gravity provides an answer to the trans-Planckian problem, since what is required to resolve that is precisely a detailed explication of what happens to the field modes. More generally, while a view like Hawking’s has some pragmatic justification in particle scattering theory, where it is very hard to probe the dynamics of the processes and we by and large settle for the overall transition probabilities, in gravitational physics we are very definitely interested in the details of the dynamics.[^16] It is true that there may well be quantum-gravitational corrections which limit the utility of the classical space–time concept in describing these, but if so we want to be able to say as explicitly as possible what these are.
In addition, of course, any quantum gravity program is speculative, and none has really been developed satisfactorily.
[*Hawking radiation arises from particles tunneling through the horizon.*]{} Essentially this same language has been used to summarize several different physical arguments.
When it is applied to describe Hawking’s original argument, it is seriously misleading: (a) Most grossly, it suggests that there is an [*acausal*]{} element to the mechanism, with some sort or physical process propagating outwards across the horizon. Nothing could be further from the truth. (b) Hawking’s prediction is that black-hole radiation is a [*quantum-field-theoretic phenomenon in which the particle number changes*]{}, from initially vacuum to a steady flux. This is clearly not the same as a quantum-[*mechanical*]{} tunneling process conserving particle number. (c) One can try to repair these defects by asserting that it is “virtual” particles which propagate across the horizon and tunnel. The difficulty with this is that the term “virtual particle” is in this context so vague that it conveys little information.
@Parikh:1999mf give a very interesting attempt to make explicit a tunneling model. However, just how this connects with quantum field theory in curved space-time is unclear. Work on this would be worthwhile, especially because the Parikh–Wilczek ideas treat the back-reaction as an essential element of the physics, in contrast to the usual Hawking picture.
Black Holes, Quantum Fields,\
Measurements and Interactions {#moreqft}
=============================
I have shown above that the usual theory of quantum fields in curved space–time is not adequate for treating black holes. This is surprising, as the theory is both natural and attractive. We should take this inadequacy very seriously, as it is giving us valuable information about what issues must be confronted in reconciling quantum theory with gravity. And since that is a deep problem, it is worthwhile exploring the information we have about it — the nature of the failures of the usual theory — in some detail.
I shall first give a simple and suggestive, but not conclusive, argument to the effect that a black hole, or more precisely what appears to be an incipient black hole, has a sort of boundary beyond which it becomes an essentially quantum object.
I shall then give three analyses which bear on the trans-Planckian problem and, more generally, on the theory’s treatment of ultraviolet virtual effects. In each of these virtual effects are promoted to real ones. (From one point of view, this is hardly surprising; a great many of the interesting effects in special-relativistic quantum field theory come from real consequences of virtual effects.) The first case involves a computation of the results of sequences of measurements of quanta in the Hawking effect; the second (closely related to the first) takes into account quantum-field-theoretic nonlinearities in the Hawking effect; and the third is an estimate of a Casimir effect of a black hole. In each of these cases, the result is unacceptable, and indeed absurd. We shall find that in each case the theory predicts [*real*]{} trans-Planckian effects.
While these failures are particularly blatant in the case of black holes, on account of the exponentially increasing red-shifts affecting mode propagation, there is no reason to think they are special to black holes. Even without these exponential increases, sufficiently high red-shifts would give rise to problems in all of these cases. Indeed, the Casimir result is problematic for even modest red-shifts. I would therefore suggest that there is a real possibility that the theory of quantum fields in curved space–time, as natural as it does appear, may turn out to be in need of serious modification.
Measuring Geometry Near a Black Hole
------------------------------------
Let us imagine a spherically symmetric distribution of matter which is collapsing and will ultimately form a black hole. Let us ask how well distant observers may measure the geometry of the space–time in the vicinity of the incipient horizon; we shall consider only spherically symmetric measurements here.
In particular, we have seen that a key element of this geometry is the fractional rate of red-shift $v''/v'$. This could, by the universality described above, be measured by looking at signals emitted by any objects falling into the hole. Suppose such signals are wave-packets emitted with known (nominal) frequency in the object’s local frame (they could be particular spectral lines), and they are received at ${{{\mathscr I}^{+}}}$ with (nominal) angular frequency $\omega$. Thus measurements of the rate of change of $\omega$ can be used to infer $v''/v'$.
Now we must have $\omega\gtrsim |v''/v'|$ in order to resolve the geometry in question. If the system is approaching a Schwarzschild solution of mass $M$, this means we should have $\omega\gtrsim c^3/GM$. On the other hand, the angular frequency in the emitted object’s frame must be $\sim k\omega /v'$, where $k$ depends on the velocity of the emitting object as it moves to cross the horizon, but $1/v'$ increases exponentially. We presumably must have $k\hbar \omega /c^2v'\leq M_{\rm Pl}$, the Planck mass. Combining the inequalities, we find $k\hbar c/GMv'\lesssim M_{\rm Pl}$, which is $$v'\gtrsim kM_{\rm Pl}/M\, .$$ This inequality represents a limitation on the portion of the geometry which can be measured by distant observers. Formally, it defines the part of the tangent bundle which is accessible to measurement. Because $v'$ approaches zero so rapidly, however, it means that for all practical purposes the close neighborhood of the event horizon cannot be investigated; we cannot operationally treat it as having a classical geometry.
Measurements of Hawking Quanta and Their Precursors
---------------------------------------------------
While a great deal of work has been done on quantum field theory in curved space–time, relatively little attention has been given to the investigation of the effects of quantum measurements, that is, the interesting possibilities arising from measuring different, in general non-commuting, operators. Yet this is a basic element of quantum theory, and any attempt to assess the plausibility of an analysis in curved space–time should consider the predictions of such measurements.
I am going to describe here the results of measurements of Hawking quanta and their precursors, according to conventional quantum theory [@Helfer:2004jx]. What we shall find is that, because the measurement process introduces a coupling between the measuring device and the field modes, measurements can promote the trans-Planckian problem from a virtual to a real one, where it is manifestly unacceptable.
[*Measurements and Unruh Detectors.*]{} Before doing this in the Hawking case, however, I want to describe something similar in the Unruh process. Let $\gamma (s)$ be the world-line of a detector responding to a linear field in Minkowski space. If the detector is strictly localized to the world-line, then it cannot really be said to be a particle detector, but the question of just what name is best for the quanta the detector responds to will not be important here, neither will strict localization to the world-line. We will also assume that the detector was initially inertial, but then is smoothly accelerated.
The accelerating detector measures what appears, along the accelerating world-line, to be a number operator $n=n(\gamma ,\omega , s_0,s_1)$ corresponding to measurements of field modes of a angular frequency $\omega$ over a particular sampling time $s_0\leq s\leq s_1$; it is determined by resolving the field $\phi (\gamma (s))$ along the world-line into Fourier components (with respect to the proper time $s$), and following the usual constructions as if $s$ were ordinary time.
Now let us suppose the state is initially the vacuum $|0\rangle$, and then a measurement of $n$ is made. Then the state will be projected into an eigenstate of $n$. This will certainly not be the vacuum, and in fact it will be a superposition of excited states. Typically, these will contain quanta of energies $\sim \omega t_a\dot\gamma ^a(s)$ with respect to the frame with timelike vector $t^a$, where $s_0\leq s\leq s_1$. (This will occur even if the measurement of $n$ yields zero.)
Note that if the detector has been accelerated to a very high boost relative to $t^a$, then detection will result in the state containing superpositions of very high energies. It is natural to ask where these energies come from. In some sense, they must come out of the measurement process, and the high relative energy of the detector with respect to the frame described by $t^a$.
[*Measurements and Hawking Detectors.*]{} Imagine two observers who follow trajectories each everywhere far from an object collapsing towards a black hole. Each carries a detector, which for convenience one may think of as a camera containing a photographic plate, and the shutter of the camera will be controlled to open at certain times. What will be critical in the subsequent analysis will be to distinguish between simply exposing the plate (which is to say, allowing a coupling to certain field modes) and inspecting it (actually doing a quantum measurement). Let the initial state of the field be $|\Psi\rangle$, say the vacuum in the distant past.
The plates of the first observer, call her H, are sensitive to quanta around the characteristic Hawking angular frequency $T_{\rm H}$. Her camera is set to have its shutter open during an interval of retarded time $\Delta u$ around a time $u$ late enough so that Hawking quanta are expected there. The plates of the second observer, call him UE, are sensitive to quanta at the angular frequency of the precursors to those measured by H, that is, at $(v'(u))^{-1}T_{\rm H}$ — UE is for “ultra-energetic.” His camera is set to have its shutter open in an interval around the advanced time $v(u)$ at which those precursors are moving inwards towards the origin.
In fact, the cameras simply enable H and UE to measure the number operators of quanta, H measuring Hawking quanta and UE the quanta at the frequency of the precursors. We now consider what happens when H and UE make their measurements, that is, examine the plates. The results depend critically on which order this is done in.
[r]{}[0.4]{}
[{width=".35\textwidth"}]{}
\[fig:HUE\]
If UE goes first, nothing remarkable happens. UE measures the particle-number for an ultra-energetic mode in the past, but this mode was not excited in the state $|\Psi\rangle$. In other words, the state is already an eigenstate of that operators. Thus UE’s measurement returns the value zero. Then H may or may not record a number of Hawking quanta, with probabilities given according to the usual Bogoliubov transformation.
But if H goes first, things are very different. She will record a finite number of Hawking quanta. The precise number does not matter in this analysis, and the most likely number (if $\Delta u$ is not too long) is zero, so let us say that is what she gets. Then the state is projected to an eigenstate of her number operator.
It is possible to work this out exactly, but a rough computation will convey the main idea. Let us ignore dispersive effects in the propagation of the mode in question, so the annihilation operator $a$ for her mode can be written as $a=\alpha b+\beta b^*$ in terms of the annihilation and creation operators for UE’s mode. Because H’s mode is a Hawking one, the Bogoliubov coefficients $\alpha$ and $\beta$ have comparable magnitudes. Then the state $|0_{\rm H}\rangle$ which H’s measurement has projected $|\Psi\rangle$ to is characterized by $a|0_{\rm H}\rangle =0$, whence $$|0_{\rm H}\rangle =\mbox{(normalization)}\cdot
\exp [ -(\beta /2\alpha ) b^*b^*] |0_{\rm UE}\rangle$$ in terms of the eigenstates of UE’s number operator. That is, the result of H’s measurement, the state $|0_{\rm H}\rangle$, appears in terms of the number basis in the past to be a superposition of excitations of ultra-energetic modes. If now UE makes a measurement, he will with finite probability record a number of these ultra-energetic quanta.
This result is unacceptable and indeed absurd, because the energies $(v'(u))^{-1}T_{\rm H}$ of the quanta supposedly measured by UE grow exponentially quickly, passing not only the Planck scale but indeed the mass–energy of the collapsing object (and even of the observed Universe).
Interacting Quantum Fields
--------------------------
Virtually all of the specific computations of quantum fields in curved space–time have been done in the case of linear field theories, because we do not yet have a practical understanding of how to treat the nonlinear case. The concern has therefore been raised that Hawking’s analysis might be inadequate because of its neglect of nonlinear effects.[^17]
The most concrete attempt to respond to this was a very interesting paper by @Gibbons:1976es. They sketched an argument to show that at late time all the Feynman propagators of the theory would have a periodicity in imaginary time resulting in the field theory being at the Hawking temperature. However, the argument was not complete. For one thing, the authors did not really spell out the issues involves with “dressing” the in and out states (and we shall see shortly this is essential). For another, while they emphasized the importance of showing that an initially vacuum state would dynamically equilibrate to a thermal one, it was in their analysis unrealistically hard to track this behavior, and instead they worked with the “Hartle–Hawking” state, which in effect builds in the assumption that an equilibrium has been reached and one can neglect all the early-time physics. In particular, issues like the trans-Planckian problem cannot be properly investigated with this assumption.
One also sometimes hears the suggestion that perhaps [*asymptotic freedom*]{} (the running of couplings towards zero at higher energy scales) may solve this problem. As far as I know, however, no careful argument has been offered to this effect, and it is far from clear how one could be made to work.
I am going to outline a computation of a nonlinear quantum-electrodynamic correction to Hawking’s analysis [@Helfer:2005wy; @Helfer:2005wz]. It will be an effect which is [*first-order in the electric charge.*]{} (In special-relativistic computations in scattering theory, one only gets contributions in even powers of the charge, that is, integral powers of the fine-structure constant; however, here we will get an amplitude which is first-order.) I am in fact going to choose the particular amplitude so that we do not have to confront renormalization theory beyond normal ordering. All of the interesting effects will be due to the change in “dressing” in the future compared to the past.
Let us begin by considering the situation in the distant past, when the matter is dispersed, and the space–time is to good approximation Minkowskian. Then the quantum-electrodynamic Hamiltonian is $$H=H_{\rm b}+H_{\rm int}$$ where $H_{\rm b}$ is the “bare” Hamiltonian, describing the linear fields, and $$H_{\rm int}=-e\int _\Sigma \tilde\psi \gamma ^a\Phi _a\psi$$ is the interaction. Here $\psi$ is the charged field operator and $\Phi _a$ is the electromagnetic potential operator, and $\Sigma$ is the Cauchy surface of interest.
The first thing we must consider is that the vacuum is altered by the interaction, the dressed vacuum being given, to first order in the electric charge, by $$|0_{\rm d}\rangle =(1-H_{\rm b}^{-1}H_{\rm int})|0_{\rm b}\rangle\, ,$$ according to standard perturbation theory. In this expression, the factor $H_{\rm int}|0_{\rm b}\rangle$ will contain bare electron, positron and photon creation operators acting on $|0_{\rm b}\rangle$; one may say that the dressed vacuum contains “bubbles” of bare electrons, positrons and photons.[^18] These bubbles involve modes with arbitrarily large wave-numbers (which ultimately contribute to the ultraviolet divergences of the theory). However, a key point is that the spatial wave-numbers of each vacuum triple sum to zero. This is represented diagrammatically by the particle lines terminating at common vertices.
[r]{}[0.35]{}
[{width=".3\textwidth"}]{}
\[fig:vacbubble\]
The next issue is how to extract the physically significant information about the state in the future. In special-relativistic theory, one would generally do this by analyzing its particle content. This involves technical issues which would take us too far afield. Instead we shall concentrate on the field aspect, and only give heuristic comments about the particle content.
A basic element of quantum electrodynamics is the [*interaction vertex*]{}, which is $\langle 0_{\rm d}|{\tilde\psi}(k')\Phi (k)\psi (k'')|0_{\rm d}\rangle$, where $k'$, $k$ and $k''$ are mode labels which include both momentum and polarization information. We will compute this for certain modes. The point will be that any difference we find between the black-hole and the Minkowskian cases will represent an excitation of the black-hole case relative to the Minkowskian vacuum — taking the difference amounts to renormalizing.
The interaction vertex for the particular modes in question will be $$\label{vert}
\langle 0_{\rm d}|{\tilde\psi}_{\rm f}^+(k')\Phi _{\rm f}^- (k)\psi _{\rm f}^+ (k'')|0_{\rm d}\rangle\, ,$$ where the postscripts indicate positive or negative frequencies in the future. For the modes, we take $k$ to correspond to a characteristic Hawking photon in the future, and $k'$, $k''$ will describe field modes propagating everywhere far from the incipient hole. (A more precise specification of $k'$ and $k''$ will be given in a moment.)
First-order contributions to the vertex function (\[vert\]) can potentially come from three places: perturbations of the bra relative to the bare vacuum, perturbations of the operators relative to the bare ones, and perturbations of the ket relative to the bare vacuum. Of these, the only non-trivial contribution arises from the perturbation of the ket, because in each of the other cases one has at least one of the bare charged field destruction operators acting on the bare vacuum. Thus to first order we have $$\label{verte}
\langle 0_{\rm d}|{\tilde\psi}_{\rm f}^+(k')\Phi _{\rm f}^- (k)\psi _{\rm f}^+ (k'')|0_{\rm d}\rangle=
-\langle 0_{\rm b}|{\tilde\psi}_{\rm f}^+(k')\Phi _{\rm f}^- (k)\psi _{\rm f}^+ (k'')H_{\rm b}^{-1}H_{\rm int}|0_{\rm b}\rangle\, ,$$ where the operators may be taken to be bare. Here the factor $H_{\rm int}$ creates vacuum bubbles of bare particles, and $H_{\rm b}^{-1}$ only changes their weightings. Since $k'$ and $k''$ are chosen to represent modes everywhere far from the collapsing region, the charged-field operators annihilate the bare quanta in the vacuum bubbles. Were we in Minkowski space, the field $\Phi ^-_{\rm f}(k)$ would annihilate the bra, and the expression (\[vert\]) would vanish. Here, however, that field, which represents creation of a bare Hawking photon in the future, will in the past represent a precursor, which will have a mixture of positive and negative frequency terms, but will be highly blue-shifted. Thus $k$, in the distant past, will correspond to an ultra-high frequency wave-vector directed inwards.
[r]{}[0.4]{}
[{width=".35\textwidth"}]{}
\[fig:brokenbubble\]
Now recall that the sum of the spatial wave-numbers of the vacuum triples is zero, so we will get non-trivial contributions to the interaction vertex when $k'$, $k''$ correspond to wave-vectors whose total spatial component is ultra-high and directed [*outwards*]{}. We thus see that there will be, in the future, non-trivial contributions to the interaction vertex corresponding to ultra-high frequencies in the charged fields.
I pointed out above that the analysis in terms of physical particles involves further technicalities, but if we ignore the niceties, then the bra $\langle 0_{\rm d}|{\tilde\psi}_{\rm f}^+(k')\Phi _{\rm f}^- (k)\psi _{\rm f}^+ (k'')$ represents a state which is a modification of the in-vacuum by deleting (in terms of the out-regime) one Hawking photon and creating an ultra-energetic electron-positron pair; the vertex function (\[vert\]) can thus be thought of as the amplitude to find this state given that the initial state was the vacuum. In other words, speaking of particles in this heuristic sense, there would be a finite probability of an ultra-energetic electron-positron pair appearing at late time. While this particulate interpretation is problematic, the interpretation at the level of three-point functions shows unequivocally that in the future the state $|0_{\rm d}\rangle$ possesses ultra-high wave-number excitations relative to the Minkowski vacuum.
This result is unphysical and absurd because the characteristic wave-numbers for the charged pair increase exponentially quickly, with $(v'(u))^{-1}$. I would like to emphasize that this computation follows very closely the original point of view from which Hawking’s predictions were derived, and is conventional as far as both quantum field theory and relativity go.
A Casimir Calculation
---------------------
One would like to understand how quantum fields can act as sources for gravity. In the case of black holes and Hawking-type proposals, this is [*the back-reaction problem:*]{} how do the effects of the quantum field change the geometry of the space–time? If in fact black holes radiate, and if in any usual sense energy is conserved, presumably the black hole (or more accurately, the incipient hole) must lose the energy emitted. But this must mean that in some sense there is a negative energy flux inwards towards the incipient hole.
We do not have a good understanding of how quantum fields act as sources for gravity. The most direct analog of a classical source term is the [*stress–energy operator*]{}. This is not at all trivial to construct, because the formal expression for it involves products of the field operators at the same point, and these are only [*distribution*]{}-valued operators and the product of two distributions is not generally defined. One has to renormalize this formal expression via an infinite subtraction to obtain a well-defined operator. This, it turns out is always possible for quantum fields in curved space–time, and the result is the [*renormalized stress–energy operator*]{} $T^{\rm ren}_{ab}$. On the other hand, there is an ambiguity in the process, because there are no general criteria known for fixing the zero-point of the subtraction, and thus the renormalized stress–energy operator is only determined up to the addition of a conserved c-number term [@Wald:1995yp].
It is probably fair to say that most theorists regard this as a fairly good state of affairs, and that hopefully at a later date we will have a better understanding of the c-number ambiguity. I think that more caution is in order. Even if we defer the issue of the c-number ambiguity, the stress–energy operator is rather problematic, giving generically measures of energy which are unbounded below due to ultraviolet problems [@Helfer:1996my]. And here I will show that trying to reconcile the c-number contributions with the standard understanding of Casimir effects leads to another serious problem.
If it is possible to argue for a particular system that we have a good way of fixing the c-number contributions to the stress–energy, and if the system possesses a lowest-energy state, then we refer the expectation of the stress–energy in that state as a [*Casimir term*]{}. Similarly, even if it may not be clear how to fix the c-number contribution to the entire stress–energy, but we can fix the contribution to the Hamiltonian, the lowest energy is the [*Casimir energy*]{}. Generally, the most secure computations of Casimir effects occur when we have a mode-by-mode comparison with a known system.
The first and most famous example is due to Casimir, following an exchange of his with Bohr. Consider the region between two parallel perfect plane conductors separated by a distance $l$. In this case, one has a mode-by-mode comparison of the finite-$l$ case with the Minkowski vacuum $l\to\infty$. It turns out that the Casimir stress–energy is $$\label{Cas}
\langle 0 |T_{ab}^{\rm ren}|0\rangle =-\frac{\pi ^2\hbar c}{1440 l^4}\left[
\begin{array}{cccc} 2&&&\\ &1&&\\ &&1&\\&&&0\end{array}\right]$$ in a Cartesian coordinate system adapted to the symmetry of the problem. Of course, the most striking thing about this is that the energy density is negative; Casimir energies may have either sign. (For questions about the detectability of negative energy densities, see [@Helfer:1998].) That the Casimir energy density comes out to be constant between the plates is a feature of the idealized nature of the problem. Also its modest size is the result of very delicate calculations: the renormalized squares of the electric and magnetic fields separately actually [*diverge*]{} with opposite signs as one approaches the plates. (The divergence is again due to the idealized, perfect-conductor, boundary conditions; more realistic situations can have large finite energy densities.) Finally, there is little chance at present for experimentally detecting Casimir energies (they are too small), but there is some prospect for observing Casimir pressures (see e.g. [@Lamoreaux:2010; @KM:2010]).
Now consider as usual a massless scalar field propagating in a spherically symmetric space–time representing a system collapsing to a black hole. We shall only be concerned with the ultraviolet asymptotics of this theory, that is, the propagation of very high frequency modes. (Whether the state is excited in these modes will not matter; it will be most natural to assume these modes are unexcited.) For high enough frequencies, these modes propagate to good approximation by geometric optics.
Let us now consider just the s-wave modes which are supported at late retarded times — these are like the Hawking modes, but we are here interested in the ultraviolet regime, where there are few Hawking quanta. Since $T_{\rm H}$ is the only relevant characteristic scale, we may say we are interested in modes here of angular frequencies $\omega\gtrsim T_{\rm H}$. Each of these modes will have a vacuum stress–energy contribution.
We may compute the s-wave contribution $\Phi _{\rm s}$ to the vacuum flux $\int T_{ab}^{\rm vac} n^an^b d^2S$ over a large sphere. The s-wave fields near ${{{\mathscr I}^{+}}}$ has the form $\phi _0(u)/r$, where $\phi _0(u)$ is effectively a quantum field in one dimension [@Helfer:2003va]. Using a standard result from this theory, we find $$\begin{aligned}
\Phi _{\rm s}&=&\langle 0|\int n\cdot\nabla\phi \Bigr|_{\rm s}
n\cdot\nabla\phi \Bigr|_{\rm s} d^2S|0\rangle\nonumber\\
&=&4\pi\langle 0|\partial _u\phi _0\partial _u\phi _0|0\rangle
\nonumber\\
&=&(2\pi )^{-1}\int _{T_{\rm H}}^\Lambda\omega d\omega\nonumber\\
&=&(4\pi )^{-1}(\Lambda ^2-T_{\rm H}^2)\, ,\end{aligned}$$ where $\Lambda$ is an ultraviolet cut-off. That is, the vacuum energy due to s-wave modes passing through an interval of retarded time of length $du$ will be $(4\pi )^{-1}(\Lambda ^2-T_{\rm H}^2) du$.
On the other hand, those same modes originated at ${{{\mathscr I}^{-}}}$ with their frequencies blue-shifted by a factor $(v'(u))^{-1}$, so the energy inwards at ${{{\mathscr I}^{-}}}$ will be $(4\pi )^{-1}(v'(u))^{-2}(\Lambda ^2 -T_{\rm H}^2) du$. Thus if we mode-by-mode compare the vacuum energies, we find $$\label{Casdif}
E_{\rm future}-E_{\rm past} =(4\pi )^{-1}(1-(v'(u))^{-2})(\Lambda ^2-T_{\rm H}^2) du\, ,$$ which is grossly negatively divergent. In other words, [*the usual arguments about Casimir energies lead to the conclusion that the black hole absorbs a divergent Casimir contribution.*]{}
This conclusion is again unacceptable and absurd. The problem is, however, to put one’s finger on what is wrong. Simply rejecting this argument wholesale would seem to require also rejecting more familiar instances of Casimir calculations. And while I have indicated above that there are some questions about those, it is hard to believe they are entirely wrong.
In fact, the divergence we have found here does not depend at all on having an incipient black hole; it is simply a consequence of the modes suffering a red-shift in their passage through the space–time. (That, in eq. (\[Casdif\]), the factor $v'(u)$ tends rapidly to zero is not needed for there to be a divergence; one only needs $v'(u)\not=1$.) So this problem with Casimir stress–energy is really a generic problem for quantum fields in curved space–time.
Conclusion {#conclusion .unnumbered}
==========
The theory of black holes has given us results of extraordinary depth and beauty. It is the resulting pushes to treat still more difficult and fundamental issues which have led to the problematic points discussed here. These points show that black holes are linked to some of the deepest problems in physics, and we have yet much to learn from them — if we are up to the challenge.
It is a pleasure to thank the organizers and participants of BSCG XIV for a stimulating and enjoyable time.
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[^1]: Space–times are always assumed to be oriented, time-oriented (there is a globally sensible distinction between past- and future-directed causal vectors), and strongly causal (there are no curves which come “arbitrarily close” to violating causality — see @Penrose:1972 for formal definitions.
[^2]: These terms have nothing to do with geodesic completeness.
[^3]: For an important feature which does not generalize, see @Hollands:2004ac.
[^4]: Here $ {\hat\nabla}_a $ is the covariant derivative operator with respect to ${\hat g}_{ab}$; subsequently, we will use the convention that hatted quantities have their indices raised and lowered with ${\hat g}^{ab}=\Omega ^{-2}g^{ab}$ and ${\hat g}_{ab}$.
[^5]: This sort of behavior was noted by @Carter:1979 and quite explicitly by @Booth:2005qc.
[^6]: The minus sign arises from the convention used in converting the two-form for angular momentum to a vector.
[^7]: The red-shift here is the ratio of the frequencies, that is, the quantity $1+z$ and not just $z$.
[^8]: If there is gravitational radiation present, it contributes interesting effects to the portion of the path along ${{{\mathscr I}^{+}}}$. However, these do not affect the main points and will be discussed elsewhere.
[^9]: Even in the massive case, the inner product is nonlocal on the scale $(\| {\bf k}\|^2+m^2)^{-1/2}c$. In the non-relativistic case, this reduces to the Compton wavelength, which is very small.
[^10]: Often one works with the time-ordered two-point function, but this is unnecessary here.
[^11]: The formal definition is based on this idea but phrased somewhat differently for technical reasons.
[^12]: The precise state used in this relation turns out not to matter very much, because in the ultraviolet limit one expects all of the modes to be unoccupied.
[^13]: Because only gauge-invariant quantities are observable, one cannot directly measure the potential.
[^14]: The mathematical technicalities in treating these wave functionals are more involved than ordinary quantum mechanics, owing to the infinite-dimensionality of their domains.
[^15]: In loop quantum gravity models, it is shown that while area is quantized the spacing is very fine except for Planck-scale holes.
[^16]: Even in the special-relativistic case, one would like to treat the full dynamical structure of the quantum theory. The restriction to scattering problems has been enormously fruitful, but one would certainly like to go further.
[^17]: In attempts to model astrophysical black holes there have also been what one might call phenomenological models, where certain species are assumed to be produced by the (linear) Hawking mechanism, and then interact as they propagate outwards (e.g., [@MacGibbon:1990zk]). However, this does not address the concern that nonlinearities have not been taken into account at a fundamental level.
[^18]: However, it should be emphasized that $|0_{\rm d}\rangle$ really is the physical vacuum, no-particle, state, and these bubbles are an artifact of examining the state in terms of the bare particles, which are mathematically convenient but not of direct physical relevance.
|
---
abstract: 'We analyze the perturbative series of the Keldysh-type sigma-model proposed recently for describing the quantum mechanics with time-dependent Hamiltonians from the unitary Wigner–Dyson random-matrix ensemble. We observe that vertices of orders higher than four cancel, which allows us to reduce the calculation of the energy-diffusion constant to that in a special kind of the matrix $\phi^4$ model. We further verify that the perturbative four-loop correction to the energy-diffusion constant in the high-velocity limit cancels, in agreement with the conjecture of one of the authors.'
author:
- 'D. A. Ivanov'
- 'M. A. Skvortsov'
date: 'March 29, 2006'
title: |
Quantum mechanics with a time-dependent random unitary Hamiltonian:\
A perturbative study of the nonlinear Keldysh sigma-model
---
Introduction
============
While spectral properties of static random-matrix Hamiltonians have been studied in detail by various methods [@Mehta; @Guhr; @Efetov-book], much less is known about quantum-mechanical evolution with a [*time-dependent*]{} random-matrix Hamiltonian: i = H(t) . \[schroedinger\] Such a system with the Hamiltonian $H(t)$ belonging to one of the three Wigner–Dyson random-matrix ensembles (unitary, orthogonal or symplectic) has been studied by Wilkinson in Ref. . The trajectory of the Hamiltonian $H(t)$ in the space of Hermitian matrices is assumed to be nearly linear on the time scales relevant for the problem. This assumption is justified in the limit of large matrix dimension $N$: in this limit the energy level spacing $\Delta$ is small, and a small variation of the Hamiltonian matrix elements (of order $\Delta$, which is much smaller than the matrix elements themselves) already shifts the energy levels by the order of the level spacing and thus changes the level correlations completely. Therefore the relevant time scales are small in $N$, and any trajectory $H(t)$ with smooth (independent of $N$) time dependence may be approximated as linear. This is the usual reasoning in the studies of parametric level statistics [@SimonsAltshuler93] which deduces that under such an assumption the spectral correlations acquire universal properties (independent of the particular choice of the trajectory).
Following the traditional notation and for the future possibility of describing different time dependencies of $H(t)$, we introduce the time dependence of the Hamiltonian in two steps. First we define the class of linear trajectories $H(\varphi)$ and then let the parameter $\varphi$ be a given function of time $t$. In the present paper we restrict our discussion to the Gaussian Unitary Ensemble, for which the linear trajectories $H(\varphi)$ may be defined by the pair-correlation functions
$$\begin{gathered}
\overline{H_{ij}(\vp) H_{kl}(\vp)}
= \frac{N\Delta^2}{\pi^2} \, \delta_{il} \delta_{jk} ,
\label{GUE}
\\
\overline{\big(H(\vp)-H(\vp')\big)_{ij} \big(H(\vp)-H(\vp')\big)_{kl}}
= \delta_{il} \delta_{jk} (\vp-\vp')^2 \Delta^2 C(0)
+ O\left( \frac{(\vp-\vp')^4}{N}\right) ,
\label{HH}\end{gathered}$$
where $\Delta$ is the mean level spacing in the center of the Wigner semicircle, and $C(0)$ is the conventional notation for the sensitivity of the energy spectrum on the parameter $\vp$ (see, e.g., Ref. ).
Two possibilities of the motion along the trajectory $H(\vp)$ are of principal importance:
- Linear time dependence $\vp(t)=vt$. This is the problem studied by Wilkinson [@wilkinson] and also the situation considered in the present paper (except in the Section \[section-phi(t)\] where more general time dependencies $\vp(t)$ are discussed).
- Periodic time dependence $\vp(t)=\cos(\omega t)$. In that case, diffusion in the energy space is suppressed by quantum interference, leading to the phenomenon of [*dynamic localization*]{} [@Casati79; @wilkinson2; @BSK03]. We do not discuss the periodic problem in the present paper.
We further specify to the case of linear motion along the trajectory $\vp(t)=vt$ and replace the parameter $C(0)$ by a more convenient for our present discussion dimensionless parameter $\alpha$ defined as $$\alpha
= \frac{\pi}{\Delta^2} C(0) v^2
= \frac{\pi}{\Delta^4}
\overline{\left( \frac{\partial E_n}{\partial t}\right)^2}
.
\label{alpha-definition}$$ Depending on whether $\alpha$ is much smaller or much larger than one, the transitions between levels may be described either as Landau–Zener transitions between the neighboring levels or as transitions in the continuum spectrum according to the linear-response Kubo formula. In both limits, the quantum-mechanical state effectively experiences a diffusion in energy, so that the energy drift over a large time $T$ is given by $$\overline{ [E(T)-E(0)]^2 } = 2 D\, T\, \Delta^3\, .
\label{diffusion-definition}$$ The dimensionless diffusion coefficient $D$ depends on $\alpha$. A remarkable result of Wilkinson [@wilkinson] is that in the case of unitary random-matrix ensemble, in both limits of [*large*]{} and [*small*]{} $\alpha$, the energy-diffusion coefficient is given by the same expression $$D(\alpha)=\alpha.$$
Recently, this problem has been analyzed further by one of the authors [@skvor; @SBK04] with the use of a $\sigma$-model constructed from the Keldysh-integral averaging over random matrices $H(t)$. The $\sigma$-model formulation of Refs. , contains, in principle, full information about the function $D(\alpha)$, and its diagrammatic expansion allows us to compute the perturbative series for $D(\alpha)$ in the limit of large $\alpha$. For the driven Gaussian [*orthogonal*]{} matrices, the one-loop correction has the form [@skvor; @GOE-comment]: $D(\alpha)=\alpha(1+d_1^{\rm(O)}\pi^{-1}\alpha^{-1/3}+\dots)$, where $d_1^{\rm(O)}=\Gamma(1/3)\,3^{-2/3}$. For the [*unitary*]{} ensemble, the number of loops in the diagrams for calculating $D(\alpha)$ must be even, which corresponds to expanding in powers of $\alpha^{-2/3}$: D() = ( 1 + + + … ) , \[D-series\] where we took into account that, according to Sec. \[section-diagrams-D\], the expansion parameter is $\pi^{-2}\alpha^{-2/3}$. In Refs. and , it has been found that $d_2=0$, and it has been further conjectured that all higher-order perturbative terms also vanish. In the present paper we shall verify that this conjecture holds up to the four-loop order: the value of $d_4$ obtained by numerical evaluation vanishes with very high accuracy ($|d_4|<3\times10^{-4}$) which is a strong argument in favor of d\_4 = 0.
In the process of developing the diagrammatic expansion for $D(\alpha)$ we observe a remarkable cancellation of diagrammatic vertices of order higher than four. This cancellation is proven to [*all*]{} orders in the perturbation theory by combinatoric means. Therefore we find that, at the level of perturbative series, the non-linear $\sigma$-model of Refs. is exactly equivalent to a matrix $\phi^4$-type theory. We show that the resulting theory can be obtained from the initial $\sigma$-model by applying a transformation analogous to the Dyson-Maleev [@Dyson56; @Maleev57] parameterization of quantum spin operators. The equivalence of the $\sigma$-model to a kind of a $\phi^4$-type theory may be a more important result than just a tool for calculating higher-order corrections in (\[D-series\]): in particular, this equivalence between two theories may have its counterparts for other types of non-linear $\sigma$-models, such as the one describing the diffusion of a quantum particle in a disordered media [@Wegner1979; @ELK1980; @Efetov-book]. We perform one straightforward generalization of this diagram cancellation to the case of arbitrary time-dependence of the control parameter $\vp(t)$.
The rest of the paper is organized as follows. In Section \[section-model\] we describe the $\sigma$-model of Refs. . In Section \[section-diagrams\], we develop the rules of the diagrammatic expansion. Further in Section \[section-diffuson\] we apply these rules for constructing the perturbative series for $D(\alpha)$. In Section \[section-cancellation\] we prove the cancellation of the vertices of order higher than four in the “rational” parameterization. We further formulate the new diagrammatic rules and the corresponding $\phi^4$ theory. In the following Section \[section-diagrams-D\], we apply the derived equivalence to explicitly write down the corrections to the diffusion coefficient (\[D-series\]). The numerical evaluation of the diagrams to the four-loop order is reported in Section \[section-four-loops\]. In Section \[section-phi(t)\] we generalize our results to a more general situation of arbitrary dependence of $\vp(t)$. In Section \[section-DM\] we demonstrate that our $\phi^4$ theory can be obtained from the initial $\sigma$-model by transformation similar to the Dyson-Maleev transformation. We conclude our discussion in Section \[section-conclusion\]. Technically complicated details of the calculations are delegated to Appendices.
Keldysh sigma-model {#section-model}
===================
The starting point of our analysis is the $\sigma$-model action derived in Ref. for the unitary ensemble. The field variable is the operator $Q$ which is the integral kernel in time domain with values in $2\times 2$ matrices in retarded-advanced Keldysh space. Technically, we write $Q$ as a $2\times 2$ matrix $Q_{tt'}$ depending on time variables $t$ and $t'$. The general expression for the action with arbitrary $\vp(t)$ has the form S\[Q\] = Q - \^2 . \[action-general\] Here ${\hat E}$ is the energy operator with the matrix elements ${\hat E}_{tt'} = i \delta_{tt'} \partial_{t'}$, the diagonal in time representation operator $\hat\vp$ is defined by the matrix elements $\hat\vp_{tt'} = \delta_{tt'} \vp(t')$, and the trace is taken both over the Keldysh and time spaces. The commutator in the last term of Eq. (\[action-general\]) vanishes if $\vp(t)=\const$, and is generally nonzero for a time-dependent $\vp(t)$.
To study the system’s dynamics for the case of the linear perturbation $\vp(t)=vt$, we prefer to simplify the notation by measuring time in the units of $\Delta^{-1}$. The resulting action in dimensionless units may be written as
\[action-sigma-model\] S\[Q\]= S\_[E]{}\[Q\] + S\_[kin]{}\[Q\], \[action-total\] where $$\begin{gathered}
S_{\rm E}[Q] = \pi i \Tr_{{\rm K},t} (\hat{E}Q) = -\frac{\pi}{2} \Tr_{\rm K}
\int dt\, (\partial_1 - \partial_2) Q_{t_1 t_2} \Big|_{t_1=t_2=t} ,
\label{action-E}
\\
S_{\rm kin}[Q]
= -\frac{\pi\alpha}{4} \Tr_{{\rm K},t} [t,Q]^2
= \frac{\pi\alpha}{4}
\Tr_{\rm K} \int\!\!\int dt\, dt'\, (t-t')^2\, Q_{tt'} Q_{t't} .
\label{action-kin}\end{gathered}$$
Here $\Tr_{{\rm K},t}$ and $\Tr_{\rm K}$ denote the traces over the full Keldysh-time space and over the two-dimensional Keldysh space only, and $\alpha$ is the same dimensionless coupling constant as in (\[alpha-definition\]) and (\[D-series\]).
The $Q$-matrix itself is subject to the constraint ${Q}^2=1$, or, more precisely, $$\int dt'\, Q_{t t'} Q_{t' t''}
=
\begin{pmatrix}
\delta^R_{t t''} & 0 \\
0 & \delta^A_{t t''}
\end{pmatrix} .
\label{constraint}$$ Where we have introduced the “retarded” and “advanced” $\delta$-functions $\delta^{R,A}_{t_1,t_2}=\delta(t_1-t_2 \mp \varepsilon)$ with an infinitesimal shift $\varepsilon$. This definition ensures the proper regularization [@KamenevAndreev99] of the functional integral of $\exp(-S[Q])$.
The functional integration in $Q$ is performed over an appropriate real submanifold of the complex manifold defined by the constraint (\[constraint\]) (this procedure is standard in the sigma-model derivation both in Keldysh [@KamenevAndreev99; @HorbachSchoen1993] and supersymmetric [@Efetov-book] formalisms). We take this manifold to be the orbit $$Q=U^{-1} \Lambda U
\label{orbit}$$ of the saddle-point solution $$\Lambda_{tt'}
=
\begin{pmatrix}
\delta^R_{t t'} & 0 \\
0 & -\delta^A_{t t'}
\end{pmatrix}
\label{lambda-def}$$ under unitary rotations $U$. The integration measure $[DQ]$ is then the usual invariant measure on the orbit.
We want to stress that the present approach slightly differs from the scheme originally proposed in Refs. . The sigma-model derived in Ref. , having the same action (\[action-general\]), was formulated on a different manifold $Q_F=U_F^{-1}U^{-1}\Lambda UU_F=U_F^{-1}QU_F$, where $U_F$ is a non-Hermitian rotation which contains the knowledge about the fermionic distribution function \[see Eq. (\[UF\]) for an explicit form of $U_F$\]. However, since calculating the energy-space diffusion coefficient $D$ is essentially a [*single-particle*]{} problem, one can get rid of the distribution function $F$ in the definition of the integration manifold and express the diffusion coefficient $D$ in terms of a certain correlation function of the fields $Q$. This refinement of the theory is presented in Appendix \[A:Keldysh\].
The quantity of our interest will be the [*diffuson*]{} ${\cal D}_{\eta}(t)$ defined by $$\langle Q^{(+)}_{t_1,t_2} \, Q^{(-)}_{t_3,t_4} \rangle = \int
[DQ]\, e^{-S[Q]} Q^{(+)}_{t_1,t_2} \, Q^{(-)}_{t_3,t_4} =
\frac{2}{\pi}
\delta(t_1-t_2+t_3-t_4)
{\cal D}_{t_1-t_2}(t_1-t_4)\, ,
\label{full-diffuson-definition}$$ where $$Q^{(\pm)}_{t_1 t_2}=\Tr_K (\sigma^\mp Q_{t_1 t_2})
\label{Q-off-diag}$$ are the off-diagonal elements of the $Q$-matrix in the Keldysh space. The form of the right-hand side in (\[full-diffuson-definition\]) follows from the invariance of the action $S[Q]$ with respect to time translations ($Q_{tt'}\mapsto Q_{t+\delta
t,t'+\delta t}$) and with respect to the energy shift ($Q_{tt'}\mapsto Q_{tt'}e^{i\omega(t-t')}$). Using the causality principles, one can further show (see Section \[section-diagrams\] and Appendix \[A:Keldysh\]) that the diffuson must have the form $${\cal D}_\eta(t)= \theta(t) \exp [-P(\eta,t)],
\label{full-diffuson}$$ where $P(\eta=0,t)=0$. Further we may expand $P(\eta,t)$ in $\eta$. The diffusion coefficient $D(\alpha)$ defined in (\[diffusion-definition\]) is given by the coefficient at $P(\eta,t)$ at $\eta^2 t$ (at small $\eta$ and at large $t$). The derivation is given in Appendix \[A:Keldysh\]. Formally we may write $$D(\alpha)= - \lim_{t\to\infty} \frac{1}{2t}
\left. \frac{\partial^2}{\partial \eta^2} \right|_{\eta=0}
{\cal D}_\eta(t) .
\label{diffusion-coefficient}$$
Diagrammatic expansion {#section-diagrams}
======================
To develop the diagrammatic technique, we explicitly parameterize the unitary rotations $U$ in (\[orbit\]) by elements of the corresponding Lie algebra. This parameterization may be chosen in many different ways, and we write generally $$U=f^{1/2}(W)\, ,
\label{parameter-general-1}$$ where the components of $W$ are given by $$W_{tt'}
=
\begin{pmatrix}
0 & b_{tt'} \\
-\bar{b}_{tt'} & 0
\end{pmatrix} ,
\label{W-b}$$ and $f(W)$ may be represented as the series f(W)=1+W+W\^2/2 + c\_3 W\^3 + c\_4 W\^4 + …\[f-series\] The unitarity of $U$ implies $\bar{b}_{tt'} = b^*_{t't}$ and $f(W)f(-W)=1$. We have chosen to write $f^{1/2}$ in the definition (\[parameter-general-1\]) so that the parameterization of $Q$ contains the first power of $f$: $$Q = \Lambda f(W)
\label{parameter-general-2}$$ (note that $W$ anticommutes with $\Lambda$).
Possible choices of the parameterization will be discussed in detail in Appendix \[A:parameterizations\]. Using the parameterization (\[W-b\]), (\[parameter-general-2\]), the integration measure $[DQ]$ may be written as $[Db\, D\bar{b}] J_f[b,\bar{b}]$, where $J_f$ is the Jacobian associated with the parameterization $f$. The explicit expression for the Jacobian in terms of the function $f$ is given in Appendix \[A:parameterizations\].
Then we do the algebra of substituting the parameterization (\[lambda-def\]), (\[f-series\]), (\[parameter-general-2\]) into the action (\[action-total\])–(\[action-kin\]) to obtain $$S[b,\bar{b}]=S^{(2)}[b,\bar{b}]+ S^{(\ge 4)}_{\rm E} [b,\bar{b}]+
S^{(\ge 4)}_{\rm kin} [b,\bar{b}],
\label{S[b]}$$ where $$S^{(2)}[b,\bar{b}] = \frac{\pi}{2} \int\!\!\int dt_1\, dt_2\,
\bar{b}_{12}\Big[ (\partial_1 + \partial_2) + \alpha(t_1-t_2)^2 \Big]
b_{21}
\label{b-action-quadratic}$$ is the quadratic part of the action, and $S^{(\ge 4)}_{\rm E} [b,\bar{b}]$ and $S^{(\ge 4)}_{\rm kin} [b,\bar{b}]$ are the higher-order terms (only even-order terms in $b$, $\bar{b}$ appear in the action).
The propagator of the quadratic action $S^{(2)}[b,\bar{b}]$ is the [*bare diffuson*]{}: b\_[t+/2,t-/2]{} |[b]{}\_[t’-’/2,t’+’/2]{} \^[(0)]{} = (-’) [D]{}\^[(0)]{}\_(t-t’) \[bare-propagator\] with $${\cal D}^{(0)}_\eta(t)=\theta(t) \exp[-\alpha\eta^2 t] .
\label{bare-diffuson}$$
Now we build up the diagrammatic expansion: by expanding the higher-order terms $S^{(\ge 4)}_{\rm E} [b,\bar{b}]$ and $S^{(\ge 4)}_{\rm kin} [b,\bar{b}]$ we generate vertices which we then connect by the propagators (\[bare-propagator\]) of the quadratic theory. Each propagator (\[bare-propagator\]) will be graphically represented as a band with two incoming legs (corresponding to the $b$-end of the diffuson) and two outgoing legs (the $\bar{b}$-end of the diffuson), see Fig. \[fig:diagram-elements\]. These two ends are not equivalent: the $b$-vertex must contain later times than the $\bar{b}$ vertex (due to the retarded $\theta(t)$ factor in the diffuson (\[bare-diffuson\])). Correspondingly, we will draw an arrow on the diffuson line pointing in the direction of decreasing time, from $b$ to $\bar{b}$.
=0.5
The nonlinear vertices arising from $S^{(\ge 4)}_{\rm E} [b,\bar{b}]$ have the form $$S^{(2n)}_{\rm E} [b,\bar{b}]
=
(-1)^{n+1} \pi \, c_{2n}
\Tr [(\hat\partial b)\bar{b} (b\bar{b})^{n-1}] ,
\label{E-vertex}$$ where we use the shorthand notation $(\hat\partial b)_{12}=
(\partial_1+\partial_2)b_{12}$, and the trace is understood as the convolution in time variables. The vertices from $S^{(\ge 4)}_{\rm kin} [b,\bar{b}]$ are of the form $$S^{(2n)}_{\rm kin} [b,\bar{b}]
=
\alpha
\int dt_1 \dots dt_{2n} \,
b_{12} \bar{b}_{23} \dots \bar{b}_{2n,1} \sum_{i<j} a_{ij}^{(n)}(t_i-t_j)^2 ,
\label{kin-vertex}$$ where $a_{ij}^{(n)}$ are numerical coefficients expressed through the coefficients $c_k$ of the expansion (\[f-series\]): a\_[ij]{}\^[(n)]{} = (-1)\^[i-j+n]{} c\_[j-i]{} c\_[i-j+2n]{} . Graphically, we order the diffusons at each vertex clockwise according to their appearance in the traces (\[E-vertex\]) and (\[kin-vertex\]).
The average value of the physical observables is given by certain correlators of the original $Q$-field. When we compute them in terms of the $b$ and $\bar{b}$ fields, those observables generate “external vertices”. The vertices generated by $S^{(\ge 4)}_{\rm E} [b,\bar{b}]$ and $S^{(\ge 4)}_{\rm kin} [b,\bar{b}]$ will be further called “internal vertices”. Finally, vertices generated by the Jacobian $J_f[b,\bar{b}]$ (unless it equals one) we shall call “Jacobian vertices” (see Appendix \[A:parameterizations\] for the explicit form of the Jacobian vertices).
There are two obvious rules for constructing the diagrams. Firstly, the diffusons may be drawn on a planar figure without “twisting” (but intersections of different diffusons are allowed, see, e.g., Fig. \[fig:diagram-examples\]c). Secondly, the diagram must not contain closed loops formed by internal lines. Such loops would immediately produce the factor $\theta(t_1-t_2)\theta(t_2-t_3)\dots
\theta(t_{n}-t_1)$ which always vanishes \[note that $\theta(t_1-t_1)=0$ due to the causality rule [@KamenevAndreev99; @AltlandKamenev00], and so loops of length one are not allowed either\]. For example, the diagrams in Figs. \[fig:diagram-examples\]ab identically vanish and should not be considered. An example of a non-vanishing two-loop diagram is shown in Fig. \[fig:diagram-examples\]c.
=0.4
From the above constraints on the diagram we can prove that only even-loop diagrams contribute to the diffuson (\[full-diffuson\]). More generally, the following [*theorem*]{} holds: Let $b^{(2n+1)}$ and $\bar{b}^{(2n+1)}$ denote the operators $$\begin{aligned}
b^{(2n+1)}_{tt'}&=&
\int b_{t t_1} \bar{b}_{t_1 t_2} b_{t_2 t_3} \dots b_{t_{2n} t'}
\, dt_1\,\dots\, dt_{2n} ,
\nonumber \\
\bar{b}^{(2n+1)}_{tt'}&=&
\int \bar{b}_{t t_1} b_{t_1 t_2} \bar{b}_{t_2 t_3} \dots \bar{b}_{t_{2n} t'}
\, dt_1\,\dots\, dt_{2n} .
\label{b(n)}\end{aligned}$$ Then any nonzero average ${\langle b^{(n_1)}\dots b^{(n_k)}\, \bar{b}^{(\bar{n}_1)}
\dots\bar{b}^{(\bar{n}_{\bar{k}})}\rangle}$ must contain equal number of $b$- and $\bar{b}$-operators and its diagrammatic expansion contains only diagrams with [*even*]{} number of loops.
Furthermore, from the same principle it is easy to prove that $\langle b^{(n)}_{t_1 t_2} \bar{b}^{(\bar{n})}_{t_3 t_4}\rangle=0$ if $t_1\le t_4$ in all orders of the diagrammatic expansion. This proves the causality of the full diffuson (\[full-diffuson-definition\]): ${\cal D}_\eta(t\le 0) =0$.
In the main body of the paper, we shall only use the [*rational*]{} parameterization: $$f(W) = \frac{1 + W/2}{1 - W/2} .
\label{parameter-rational}$$ The Jacobian for the rational parameterization is known [@Efetov1983; @Efetov-book] to be equal to 1. (In fact, such a parameterization is not unique. As shown in Appendix \[A:parameterizations\], the class of parameterizations with unit Jacobian is given by a one-parameter family, with the rational parameterization being a particular representative.)
In the rational parameterization (\[parameter-rational\]), the coefficients of the series (\[f-series\]) are given by $c_k=2^{1-k}$ ($k\geq1$) and the coefficients $a_{ij}^{(n)}$ in the nonlinear vertices (\[kin-vertex\]) have the form a\_[ij]{}\^[(n)]{} = (-1)\^[i-j+n]{} 2\^[1-2n]{}. \[a-rational\]
Diagrammatic series for the diffusion coefficient {#section-diffuson}
=================================================
In this section we apply the developed diagrammatic technique to calculating the full diffuson (\[full-diffuson-definition\]) and to further extracting the diffusion coefficient (\[diffusion-coefficient\]).
The observables in (\[full-diffuson-definition\]) have the form $$\begin{aligned}
Q^{(+)} &=& b^{(1)} - c_3 b^{(3)} + c_5 b^{(5)} - \dots , \cr
Q^{(-)} &=& \bar{b}^{(1)} - c_3 \bar{b}^{(3)} + c_5 \bar{b}^{(5)} - \dots ,\end{aligned}$$ where $c_3$, $c_5$, …are the coefficients in the Taylor expansion of the parameterization (\[f-series\]), and the operators $b^{(n)}$ and $\bar b^{(n)}$ are defined in Eq. (\[b(n)\]). Respectively, the average $\langle Q^{(+)} Q^{(-)} \rangle$ contains averages of different powers of $b$ and $\bar{b}$.
Let us first discuss the simplest of those averages: b\_[t+/2,t-/2]{} |[b]{}\_[t’-’/2,t’+’/2]{} = (-’) [D]{}\^\*\_(t-t’) . \[D-star-propagator\] Eq. (\[D-star-propagator\]) defines the “pro-diffuson” ${\cal D}^*_\eta(t)$ which should not be confused with the full diffuson given by Eq. (\[full-diffuson-definition\]). The pro-diffuson ${\cal D}^*_\eta(t)$ may be written in the perturbative series as shown in Fig. \[fig:diffuson-diagram\]b. Defining the irreducible “self-energy” block $\Sigma_\eta(t)$ (shown pictorially in Fig. \[fig:diffuson-diagram\]a) by the condition that it cannot be split in two pieces by cutting any one diffuson, we can resum the diagrammatic series for the pro-diffuson as \^[-1]{} = \^[-1]{} - \_() . \[D-star-Sigma\] Here we adopted the frequency representation, with \^[(0)]{}\_() = \_[-]{}\^[+]{} e\^[it]{} [D]{}\^[(0)]{}\_() dt = . \[D0-freq\] The role of the self-energy $\Sigma_\eta(\omega)$ is shifting the pole of the pro-diffuson ${\cal D}^*_\eta(\omega)$.
One can see that the bare diffuson (\[D0-freq\]) in the frequency representation looks exactly as the diffuson propagator $1/(-i\omega+Dq^2)$ for a particle in a disordered medium. However, the analogy between these problems does not extend beyond the formal coincidence of the bare propagators. Indeed, for a particle in a random potential, the frequency $\omega$ and momentum $q$ are “decoupled” from each other: each diffuson in a diagram has the same frequency, while the choice of their momenta $q_i$ is dictated by the momentum conservation law in the vertices. Contrary, for the dynamic problem, the times $\eta_i$ and $t_j$ of the internal diffusons ${\cal D}^{(0)}_{\eta_i}(t_i)$ which constitute the self-energy block $\Sigma_\eta(t)$ are linearly related to each other (see Section \[section-diagrams-D\] for their explicit form). Therefore, in the frequency representation, the internal diffusons in $\Sigma_\eta(\omega)$ will enter with different frequencies $\omega_i$ coupled to the time arguments $\eta_j$. Thus, the frequency representation (\[D0-freq\]) is not a way to calculate $\Sigma_{\eta}(t)$, but is a convenient tool for summing the geometric series (\[D-star-Sigma\]).
Note that the full diffuson ${\cal D}_\eta(t)$ differs from the pro-diffuson ${\cal D}^*_\eta(t)$ by higher-power averages $\langle b^{(n)} \cdot \bar{b}^{(\bar{n})} \rangle$ (see Fig. \[fig:diffuson-diagram\]c). We observe that [*all*]{} irreducible parts \[both $\Sigma_\eta(t)$ and the irreducible blocks in Fig. \[fig:diffuson-diagram\]c\] remain exponentially decaying with time even at $\eta=0$ (in the frequency representation, they are non-singular functions of $\eta$ and $\omega$ at $\eta \to 0$ and $\omega\to 0$) [@convergence]. Therefore we may sum all the diagrams to the form $${\cal D}_\eta(\omega)=\frac{Z(\eta,\omega)}{-i\omega + \alpha\eta^2
- \Sigma_\eta(\omega)} ,$$ where $Z(\eta,\omega)$ is a regular function at $\eta \to 0$ and $\omega\to 0$. As we shall see from the further explicit calculations, $\Sigma_\eta(\omega)\to 0$ as $\eta\to 0$. Furthermore, the full diffuson ${\cal D}_\eta(\omega)$ remains unrenormalized at $\eta=0$ (see Appendix \[A:Keldysh\] for a proof). Therefore, $Z(\eta=0,\omega)=1$.
Thus the correction to the diffusion coefficient is determined by the leading term in the expansion of $\Sigma_\eta(\omega)$ in $\eta$ and $\omega$: D() = - \_[0]{} = - \_[0]{} \_0\^\_(t) dt , \[D-correction\] where the lower limit of $t$-integration follows from the causality of $\Sigma_\eta(t)$ which can be proved analogously to the causality of the diffuson ${\cal D}_\eta(t)$.
The diagrams representing the irreducible block $\Sigma_\eta(t)$ may be classified in the number of loops $L$. From simple power counting (as shown in Ref. ) the $L$-loop diagrams give a contribution to $D(\alpha)$ proportional to $\alpha^{1-L/3}$, cf. Eq. (\[D-series\]). Therefore, the diagrammatic expansion is a series in the number of loops, and (since the number of loops $L$ must be even) the expansion small parameter is $\alpha^{-2/3}$, up to some unknown number. A more accurate analysis of Sec. \[section-diagrams-D\] shows that the actual small parameter is $\pi^{-2}\alpha^{-2/3}$.
In the following section we shall see that certain different diagrams with the same number of loops $L$ may partially cancel each other.
Canceling vertices of order higher than four {#section-cancellation}
============================================
In this section we show that, in the [*rational*]{} parameterization (\[parameter-rational\]), in diagrams containing only internal vertices (e.g., in $\Sigma_\eta(t)$), [*all*]{} vertices of order higher than four are cancelled [*in all orders*]{} of the diagram series.
The outline of the calculation is as follows. The vertices with derivatives originating from $S_{\rm E}^{(\geq4)}$ (Fig. \[fig:diagram-elements\]b) generate terms $\partial_t {\cal D}^{(0)}_\eta (t)$. We transform those terms according to the [*equation of motion*]{} for the diffuson: $$\partial_t {\cal D}^{(0)}_\eta (t) = \delta(t) - \alpha \eta^2
{\cal D}^{(0)}_\eta (t) .
\label{eq-motion}$$ The resulting expressions have the form of $(t_i-t_j)^2$ vertices in Fig. \[fig:diagram-elements\]c and may then be combined with the vertices originally generated by $S^{(\ge 4)}_{\rm kin} [b,\bar{b}]$.
When performed in the rational parameterization (\[parameter-rational\]), this procedure leads to the cancellation of all vertices of order higher than four in all orders of the perturbation series. The calculation is somewhat technical with the combinatoric counting of diagrams and coefficients. This calculation is reported in detail in Appendix \[A:rational\].
Of course, at the end of the above simplification procedure, the fourth-order vertices get modified from their original form. The final form of the fourth-order vertex is S\^\*\_4\[b,|[b]{}\] = - dt\_1 dt\_2 dt\_3 dt\_4 (t\_1-t\_2)(t\_3-t\_4) b\_[12]{}|[b]{}\_[23]{}b\_[34]{}|[b]{}\_[41]{} . \[S-star\] The resulting theory in terms of $b$-fields (which we will further call “$b$-theory”) has the action S\_[eff]{}\[b,|b\] = S\^[(2)]{}\[b,|[b]{}\] + S\^\*\_4\[b,|[b]{}\] , \[b-action-rational\] where $S^{(2)}[b,\bar{b}]$ is given by (\[b-action-quadratic\]).
We need to make two comments on the above derivation.
First, up to now the correspondence is established only between diagrams containing [*internal*]{} vertices. The correspondence between [*physical observables*]{} (i.e., external vertices) will be discussed in Section \[section-DM\]. In the further discussion, we employ the $b$-theory for calculating $\Sigma_\eta(t)$ which contains only internal vertices.
Second, when constructing the diagrams for $\Sigma_\eta(t)$ in the $b$-theory, one of the $(t_{2i-1}-t_{2i})$ factors in $S^*_4[b,\bar{b}]$ always coincides with the external time difference $\eta$. The integrals over times $t_i$ converge [@convergence] and therefore $$\Sigma_{\eta=0}(t)=0 .$$ Obviously $\Sigma_\eta(t)$ is an even and regular function of $\eta$. Thus the Taylor expansion of $\Sigma_\eta(t)$ in small $\eta$ starts with $\eta^2$. The leading term in this expansion determines the correction to the diffusion coefficient, according to (\[D-correction\]). We shall perform an explicit calculation in the next section.
Diagrammatic expansion for $D(\alpha)$ {#section-diagrams-D}
======================================
In this section we classify the diagrams for $\Sigma_\eta(t)$ in the $b$-theory (\[b-action-rational\]) and write down explicit integral expressions for the correction to the diffusion coefficient (\[D-correction\]).
Consider any diagram (in the $b$-theory) with $L$ loops contributing to $\Sigma_\eta(t)$. As mentioned previously, such a diagram must contain no closed threads, which greatly restricts the number of allowed diagrams (in particular, the number of loops $L$ must be even). Since all the vertices in the $b$-theory have valency four, the number of diffusons in the diagram is $2L-1$. We enumerate those diffusons in two different sequences which we shall call the “left-hand” and “right-hand” numberings. Start at the “entry point” (external diffuson leading into the diagram) and go along the diffusons in two different ways: using the right-hand rule (following the right edges of the diffusons, i.e., turning right at every vertex) and using the left-hand rule (following the left edges of the diffusons and turning left at every vertex). The relative re-numbering of the diffusons in the two sequences defines a permutation $\sigma$ of $(2L-1)$ elements. In other words, let us number the diffusons following the [*right-hand*]{} route. Then reading off the diffuson numbers along the [*left-hand*]{} route produces the sequence of numbers \[$\sigma(1)$, $\sigma(2)$,…, $\sigma(2L-1)$\]. We shall further label the diagrams by such sequences (see Table \[table:four-loop\]). An example of the 4-loop diagram corresponding to the permutation $\sigma=[3764215]$ (numbered 2 in the Table \[table:four-loop\]) is shown in Fig. \[fig:loop4example\].
--------------------------------------------------------------
No. $\sigma$ comments
------------------------------------ ------------- -----------
1 \[3214765\] reducible
**2 & \[3764215\] &\
**3 & \[3752164\] &\
**4 & \[3621754\] &\
**5 & \[3654721\] &\
6 & \[4276315\] & = No. 3 (T)\
**7 & \[4731652\] &\
8 & \[4317625\] & = No. 4 (LR/T)\
**9 & \[4726531\] &\
**10& \[5274163\] &\
11& \[5417362\] & = No. 3 (LR)\
**12& \[5372641\] &\
13& \[6327514\] & = No. 7 (T)\
14& \[6514732\] & = No. 2 (LR)\
15& \[6251743\] & = No. 3 (LR+T)\
16& \[6437251\] & = No. 9 (LR+T)\
17& \[7614325\] & = No. 5 (LR/T)\
18& \[7532614\] & = No. 9 (T)\
19& \[7426153\] & = No. 12 (LR/T)\
20& \[7361542\] & = No. 9 (LR)\
**21& \[7254361\] &\
******************
--------------------------------------------------------------
: The list of all four-loop diagrams for the self-energy part $\Sigma$. Each diagram is characterized by a permutation $\sigma$, see text. The diagram No. 1 is reducible: it can be written as $\Sigma_2 {\cal D}^{(0)} \Sigma_2$, where $\Sigma_2$ is the two-loop contribution. The other diagrams are irreducible. The diagrams related by the “left-right” (LR) or time-reversal (T) symmetries give the same contribution. LR+T means that two diagrams are related by the combined action of the two symmetries. LR/T indicate that two symmetries applied to a permutation give the same result. Graphical representation of the diagram No. 2 is given in Fig. \[fig:loop4example\].[]{data-label="table:four-loop"}
=0.4
Note that not every permutation of $2L-1$ elements defines a diagram. We do not discuss here the combinatoric problem of enumerating all diagrams in all orders. For our modest purpose of calculating the four-loop correction to the diffusion coefficient, the enumeration of the diagrams may be easily done “by hand”. Recall that for $L=2$ there is only one allowed diagram (shown in Fig. \[fig:diagram-examples\]) corresponding to the permutation \[3,2,1\]. For $L=4$, the number of allowed diagrams (and the corresponding permutations) is 21, see Table \[table:four-loop\]. Note that one of those diagrams (No. 1 in Table \[table:four-loop\]) is [*reducible*]{}: it consists of two independent two-loop blocks.
Given a diagram characterized by a certain permutation $\sigma$, let $\eta_i$ and $t_i$ denote the parameters of the diffusons ${\cal D}^{(0)}_{\eta_i} (t_i)$ involved in the diagram, with the diffusons enumerated along the right-hand route. For calculating the diagram, it is convenient to take $t_1,\dots,t_{2L-1}$ as independent integration variables. The parameters $\eta_i$ may be expressed via $t_i$ as $$\eta_i
= \eta + \sum_{j<i} t_j - \sum_{\sigma(j)<\sigma(i)} \!\!\!\! t_j
= \eta + \sum_j \gamma^{(\sigma)}_{ij} t_j .
\label{eta-i}$$ Here the first (second) sum contains all times $t_j$ encountered before $t_i$ along the right-hand (left-hand, respectively) route. The last equality is the definition of the coefficients $\gamma^{(\sigma)}_{ij}$. The matrix $\gamma^{(\sigma)}_{ij}$ contains only entries $0$, $1$, and $-1$, and is antisymmetric: $\gamma^{(\sigma)}_{ij}=-\gamma^{(\sigma)}_{ji}$. Moreover, the elements $\gamma^{(\sigma)}_{ij}$ with $i>j$ are non-negative (either 0 or 1), and those with $i<j$ are non-positive (either 0 or $-1$).
Finally, calculating the self-energy block $\Sigma_\eta(t)$ in the $b$-theory (\[b-action-rational\]) and employing Eq. (\[D-correction\]), we may write down the contribution of any given irreducible diagram (corresponding to a given permutation $\sigma$) to the diffusion coefficient $D(\alpha)$: $$\delta^{(\sigma)} D(\alpha)
= - \frac{K^{(\sigma)}}{\pi^L} \alpha^{1-L/3} ,$$ where K\^[()]{} = \_0\^…\_0\^ ( \_[i=1]{}\^[2L-1]{} dT\_i ) P\^[()]{} (T\_1, …, T\_[2L-1]{}) e\^[-S\_3\^[()]{}(T\_1, …, T\_[2L-1]{})]{} . \[K-sigma\] Here $P^{(\sigma)}(T_1, \dots, T_{2L-1})$ and $S_3^{(\sigma)}(T_1, \dots, T_{2L-1})$ are homogeneous polynomials of degrees $2L-2$ and three, respectively. They are defined as
$$\begin{aligned}
P^{(\sigma)}(T_1, \dots, T_{2L-1}) &=
\left(\prod_{i=1}^{2L-1} {\tilde\eta}_i \right)
\sum_{i=1}^{2L-1} \frac{1}{{\tilde\eta}_i} ,
\\
S_3^{(\sigma)}(T_1, \dots, T_{2L-1}) &=
\sum_{i=1}^{2L-1} {\tilde\eta}_i^2 T_i ,
\\
{\tilde\eta}_i &=
\sum_j \gamma^{(\sigma)}_{ij} T_j ,\end{aligned}$$
where the coefficients $\gamma^{(\sigma)}_{ij}$ are constructed from the permutation $\sigma$ according to Eq. (\[eta-i\]).
--------------------------------------------------------------------------------
No. $M^{(\sigma)}$ $K^{(\sigma)}$ st. dev.
----------------------------------- ---------------- ---------------- ----------
**2 & 2 & 0.066945 & 0.000015\
**3 & 4 & $-0.123205$ & 0.000012\
**4 & 2 & 0.082358 & 0.000018\
**5 & 2 & 0.008234 & 0.000005\
**7 & 2 & 0.006030 & 0.000007\
**9 & 4 & $-0.006567$ & 0.000005\
**10& 1 & 0.206687 & 0.000013\
**12& 2 & $-0.001646$ & 0.000006\
**21& 1 & $-0.011374$ & 0.000003\
******************
--------------------------------------------------------------------------------
: Results of numerical evaluation of the diagrams from Table \[table:four-loop\]. $M^{(\sigma)}$ denote the multiplicities of the diagrams. $K^{(\sigma)}$ is given by Eq. (\[K-sigma\]). The last column is the standard deviation of $K^{(\sigma)}$.[]{data-label="table:numerics"}
Numerical calculation of the 4-loop diagrams {#section-four-loops}
============================================
The list of allowed four-loop diagrams (permutations) is given in Table \[table:four-loop\]. The first diagram in the list is not irreducible: it consists of two blocks of two loops each and should not be included in the irreducible part $\Sigma_\eta(t)$. Out of the remaining 20 diagrams, some are related by symmetries and produce the same contributions (\[K-sigma\]). Namely, there are two possible symmetries: the “left-right” (LR) reflection of the diffusons (corresponding to the transformation $\sigma \mapsto \sigma^{-1}$) and the time-reversal (T) symmetry (corresponding to replacing $\sigma(i)\mapsto 2L-\sigma(2L-i)$). The diagrams related by the symmetries are identified in the above table. It remains to calculate the nine different diagrams and add them with the corresponding multiplicities $M^{(\sigma)}$, see Table \[table:numerics\].
The antisymmetric matrices $\gamma^{(\sigma)}$ defined for nine permutations $\sigma$ from Table \[table:numerics\] are listed below, with the elements $+$ ($-$) standing for 1 ($-1$) for brevity:
$$\begin{aligned}
\gamma^{(2)} = \left[ \begin {array}{ccccccc} 0&-&-&-&0&-&-\\+&0&
-&-&0&-&-\\+&+&0&0&0&0&0\\+&+&0
&0&0&-&-\\0&0&0&0&0&-&-\\+&+&0&+&+&0&-\\+&+&0&+&+&+&0\end {array} \right];
\quad
\gamma^{(3)} = \left[ \begin {array}{ccccccc} 0&-&-&0&-&0&-\\+&0&
-&0&-&0&-\\+&+&0&0&0&0&0\\0&0&0&0
&-&-&-\\+&+&0&+&0&0&-\\0&0&0&+
&0&0&-\\+&+&0&+&+&+&0\end {array} \right];
\quad
\gamma^{(4)} = \left[ \begin {array}{ccccccc} 0&-&-&0&0&-&0\\+&0&
-&0&0&-&0\\+&+&0&0&0&0&0\\0&0&0&0
&-&-&-\\0&0&0&+&0&-&-\\+&+&0&+
&+&0&0\\0&0&0&+&+&0&0\end {array} \right];
\nonumber \\
\gamma^{(5)} = \left[ \begin {array}{ccccccc}
0&-&-&-&-&-&-\\
+&0&-&-&-&-&-\\
+&+&0&0&0&0&0\\
+&+&0&0&-&-&0\\
+&+&0&+&0&-&0\\
+&+&0&+&+&0&0\\
+&+&0&0&0&0&0
\end {array} \right];
\quad
\gamma^{(7)} = \left[ \begin {array}{ccccccc} 0&0&-&-&0&0&-\\0&0&-
&-&-&-&-\\+&+&0&-&0&0&-\\+&+&+&0&0&0&0\\0&+&0&0&0&-&-\\0&+&0&0
&+&0&-\\+&+&+&0&+&+&0\end {array} \right];
\quad
\gamma^{(9)} = \left[ \begin {array}{ccccccc} 0&-&-&-&-&-&-\\+&0&0
&-&0&0&-\\+&0&0&-&-&-&-\\+&+&+
&0&0&0&0\\+&0&+&0&0&-&-\\+&0&+
&0&+&0&-\\+&+&+&0&+&+&0\end {array} \right];
\\
\gamma^{(10)} = \left[ \begin {array}{ccccccc} 0&-&0&-&-&0&-\\+&0&0
&0&-&0&0\\0&0&0&-&-&-&-\\+&0&+&0
&-&0&-\\+&+&+&+&0&0&0\\0&0&+&0
&0&0&-\\+&0&+&+&0&+&0\end {array} \right];
\quad
\gamma^{(12)} = \left[ \begin {array}{ccccccc} 0 & - & - & - & - & - & - \\
+ & 0 & - & 0 & - & 0 & - \\ + &
+ & 0 & 0 & - & 0 & 0 \\
+ & 0 & 0 & 0 & - & - & - \\ + & + & + &
+ & 0 & 0 & 0 \\ + & 0 & 0 &
+ & 0 & 0 & - \\ + & + & 0 & + & 0 &
+ & 0 \end {array} \right] ;
\quad
\gamma^{(21)} = \left[ \begin {array}{ccccccc} 0&-&-&-&-&-&-\\+&0&0
&0&0&0&-\\+&0&0&-&-&0&-\\+&0&+&0
&-&0&-\\+&0&+&+&0&0&-\\+&0&0&0
&0&0&-\\+&+&+&+&+&+&0\end {array} \right].
\nonumber\end{aligned}$$
Here, the matrix $\gamma^{(m)}$ denotes the matrix $\gamma^{(\sigma)}$ for the permutation No. $m$ from the Table \[table:numerics\].
The results of Monte Carlo numeric evaluation of the seven-fold integrals $K^{(\sigma)}$ given by Eq. (\[K-sigma\]) are summarized in Table \[table:numerics\]. Performing summation with the multiplicities $M^{(\sigma)}$ we get finally the estimate for the coefficient $d_4$ in the Taylor expansion (\[D-series\]): d\_4 = - (7 7) 10\^[-5]{}. \[d4-result\] The numerical uncertainty indicated in Eq. (\[d4-result\]) corresponds to one standard deviation. Thus, we cannot distinguish $d_4$ from zero and can estimate the upper bound for its absolute value as $|d_4|<3\times10^{-4}$.
Arbitrary dependence of $\vp(t)$ {#section-phi(t)}
================================
In this section we discuss to what extent the results obtained above can be generalized to an arbitrary time dependence of the control parameter $\vp(t)$.
We start by summarizing the modifications of the theory introduced by an arbitrary $\vp(t)$. For a generic $\vp(t)$, the diffuson becomes a function of [*three*]{} times \[cf. Eq. (\[full-diffuson-definition\])\]: Q\^[(+)]{}\_[t+/2,t-/2]{} Q\^[(-)]{}\_[t’-’/2,t’+’/2]{} = (-’) [D]{}\_(t,t’) . \[full-diffuson-phi\] To study the action (\[action-general\]) one can develop the standard perturbation theory described in Section \[section-diagrams\]. In terms of the $b$-fields, the action will have the form (\[S\[b\]\]) with the quadratic part ($\vp_i\equiv\vp(t_i)$) S\^[(2)]{}\[b,|[b]{}\] = dt\_1 dt\_2 |[b]{}\_[12]{}b\_[21]{} \[S2\[b\]-general\] and infinite number of nonlinear terms $S^{\geq4}[b,\bar b]$. In Eq. (\[S2\[b\]-general\]) we introduced $\Gamma=\pi\Delta C(0)$. The bare diffuson defined as the propagator of $S^{(2)}[b,\bar b]$ is given by [@AAK1982; @VavilovAleiner1999; @SBK04] \^[(0)]{}\_(t,t’) = (t-t’) { - \_[t’]{}\^t \[(+/2)-(-/2)\]\^2 d } . \[bare-diffuson-phi\]
It is remarkable that with such a modification of the theory one can still prove the cancellation of all [*internal*]{} vertices of the order higher than 4 in the rational parameterization. Indeed, the proof presented in Section \[section-cancellation\] was based on the equation of motion (\[eq-motion\]) for the diffuson, which is now replaced by an analogous equation \_t [D]{}\^[(0)]{}\_(t,t’) = (t-t’) - \^2 [D]{}\^[(0)]{}\_(t,t’) , \[eq-motion-general\] and the combinatorial counting of coefficients which is insensitive to time dependence of $\vp(t)$.
The resulting $b$-theory has the action S = S\^[(2)]{}\[b,|[b]{}\] - dt\_1 dt\_2 dt\_3 dt\_4 (\_1-\_2)(\_3-\_4) b\_[12]{}|[b]{}\_[23]{}b\_[34]{}|[b]{}\_[41]{} , \[b-theory-general\] which is a generalization of Eq. (\[b-action-rational\]) for an arbitrary dependence of the control parameter $\vp(t)$. Thus, the diagrams for the pro-diffuson ${\langle b\bar{b}\rangle}$: b\_[t+/2,t-/2]{} |[b]{}\_[t’-’/2,t’+’/2]{} = (-’) [D]{}\^\*\_(t,t’) \[star-diffuson-phi\] with an arbitrary $\vp(t)$ are exactly the same as the diagrams in the linear case $\vp(t)=vt$ considered in the previous Sections (but with the new diffusons (\[bare-diffuson-phi\])).
The energy absorption rate is expressed through the full diffuson ${\cal D}_\eta(t,t')$ defined in terms of the field $Q$ by Eq. (\[full-diffuson-phi\]). It differs from the pro-diffuson ${\cal D}^*_\eta(t,t')$ in the $b$-theory since $Q$ is a nonlinear function of $b$ and $\bar b$. In studying the linear perturbation this difference was irrelevant for the calculation of the diffusion coefficient (see Section \[section-diagrams-D\]). For a generic perturbation $\vp(t)$, the difference between the full diffuson ${\cal D}_\eta(t,t')$ and the pro-diffuson ${\cal D}^*_\eta(t,t')$ becomes important. In particular, the two-loop analysis of the quantum interference correction under the action of a harmonic perturbation [@SBK04] has shown that the average ${\langle b\bar{b}b\cdot \bar{b}\rangle}$ (the diagram (c) in Ref. ) has a contribution comparable to that of the average ${\langle b\cdot \bar{b}\rangle}$, both being negative and growing with time $\propto t$.
Relation to the Dyson-Maleev transformation {#section-DM}
===========================================
In this section we elucidate the meaning of the effective $b$-theory (\[b-action-rational\]) and discuss its relation to the Dyson-Maleev transformation widely used in dealing with quantum spin ferromagnets.
The action (\[b-action-rational\]) has been obtained in the previous sections from an analysis of mutual cancellations of higher-order vertices in the perturbation theory for the rational parameterization. However, it turns out that the same action may be obtained directly from the initial $\sigma$-model action (\[action-sigma-model\]) if one adopts the following parameterization of the $Q$-matrix in terms of the fields $b$ and $\bar b$: Q =
1-b|b/2 && b-b|bb/4\
|b && -1+|bb/2
. \[Q-DM\] Indeed, this parameterization respects the nonlinear constraint $Q^2=1$, and the trivial algebra of substituting (\[Q-DM\]) into the initial $\sigma$-model action (\[action-sigma-model\]) immediately leads to the effective action (\[b-action-rational\]) of the $b$-theory. With the explicit parameterization (\[Q-DM\]), [*external*]{} vertices (matrix elements of the $Q$-matrix) also become finite polynomials in $b$ and $\bar b$. We have checked that the two approaches (direct use of the parameterization (\[Q-DM\]) and the vertex cancellation by the technique developed in Appendix \[A:rational\]) produce the same effective external vertices.
Thus we come to an important conclusion about the $\sigma$-model (\[action-sigma-model\]): The rational parameterization (\[parameter-rational\]) (which contains an infinite series of higher-order interaction vertices) after mutual cancellation of higher-order vertices in the diagrammatic expansion is perturbatively equivalent to the parameterization (\[Q-DM\]).
In the rational parameterization \[more generally, in any parameterization of the form (\[W-b\])–(\[parameter-general-2\])\], the matrices $b$ and $\bar b$ are Hermitian conjugate of each other: $\bar b=b^\dagger$. The same remains therefore true for the $b$-theory (\[b-action-rational\]) obtained from the rational parameterization. On the other hand, the parameterization (\[Q-DM\]) with $\bar b=b^\dagger$ violates the hermiticity of the $Q$ matrix. Nevertheless, our derivation indicates that this violation is inessential at the perturbative level and can be taken into account by a proper deformation of the integration contour over the elements of the $Q$ matrix.
The parameterization (\[Q-DM\]) is closely related to the famous Dyson-Maleev [@Dyson56; @Maleev57] parameterization for quantum spins. In that representation, the spin-$S$ operators are expressed by the boson creation and annihilation operators $\hat a^\dagger$ and $\hat a$ as S\^+ = (2S-a\^a) a, S\^- = a\^, S\^z = S-a\^a . \[DM\] The Dyson-Maleev transformation conserves the spin commutation relations but violates the property $(\hat S^-)^\dagger=\hat S^+$ rendering the spin Hamiltonian manifestly non-Hermitian. This is the expense one has to pay for making the spin Hamiltonian a finite-order polynomial in boson operators \[as opposed to an infinite series in the Holstein-Primakoff parameterization [@HolstPrim40]; note that the Holstein-Primakoff parameterization is analogous to the square-root-even parameterization (\[param-sqrt-even\])\]. The Dyson-Maleev transformation has proven to be the most convenient tool for studying spin wave interaction [@Canali92; @Hamer92-93]: it reproduces all the perturbative results obtained with the Holstein-Primakoff parameterization in a much faster and compact way.
A classical analogue of the Dyson-Maleev transformation was recently used by Kolokolov [@Kolokolov2000] in studying two-dimensional classical ferromagnets. To the best of our knowledge, there exists just one article by Gruzberg, Read and Sachdev [@Gruzberg1997] where the Dyson-Maleev parameterization was applied for the perturbative treatment of a $\sigma$-model (in the replica form).
In the Dyson-Maleev representation, the Hilbert space of free bosons should be truncated in order for the operator $\hat S^z$ to have a bounded spectrum. In Ref. , this truncation corresponds to integrating over a bounded region in bosonic variables. We expect that a similar constraint on the matrices $b$ and $\bar b$ may be required in order to achieve a nonperturbative equivalence between the initial $\sigma$-model in the $Q$-representation and the $b$-theory. This question is of importance for studying non-perturbative effects in $\sigma$-models, but goes beyond the scope of the present paper.
Discussion {#section-conclusion}
==========
This work has appeared as a result of our attempt to prove the conjecture formulated in Ref. that the Kubo formula for the energy absorption rate of a linearly driven unitary random Hamiltonian gives an exact result in the whole range of driving velocities. In the notation of the present paper, this conjecture implies that the relation $D(\alpha)=\alpha$ is exact.
Being unable to verify the conjecture nonperturbatively, we calculate the four-loop correction to the Kubo formula $D(\alpha)=\alpha$ in the limit of large velocities of the driving field, $\alpha\gg 1$. In the process of our derivation, we have refined the $\sigma$-model approach of Ref. : our improved method does not involve the fermionic distribution function and expresses the energy diffusion coefficient (\[diffusion-coefficient\]) in terms of the full diffuson (\[full-diffuson-definition\]) of the field theory (\[action-sigma-model\]).
We have further proven that the resulting $\sigma$-model is [*perturbatively equivalent*]{} to the specific matrix $\phi^4$-theory (\[b-action-rational\]). The lack of higher nonlinearities makes it possible to classify all four-loop diagrams and write down analytic expressions for them without resorting to computer symbolic computations. The final evaluation of emerging 7-dimensional integrals (\[K-sigma\]) cannot be done analytically, and we calculate them numerically. We find that the coefficient $d_4$ in the expansion (\[D-series\]) is indistinguishable from zero within the precision of our calculation, with its absolute value bounded by $|d_4|<3\times10^{-4}$.
We believe that this conclusion acts in favor of the conjecture $D(\alpha)=\alpha$.
This result may appear less surprising if we recall that the static (time-independent) unitary random-matrix ensemble is also known to possess some peculiar properties. In particular, its spectral statistics can be mapped onto the problem of one-dimensional noninteracting fermions; for a certain class of integrals over the unitary group the saddle-point approximation is exact (the Duistermaat-Heckman theorem [@Duistermaat-Heckman], see Ref. for a discussion). However, in the time-dependent problem considered in the present work, we are not able to perform mapping onto free fermions, and the Duistermaat-Heckman theorem does not help to evaluate the functional integral of the Keldysh $\sigma$-model. The $\sigma$-model is nontrivial, and its diffuson ${\cal D}_\eta(t)$ is a complicated function of $\eta$ and $t$ at arbitrary value of the coupling $\alpha$. The diffuson ${\cal D}_\eta(t)$ is renormalized from the simple diffusive form (\[bare-diffuson\]), and our result indicates that only its long-time asymptotics \[which determines the energy diffusion coefficient (\[diffusion-coefficient\])\] is free from perturbative corrections up to the order $\alpha^{-4/3}$.
The verification of the original nonperturbative conjecture $D(\alpha)=\alpha$ remains an open question.
As a byproduct of our analysis, we have rederived the $Q$-matrix parameterization (\[Q-DM\]) which is analogous to the Dyson–Maleev parameterization for spin operators [@Gruzberg1997]. The parameterization (\[Q-DM\]) is not specific to the Keldysh formalism, and can be applied to a wide class of [*unitary*]{} $\sigma$-models (e.g., to that describing diffusion of a particle in random media, both in the supersymmetric or replica [@Gruzberg1997] approaches), leading to a considerable simplification of the perturbative expansion. We could not extend this parameterization to the orthogonal and symplectic $\sigma$-models, since those involve additional linear constraints on the $Q$-matrix, apparently incompatible with our parameterization.
We thank M. V. Feigel’man and I. V. Kolokolov for drawing our attention to the Dyson-Maleev parameterization, and I. Gruzberg for pointing out Ref. . This research was partially (M. A. S.) supported by the Program “Quantum Macrophysics” of the Russian Academy of Sciences, RFBR under grant No. 04-02-16998, the Dynasty foundation and the ICFPM. M. A. S. acknowledges the hospitality of the Institute for Theoretical Physics at EPFL, where the main part of this work was performed.
Ward identity and energy absorption rate {#A:Keldysh}
========================================
In this Appendix we show that in the many-particle formulation the energy absorption rate under the action of an arbitrary perturbation $\vp(t)$ is expressed through the generalized diffuson (\[full-diffuson-phi\]). In particular, for the linear perturbation $\vp(t)=vt$, the diffusion coefficient in the energy space may be extracted from the decay rate of the diffuson (\[full-diffuson-definition\]). As a byproduct of our discussion, we also prove that, at $\eta=0$, the diffuson reduces to the step function: \_[=0]{}(t)=(t) .
Two approaches to energy diffusion
----------------------------------
The diffusion coefficient in the energy space may be defined in two different ways.
In the original sigma-model derivation [@skvor], the states of the time-dependent Hamiltonian $H(t)$ were occupied by noninteracting fermions. In such a multi-particle formulation, the step-like structure of the fermionic distribution function $f(E)$, \_[E-]{}f(E)=1, \_[E]{}f(E)=0, \[fermi-bc\] generates a spectral flow of fermions from low to high energies leading to the increase of the total energy of the system with time. At large time and energy scales, this many-particle process may be described by a diffusion equation on the distribution function $f(E)$. The corresponding diffusion coefficient expressed as $D\Delta^3$ (where $\Delta$ is the average interlevel spacing and $D$ is dimensionless) translates into the energy pumping rate $W=D\Delta^2$ (the density of states is $\Delta^{-1}$).
On the other hand, the same diffusion process may be observed in a single-particle problem. In the single-particle quantum mechanics (\[schroedinger\]), the energy of the system $E(t)$ defined with the instantaneous Hamiltonian E(t)=(t)|H(t)|(t)diffuses with time as described by (\[diffusion-definition\]). One can expect that in the process of time evolution of $H(t)$, the relative phases of the wave function components corresponding to widely separated energies become uncorrelated, and therefore the multi-particle diffusion evolution of $f(E)$ and the single-particle diffusion process give equivalent definitions of the diffusion coefficient $D$. The above reasoning is justified for a non-periodic evolution of $H(t)$ (for example, for the linear evolution of the parameter $\varphi(t)=vt$); for a periodic perturbation studied in Ref. , the phase correlations become important which leads to dynamic localization. Nevertheless, we show below that the energy pumping rate $W$ in the multi-particle problem may be expressed in terms of the single-particle diffuson for a rather general time dependence $\varphi(t)$.
Ward identity
-------------
A helpful tool for our further derivation are the Ward identities generated by rotations of the integration variables $Q$ in the functional integral of the sigma-model.
Consider the functional = V\^[-1]{}QV e\^[-S\[V\^[-1]{}QV\]]{} \[DQ\] , \[Ward0\] where $V$ is an arbitrary matrix in the time and Keldysh spaces (not necessarily unitary), and the action is given by Eq. (\[action-general\]). The integration is performed over the matrices $Q$ of the form (\[orbit\]), (\[lambda-def\]). If the rotation by the matrix $V$ is local (with $V_{tt'}\to\delta_{tt'}$ at $|t|,|t'|\to\infty$), it may be compensated by changing the integration variable $Q\mapsto V^{-1}QV$ in Eq. (\[Ward0\]) producing no anomalous contribution, and therefore $\Pi[V]$ is independent of such rotations $V$.
The multi-particle approach described in the previous subsection may also be introduced with the same rotated path integral (\[Ward0\]), but with the [*anomalous*]{} rotation matrix $V$ (involving the distribution function $f(E)$) [@skvor]. This anomalous rotation will be discussed in detail below in the subsequent subsection, and in this subsection we derive the Ward identity generated by infinitesimal non-anomalous rotations $V$.
Expanding $V=1+A+\dots$ and taking the variation of $\Pi[V]$ with respect to the infinitesimal generator $A$ we get = \[[Q]{},A\] + [Q (\[Q,E\] A)]{} + \^2 C(0) [Q (\[, Q Q\] A) ]{} = 0 . \[Ward1\] In deriving the second term with $[Q,\hat E]$ we performed a cyclic permutation under the trace, which is equivalent to integration by parts and omitting the resulting boundary terms. This procedure is justified for local $A_{tt'}$.
Various components of Eq. (\[Ward1\]) associated with different elements of the matrix $A$ give the set of Ward identities related to the $V$-invariance of $\Pi[V]$. Of particular importance is its off-diagonal (Keldysh) component $\Pi^{(+)}$ generated by the Keldysh element $A^{(+)}$: \_[t\_1t\_4]{}\_[t\_2t\_3]{} + \_[t\_1-t\_2,t\_4-t\_3]{} (\_3+\_4) [D]{}\_[t\_1-t\_2]{} ( , ) + C(0) (\_3-\_4) dt\_5 \_5 [Q\^[(+)]{}\_[12]{} (Q\_[35]{}Q\_[54]{})\^[(-)]{} ]{} = 0 , \[Ward2\] where the diffuson with three times ${\cal D}_\eta(t,t')$ is defined in Eq. (\[full-diffuson-phi\]) for an arbitrary time dependence $\varphi(t)$, and we have used that, due to causality, ${\langle Q_{tt'}\rangle}=\Lambda=\sigma_3\delta_{tt'}$.
Causality of the diffuson
-------------------------
The diffuson ${\cal D}_\eta(t,t')$ may be calculated by summing the diagrammatic series as described in Section \[section-diagrams\]. Every line in the diagram is the bare retarded diffuson (\[bare-diffuson-phi\]), and therefore the full diffuson ${\cal D}_\eta(t,t')$ also equals zero when $t<t'$.
Furthermore, using the Ward identity (\[Ward2\]) at $\eta=0$, we immediately find that the diffuson ${\cal D}_\eta(t,t')$ reduces to the step function of $t-t'$. Indeed, setting $t_3=t_4$ nullifies the last term yielding $\partial_{t'} {\cal D}_{\eta=0}(t,t') = -\delta(t-t')$. Integrating this equation and using the causality of the diffuson (${\cal D}_\eta(t,t')=0$ for $t<t'$), we obtain ${\cal D}_{\eta=0}(t,t')=\theta(t-t')$.
For our discussion in the next subsection, in order to regularize the diffusion process at $t \to -\infty$, we consider the situation where the evolution of the Hamiltonian $H(t)$ switches on at a certain time moment $t_0$. This can be modelled by a time dependence of $\vp(t)$ such that it remains constant at earlier times $\vp(t<t_0)={\rm const}$. For constant $\vp(t)$, the last term in (\[Ward2\]) vanishes and (similarly to the case $\eta=0$) we obtain \_(t,t’) + (t-t’) =0 t’<t\_0-. We can integrate this equation in two domains: \_(t,t’) = (t-t’) t,t’<t\_0- , \[caus-1\] and \_(t,t’) = [D]{}\_(t,-) t’<t\_0-<t . \[caus-2\] The first property (\[caus-1\]) guarantees that the diffuson “switches on” only after the moment $t_0$ when the evolution of the $H(t)$ starts. The second property (\[caus-2\]) states that the diffuson with the initial time before $t_0$ does not actually depends on this initial time, which allows us to define the diffuson originating at $t=-\infty$.
Distribution function and energy absorption rate
------------------------------------------------
To prove the relation between the multi-particle and single-particle definitions of the energy diffusion (see the first subsection of this Appendix), we consider the multi-particle kinetic problem in which the system is initially in a stationary state characterized by an arbitrary fermionic distribution function $f^{(0)}(E)$. The evolution of the Hamiltonian starts at a time moment $t_0$, which is described by $\vp(t<t_0)={\rm const}$. Within the Keldysh formalism [@RammerSmith1986; @KamenevAndreev99], occupation of states by non-interacting fermions may be taken into account by rotating the matrix $Q$ by the upper-triangular matrix [@UF-comment] (V\_[F\^[(0)]{}]{})\_[tt’]{} =
\_[tt’]{} & F\_[tt’]{}\^[(0)]{}\
0 & \_[tt’]{}
, \[UF\] where $F_{tt'}^{(0)}$ is the Fourier transform (in $t-t'$) of $F^{(0)}(E)=1-2f^{(0)}(E)$. The evolution of the distribution function with time may be read off as F= \^[(+)]{}\[V\_[F\^[(0)]{}]{}\] . Note that $V_{F^{(0)}}$ is not a local rotation (it does not vanish at $t\to -\infty$), and therefore $F$ undergoes a non-trivial time evolution.
We may calculate the time evolution of $F$ in the same procedure as the derivation of the Ward identity (\[Ward2\]). The only difference is that the second term in Eq. (\[Ward1\]) originating from $S_\text{E}[V^{-1}QV]$ vanishes for $V=V_{F^{(0)}}$. This can be most easily seen in the energy representation, where $F^{(0)}$ is diagonal and evidently commutes with $\hat E$. Then, applying the Ward identity (\[Ward2\]) one arrives at the result \^[(+)]{}\[V\_[F\^[(0)]{}]{}\]\_[12]{} = - 2 dt\_3 dt\_4 F\^[(0)]{}\_[t\_4-t\_3]{} \_[t\_1-t\_2,t\_4-t\_3]{} (\_3+\_4) [D]{}\_[t\_1-t\_2]{} ( , ) , which translates into F\_[t+/2,t-/2]{} = [D]{}\_(t,-) F\^[(0)]{}() . \[F2\] This equation proves that the diffuson ${\cal D}_{\eta}(t,t')$ formally defined as a ${\langle QQ\rangle}$ correlation function in the field theory (\[action-general\]) indeed plays the role of the evolution kernel for the distribution function $F(t,E)$.
The energy absorption rate can be expressed in terms of the distribution function as [@skvor; @BSK03] W(t) = - \_[0]{} \_[t]{}\_ F\_[t+/2,t-/2]{} . Using Eq. (\[F2\]) and the asymptotics $F^{(0)}(\eta)\sim1/(i\pi\eta)$ at $\eta\to0$ which follows from the fermionic boundary conditions (\[fermi-bc\]), we get W(t) = - 1[2]{} . |\_[=0]{} [D]{}\_(t,-) . \[W-general\]
If we consider the situation of a linear time evolution $\vp=vt$ switched on at a certain time moment $t_0$, the linearly growing contribution to the three-time diffuson ${\cal D}_\eta(t,-\infty)$ equals that of the two-time diffuson ${\cal D}_\eta(t-t_0)$ of the translationally invariant theory (\[full-diffuson-definition\]). This can be easily seen from the diagrammatic expansion of the diffuson ${\cal D}_\eta(t,-\infty)$: the characteristic decay time of diagrams is $t^*=\Delta^{-1}\alpha^{-1/3}$, and switching on at time $t_0$ plays the role of a soft cut-off for the vertices (\[b-theory-general\]). Therefore, at time scales $t-t_0\gg t^*$, the details of the switching-on process become unimportant, and we may replace ${\cal D}_\eta(t,-\infty)$ in (\[W-general\]) by ${\cal D}_\eta(t-t_0)$. This proves the equivalence of our single-particle definition (\[diffusion-coefficient\]) of the diffusion coefficient to the earlier multi-particle approach of Refs. .
Parameterizations of the $Q$ matrix {#A:parameterizations}
===================================
The Jacobian
------------
In this Appendix we discuss different parameterizations of matrices $Q$ by elements $W$ of the corresponding tangent space, according to (\[parameter-general-2\]). With such parameterizations, the integration over the non-linear manifold of matrices $Q$ reduces to that over the linear space of $W$ (i.e., over the fields $b_{tt'}$ and ${\bar b}_{tt'}$): = e\^[-S\_J(W)]{} \[DW\] , where $J_f[W]=\exp[-S_J(W)]$ is the Jacobian depending on the function $f(W)$ in (\[parameter-general-2\]).
The integration measure $[DQ]$ may be defined from the invariant metric on the space of $Q$ matrices: (dQ,dQ) -dQ dQ . Similarly, the integration measure $[DW]=[Db\, D{\bar b}]$ may be defined with the metric (dW,dW) - dW dW = 2 db d[|b]{} . The Jacobian $\exp[-S_J(W)]$ is the square root of the determinant of the metric $(dQ,dQ)$ relative to $(dW,dW)$. Further in the Appendix, we compute explicitly the determinant action $S_J(W)$ for any given parameterization $f(W)$ and find those parameterizations which have the unit Jacobian ($S_J(W)=0$).
The function $f(W)$ satisfies the nonlinear constraint: f(W) f(-W)=1 . \[W-constraint\] This constraint is solved by the condition that $\ln f(W)$ is an odd function of $W$.
The function $f(W)$ must be regular at $W=0$, with the Taylor expansion f(W)=1+c\_1 W + c\_2 W\^2 + …\[appendix-f-expansion\] Without loss of generality, we can normalize $W$ by the condition $c_1=1$. Then necessarily $c_2=1/2$. Other coefficients have a certain freedom. For example, one can chose $c_{2n+1}$ arbitrarily, then $c_{2n+2}$ are uniquely determined by the constraint (\[W-constraint\]). We naturally define $c_0=1$.
The quadratic form $(dQ,dQ)$ may now, using the cyclic trace property, be written as (dQ,dQ)= - df(W) df(-W) = \_[n\_1,n\_2]{} \_[n\_1,n\_2]{} (W\^[n\_1]{} dW W\^[n\_2]{} dW) with symmetric coefficients $\alpha_{n_1,n_2}=\alpha_{n_2,n_1}$. Using the tensor-product notation, we can write this as (dQ,dQ)=(dWdW, [**R**]{}), \[metric-tensor-product\] where = \_[n\_1,n\_2]{} \_[n\_1,n\_2]{} W\^[n\_1]{} W\^[n\_2]{} and $( \cdot , \cdot )$ in the right-hand side of (\[metric-tensor-product\]) is understood as $(A \otimes B, C \otimes D) = \Tr (ACBD)$. The Jacobian equals $(\det {\bf R})^{1/2}$ and can be generated as the “Jacobian action” $S_J(W)$: S\_J = - .
The final step of the derivation is to express ${\bf R}$ in terms of $f$. Since ${\bf R}$ contains only commuting matrices $W$, it is convenient to replace ${\bf R}$ by the function $R(x,y)$ defined as R(x,y) = \_[n\_1,n\_2]{} \_[n\_1,n\_2]{} x\^[n\_1]{} y\^[n\_2]{} , so that = R(W ,[**1**]{} W) . By inspection, \_[n\_1,n\_2]{}=\_[k\_1=0]{}\^[n\_1]{} \_[k\_2=0]{}\^[n\_2]{} (-1)\^[k\_1+k\_2]{} c\_[k\_1+k\_2+1]{} c\_[n\_1+n\_2-k\_1-k\_2+1]{} . In terms of generating functions $f(x)$ and $R(x,y)$, this can be written as R(x,y)= - = \^2 .
Now we can write down the explicit formula for $S_J(W)$ in terms of $f(W)$. For a given $f(W)$, we may expand = \_[n\_1,n\_2]{} \_[n\_1,n\_2]{} x\^[n\_1]{} y\^[n\_2]{} , \[beta-definition\] and then S\_J(W)= - \_[n\_1,n\_2]{} \_[n\_1,n\_2]{} (W\^[n\_1]{}) (W\^[n\_2]{}) . \[Jacobian-action\]
Potentially useful parameterizations
------------------------------------
For doing perturbative calculation in the sigma-model one has to chose a certain parameterization. We want to mention four possibilities:
- [*Square-root-even*]{} parameterization (in this parameterization, $f(W)$ does not contain odd powers of $W$ higher than one): f(W) = +W . \[param-sqrt-even\] This parameterization is widely used in literature since perturbative expansion in this parameterization directly corresponds [@HikamiBox] to the standard diagrammatic technique with cross averaging over disorder [@diagrammatics]. The Jacobian in the square-root-even parameterization in not equal to 1.
- [*Exponential*]{} parameterization: f(W) = (W) . This form was used by Efetov to construct his famous parameterization [@Efetov1983] of the integration manifold for the zero-dimensional supersymmetric sigma-model, that opened a way to exact evaluation of the two-level correlation function. The Jacobian in the exponential parameterization is not equal to 1.
- [*Rational*]{} parameterization: f(W) = . \[rational\] This parameterization was suggested by Efetov [@Efetov1983] in studying the supersymmetric sigma-model. The rational parameterization is frequently used for its Jacobian is equal to 1 [@Efetov1983; @Efetov-book].
- [*Square-root-odd*]{} parameterization (in this parameterization, $f(W)$ does not contain even powers of $W$ higher than two): f(W) = 1+ + W = (+ )\^2 . \[param-sqrt-odd\] This parameterization was used in Refs. for the study of the level statistics of random Hamiltonians within the replica formalism. The square-root-odd parameterization is known to have Jacobian equal one. In addition to that, the choice of the parameterization (\[param-sqrt-odd\]) renders the action of the zero-dimensional sigma-model $S[Q]\propto\Tr\Lambda Q$ to be Gaussian in $W$.
Below we shall prove that the class of parameterizations with unit Jacobian (e.g., $S_J(W)=0$) consists of a one-parametric family (\[unit-Jacobian-parameter\]), with the [*rational*]{} and [*square-root-odd*]{} parameterization being particular cases.
Parameterizations with unit Jacobian
------------------------------------
Now we shall solve the problem of finding all parameterizations $f(W)$ leading to the trivial Jacobian $S_J(W)=0$. Some powers of $W$ in the Jacobian action (\[Jacobian-action\]) give zero traces: $\Tr(W^0)=0$ due to the causality and $\Tr(W^n)=0$ for all odd $n$. Therefore we look for functions $f(x)$ producing in (\[beta-definition\]) only nonzero $\beta_{n_1,n_2}$ with one of the indices zero or odd.
Thus, a parameterization with unit Jacobian must satisfy the condition = F\_1(x) + F\_1(y) + F\_2(x,y) \[unit-J-cond\] with antisymmetric $F_2(x,y)=F_2(-x,-y)=-F_2(-x,y)=-F_2(x,-y)$. By setting $x=0$ or $y=0$ we obtain F\_1(x) = . Finally, by symmetrizing the left-hand side of Eq. (\[unit-J-cond\]), we get rid of $F_2(x,y)$ and obtain a closed equation on $f(x)$: = . The latter equation may be simply solved as f(x)+f(-x) = 2 + with an arbitrary constant $\lambda$. For $f(x)$ this gives f(x) = . \[unit-Jacobian-parameter\] Thus the parameterization with the unit Jacobian form a one-parameter family. Out of the four examples of parameterizations mentioned above, two belong to this family: at $\lambda=1$, the expression (\[unit-Jacobian-parameter\]) gives the [*rational*]{} parameterization, and at $\lambda=0$ it gives the [*square-root-odd*]{} parameterization. The [*square-root-even*]{} and [*exponential*]{} parameterizations do not belong to this family and have non-trivial Jacobian contributions $S_J(W)$.
The main calculation of the paper (cancellation of higher-order vertices) is done in the rational parameterization. We could not obtain similar results in other parameterizations.
Cancelation of higher-order vertices in rational parameterization {#A:rational}
=================================================================
Combinatoric coefficients and diagrammatic rules
------------------------------------------------
The goal of this section is to show that all internal vertices of order higher than four cancel in the rational parameterization.
=0.5
In this Appendix, to simplify the figures, we shall pictorially denote the diffusons by single lines (with arrows), instead of double lines as in Figs. \[fig:diagram-elements\] and \[fig:diagram-examples\]. The order of the arrows at any vertex remains important: it represents the order of $b$-operators in the corresponding product $\Tr (b{\bar b}b \dots {\bar b})$. In particular, arrows going to/from any vertex alternate between incoming and outgoing directions. Only vertices of even order are allowed. The diagram from Fig. \[fig:diagram-examples\]c is again shown in Fig. \[fig:appendix-1\]a in the new notation. The vertices may contain differentiations represented by a cross (Fig. \[fig:appendix-1\]b) and $(\vp_i-\vp_j)^2$ terms (Fig. \[fig:appendix-1\]c). When we transform a diffuson according to the equation of motion (\[eq-motion\]), there appear terms with the diffusons replaced by the $\delta$-function. Such a replacement will be denoted further as the double-crossed diffuson, Fig. \[fig:appendix-1\]d.
Before turning to canceling vertices, we calculate the numerical prefactors at each diagram. For simplicity, we perform now the calculation only for diagrams representing corrections to the pro-diffuson (\[D-star-propagator\]) defined as the $\langle b{\bar b}\rangle$ propagator, with one incoming and one outgoing diffusons. Let $N_D$ denote the number of diffusons, $N_V$ the number of vertices, and $2n_i$ be the vertex valencies. Then $N_D=1+\sum n_i$ and the number of loops in the diagram is $L=N_D-N_V-1$.
Every diffuson in the diagram brings in an additional factor of $(2/\pi)$. In the rational parameterization (\[parameter-rational\]), collecting the coefficients of the vertices $S_{\rm E}^{(\ge 4)}$ \[Eq. (\[E-vertex\])\] and $S_{\rm kin}^{(\ge 4)}$ \[Eqs. (\[kin-vertex\]) and (\[a-rational\])\] (and taking into account the $n_i$-fold symmetry of vertices (\[kin-vertex\])), the total numerical prefactor in front of the diagram for the pro-diffuson ${\cal D}^*=(\pi/2){\langle b\bar b\rangle}$ \[Eq. (\[D-star-propagator\])\] becomes $(-1)^{N_D-1}(2\pi)^{-L}$. With this prefactor, each vertex coming from $S_{\rm E}^{(\ge 4)}$ enters with the coefficient 1, and each vertex originating from $S_{\rm kin}^{(\ge 4)}$ enters with the coefficient $\alpha a_{ij}$, where a\_[ij]{} = (-1)\^[i-j+1]{} . \[a-ij\]
Finally, by rescaling the time in the units of $\alpha^{-1/3}$, the diagram acquires the overall factor $\alpha^{-L/3}$ ($\alpha^{1/3}$ for every vertex and $\alpha^{-(N_D-1)/3}$ from $N_D-1$ independent time integrations). Now we may factor out the common coefficient for all the diagrams of a given order $L$, and the remaining coefficients equal plus or minus one. Note that within a given order $L$, the relative sign alternates between diagrams with different number of vertices.
=0.5
The diagrammatic rules now may be formulated as follows.
1. First we draw all the diagrams of a given order $L$ allowed by the no-loop rule (see Section \[section-diagrams\]). The sign of the diagram is $(-1)^{N_D-1}$.
2. In every vertex, we put the sum of all differentiations on outgoing diffusons ($S_{\rm E}$-generated) and the full graph of $a_{ij}(\varphi_i-\varphi_j)^2$ ($S_{\rm kin}$-generated), see Fig. \[fig:appendix-1\]e.
3. We expand every differentiation according to the diffuson equation of motion (Fig. \[fig:appendix-1\]f).
4. During this differentiation, some of the diffusons are replaced by $\delta$ functions, which leads to merging or splitting some vertices (discussed in details below). Since every vertex has at most one differentiation on outgoing diffusons, the connected graph of “contracted” diffusons contain no more than one loop. We further distinguish two possibilities:
- the connected graph of “contracted” diffusons is a tree (contains no loops), Fig. \[fig:appendix-2\]a. In this case, all the vertices of the graph are contracted to a single vertex (“root” of the tree; point $A$ in Fig. \[fig:appendix-2\]a). As the result of summing such diagrams, the coefficients $a_{ij}$ at the “root” vertex get modified.
- the connected graph of “contracted” diffusons contains one loop, Fig. \[fig:appendix-2\]b. Then upon contraction, we obtain two disconnected vertices without any $(\varphi_i-\varphi_j)^2$ factors (vertices of this type would also be generated by the Jacobian action as described in Appendix \[A:parameterizations\], if the Jacobian were not equal to one).
In the subsequent subsections we sum separately those two series of diagrams and show that in the case (i) vertices of order higher than four cancel, and in the case (ii) all such loop-contractions cancel each other.
=0.4
=0.6
=0.6
Cancellations in tree-like contractions
---------------------------------------
Consider now the case when the connected graph of contracted diffusons is a tree (Fig. \[fig:appendix-2\]a). This tree has one distinguished vertex (denote it $A$) which does not contain differentiation. All the $(\vp_i-\vp_j)^2$ terms of the resulting “contracted” vertex come from the vertex $A$. Now we represent the contracted vertex as a $2n$-polygon, and the possible graph configurations leading to this vertex as dividing this polygons by several diagonals into smaller polygons (Fig. \[fig:appendix-3\]). Each such diagonal (a “separator”) should be thought of as a “wall” which represents a diffuson to be contracted. We “paint” such a wall from the side of the $b$-operator (the side of differentiation). The “outside” walls of the big $2n$-polygon are also originally painted in the alternating order (Fig. \[fig:appendix-3\]). With this representation, the different contractible graphs generating a given vertex are in one-to-one correspondence with “admissible” partitions of $2n$-polygons into smaller “rooms”. A partition is called “admissible” if after painting all the walls in alternating order, no room has more than one painted separator. Obviously, with this rule, exactly one room of the partition has [*no*]{} painted separator. This room is called the “distinguished” room of the partition. As an illustration, in Fig. \[fig:appendix-4a\]a we show all the admissible partitions of the 8-wall room. From our discussion of the diagram rules and prefactors, each partition contributes to the contracted vertex the “full graph of the distinguished room” minus “painted walls of the distinguished room” with the relative sign of the parity number of rooms in the partition. Here the “full graph” means the full graph with the sign $\sum (-1)^{i-j+1}(\vp_i-\vp_j)^2$, and the painted walls are subtracted when expanding the corresponding differentiations (see fig. \[fig:appendix-4a\]b for an example).
Before we compute the total vertex as a sum of all room partitions, we define a combinatoric quantity $p(n)$. Consider a $2n$-wall room with external walls painted in the alternating order, and one of the painted walls is “special” (say, it has a door). Let $p(n)$ denote the “algebraic” number of all admissible partitions of the $2n$-room such that the distinguished room contains special wall. By the “algebraic” number we mean that every partition comes with plus or minus sign depending on the parity of the rooms in the partition (and the trivial partition with only one room comes with plus sign). For example, $p(2)=1$, $p(3)=-1$, $p(4)=1$ (Fig. \[fig:appendix-4b\]), etc.
=0.7
One can prove by induction that $p(n)=(-1)^n$. Indeed, all partitions may be classified by “external” blocks (shaded in fig. \[fig:appendix-5\]a) adjacent to the distinguished room. Each of those external blocks contributes $-p(n_i)$ possibilities to subdivide it, where $n_i$ is the number of [*unpainted*]{} external walls of the block. All possible configurations of external blocks may be enumerated by dividing the total of $n$ unpainted external walls into non-intersecting sequences of length $n_i$. Thus, p(n)=(-p(n\_i)) , \[pn-computation\] where the sum is taken over all possibilities to choose a set of non-intersecting blocks of adjacent elements from a sequence of total $n$ elements (Fig. \[fig:appendix-5\]b). The only element excluded from the sum is the set where the whole sequence belongs to a single block. In Fig. \[fig:appendix-5\]c we illustrate the possible choices of such block sets for $n=4$. Now we perform the induction step assuming that for all $n_i<n$, $p(n_i)=(-1)^{n_i}$. Then the same sum (\[pn-computation\]) may be rewritten as $p(n)=\sum (-1)^{n_{\rm adj}}$, where $n_{\rm adj}$ is the number of adjacent pairs belonging to the same block. Finally, we note that we may parameterize block sets by independently choosing two neighboring elements either to belong or not to belong to the same block. This easily gives us the total sum. If all the sets were allowed, the sum would be zero. The only set which is not allowed is the one with all neighboring elements (total $n-1$ pairs) belonging to the same block. This gives $p(n)=(-1)^n$, which completes the proof.
=0.5
Now we do the final step of the calculation: we calculate the renormalized coefficient $\tilde a_{ij}$ with which the combination $(\varphi_i-\varphi_j)^2$ enters the contracted vertex. First, such terms with $i$ adjacent to $j$ through a [*painted*]{} external wall never enter (they are always subtracted by expanding differentiations). So we may assume that $i$ and $j$ belong to different painted external walls (let us call those walls $w_i$ and $w_j$ for future reference).
A room partition gives a contribution to $\tilde a_{ij}$ if and only if $w_i$ and $w_j$ both belong to the distinguished room. Similarly to our discussion above, for each room partition, we may consider external “blocks” complementary to the distinguished room. Now the walls $w_i$ and $w_j$ divide the unpainted walls into two sequences (define their lengths as $m^{(1)}$ and $m^{(2)}$), and the contributions from these two sequences factorize: a\_[ij]{}=(-1)\^[i-j-1]{} \_[(1)]{} (-p(m\^[(1)]{}\_i)) \_[(2)]{} (-p(m\^[(2)]{}\_i)) (see Fig. \[fig:appendix-6\]). The only difference from the previous calculation is that now the whole sequence $m^{(1)}$ (or $m^{(2)}$) is [*allowed*]{} to be placed in one block. Therefore the sum gives zero unless [*both*]{} $m^{(1)}$ and $m^{(2)}$ equal one. But this is possible only if the total number of unpainted external walls is $n=m^{(1)}+m^{(2)}=2$, i.e. for the four-valent vertex. This completes the proof that all higher-order vertices cancel in the process of tree-like contractions.
The only surviving four-leg vertex is shown in Fig. \[fig:appendix-7\], with \_[i<j]{} a\_[ij]{} (\_i-\_j)\^2 = (\_2-\_3)\^2 + (\_1-\_4)\^2 - (\_1-\_3)\^2 - (\_2-\_4)\^2 = 2 (\_1-\_2) (\_3-\_4) . The diagrammatic series with such a vertex may be considered as a perturbative expansion of the field theory (\[S-star\]), (\[b-action-rational\]).
=0.15
Cancellation of loop-like contractions
--------------------------------------
In this section we prove cancellation of vertices obtained in the process of contracting a connected graph of diffusons containing a closed loop (Fig. \[fig:appendix-2\]b). After such a contraction, two vertices appear without any $(\phi_i-\phi_j)^2$ terms. This is the same type of vertices as those appearing from the Jacobian. We consider the rational parameterization which contains no Jacobian vertices, so the only vertices of this type are those produced by the closed-loop diffuson contractions.
=0.6
Here we make two important remarks. First, there is no internal structure of those vertices, and the only parameter of the vertex is its numerical prefactor. Second, it is sufficient to prove cancellation of the “loop part” of the diagram (Fig. \[fig:appendix-8\]) containing only the closed loop of contracted diffusons, without any “branches” which do not depend on the internal structure of the loop part.
So we consider loops of length $l\ge 1$ of contracted diffusons. If we go along the loop, the legs on the left and on the right sides of our path are collected into two new “contracted” vertices. We fix the valencies of these vertices to be $2n$ and $2m$ and count the total combinatoric factor $p(n,m)$ as the number of ways to obtain these vertices from loops of different lengths. Taking into account the sign alternating with the total vertex number, we obtain p(n,m)=\_[l=1]{}\^ (-1)\^l p\_l(n,m) , where $p_l(n,m)$ is the positive integer number counting the number of ways to create the two vertices of valencies $2n$ and $2m$ from the loop of length $l$. Note that from the left-right hand rule both $n$ and $m$ must be positive. Also, since the two-leg vertices are not allowed in the original action, $p_l(n,m)=0$ for $l>n+m$. An example of calculating $p(1,2)$ is shown in Fig. \[fig:appendix-9\].
=0.7
The combinatoric problem of computing $p_l(n,m)$ may be related to computing another combinatoric quantity $q_l(n,m)$ defined as [*the number of ways to distribute $n$ identical black and $m$ identical white balls among $l$ different boxes so that no box remains empty*]{}: p\_l(n,m)=n m q\_l(n,m) (here factors $n$ and $m$ come from different ways to circularly re-label external legs, and $1/l$ factors reflects equivalence of diagrams obtained by the circular permutation of vertices). The example of calculating $q_l(1,2)$ is shown in Fig. \[fig:appendix-10\].
=0.4
Finally, we use the generating-function method to compute $q_l(n,m)$. Consider the series f\_l(x,y)=(x+y+x\^2+xy+y\^2+…)\^l , where the expression in the brackets contains all terms $x^{n_x} y^{n_y}$ except for $n_x=n_y=0$. Then the coefficient at $x^n y^m$ in the Taylor expansion of $f(x,y)$ equals $q_l(n,m)$. Now a straightforward calculation gives: f\_l(x,y)=(-1)\^l , and then \_[l=1]{}\^ (-1)\^l f\_l(x,y) = (1-x) + (1-y) \[generating\_pl\] (note the factorization!). Now the coefficient $p(n,m)$ equals $n m$ times the coefficient at $x^n y^m$ in the Taylor expansion of the expression (\[generating\_pl\]). The latter however equals 0 unless one of the two numbers $n$ or $m$ is zero. This finishes the proof: all vertices generated from loop contractions cancel.
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---
abstract: 'In this work, we simulate the dynamics of varying density quasi-2D spin-ensembles in solid state systems, focusing on the Nitrogen-Vacancy (NV) centers in diamond. We consider the effects of various control sequences on the averaged dynamics of large ensembles of spins, under a realistic “spin-bath" environment. We reveal that spin-locking is efficient for decoupling spins initialized along the driving axis, both from coherent dipolar interactions, and from the external spin-bath environment, when the driving is two orders of magnitude stronger than the relevant coupling energies. Since the application of standard pulsed dynamical decoupling (DD) sequences leads to strong decoupling from the environment, while other specialized pulse sequences can decouple coherent dipolar interactions, such sequences can be used to identify the dominant interaction type. Moreover, a proper combination of pulsed decoupling sequences could lead to the suppression of both interaction types, allowing additional spin manipulations. Finally, we consider the effect of finite-width pulses on these control protocols, and identify improved decoupling efficiency with increased pulse duration, resulting from the interplay of dephasing and coherent dynamics.'
author:
- 'D. Farfurnik'
- 'Y. Horowicz'
- 'N. Bar-Gill'
bibliography:
- 'nvbibliography.bib'
title: 'Identifying and decoupling many-body interactions in spin ensembles in diamond'
---
I. introduction
===============
The studies of many-body dynamics of spin ensembles in the solid state have attracted significant attention. In particular, ensembles of negatively-charged nitrogen-vacancy (NV) centers were recently used for the demonstration of many-body depolarization dynamics [@Choi2017; @Kucsko2017]. A proper application of microwave (MW) control sequences on the spin ensemble of interest may lead to a variety of applications in quantum sensing and quantum information processing. For example, a proper modification of traditional sequences in NMR such as WAHUHA [@Waugh1968; @Rhim1973] and MREV [@Mansfield1971], could result in engineered Hamiltonians for the interacting spins within the ensemble. Such engineered Hamiltonians could pave the way toward the creation of non-classical states, e.g. spin squeezed-states, which could eventually lead to magnetic sensing beyond the shot-noise limit [@Rey2008; @Cappellaro2009]. Studies of such control sequences on interacting spin ensembles, in the presence of noise and for various parameter regimes, are still lacking.
####
In this work, we use a cluster-based simulation method to estimate the dynamics of a quasi-2D ensemble consisting of more than 400 spins under various control sequences. The simulations consider a realistic environment, consisting of a “spin-bath" noise, representing the typical scenario of dense ensembles of NV centers in diamond. Our analysis identifies techniques for controllably decoupling specific interactions, and clarifies the effects of finite pulse durations originating from the interplay of different interaction sources.
####
The electronic structure of the negatively-charged NV center has a spin-triplet ground state, in which the $m_s=\pm 1$ sublevels experience a zero-field splitting ($\sim 2.87$ GHz) from the $m_s = 0$ sublevel due to spin-spin interactions. Application of an external static magnetic field along the NV symmetry axis Zeeman shifts the $m_s=\pm 1$ levels. If MW driving is applied at a frequency $\omega_0$ resonant with the $m_s= 0\leftrightarrow+1$ transition (for example), this spin manifold can be treated as a two-level subspace of the spin-triplet [@Taylor2008].
II. Interactions and decoupling
===============================
The effective Hamiltonian representing dipolar interactions within such an ensemble is [@Kucsko2017]: $$H_{dipolar}=\sum_{ij} w_{ij}[\vec{\sigma}_i \cdot \vec{\sigma}_j-2\sigma^z_i\sigma^z_j],
\label{eq:dipolar}$$ with $\omega_{i,j}=\frac{J_0}{r_{ij}^3}$, where $\sigma^{x,y,z}_{i,j}$ are the Pauli spin operators, $r_{ij}^3$ is the distance between spins $i$ and $j$, and $J_0\approx 52$ MHz$\cdot$nm$^3$. The environment of NV centers corresponds to a “spin-bath", typically dominated by $^{13}$C nuclear and nitrogen paramagnetic spin impurities, which create time-varying random local magnetic fields in the crystal [@Pham2012; @Acosta2009; @deSousa2009; @BarGill2012]. In typical samples, the interaction between such a bath and the spins of interest is modeled as an Ornstein-Uhlenbeck (OU) process $B(t)$ [@deSousa2009; @BarGill2012], with a typical correlation time $\tau_c$ of the bath, and coupling strength between the bath and the spins of interest $b$. The effect of the resulting fluctuating fields on the spin ensemble (and assuming that these fields are uniform within the measurement volume) can be modeled as the interaction Hamiltonian $$H_{bath}=B(t)\sum_{i}\sigma^z_{i}.
\label{eq:bath}$$ The amplitude of the random bath noise as a function of time can be simulated by an exact algorithm [@Gillespie1996]: $$B(t+\Delta t)=B(t)e^{-\Delta t/\tau_c}+\frac{b}{2} n \sqrt{1-e^{-2\Delta t/\tau_c}},
\label{eq:OU}$$ where $n$ is a randomly generated number from a normal distribution with mean 0 and standard deviation of 1.
####
For many decades, dynamical decoupling (DD) MW sequences were used in NMR for decoupling spin interactions, and thus controlling their dynamics [@Hahn1950; @Meiboom1958; @Mansfield1971; @Rhim1973; @Gullion1990; @Khodjasteh2005; @Ryan2010; @Hirose2012]. In the case of spin ensembles, the simplest decoupling method method from the bath is to apply a continuous driving at the resonant frequency $\omega_0$ [@Hirose2012] which (in the rotating frame) takes the form $$H_{SL}=\Omega \sum_{i}\sigma^x_{i},
\label{eq:spinlock}$$ where all the spins are assumed to be driven by the same strength $\Omega$. If all spins are initialized along the driving (“x") axis, the driving overcomes the effects of frequency terms in the spin-bath with frequencies lower than $\Omega$ (“spin-locking"). This enhances the fidelity of the initial state with time up to a timescale usually referred to as $T_{1\rho}$ which, for sufficiently high $\Omega$, is typically limited by phononic interactions and experimental imperfections [@Cai2012; @Hirose2012; @Farfurnik2017cont], but may also be limited by dipolar interactions in extremely-dense ensembles [@Choi2017]. Another method for decoupling from the bath is the repetitive application of resonant $\pi$-pulses. In the simplest implementation, the Carr-Purcell-Meiboom-Gill (CPMG) sequence, all pulses are applied along the initialization (“x") axis [@Meiboom1958], while other sequences for overcoming pulse imperfections are available [@Khodjasteh2005; @Ryan2010; @Wang2012a; @Farfurnik2015]. Another useful decoupling sequence is WAHUHA [@Waugh1968; @Rhim1973; @Mansfield1971], consisting of four unequally spaced resonant $\pi/2$ pulses, which was designed to decouple collective dipolar interactions between spin-1/2 particles at times determined by the sequence length. Since the NV dipolar Hamiltonian differs from the spin-1/2 dipolar Hamiltonian by the term $-\sigma^z_i\sigma^z_j$, the WAHUHA sequence is expected to decouple the dipolar interactions only partially. In the average Hamiltonian picture, the remaining isotropic Hamiltonian $\sum \omega_{ij}\vec{\sigma}_i \cdot \vec{\sigma}_j$ conserves the total angular momentum $J^2$, providing advantages toward the creation of high-fidelity non-classical states over the scenario of spin-1/2 dipolar coupling in which the effective Hamiltonian is just 0 [@Rey2008; @Cappellaro2009] (Appendix A), highlighting the importance of the current studies of spin dynamics under these unique interactions. In this work, we simulate the effect of continuous and pulsed decoupling techniques on the dynamics of a spin ensemble under spin-bath and collective dipolar interactions with different strengths. We demonstrate procedures for distinguishing between these types of interactions, for decoupling from them, and identify the physical mechanisms underlying effects resulting from finite pulse durations.
III. Simulations
================
We use a cluster-based simulation [@Maze2008b] (Appendix B) to estimate the dynamics of a quasi-2D spin ensemble consisting of more than 400 spins, interacting by dipolar interactions according to eq. . All spins in the ensemble are initialized along the $x$ axis, and the spin polarization along the initialization axis (expectation value $\langle S_x \rangle$) is extracted as a function of time (Fig. \[fig:dipolar\]). Since simulating the evolution of the total density matrix in such a large ensemble is impractical, we follow the general idea of dividing the ensemble into independent clusters [@Maze2008b], and perform the simulation by combining the results for small clusters in the following way (Appendix B): First, we obtain a “representative probability distribution" for describing dipolar interaction strengths in a typical experimental scenario, by randomly generating the positions of 464 spins inside a circle with a surface of $\approx 4.5$ $\SI{}{\micro\meter}^2$. Afterwards, we use this distribution for evaluating the exact dynamics of small spin clusters: a few (4-10) spins are generated under the distribution, and their dynamics under the Hamiltonian is simulated exactly. We repeat this spin generation process with newly generated clusters, to take into account all possible scenarios for the proximity between spins within clusters, until the number of realizations is sufficient for the convergence of the resulting dynamics. In such a way, each realization samples the dynamics within a certain cluster consisting of a few spins, and the averaged dynamics of all realizations provides insights into the combined dynamics of the whole ensemble. Similar procedures could be done for 3D-ensembles, with the only difference being that the distribution of spin couplings should take into account their different orientations.
A. Dipolar Dynamics
-------------------
The Spin dynamics simulations of NV ensembles under the Hamiltonian are shown in Fig. \[eq:dipolar\]. Although our simulation method does not take into account collective phenomena such as extended, long-range many-body correlations, our detailed convergence analysis (Appendix B) indicates that such phenomena do not significantly affect the spin dynamics of driven systems studied in this work. The significant frequencies (4,12 and 20 times the typical interaction strength) contributing to the spin dynamics (Fig. \[fig:dipolar\]) can be predicted from an analytical expression for all-to-all equal interactions, with $25\%$ accuracy of evaluating the typical interaction strength. Additional convergence analysis emphasizes that the choice of six-spin clusters provides a converged quantitative estimate for spin dynamics even for large numbers of spins (Appendix B). As a result, and in order to optimize run times, the simulations in this work utilize clusters with six spins only. Qualitatively, the resulting decay structures are similar for different dipolar interaction strengths, with only the decay timescales changing by the same factor as the interaction strength ratios (stronger interactions lead to a faster decay). Small quantitative variations remain, though, due to the difference in the spin generation probability distributions for different spin concentrations (Appendix C) \[Fig. \[fig:dipolar\](b)\].
![(Color online) Cluster-based simulations of the dynamics of a spin ensemble, dominated by internal dipolar interactions. (a) Spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\sim 60$ Hz. Different curves represent different numbers of interacting spins taken into account in a cluster. (b) Varying typical dipolar strengths, using clusters of six spins: 60 Hz and 10 kHz typical interactions within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (464 and 9980 spins), and 1 MHz within a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins).[]{data-label="fig:dipolar"}](20180510_Fig1a_dipolar_50Hz_different_spin_nums.eps "fig:"){width="1\columnwidth"} ![(Color online) Cluster-based simulations of the dynamics of a spin ensemble, dominated by internal dipolar interactions. (a) Spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\sim 60$ Hz. Different curves represent different numbers of interacting spins taken into account in a cluster. (b) Varying typical dipolar strengths, using clusters of six spins: 60 Hz and 10 kHz typical interactions within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (464 and 9980 spins), and 1 MHz within a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins).[]{data-label="fig:dipolar"}](20180510_Fig1b_dipolar_6spins_different_strengths.eps "fig:"){width="1\columnwidth"}
B. Spin-Locking
---------------
Next, we consider the effect of spin-lock driving applied along the initialization axis, on the dynamics of the ensemble. Our simulations considering different spin concentrations, which correspond to different dipolar coupling strengths (Fig. \[fig:spinlock\]), demonstrate that without including the effect of the bath \[Fig. \[fig:spinlock\] (a),(b)\], when the driving intensity is two orders of magnitude stronger than the dipolar coupling, the dipolar interactions are fully decoupled. In particular, for a dipolar coupling of $\sim 60$ Hz, which could be achieved for NV ensembles using standard CVD procedures and TEM irradiation [@Farfurnik2015; @Farfurnik2017], even a driving as weak as 0.1 MHz results in a complete preservation of the initial state (Appendix C). These results agree with our theoretical model: consistent with the results of Fig. \[fig:spinlock\], an analytical expression for the spin dynamics within a six-spin cluster (Appendix B) correctly predicts full decoupling under spin-lock two orders of magnitude stronger than the typical dipolar strength. Moreover, for spin-locking one order of magnitude stronger than the typical interaction strength, the simulations in Fig. \[fig:spinlock\] result in the converged baseline of 0.945, in agreement with the conserved population 0.9375 predicted from this theoretical model (Appendix B).
####
In a more realistic scenario, one has to take into account the additional effects of the spin-bath environment interacting with the spin ensemble. By considering the OU model within a time $T$, the simulation of spin dynamics under such an environment with $\tau_c=5$ $\SI{}{\micro\second}$ and $b=20$ kHz using an exact algorithm according to eq. leads to a free induction decay (FID) time of $T_2^*\approx 0.5$ ms, consistent with theoretical calculations \[$T_2^*=1/(b^2\tau_c)=0.5$ ms\] [@deSousa2009]. Since the simulation of OU processes requires evolution in small time increments $\Delta t$ and averages over noise realizations, it is significantly more time-consuming, and therefore for the combined simulation under spin-bath and dipolar interactions only five realizations of the dipolar interactions were considered. Although the exact dynamics varies with the specific interactions in the generated cluster (Appendix C), the total trend remains clear: Fig. \[fig:spinlock\](c) demonstrates that by considering the evolution of a spin ensemble with a dipolar strength of 60 Hz, spin-lock driving two orders of magnitude stronger than the interactions with the bath ($5$ $\SI{}{\micro\second}$ $\leftrightarrow 0.2$ MHz) decouples both the spin-bath and dipolar interactions within the ensemble, resulting in a unity evolution on a timescale of $50$ ms. Such long coherence times can be experimentally observed at cold temperatures ($\sim 77$ K), for which the longitudinal relaxation time ($T_1$) is much longer [@Jarmola2012; @Farfurnik2015].
![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 9980 spins, under spin-lock driving at different intensities, for dipolar interactions of (a) 10 kHz (a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface) and (b) 1 MHz (a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface) within the ensemble. For a dipolar interaction of 60 Hz, the evolution was unity even for the weakest examined spin-lock intensity. (c) Effect of spin-locking on an ensemble with dipolar interactions of 60 Hz (464 spins), in a spin-bath environment with $\tau_c=5$ $\mu$s correlation time and $b=20$ kHz coupling strength to the ensemble, averaging only 5 dipolar realizations.[]{data-label="fig:spinlock"}](20180510_Fig2a_spinlock_different_strengths_dipolar_5kHz.eps "fig:"){width="0.85\columnwidth"} ![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 9980 spins, under spin-lock driving at different intensities, for dipolar interactions of (a) 10 kHz (a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface) and (b) 1 MHz (a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface) within the ensemble. For a dipolar interaction of 60 Hz, the evolution was unity even for the weakest examined spin-lock intensity. (c) Effect of spin-locking on an ensemble with dipolar interactions of 60 Hz (464 spins), in a spin-bath environment with $\tau_c=5$ $\mu$s correlation time and $b=20$ kHz coupling strength to the ensemble, averaging only 5 dipolar realizations.[]{data-label="fig:spinlock"}](20180510_Fig2b_spinlock_different_strengths_dipolar_500kHz.eps "fig:"){width="0.87\columnwidth"} ![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 9980 spins, under spin-lock driving at different intensities, for dipolar interactions of (a) 10 kHz (a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface) and (b) 1 MHz (a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface) within the ensemble. For a dipolar interaction of 60 Hz, the evolution was unity even for the weakest examined spin-lock intensity. (c) Effect of spin-locking on an ensemble with dipolar interactions of 60 Hz (464 spins), in a spin-bath environment with $\tau_c=5$ $\mu$s correlation time and $b=20$ kHz coupling strength to the ensemble, averaging only 5 dipolar realizations.[]{data-label="fig:spinlock"}](20170725_Fig2c_spin_bath_5us_10kHz_dipolar50Hz_spinlock_dfferentstrengths.eps "fig:"){width="1\columnwidth"}
C. Pulsed Decoupling
--------------------
We now consider the effects of pulsed DD sequences on the spin dynamics dominated by these interaction sources. Using such sequences with optimally chosen phases along the Bloch Sphere, arbitrary spin states of the ensemble could be preserved [@Farfurnik2015], and their modification could contribute to the engineering of unique interaction Hamiltonians, potentially leading to the creation of useful non-classical states of the spin ensemble [@Cappellaro2009]. We study the effect of the CPMG sequence, consisting of $\pi$-pulses applied along the initialization axis, the similar XY8 DD sequence, which is more robust to pulse imperfections, as well as the WAHUHA sequence, which was designed to decouple dipolar interactions of spin-$1/2$ systems (Fig. \[fig:pulsed\]). As expected, in the ideal case with no pulse imperfections, the phases of the pulses do not affect the decoupling efficiency, and the CPMG and XY8 sequences produce similar results [@Khodjasteh2005; @Ryan2010; @Wang2012a; @Farfurnik2015]. While these DD sequences dramatically improve coherence in a spin-bath dominated environment \[Fig. \[fig:pulsed\](a)\], our simulations show that they do not affect dipolar interactions within the ensemble at all \[Fig. \[fig:pulsed\](b)\]. However, by applying 100 repetitions of the WAHUHA sequence (a total of 400 pulses), dipolar interactions of 60 Hz are decoupled up to a timescale of $50$ ms, which could be observed at cold temperatures. [@Jarmola2012; @Farfurnik2015]. Similar results for other dipolar interaction strengths (Appendix C) demonstrate that the typical decay time under WAHUHA with 100 repetitions is an order of magnitude longer than the typical dipolar interaction time, and more repetitions of this sequence will result in even higher fidelities.
####
Since the WAHUHA sequence is not efficient for decoupling the ensemble from the spin-bath environment \[Fig. \[fig:pulsed\](a)\], in order to achieve full decoupling in a realistic scenario with both types of interactions, one has to combine DD pulses and WAHUHA. Figure \[fig:pulsed\](c) depicts the dynamics under a combined sequence, for which 5 WAHUHA repetitions are applied between every adjacent pair of CPMG $\pi$-pulses, compared to the application of the same amount (21,000) of CPMG or WAHUHA pulses only. In contrast to the results of Fig. \[fig:pulsed\](a)-(b), which were completely dominated by a single interaction source, in the scenario of Fig. \[fig:pulsed\](c), disregarding their different final baselines, both WAHUHA and CPMG result in comparable decay timescales, indicating that spin-bath and dipolar interactions have a nearly-equal contribution. These results with a total of 21,000 applied pulses, demonstrate that such a combination could lead to the preservation of the spin state up to $5$ ms, while the application of DD alone ($\pi$-pulses) or WAHUHA alone is no longer effective. The modification of such sequences could lead to the creation of useful non-classical states of the ensemble [@Cappellaro2009]. For a given sample, if the origin of the dominant interactions of the ensemble is not known (internal dipolar/spin-bath), the separate application of both sequences could identify this dominant term: a train of $\pi$-pulses significantly enhances the fidelity only for interactions dominated by the bath, while WAHUHA significantly enhances the fidelity only when the internal dipolar interactions dominate.
![(Color online) Cluster-based simulations of the spin dynamics of a spin ensemble, under DD and the WAHUHA sequence consisting of 1000 pulses. The dominant interactions are: (a) A spin-bath environment, correlation time $\tau_c=5$ $\mu$s and coupling strength $b=20$ kHz, and (b) dipolar interactions among the spins in the ensemble, with a typical interaction strength of 60 Hz (464 spins within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface). (c) Combined interaction of spin-bath and dipolar couplings with the same parameter is in (a)-(b), utilizing a total number of $\sim 21000$ pulses: total $21000$ CPMG, WAHUHA, and a combination of 5 WAHUHA within $1000$ CPMG pulses, averaging only 5 dipolar realizations.[]{data-label="fig:pulsed"}](20180510_Fig3a_1000DDpulses_spinbath5us_10kHz.eps "fig:"){width="0.85\columnwidth"} ![(Color online) Cluster-based simulations of the spin dynamics of a spin ensemble, under DD and the WAHUHA sequence consisting of 1000 pulses. The dominant interactions are: (a) A spin-bath environment, correlation time $\tau_c=5$ $\mu$s and coupling strength $b=20$ kHz, and (b) dipolar interactions among the spins in the ensemble, with a typical interaction strength of 60 Hz (464 spins within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface). (c) Combined interaction of spin-bath and dipolar couplings with the same parameter is in (a)-(b), utilizing a total number of $\sim 21000$ pulses: total $21000$ CPMG, WAHUHA, and a combination of 5 WAHUHA within $1000$ CPMG pulses, averaging only 5 dipolar realizations.[]{data-label="fig:pulsed"}](20180510_Fig3b_1000DDpulses_dipolar50Hz.eps "fig:"){width="0.87\columnwidth"} ![(Color online) Cluster-based simulations of the spin dynamics of a spin ensemble, under DD and the WAHUHA sequence consisting of 1000 pulses. The dominant interactions are: (a) A spin-bath environment, correlation time $\tau_c=5$ $\mu$s and coupling strength $b=20$ kHz, and (b) dipolar interactions among the spins in the ensemble, with a typical interaction strength of 60 Hz (464 spins within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface). (c) Combined interaction of spin-bath and dipolar couplings with the same parameter is in (a)-(b), utilizing a total number of $\sim 21000$ pulses: total $21000$ CPMG, WAHUHA, and a combination of 5 WAHUHA within $1000$ CPMG pulses, averaging only 5 dipolar realizations.[]{data-label="fig:pulsed"}](20180224_Fig3c_CPMGwithWAHUHA.eps "fig:"){width="1\columnwidth"}
D. Finite Width Pulses
----------------------
Finally, we consider the effect of $\pi$-pulses with realistic finite durations on the spin dynamics of the ensemble utilizing the CPMG and XY8 protocols (Fig. \[fig:duration\]).
![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under a realistic spin-bath environment and dipolar coupling (same parameters as before), utilizing 4000 realistic CPMG pulses with various finite durations. Since weaker driving results in shorter free evolution times for dephasing between pulses, the spin state is better-initialized along the driving axis, which leads to a more efficient decoupling of the MW driving from dipolar interactions and longer decay times.[]{data-label="fig:duration"}](20180226_Fig4_CPMG4000_different_durations_50Hz_SpinBath_5_20.eps "fig:"){width="1\columnwidth"} ![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under a realistic spin-bath environment and dipolar coupling (same parameters as before), utilizing 4000 realistic CPMG pulses with various finite durations. Since weaker driving results in shorter free evolution times for dephasing between pulses, the spin state is better-initialized along the driving axis, which leads to a more efficient decoupling of the MW driving from dipolar interactions and longer decay times.[]{data-label="fig:duration"}](ShortAndLongPulses.png "fig:"){width="1\columnwidth"}
When considering spin-bath only, we find that the spin dynamics are not affected by varying pulse durations. However, the decay time significantly grows with the pulse durations in realistic scenarios for which spin-bath and internal dipolar interactions exist. We explain this outcome in the following way: sufficiently strong MW driving decouples from the bath completely, but it is efficient for decoupling dipolar interactions only when the state is initialized along the driving axis. While applying a DD sequence, the spin state is no longer initialized along the same axis, both due to coherent dipolar dynamics and dephasing caused by the interactions with the bath, within the free evolution times between pulses. For longer pulses, the free evolution times are shorter, leading to a better preservation of the state initialized along the driving axis before the beginning of the next pulse. This results in an enhanced decoupling over the case of short pulses, in which dephasing within the free evolution times is more significant. This effect is more significant for the CPMG sequence (Fig. \[fig:duration\]), in which the driving is always along the initialization axis, than for the XY8 sequence (Appendix C), in which the driving is along this axis only half of the time.
IV. Conclusions
================
To summarize, by simulating the dynamics of an ensemble of 464 spins in a spin-bath environment using a cluster approach and an exact OU algorithm, we showed that a strong enough spin-lock driving (two orders of magnitude stronger than the interactions) could decouple the related interactions for states initialized along the driving axis. The separate application of the CPMG and WAHUHA sequences could identify the dominant interaction source, while their combined application could decouple both types of interactions, preserving the spin state of the ensemble. Additional modification of such sequences could lead to the generation of engineered interaction Hamiltonians, creating non-classical states of the ensemble, and contributing to quantum sensing and quantum information prcessing [@Cappellaro2009]. Finally, the duration of the applied pulses may affect the decoupling efficiency, due to imperfect initialization within the free evolution times, as well as the interplay between the spin-bath, internal dipolar interactions and the MW driving within the durations of the pulses.
acknowledgements
================
We thank Yonatan Hovav, Nati Aharon, Connor Hart, Erik Bauch, Jennifer M. Schloss, Matthew Turner, Emma Rosenfeld, Ronald L. Walsworth, Joonhee Choi, Hengyun Zhou, Mo Chen and Paola Cappellaro for the fruitful discussions. This work has been supported in part by the Minerva ARCHES award, the CIFAR-Azrieli global scholars program, the Israel Science Foundation (grant No. 750/14), the Ministry of Science and Technology, Israel, and the CAMBR fellowship for Nanoscience and Nanotechnology.
Appendix A: Two-level subspace of spin one dipolar interactions
===============================================================
####
When MW driving is applied resonant with the $m_s= 0\leftrightarrow+1$ transition (for example), the spin manifold can be treated as a two-level subspace of the spin-triplet [@Taylor2008]. The effective Hamiltonian representing an ensemble of such spins interacting with each other by dipolar interactions is given by . Note that expression slightly differs from the interaction Hamiltonian among spin-1/2 particles, for which the coefficient of $\sigma^z_i\sigma^z_j$ is $-3$. Although this difference may seem incremental, it can lead to observations of unique physical phenomena when the Hamiltonian is modified by pulsed MW control. For example, the WAHUHA sequence [@Waugh1968; @Rhim1973; @Mansfield1971], consisting of four $\frac{\pi}{2}$-pulses with unequal timings, was designed to decouple spin-1/2 dipolar interactions.
{width="2\columnwidth"}
Indeed, in the first order average Hamiltonian picture [@Waugh1968], the effective Hamiltonian of such interactions yields zero. In the NV-NV dipolar interaction Hamiltonian, however, the extra $\sigma^z_i\sigma^z_j$ term results in the effective Hamiltonian $\frac{\vec{\sigma}_i \cdot \vec{\sigma}_j}{3}$ \[Fig. \[fig:WAHUHA\](a)\].
Such an additional term can contribute to the creation of robust non-classical states of the spin ensemble [@Rey2008; @Cappellaro2009]. For example, if one of the WAHUHA pulses is slightly shifted by a small time increment $\epsilon \tau$ \[Fig. \[fig:WAHUHA\](b)\], the resulting interaction Hamiltonian yields $\epsilon \frac{\sigma^x_i\sigma^x_j-\sigma^y_i\sigma^y_j}{2}$ which, under certain conditions, can generate spin-squeezed states $45^0$ along the x-y plane of the Bloch Sphere [@Cappellaro2009]. Such states, not yet demonstrated in the solid state, could eventually break the standard quantum limit (SQL), leading to novel directions in quantum sensing. However, in order for these generated non-classical states to remain robust, the total angular momentum number $J^2$ has to be conserved. Such a condition is fulfilled when the squeezing-generating Hamiltonian acts as a small perturbation to another $J^2$-conserving Hamiltonian [@Rey2008; @Cappellaro2009]. In our case, the effective Hamiltonian $\frac{\vec{\sigma}_i \cdot \vec{\sigma}_j}{3}$, appearing in the unique case of a two-level subspace of spin one systems such as NV ensembles, acts as such an $J^2$-conserving terms. This highlights the potential of these systems for the generation of robust non-classical states, over well-studied spin-1/2 systems. It is therefore interesting to estimate the dynamics of such systems, which is the subject of this work.
Appendix B: Simulation Methods
==============================
In this Appendix, we present our simulation methods and justify their validity for evaluating the exact dipolar dynamics of systems up to 10 spins, as well as the expected dynamics for large (hundreds or thousands) numbers of spins. Due to computational cost limitations, the dynamics of large spin ensembles cannot be simulated explicitly from the interaction Hamiltonian, our method is cluster-based [@Maze2008b]. Our analysis below, exhibiting convergence of the dynamics with the use clusters consisting of six spins, emphasizes the effectiveness of the simulation method in estimating the dynamics of large ensembles. The Appendix is organized as follows: In section (i), we consider a simplified (non-realistic) model of all-to-all equal couplings to obtain an analytical expression for the spin dynamics, which can be solved analytically to provide intuition for the simulated dynamics later on. Section (ii) describes our method of cluster simulations and its obtained results for the dynamics for different cluster sizes in the realistic case of random spin positions. Using Fourier and convergence analysis on driven and non-driven dynamics, the convergence of six-cluster spins is justified in section (iii).
(i) Analytical all-to-all equal interactions model
--------------------------------------------------
####
In the simplest (and non-realistic) scenario, we consider an ensemble consisting of n spins with all-to-all dipolar interactions with equal strengths $\omega_{ij} \equiv \omega_0$. All spins are initialized to the same (“x") axis in the Bloch sphere, $$|+\rangle=\frac{1}{\sqrt{2}^n} \sum^n_{k=0} |\uparrow^{(k)}\downarrow^{(n-k)}\rangle,
\label{eq:upx}$$ where $|\uparrow^{(k)}\downarrow^{(n-k)}\rangle$ denotes a sum over all ${n}\choose{k}$ combinations in which $k$ spins point up and $n-k$ spins point down. Since the left term in eq. is isotropic, the dynamics is dominated by the Hamiltonian $H_0=-2 w_{0}\sum_{i,j}\sigma^z_i\sigma^z_j$. For a given value of $k$ spins pointing up, there are $k(n-k)$ different combinations in which spins $i$, $j$ have opposite signs such that $\sigma^z_i\sigma^z_j\rightarrow -1$, and ${n\choose2}-k(n-k)$ combinations in which these spins share the same sign, such that $\sigma^z_i\sigma^z_j\rightarrow 1$. Therefore, the application of the time evolution operator $U=e^{-iH_0 t}$ on the initial state yields $$|\psi(t)\rangle=\frac{1}{\sqrt{2}^n} \sum^n_{k=0} e^{2i\omega_0 t \left[{{n}\choose{2}}-2k(n-k)\right]}|\uparrow^{(k)}\downarrow^{(n-k)}\rangle.
\label{eq:wavefunc}$$ Let us now calculate the spin polarization along the initialization axis, $\langle S_x \rangle = \frac{1}{n}\sum_k \langle S_{xk} \rangle = \frac{1}{n}\left(\langle \sigma_x\otimes \mathbb{I}\otimes \cdots \otimes\mathbb{I}\rangle+\cdots +\langle\mathbb{I}\otimes \mathbb{I}\otimes \cdots \otimes \sigma_x\rangle\right)$. Since all spins are equivalent, it is sufficient to calculate the expectation value of one spin operator only, $\langle S_x \rangle=\langle S_{x1} \rangle=\langle \left(|\uparrow\rangle \langle \downarrow|+|\downarrow \rangle \langle \uparrow| \right)\otimes \mathbb{I}\otimes \cdots \otimes\mathbb{I}\rangle$. After applying $|\psi(t)\rangle$ from both sides, this expectation value yields $$\langle S_x\rangle=\frac{1}{2^{n-1}} \left(e^{4i\omega_0 t (n-1)}+\sum^{n-1}_{k=1} {{n-1}\choose{k-1}} e^{4i\omega_0 t (2k-n-1)}\right),
\label{eq:Sx1}$$ where the first term represents the contribution from the states in which all spins point up / down, the sum represents the states in which $k$ spins point up (given that the first spin points up, ${n-1}\choose{k-1}$ is the number of combinations for which $k-1$ of the other spins point up), and a factor of 2 arises from the symmetry between up and down. Expression is always real, and can be rewritten for even numbers of spins in the form: $$\begin{split}
\langle S_x\rangle=\frac{1}{2^{n-2}} \Bigg[ & \cos\left(4\omega_0 t (n-1)\right)\\ &+\sum^{\ceil{(n-1)/2}}_{k=2} {{n-1}\choose{k-1}} cos\left(4\omega_0 t (2k-n-1)\right)\Bigg],
\label{eq:Sx2}
\end{split}$$ with an additional DC term for odd numbers of spins, $\frac{1}{2^{n-2}}{{n-1}\choose{(n-1)/2}}$. Using the analytical expressions , the spin dynamics can be easily calculated in the special case of equal couplings up to thousands of spins, in contrast to our ability to perform exact simulations on the Hamiltonian , which is limited to 10 spins. The resulting oscillation frequencies and their populations, identical to those obtained by a direct simulation with equal couplings, are shown in Fig. \[fig:equal\].
![(Color online) Oscillation frequencies and their populations for different numbers of spins in the limit of couplings with equal strength $\omega_0$, calculated from eq. . Larger numbers of spins introduce additional higher frequencies, and a significant drop in the populations of all terms.[]{data-label="fig:equal"}](20180220_EqualCouplings_frequencies_and_intensities_Sx.eps){width="1\columnwidth"}
For growing numbers of spins, oscillating terms with higher frequencies are added. However, due to the factor $1/2^{n-2}$, the relative population of each oscillating component drops significantly with the number of spins. As will be demonstrated in the following section, this picture does not represent the realistic scenario, in which interactions vary due to the different positions of the spins, thus a cluster approach is necessary.
(ii) Cluster simulations using the explicit Hamiltonian
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####
In the previous section we considered a scenario in which all dipolar interactions have equal strengths. This scenario is non-realistic since varying distances between spins correspond to different strengths $\omega_{ij}$. As a result, in order to get a more relevant estimate of the dynamics arising from Hamiltonian , interaction strengths should be generated from a distribution, which correctly represents the experimental measurement scenario. In our following more-realistic simulations, for an ensemble with a given spin concentration, the proper amount of spins is generated inside the measurement surface / volume, and the interaction of each spin with its nearest neigbor is taken into account. This process is repeated for many realizations until convergence, to form a histogram representing the interaction distribution. Such a typical distribution, considering a quasi-2D NV ensemble with a spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$, which can be achieved by TEM irradiation [@Farfurnik2017], within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (total 464 spins) is shown in Fig. \[fig:dist\].
![(Color online) Probability distribution generated for the dominant interactions in a quasi-2D NV ensemble with a spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, obtained by averaging over many realizations of randomly-generated 464 spins. The resulting average interaction strength is $\omega_0\approx 60$ Hz.[]{data-label="fig:dist"}](dist.eps){width="1\columnwidth"}
####
In order to gain some intuition for the simulated results, Fourier transforms of the dipolar dynamics considering different cluser sizes \[Fig. \[fig:dipolar\](a)\] are shown in Fig. (\[fig:dipdiffn\]).
![(Color online) Fourier transform of the dynamics of a spin ensemble consisting of 464 spins with a typical interaction strength $\omega_0\approx 60$ Hz, under the dipolar Hamiltonian . The interactions within clusters consisting of different numbers of spins are generated from the probability distribution in Fig. \[fig:dist\].[]{data-label="fig:dipdiffn"}](20180215_Fig1a_dipolar_50Hz_different_spin_nums_Fourier.eps){width="1\columnwidth"}
The oscillating frequencies presented in these dynamics are compatible with those obtained from the equal-coupling model \[eq. and Fig. \[fig:equal\]\], with the two most significant terms centered at $4\omega_0$ and $12\omega_0$. Note that the uncertainty in the main frequency $4\omega_0$ is $\pm \omega_0$, which enables to extract the typical interaction strength up to an accuracy of $25\%$. However, in contrast with the equal coupling calculation, the varying spin positions randomly generated from the measurement surface result in the broadening of the frequency peaks, as well as in the significant drop in the contribution of high-frequency terms. For example, the population ratio between the two slowest frequencies is $\sim 2.5$ for the explicit simulations and only $\sim 1.5$ for the equal-coupling model considering 10 spins, emphasizing that the contribution of high frequency terms is greatly diminished in the realistic case. Therefore, we conclude that in spite of the correct prediction of the oscillation frequencies, the equal-coupling model is not suitable for simulating the realistic dynamics for large spin ensembles, which indeed requires the use of a cluster approach.
(iii) Determining the cluster size
----------------------------------
####
The sufficient amount of spins in a cluster for estimating spin ensemble dynamics can be estimated by considering the population ratio between different frequency terms, and justified by convergence analysis on driven spin dynamics. The ratios between the three main frequencies in the explicit dynamics (Fig. \[fig:dipdiffn\]) are $\sim 1:2.5:6$ which, even in the simplified model of equal couplings, is mostly compatible with the choice of six-spin clusters, exhibiting the ratios $\sim 1:2:10$. Although in general, the equal coupling model is not suitable for describing the realistic scenario, our choice of six-spin clusters exhibits the same main oscillation frequencies dominating the dynamics. However, such a comparison is rather rough, and the choice of cluster size has to be justified quantitatively. Indeed, we further justify this chosen number of spins in a cluster by performing convergence analysis on the dynamics of a driven system: the spin dynamics under Hamiltonian is simulated together with a continuous driving along the initialization axis (spin-lock), which is an order of magnitude stronger than the typical interaction strength, $\Omega\approx 12 \omega_0$ \[Fig. \[fig:SL\]\].
![(Color online) Cluster-approximation simulations of the dynamics of a spin ensemble consisting of 464 spins with a typical interaction strength $\omega_0\approx 60$ Hz , under a spin-lock driving with an intensity of $\Omega\approx 12\omega_0$. The interactions within clusters consisting of different numbers of spins are generated from the probability distribution in Fig. \[fig:dist\].[]{data-label="fig:SL"}](20180207_spinlock750Hz_dipolarfromtypicaldist_Sx.eps){width="1\columnwidth"}
Theoretically, such a driving is expected to cancel out effects of components oscillating slower than the driving strength [@Hirose2012]. In the case of Fig. \[fig:dist\], a polarization of $\approx 0.95$ is achieved by driving the system with $\Omega\approx 12\omega_0$, which is expected to result in the cancelation of the two first frequency terms $4\omega_0$ and $12\omega_0$. These results charactarize the natural physical dynamics under the dipolar interaction, which will remain the same for any external control applied, making our choice of spins within clusters relevant for simulating any sequence. The difference between this converged value and the steady-state polarization obtained by considering clusters of six spins is only about 1 percent, indicating that such a cluster size is suitable for simulating large ensembles. Furthermore, for such a cluster size, even the equal-coupling model (Fig. \[fig:equal\]) provides a close prediction for the degree of polarization: the population of the first two frequencies sums up to $0.9375$, very close to the converged degree of polarization, indicating that such a choice for the cluster size incorporates the effects of randomness caused by varying dipolar interaction strengths.
Appendix C: Supplemental simulation results
===========================================
(i) Dipolar interactions for different spin concentrations
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####
The simulations in this work were performed for several different experimental scenarios (concentrations and number of spins): spins of an ensemble with a defined concentration were generated randomly within the a defined measurement surface. As can be seen in Fig. \[fig:dipolar\](b), as well as Fig. \[fig:diffdist\](b) and Fig. \[fig:WAHUHAdiff\] below, the qualitative dynamics and decoupling results for diffrerent spin concentrations are similar, with the timescales changing according to the coupling strength ratios. Small Quantitative differences \[Fig. \[fig:diffdist\](b)\] correspond to slightly different structures from the interaction distributions \[Fig. \[fig:diffdist\](a)\].
![(Color online) (a) Interaction strength distributions and (b) cluster-based simulations of the spin dynamics of spin ensembles with different properties: Spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\omega_0 \approx 60$ Hz, $\omega_0\approx 10$ kHz typical interaction within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins), and $\omega_0 \approx 1$ MHz within a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins). The horizontal axes are normalized to $\omega_0$ to emphasize similarities between the results, with small differences due to the slightly-different strength distributions.[]{data-label="fig:diffdist"}](20180226_Normalized_Distributions.eps "fig:"){width="1\columnwidth"} ![(Color online) (a) Interaction strength distributions and (b) cluster-based simulations of the spin dynamics of spin ensembles with different properties: Spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\omega_0 \approx 60$ Hz, $\omega_0\approx 10$ kHz typical interaction within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins), and $\omega_0 \approx 1$ MHz within a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins). The horizontal axes are normalized to $\omega_0$ to emphasize similarities between the results, with small differences due to the slightly-different strength distributions.[]{data-label="fig:diffdist"}](20180226_Normalized_Dipolars.eps "fig:"){width="1\columnwidth"}
(ii) Spin-locking on dipolar interaction of $60$ Hz
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####
Figure \[fig:SL60\] demonstrates the dynamics of an NV ensemble with a typical dipolar NV-NV interaction strength of 60 Hz under spin-lock driving with the strength of 0.1 MHz. In consistence with the results given in the main text for other intensities, driving two orders of magnitude stronger than the typical dipolar interactions decouples the interactions completely and results in unity evolution for over more than 10 ms.
![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under spin-lock driving at with the strength of 0.1 MHz, for dipolar NV-NV interactions with a typical interaction strength of 60 Hz.[]{data-label="fig:SL60"}](20170725_spinlock_0_1MHz_dipolar_50Hz.eps){width="1\columnwidth"}
(iii) WAHUHA on ensembles with different dipolar coupling strengths
-------------------------------------------------------------------
####
Figure \[fig:WAHUHAdiff\] demonstrates the dynamics of NV ensembles with various dipolar NV-NV interaction strengths under the application of 100 repetitions of the WAHUHA sequence. The resulting decay times are one order of magnitude longer than the typical interaction time. Furthermore, a comparison to the spin dynamics without external control \[Fig. \[fig:dipolar\](b)\] shows that this amount of repetitions enhances the decay time by two orders of magnitude over the case without any control. Qualitatively, By utilizing a constant amount of repetitions, the decay structure exhibited in different samples is similar, with the timescales changing according to the coupling strength ratios, while small quantitative differences correspond to slightly-different structures of the interaction distributions.
![Cluster-based simulations of the spin dynamics under the application of the WAHUHA sequence with 100 repetitions on a spin ensemble dominated by NV-NV dipolar interactions at various concentrations: Spin concentration of $10^{10}$ $\SI{}{\centi\meter}^{-2}$ within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface, representing 464 spins with a typical interaction strength of $\omega_0 \approx 60$ Hz, $\omega_0\approx 10$ kHz typical interaction within a $\approx 4.5$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins), and $\omega_0 \approx 1$ MHz within a $\approx 0.46$ $\SI{}{\micro\meter}^2$ measurement surface (9980 spins).[]{data-label="fig:WAHUHAdiff"}](20180226_WAHUHA100_dipolar_6spins_different_strengths.eps){width="1\columnwidth"}
(iv) XY8 with finite pulse durations
------------------------------------
####
As described in section III D and emphasized in Fig. \[fig:duration\], the decoupling efficiency from simultaneous spin-bath and internal dipolar interactions within an NV ensemble might increase with as a function of the pulse durations of the applied DD sequence. Here we show (Fig. \[fig:XY8duration\]) that this effect is also expressed by utlizing the XY8 sequence. The effect is much more significant for CPMG (Fig. \[fig:duration\]), where all pulses are applied along the initialization axis, than for XY8 (Fig. \[fig:XY8duration\]), where only half of the pulses are applied along the initialization axis.
![(Color online) Cluster-based simulations of the spin dynamics of an ensemble consisting of 464 spins, under a realistic spin-bath environment and dipolar coupling (same parameters in Fig. \[fig:duration\]), utilizing 500 repetitions of the XY8 sequence (total of 4000 pulses) with various finite durations. Since weaker driving results in shorter free evolution times for dephasing between pulses, the spin state is better-initialized along the driving axis, which leads to a more efficient decoupling of the MW driving from dipolar interactions and longer decay times.[]{data-label="fig:XY8duration"}](20180224_DipolarAndBath_XY8_DifferentPulseDurations.eps){width="1\columnwidth"}
(v) Variations between different realizations in combined spin-bath and dipolar simulations
-------------------------------------------------------------------------------------------
####
Since the simulation of OU process requires evolution in small time increments $\Delta t$ and averages over noise realizations, it is very time-consuming, thus for the combined simulations under spin-bath and dipolar interactions \[Fig. \[fig:spinlock\](c) and Fig. \[fig:pulsed\](c)\] only five realizations of the dipolar interactions were considered. The explicit results for individual realizations leading to the results of Fig. 3 \[fig:pulsed\](c) are presented here in Fig. \[fig:realizations\]. For the application of WAHUHA \[Fig. \[fig:realizations\] (a)\], internal dipolar interactions are fully decoupled up to the simulated timescales, thus different realizations result in the same dynamics dominated solely by interactions with the bath. Since the CPMG sequence does not decouple the effects of internal dipolar interactions, different realizations of different randomly-generated spins in the cluster correspond to different decay scales \[Fig. \[fig:realizations\] (b)\]. Qualitatively, however, the total decay trend expressed here and in the main text is very clear even after considering only several realizations. More importantly, by applying a control scheme that simultaneously decouples interactions with the bath and internal dipolar interactions (such as by the combined application of WAHUHA and CPMG expressed in Fig. \[fig:realizations\] (c)\], the resulting dynamics is similar for different realizations.
.[]{data-label="fig:realizations"}](20180213_DipolarDistributions50Hz_withbath_WAHUHA5250_6spins.eps "fig:"){width="1\columnwidth"} .[]{data-label="fig:realizations"}](20180213_DipolarDistributions50Hz_withbath_CPMG21000_6spins.eps "fig:"){width="1\columnwidth"} .[]{data-label="fig:realizations"}](20180213_DipolarDistributions50Hz_withbath_WAHUHAandCPMG_6spins.eps "fig:"){width="1\columnwidth"}
|
---
abstract: 'In a purely relational theory there exists a tension between the relational character of the theory and the existence of quantities like distance and duration. We review this issue in the context of the Leibniz-Clarke correspondence. We then address this conflict by showing that a purely relational definition of length and time can be given, provided the dynamics of the theory is known. We further show that in such a setting it is natural to expect Lorentz transformations to describe the mapping between different observers. We then comment on how these insights can be used to make progress in the search for a theory of quantum gravity.'
author:
- Olaf Dreyer
title: Relational Physics and Quantum Space
---
Introduction
============
In our current search for a quantum theory of gravity it is widely believed that the final theory should be purely relational. A long-standing thorny issue for a relational theory is the question of how quantities like distances and duration can be defined or emerge in a purely relational manner.
This tension first became apparent in the correspondence between Clarke and Leibniz[@alex]. We will review that part of the correspondence that is concerned with the nature of space and time and let it be our introduction to the problem of recovering the notions of distance and duration in a relational theory. Leibniz stated his position in [@alex Third paper, §4]:
> “…I hold space to be something merely relative, as time is; …I hold it to be an order of coexistences, as time is an order of succesions.”
Using his principle of the *identity of indiscernibles* Leibniz then goes on to demonstrate that an absolute view of space and time is untenable and that the relative view is the only sensible one. Clarke, not at all convinced, offers the following refutation of Leibniz’s position [@alex Third reply, §4]:
> “If space was nothing but the order of things coexisting; it would follow, that if God should remove in a straight line the whole world entire, with any swiftness whatsoever; yet it would still always continue in the same place: and that nothing would receive any shock upon the most sudden stopping of that motion. And if time was nothing but the order of succession of created things; it would follow, that if God had created the world millions of ages sooner than he did, yet it would not have been created at all sooner. Further: space and time are quantities; which situations and order are not.”
To a modern mind this argument given by Clarke looks rather vacuous and Leibniz’s reply could be given by a physicist trained today [@alex Fourth paper, §13]:
> “To say that God can cause the whole universe to move forward in a straight line, or in any other line, without making otherwise any alteration in it; is another chimerical supposition. For, two states indiscernible from each other, are the same state; and consequently. ’tis a change without any change. …”
Clarke does not acknowledge this argument. Instead he concludes that Leibniz’s position is disproved [@alex Fourth reply, §16 and §17]:
> “That space and time are not the mere order of things, but real quantities has been proven above, and no answer yet given to those proofs. And till an answer be given to those proofs, this learned author’s assertion is a contradiction.”
Having held to his position for four papers Leibniz now commits two grave mistakes within the space of two pages. The first one is the admission that there is ’…an absolute true motion …’ [@alex Fifth paper, §53]
> “…However, I grant there is a difference between an absolute true motion of a body, and a mere relative change of its situation with respect to another body. …”
If that was not enough Leibniz goes on in the next paragraph to say that distances are fundamental [@alex Fifth paper, §54]
> “…As for the objection that space and time are quantities, or rather things endowed with quantity; and that situation and order are not so: I answer, that order also has its quantity; there is in it, that which goes before and that which follows; there is distance or interval. …”
All Clarke has to do now is to collect his trophy. With the magnanimity of the victor he points out [@alex Fifth reply, §53]
> “Whether this learned author’s being forced here to acknowledge the difference between absolute real motion and relative motion, does not necessarily infer that space is really a quite different thing from the situation or order of bodies; I leave to the judgement of those who shall be pleased to compare what this learned writer here alleges, with what Sir Isaac Newton has said in the Principia, …”
Somewhat more triumphantly he continues in the next paragraph [@alex Fifth reply, §54]:
> “I had alleged that time and space were <span style="font-variant:small-caps;">quantities</span>, which situation and order are not. To this, it is replied; that *order has its quantity; there is that which goes before, and that which follows; there is distance and interval*. I answer: going before, and following, constitutes situation or order: but the distance, interval, or quantity of time or space, wherein one thing follows another, is entirely a distinct thing from the situation or order, and does not constitute any quantity of situation or order: the situation or order may be the same, when the quantity of time or space intervening is very different.”
Thus ends the correspondence between Leibniz and Clarke with a clear defeat for the relativists. If one looks at the arguments that have been presented it is not so much a defeat but more of a self-destruction of Leibniz.
In this article we will propose to resolve the tension between quantity and relation using a simple model from solid state physics. In section \[sec:relation\] we use a background-independent formulation of the Heisenberg spin chain as a simple model of the universe. In section \[sec:poincare\] we show that observers *inside* the system can use the excitations of the model (without reference to a lattice spacing) to define distances purely relationally. We also show that the maps between observers are naturally given by Poincaré transformations. This leads us to interpret this model as a “quantum Minkowski space”. We briefly discuss consequences of this argument for the problem of quantum gravity, as well as certain observations about the precise relationship of our model to Minkowski space in the Conclusions.
A Relational Solid State Model {#sec:relation}
==============================
How else could the Leibniz-Clarke correspondence have gone? How could the tension between quantity and relation have been resolved? How is one to obtain the notion of distance in a purely relational manner?
The first thing to realize is that the tension between quantity and relation can not be resolved by relying on kinematics alone. Given a dynamical degree of freedom like a traveling mode one can use it to define the notion of distance by defining how much it travels in a certain amount of time, i.e. by defining its velocity. This is just how we define the unit of length today, namely by setting the speed of light (see [@nist]). What is needed is a distinctive set of traveling degrees of freedom or excitations, that can be used to define the notion of distance in this way.
To be concrete we will look at a particular model, the Heisenberg spin chain and its higher dimensional generalizations. Its Hamiltonian is given by $$\label{eqn:hamil}
{\mbox{\boldmath $H$}}= \sum_{(i j)} {\mbox{\boldmath $\sigma$}}_i\cdot{\mbox{\boldmath $\sigma$}}_j,$$ where ${\mbox{\boldmath $\sigma$}}= (\sigma_x, \sigma_y, \sigma_z)$, and the $\sigma$’s are the Pauli matrices. For nearest neighbor interactions the lowest lying excitations are of the form $$\label{eqn:ansatz}
{\vert \psi\rangle} = \sum_{n} a_n {\vert 0\cdots 1\cdots 0\rangle},$$ where the $0$’s and $1$’s denote eigenvectors of $\sigma_z$ and the $1$ occurs at the $n$-th position. The $a_n$’s take the form $$\label{eqn:as}
a_n = e^{i\delta_k n},$$ and $$\delta_k = 2\pi \frac{k}{N},\ \ \ k=1,\ldots,N,$$ where $N$ is the number of lattice sites. Solving the Schrödinger equation gives the eigenvalues of ${\mbox{\boldmath $H$}}$ as a function of $k$: $$\label{eqn:energyk}
E = -NA + 4 A \left( 1 - \cos \left(2\pi \frac{k}{N}\right)\right).$$ We will denote the corresponding eigenvectors by $\vert k\rangle$.
This is a set of traveling degrees of freedom of the model. In the next section we will use these to give a purely relational definition of distance. Note that the excitations we have introduced are perfectly well-defined *without the introduction of a lattice spacing*.
Poincaré Transformations {#sec:poincare}
========================
Using the excitations of the model described above we can now proceed to define quantities like distance in a completely relational manner. This can be done by picking one particular excitation and assigning a speed to it. A length is then defined to be the amount the excitation has travelled in a certain time interval.
A distinctive excitation in our model is given by the fastest wave packets. These excitations are of the form $$\sum_k f(k)\,\vert k\rangle,$$ where $f(k)$ is peaked around that value of $k$ for which $dE/dk$ is maximal. These wave packets are distinguished by the fact that no other excitation can overtake them. All observers, which can also be thought of as excitations of the system, will agree on that, independent of the way they themselves move. This characterization is thus completely relational.
Since observers in the spin model have only the above excitations at their disposal to explore their world there is no way for them to tell whether they are moving or resting with respect to the lattice. It is thus consistent and natural for all of them to assign the *same* speed to these excitations.
![ A view of the system that is not available to observers confined inside the system. The observers **** and ****, here represented by the large Gaussian excitations, have no way of telling what their motion is with respect to the lattice. This is why it is consistent for both observers to assign the *same* speed to the excitation. There exists a map $\phi$ between the two coordinate systems given by the mapping of physical events onto each other. This map $\phi$ will have the property that it maps the fastest excitations onto fastest excitations. We find then that this map $\phi$ must be a Poincaré transformation. \[fig:relativ\]](relativ.eps){height="6cm"}
What will be the map between the coordinate systems of two observers? In the limit that the spin system looks smooth to the observers we can answer this question. The map can be constructed by mapping physical events onto each other. This map will in particular map the fastest excitations in one coordinate system onto the fastest excitations in the other system. Since these excitations have the same speed in both systems the map can only be a Poincaré transformation (see figure \[fig:relativ\]). One thus obtains a “quantum Minkowski space”.
Conclusion
==========
As it was pointed out by Clarke in his correspondence with Leibniz, in a completely relational theory there exists a tension between quantities and relations. We have seen here how this tension can be resolved provided one has access to excitations that can be used to define the notion of distance by defining the speed of these excitations. A consequence of this definition is that the natural mapping between two observers is given by a Poincaré transformation.
To define a notion of distance in a relational way it was necessary to have access to the dynamics of the theory. A purely kinematic approach is not sufficient. It is here where some of the candidate theories of quantum gravity, like Loop Quantum Gravity[@thie] or Causal Set Theory[@causal], face their greatest problems. The arguments presented here suggest that the dynamics of the theory is required to make progress on important issues like the question of the semi-classical limit of the theory.
We conclude by remarking that the model presented above does deviate from usual Minkowski space in two ways. In order for us to find Poincaré transformations we assumed that the excitations involved are all well separated from each other. If this is not the case an operational definition of distance and duration can not be given anymore. Another deviation occurs when the observers have access to excitations with very high values of $k$ (these excitations are not to be confused with the fastest excitations used above). In this case the observers would notice the spin lattice and would find measurable deviations from Poincaré invariance.
[MM]{} H. G. Alexander, *The Leibniz-Clarke Correspondence*, Manchester University Press (1956). B. N. Taylor (Ed.), *The International System of Units (SI)*, National Institute of Standards and Technology, Special Publication 330 (2001). T. Thiemann, *Introduction to modern canonical quantum general relativity*, gr-qc/0110034. L. Bombelli, J. H. Lee, D. Meyer and R. Sorkin, *Space-Time As A Causal Set*, Phys. Rev. Lett. [**59**]{}, 521 (1987).
|
---
author:
- 'R.I. Uklein[^1]'
- 'D.I. Makarov'
- 'S. Roychowdhury'
title: 'Rendez-vous of dwarfs'
---
The main parameters of the group {#intro}
================================
During the work on the catalog of groups of galaxies Makarov et al. (2010) [@makarov] have been found interesting groups of dwarf galaxies. One of these groups we have studied in detail at the 6-m telescope and GMRT.
The SDSS image of the group is presented in Fig. \[g33image\].
The parameters of individual galaxies are indicated in Table \[tab:1\]. The columns contain the following data: the number in the group, the name of the galaxy (for SDSS galaxies the coordinates in the name are omitted), the coordinates at the epoch J2000.0, mean radial velocity of the group relative to the centroid of the group, total magnitude in the SDSS *g*-band, absolute *g*-band magnitude, and luminosity in units of $10^8 L_\odot$.
[ cccccc ]{} No.& RaDec(J2000) & $V_{LG}$, & g, & $M_{abs}$, & $L_g$,\
& & km/s & mag & mag & $10^8L_\odot$\
1 & 124412.1+621019 & 2682 & 17.78 & -15.15 & 1.28\
2 & 124418.0+621007 & 2650 & 18.08 & -14.82 & 0.95\
3 & 124423.2+620306 & 2660 & 17.53 & -15.38 & 1.58\
4 & 124412.0+621451 & 2614 & 15.82 & -17.05 & 7.38\
5 & 124359.9+621960 & 2602 & 16.16 & -16.70 & 5.35\
The projection distance between the outermost galaxies, determining the size of the group amounts to 190 kpc. The mean velocity of the group, weighted over the velocity errors, amounts to 2675 km/s. Total luminosity of all galaxies in the SDSS *g*-band is approximately $1.6\cdot 10^9 L_\odot$. The velocity dispersion for this group is approximately 20 km/s. Total mass, calculated via the projected mass estimator method, is equal to $1.5\cdot 10^{11} M_\odot$ and the $M/L$ ratio is about 110 in solar units in *g*-band.
The groups of our sample have small dispersions of velocity and projected distances between galaxies. It makes them look like the assosiations of dwarf galaxies from Tully et al. (2006) [@tully].
Observations {#Astr_telesc}
============
BTA
---
We observed 1st and 2nd galaxies on the 6-m BTA telescope on August, 2009 with the SCORPIO focal reducer [@scorpio]. We used the VPHG550G grism and a long slit with the size of $1^{\prime\prime} \times 360^{\prime\prime}$. The wavelength range of the grism is $3100-7300$ Å. The long slit was simultaneously centered on the brightest regions of both galaxies. After the standard processing of two-dimensional spectra in the ESO-MIDAS package we extracted 3 one-dimensional spectra. The spectra are shown in Fig. \[spectra\_fig\].
In general the spectra are emission dominated but the spectrum of 1st galaxy has appreciable absorptions in the hydrogen lines. Apart from emissions in the hydrogen lines, the spectra contain the emission lines of \[OIII\], \[SII\] and others. We determined the metallicity (oxygen abundance) with a reasonable accuracy for only one spectrum (Fig. \[spectra\_fig\], middle panel). The value of 12+log(O/H) is 7.2 $\pm{}$ 0.1. To find the metallicity we use a direct method based on \[OIII\] $\lambda 4363/(\lambda 4959+\lambda 5007)$ ratio [@izotov].
Hence, it is a rather low-metallicity galaxy (a little higher metallicity than that of I Zw 18). Spectrum of 1st galaxy shows sign of low metallicity too. An important feature of the 2nd galaxy is a difference between the radial velocities of parts of this galaxy. It is amounts to 150 km/s.
GMRT
----
Fig. \[gmrt\_fig\] present GMRT data obtained on April 2010.
Three clouds of HI were observed in the field, one covering galaxies numbered 1 and 2, one covering galaxy no 3, and another covering galaxy no 4. Basic parameters for these three clouds presented in Table \[tab:2\].
[ccc]{} Galaxy & Velocity spread, km/s & HI mass,$10^8 M_\odot$\
1, 2 & 2460-2543 & 2.15\
3 & 2361-2523 & 4.81\
4 & 2369-2488 & 3.74\
Conclusions
===========
The mass-to-luminosity ratio for the group is more than 100 $M_\odot/L_\odot$. It is indicates the presence of significant amount of the dark matter.
The objects 1 and 2 are on stage of merging of two dwarf galaxies. It is indicated by structure of distribution of HI cloud and high difference in the velocity in the clumps in the galaxy.
The dwarfs in the group form the chain of the galaxies. We see the group in the process of the formation. Low metallicity of the gas in the galaxies support idea of “youth” of the galaxies. In addition we have found another candidate for very low-metallicity galaxy. The chain shape of galaxies indicates that the group had not yet virialized. Thus we see young emerging group of dwarf galaxies.
This work was supported by the RFBR grants 08–02–00627 and 10–02–09373.
D. I. Makarov et al. Proc. of the conference “A Universe of Dwarf Galaxies”, (2010).
R. Brent Tully et al. AJ **132**, (2006) 729–748.
V. L. Afanasiev, A. V. Moiseev. AstL, **31**, (2005) 194–204.
Y. I. Izotov et al. A&A **448**, (2006) 955–970.
[^1]:
|
---
abstract: 'We investigate the ground-state phase diagram of the one-dimensional half-filled Hubbard model with an alternating potential—a model for the charge-transfer organic materials and the ferroelectric perovskites. We numerically determine the global phase diagram of this model using the level-crossing and the phenomenological renormalization-group methods based on the exact diagonalization calculations. Our results support the mechanism of the double phase transitions between Mott and a band insulators pointed out by Fabrizio, Gogolin, and Nersesyan \[Phys. Rev. Lett. [**83**]{}, 2014 (1999)\]: We confirm the existence of the spontaneously dimerized phase as an intermediate state. Further we provide numerical evidences to check the criticalities on the phase boundaries. Especially, we perform the finite-size-scaling analysis of the excitation gap to show the two-dimensional Ising transition in the charge part. On the other hand, we confirm that the dimerized phase survives in the strong-coupling limit, which is one of the resultants of competition between the ionicity and correlation effects.'
address: |
$^1$Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397 Japan\
$^2$Department of Applied Physics, Faculty of Science, Tokyo University of Science, Tokyo 162-8601 Japan
author:
- 'Hiromi Otsuka$^{1}$ and Masaaki Nakamura$^{2}$'
title: ' Ground-state phase diagram of the one-dimensional Hubbard model with an alternating potential '
---
INTRODUCTION {#sec_INTRO}
============
The electronic and/or magnetic properties of the low-dimensional interacting electrons have attracted great interest in researches of materials, such as the quasi one-dimensional (1D) organic compounds and the two-dimensional (2D) high-$T_{\rm c}$ cuprates, where a variety of generalized Hubbard-type models have been introduced.[@Kana63] For the 1D case, a concept of the Tomonaga-Luttinger liquid (TLL) has been widely accepted and intensively used not only for the descriptions on the low-energy and long-distance behaviors of the critical systems,[@Tomo50; @Hald81; @Kawa90] but also for the prediction of its instabilities to, for instance, various types of density-wave phases observed in the models.[@Voit92]
The 1D Hubbard model with an alternating potential (also called the ionic Hubbard model) is one of the models for the $\pi$-electron charge-transfer organic materials, such as TTF-Chloranil,[@Naga86] and/or the ferroelectric transition metal oxides as BaTiO$_3$.[@Egam93; @Tsuc99] It is defined by the Hamiltonian $$\begin{aligned}
H=-t\sum_{j,s}
\left(
c^\dagger_{j,s}c^{}_{j+1,s}+{\rm H.c.}
\right)
&+&\sum_j U{n_{j,\uparrow}}{n_{j,\downarrow}} \nonumber\\
&+&\sum_{j}\Delta (-1)^j n_{j},
\label{eq_HAMIL}
\end{aligned}$$ where $c_{j,s}$ annihilates an $s$-spin electron ($s=\uparrow$ or $\downarrow$) on the $j$th site and the number operator ${n_{j,s}}=c^\dagger_{j,s}c^{}_{j,s}$ and $n_j=n_{j,\uparrow}+n_{j,\downarrow}$. While $t$ and $U$ terms stand for the electron transfer among sites and the Coulomb repulsion on the same site, respectively, the $\Delta$ term represents an energy difference between the donor and acceptor molecules (or between the cation and oxygen atoms), and it introduces ionicity effects into the correlated electron systems (we set $t=1$ in the following discussion).
The understandings on the model have been accumulated in the literature, where the theoretical investigations including numerical calculations have been performed mainly at the half filling: Nagaosa and Takimoto calculated the magnetic and charge-transfer gaps as functions of $\Delta$ ($U$ fixed) by using the quantum Monte Carlo (QMC) simulation.[@Naga86] Resta and Sorella, using the exact-diagonalization calculations of finite size systems, reported, for instance, the divergence of the average dynamical charge.[@Rest95] By applying the renormalization-group (RG) method to the bosonized Hamiltonian, Tsuchiizu and Suzumura estimated a boundary line between the Mott insulator (MI) and a band insulator (BI) phases in the weak-coupling regions.[@Tsuc99] On the other hand, Fabrizio, Gogolin, and Nersesyan (FGN) predicted an existence of the “spontaneously dimerized insulator” (SDI) phase between them.[@Fabr99; @Fabr00] After their proposal, various numerical calculation methods have been so far applied to confirm it: Wilkens and Martin performed the QMC simulations to evaluate, e.g., the bond order parameter, and reported the transition between the BI and SDI phases and stated an absence of MI phase for $\Delta>0$.[@Wilk01] By the combined use of the method of topological transitions (jumps in charge and spin Berry phases)[@Rest95; @Resta98; @Alig99; @NV02] and the method of crossing excitation levels, Torio [*et al.*]{} provided a global ground-state phase diagram, which is in accord with the FGN scenario.[@Tori01] And an existence of the SDI phase for all $U>0$ regions was first exhibited there. The density matrix renormalization group (DMRG) calculations [@Taka01; @Lou_03; @Zhan03; @Kamp03] have been performed by several groups. For instance, Zhang [*et al.*]{} provided the data on the structure factors of relevant order parameters in the weak- and intermediate-coupling region, which supports an existence of intermediate SDI phase between the BI and MI phases.[@Zhan03] On one hand, Kampf [*et al.*]{} estimated the excitation gaps up to 512-site system and found the boundary of the BI phase while the existence of the second boundary was not resolved.[@Kamp03] Therefore, some controversy as well as points of agreement exists in these recent investigations.
In this paper using the standard numerical techniques, we shall provide both the global structure of the ground-state phase diagram and the evidences to show the criticalities of the massless spin and charge parts. For this purpose, it is worthy of noting that the FGN scenario consists of two types of instabilities commonly observed in the TLL, i.e., the transition with the SU(2)-symmetric Gaussian criticality in the spin part, and that with the 2D-Ising criticality in the charge part (see Sec. \[sec\_GROUN\]). Furthermore, these types of phase transitions have been numerically treated by the level-crossing (LC) method, and the phenomenological renormalization-group (PRG) method.[@Room80] The LC method has been applied to the frustrated XXZ chain,[@Okam92; @Nomu94] and also used in the research of higher-$S$ spin chains,[@Kita97] spin ladders,[@Ladder] and 1D correlated electron systems.[@NakaTJ; @NakaEX] The advantage of using the LC method is not restricted to its accuracy in estimating the continuous phase transition points including the Berezinskii-Kosterlitz-Thouless type one; it also provides a means to check their criticalities (see Sec. \[sec\_NUMER\]).[@Nomu94] Both of these are important in order to settle the controversy mentioned above, and, in fact, the precise estimation of the spin-gap transition point of the $S=\frac12$ $J_1$-$J_2$ chain was first given by the LC method,[@Okam92] while numerical investigations including the DMRG work were performed. On the other hand, the PRG method is also a reliable numerical approach to determine second-order phase transition point, especially for the 2D-Ising transition where the LC method is not available. Analysis based on the PRG method for the 2D-Ising transition is successful in the spin systems.[@Kita97] Furthermore, one of the authors treated the 2D-Ising transition in the $S=\frac12$ $J_1$-$J_2$ model under a staggered magnetic field, where the critical phenomena in the vicinity of the phase boundary line were argued.[@Otsu02] Therefore, based on these recent developments, we shall perform the numerical calculations; especially, to our knowledge, this is the first time that the PRG method successfully applied to the 2D-Ising transition observed in one part of the two-component systems like the interacting electrons.
The organization of this paper is as follows. In Sec. \[sec\_GROUN\], we shall briefly refer to the effective theory based on the bosonized Hamiltonian and order parameters of expected density-wave phases, and mention the FGN scenario. In Sec. \[sec\_NUMER\], we explain procedures of the numerical calculation to determine transition lines, where connections between the methods and instabilities of the TLL systems will be explained. After that, we shall give a ground-state phase diagram in whole parameter region. Furthermore, to confirm the criticalities and to serve a reliability of our calculations, we check the consistency of excitation levels in finite-size systems. A finite-size scaling analysis of the charge excitation gaps is also performed in the vicinity of the phase boundary line. Section \[sec\_DISCU\] is devoted to discussions and summary of the present investigation. A short comment on the Berry phase method [@Rest95; @Resta98; @Alig99; @NV02; @Tori01] will also be given there. We will provide the comparison with that method, which is helpful to exhibit a reliability of our approach as well as the results.
GROUND STATES AND PHASE TRANSITIONS {#sec_GROUN}
===================================
The bosonization method provides an efficient way to describe low-energy properties of the 1D quantum systems:[@Gogo98] Linearizing the $\cos$-band at two Fermi points $\pm k_{\rm F}=\pm\pi
n/2a$ \[an electron density $n:=N/L=1$ and a number of sites (electrons) $L$ ($N$)\], and according to standard procedure, the effective Hamiltonian [@Tsuc99; @Fabr99; @Fabr00] is given as $H\rightarrow{\cal
H}={\cal H}_\rho+{\cal H}_\sigma+{\cal H}_2$ with $$\begin{aligned}
{\cal H}_\nu
&=&
\int {d}x~\frac{v_\nu}{2\pi}
\left[
{ K_\nu} \left(\partial_x \theta_\nu\right)^2 +
{1\over K_\nu} \left(\partial_x \phi_\nu\right)^2
\right]\nonumber\\
&+&
\int {d}x~\frac{2g_\nu}{(2\pi\alpha)^2}~
{\cos\sqrt8\phi_\nu},~~~(\nu=\rho, \sigma),
\label{eq_Hubba}\\
{\cal H}_2
&=&
\int {d}x~\frac{-2\Delta}{\pi\alpha}~
{\sin\sqrt2\phi_\rho}~{\cos\sqrt2\phi_\sigma}.
\label{eq_alter}
\end{aligned}$$ The operator $\theta_\nu$ is the dual field of $\phi_\nu$ satisfying the commutation relation $
\left[\phi_\nu(x),\partial_y\theta_{\nu'}(y)/\pi\right]=
{i}\delta(x-y)\delta_{\nu,\nu'}.
$ $K_\nu$ and $v_\nu$ are the Gaussian coupling and the velocity of elementary excitations. Coupling constants $g_\rho~(<0)$ and $g_\sigma$ stand for the 4$k_{\rm
F}$-Umklapp scattering and the backward scattering bare amplitudes, respectively, and ${\cal H}_2$ expresses a coupling between the spin and charge degrees of freedom. In Table \[TAB\_I\], we summarize the order parameters for the relevant $2k_{\rm F}$ density-wave phases, i.e., the charge-density-wave (CDW), bond charge-density-wave (BCDW), and spin-density-wave (SDW) phases, where the electron’s spin and the bond charge are given as $
{\bf S}_j=\sum_{s,s'}
c^\dagger_{j,s}[\frac12\mbox{\boldmath$\sigma$}]^{}_{s,s'}c^{}_{j,s'}
$ and $\overline{n}_j=\sum_s (c^\dagger_{j,s} c^{}_{j+1,s}+{\rm H.c.})$, respectively ( are the Pauli matrices). Their bosonized expressions are given in the second column. In the third column, we give the locking points of phase fields. As discussed in Ref. , there are two locking points of $\phi_\rho$, i.e., $\langle\sqrt8\phi_\rho\rangle=\pm\phi_0$ in the BCDW state. The phase $\phi_0$, a function of $U$ and $\Delta$, continuously varies from 0 to $\pi$.
Let us see the system with increasing $\Delta$ for fixed $U$. At $\Delta=0$, the ground state is in the MI phase with the most divergent SDW fluctuation (the third row of Table \[TAB\_I\]). According to the arguments,[@Tsuc99; @Naga86] the MI phase may survive for $U\gg2\Delta$. For $2\Delta\gg U$, ${\cal H}_2$ becomes relevant, and leads to the BI phase with the long-range CDW order without degeneracy (the first row). For this issue, FGN argued that under the uniform charge distribution a renormalization effect of ${\cal H}_2$ to $g_\sigma$ brings about the spin-gap transition in the spin part at a certain value of $\Delta_\sigma(U)$, which is described by the sine-Gordon (SG) theory. This is qualitatively in accord with the perturbation calculation in the strong-coupling region,[@Naga86] and leads to the SDI phase with the long-range BCDW order (the second row). Further with the increase of $\Delta$, a transition in the charge part occurs on a separatrix $\Delta_\rho(U)$ between two different types of charge-gap states. This line corresponds to the massless RG flow connecting the Gaussian (the central charge $c=1$) and the 2D-Ising ($c=\frac12$) fixed points,[@Zamo86] and its description is given by the double-frequency sine-Gordon (DSG) theory.[@Delf98] Our main task is thus to estimate $\Delta_\nu(U)$ for $U>0$ and to check the criticalities based on their prediction.
-- ------------------------------------------------- ---------------------------------------------- --------------------- --
Order parameters Bosonized forms Locking points
${\cal O}_{\rm CDW}= (-1)^jn_j$ $2\sin\sqrt2\phi_\rho \cos\sqrt2\phi_\sigma$ $( \pi , 0)$
${\cal O}_{\rm BCDW}= (-1)^j\overline{n}_j$ $2\cos\sqrt2\phi_\rho \cos\sqrt2\phi_\sigma$ $(\pm \phi_0, 0)$
${\cal O}^{\parallel}_{\rm SDW}= (-1)^jS^{z}_j$ $2\cos\sqrt2\phi_\rho \sin\sqrt2\phi_\sigma$ $( 0,\ast)$
-- ------------------------------------------------- ---------------------------------------------- --------------------- --
: The order parameters. The bosonized forms and the locking points of phase variables $
(\protect\langle\protect\sqrt8\protect\phi_{\protect\rho }\protect\rangle,
\protect\langle\protect\sqrt8\protect\phi_{\protect\sigma}\protect\rangle)$ are given in the second and third columns. $\phi_0$ is a function of $U$ and $\Delta$, and $\ast$ denotes a phase not to be locked.
\[TAB\_I\]
NUMERICAL METHODS AND CALCULATION RESULTS {#sec_NUMER}
=========================================
Low-lying excitations observed in the finite-size systems are expected to serve for the determinations of transition points. Here, we take a look at the following operators with lower scaling dimensions: $$\begin{aligned}
{\cal O}_{\nu,1}&=&\sqrt2\cos\sqrt2\phi_\nu,
\label{eq_COS} \\
{\cal O}_{\nu,2}&=&\sqrt2\sin\sqrt2\phi_\nu,
\label{eq_SIN} \\
{\cal O}_{\nu,3}&=&{\rm exp(\pm i}\sqrt2\theta_\nu).
\label{eq_EXP}
\end{aligned}$$ According to the finite-size-scaling argument based on the conformal field theory, corresponding energy levels for these operators $\Delta
E_{\nu,i}$ (taking the ground-state energy $E_0$ as zero) are expressed by the use of their scaling dimensions $x_{\nu,i}$:[@Card84] $$\Delta E_{\nu,i}\simeq \frac{2\pi v_\nu}{L}x_{\nu,i}.
\label{eq_SCALING}$$ Then these excitations can be extracted under the antiperiodic boundary condition with respect to the ground state due to the selection rule of the quantum numbers.[@NakaTJ; @NakaEX] In the numerical calculations using the Lanczos algorithm we can identify $\Delta E_{\nu,i}$ according to the discrete symmetries of the wave functions, e.g., translation ($c_{j,s}\to c_{j+2,s}$), charge conjugation \[$c_{j,s}\to (-1)^jc_{j+1,s}^{\dag}$\], spin reverse ($c_{j,s}\to c_{j,-s}$), and space inversion ($c_{j,s}\to c_{L-j,s}$). Here note that, except for the spin-reversal operation, definitions of these transformations are different from those of the uniform systems, such as the extended Hubbard model.[@NakaEX]
First, we treat the spin-gap transition in the spin part following Refs. , and . In the SDW phase, due to the marginal coupling in the SU(2)-symmetric spin part, the singlet ($x_{\sigma,1}$) and triplet ($x_{\sigma,2}=x_{\sigma,3}$) excitations split as $x_{\sigma,1}>x_{\sigma,2}=x_{\sigma,3}$ satisfying a universal relation $$\frac{x_{\sigma,1}+3 x_{\sigma,2}}{4}=\frac12.
\label{eq_SPN13}$$ Then, the degeneracy condition $$x_{\sigma,1}=x_{\sigma,2}=x_{\sigma,3}
\label{eq_SPNCR}$$ stands for the vanishing of the coupling, and provides a good estimation of the spin-gap transition point. Note that Torio [*et al.*]{} used the crossing of these excitation levels for the determination of the MI-SDI transition,[@Tori01] while the consistency check of the levels to confirm the universality of transition is still absent. Figure \[FIG1\] shows an example of the $\Delta$ dependences of $x_{\sigma,i}$ for the 16-site system at $u=0.6$ \[here we introduce the reduced Coulomb interaction parameter $u=U/(U+4)$\]. For this plot, we estimated the spin-wave velocity $v_\sigma$ from a triplet excitation with the wave number $4\pi/L$ as $v_{\sigma}=\lim_{L\to\infty}\Delta E(S=1,k=4\pi/L)/(2\pi/L)$ and normalized the excitation gaps $\Delta E_{\sigma,i}$ according to Eq. (\[eq\_SCALING\]). The singlet (triplet) level corresponding to the operator ${\cal
O}_{\sigma,1}$ \[${\cal O}_{\sigma,2}~({\cal O}_{\sigma,3})$\] is denoted by circles (triangles) with a fitting curve. Their behaviors reflect the TLL properties: For instance, the amplitude of the level splitting decreases with the increase of $\Delta$ due to its renormalization effect, and eventually the level crossing occurs at $\Delta_\sigma(U,L)$. More precisely, in order to confirm the universality, we plot the averaged scaling dimension $x_{\rm av}$, i.e., the left-hand side of Eq. (\[eq\_SPN13\]) in Fig. \[FIG1\] (squares). We also exhibit the $L$ dependence of $x_{\rm av}$ at $\Delta=1.0$ as an example (see the inset). The result shows that the condition imposed on $x_{\sigma,i}$ is accurately satisfied for $\Delta\le\Delta_\sigma(U,L)$; in particular, the extrapolated value of $x_{\rm av}$ is almost $\frac12$. Consequently, the level crossing at which Eq. (\[eq\_SPNCR\]) is satisfied can be regarded as an indication of the spin-gap transition in the spin part of the Hamiltonian (\[eq\_HAMIL\]). On the other hand, the spin part is dimerized for $\Delta>\Delta_\sigma(U,L)$.
![ The $\Delta$ dependence of $x_{\sigma,i}$ at $u=0.6$ for the 16-site system \[$u=U/(U+4)$\]. The spin-gap transition point $\Delta_\sigma(U,L)$ is estimated from the level crossing between the singlet (circles) and triplet (triangles) spin excitations. The squares plot $x_{\rm av}=(x_{\sigma,1}+3 x_{\sigma,2})/4$, and the inset shows the $L$ dependence of $x_{\rm av}$ at $\Delta=1.0$, where a least-square-fitting line to the data of $L=12$-$16$ is given. []{data-label="FIG1"}](FIG1.eps){width="3.2in"}
Next, we discuss the 2D-Ising transition in the charge part. Recently, we have treated the crossover behavior into the 2D-Ising criticality in the study of the frustrated quantum spin chain,[@Otsu02] so we shall here employ the same approach to determine $\Delta_\rho(U)$. Since there are two critical fixed points connected by the RG flow, a relationship between lower-energy excitations on these fixed points is quite important. For this, the so-called ultraviolet-infrared (UV-IR) operator correspondence provides significant informations:[@Fabr00; @Bajn01] Along the RG flow, the operators on the Gaussian fixed point (UV) are transmuted to those on the 2D-Ising fixed point (IR) as $${\cal O}_{\rho,1} \to \mu,~~~
{\cal O}_{\rho,2} \to I+\epsilon,
\label{eq_UVIRC}$$ where $\mu$ is the disorder field (Z$_2$ odd), and $\epsilon$ is the energy density operator (Z$_2$ even) with scaling dimensions $x_\mu=\frac18$ and $x_\epsilon=1$, respectively. Furthermore, since a deviation from the transition point $\Delta-\Delta_\rho(U)$, which is the coupling constant of the ${\cal O}_{\rho,2}$ term in the DSG Hamiltonian,[@Fabr99] plays a role of the thermal scaling variable, anomalous behaviors in the vicinity of $\Delta_\rho(U)$ are to be related to the divergent correlation length of the form $\xi\propto
[\Delta-\Delta_\rho(U)]^{-\nu}$ with the exponent $1/\nu=2-x_\epsilon=1$. On one hand, the excitation $\mu$ corresponding to ${\cal O}_{\rho,1}$ provides a lower-energy level, so we shall focus our attention on it. In order to determine the transition point, we shall numerically solve the following PRG equation for a given value of $U$ with respect to $\Delta$:[@Room80; @Otsu02] $$(L+2)\Delta E_{\rho,1}(U,\Delta,L+2)=
L \Delta E_{\rho,1}(U,\Delta,L ).
\label{eq_PRGEQ}$$ Since this is satisfied by the gap $\Delta E_{\rho,1}(U,\Delta,L)\propto 1/L$, the obtained value can be regarded as the $L$-dependent transition point, say $\Delta_\rho(U,L+1)$. We plot $L$ and $\Delta$ dependences of the scaled gap $L\Delta
E_{\rho,1}(U,\Delta,L)$ in Fig. \[FIG2\], and find that the size dependence of the crossing point is small for large values of $U$, but it is visible in the weak coupling case.
![ The $L$ and $\Delta$ dependences of the scaled gap $L\Delta E_{\rho,1}$. From left to right, $u=$0.12, 0.60 and 0.72, respectively. The correspondence between marks and system sizes is given in the figure. Crossing points give the $L$-dependent transition points $\Delta_\rho(U,L+1)$. []{data-label="FIG2"}](FIG2.eps){width="3.2in"}
While the results in the thermodynamic limit will be given in the last part of this section, we shall check first the criticality on and in the vicinity of the phase boundary using the extrapolated data $\Delta_\rho(U)$. For this aim, an evaluation of the central charge $c$ through the size dependence of the ground-state energy provides a straightforward way.[@Blot86] However, as exhibited in the following, the critical line in the charge part is close to the spin-gap transition line, so that influences from the spin part with the small dimer gap prohibit a reliable estimation of $c$ from the data of the finite-size systems. Alternatively, we shall evaluate a ratio of the charge-excitation gaps $\Delta E_{\rho,1}(U,\Delta,L)$ and $\Delta E_{\rho,2}(U,\Delta,L)$ on the phase boundary to check the UV-IR operator correspondence. According to Eqs. (\[eq\_SCALING\]) and (\[eq\_UVIRC\]), it is expressed by the scaling dimensions of operators $\epsilon$ and $\mu$ as $$R=
\frac
{\Delta E_{\rho,1}(U,\Delta_\rho(U),L)}
{\Delta E_{\rho,2}(U,\Delta_\rho(U),L)}
\to
\frac
{x_\mu}
{x_\epsilon}=\frac18$$ for large $L$. Figure \[FIG3\] plots the $\Delta$ dependence of $R$ for $L=10$-$16$ ($u=0.72$). The transition point in the thermodynamic limit is denoted by the arrow near the $x$ axis. While the ratio exhibits a subtle $\Delta$ dependence around the point, we interpolate these data, and estimate the $L$ dependence of $R$ at $\Delta_\rho(U)$, which is given with a least-square-fitting line in the inset. The plot shows that the extrapolated value is fairly close to $\frac18$. Therefore we conclude that the boundary line $\Delta_\rho(U)$ belongs to the 2D-Ising universality class.
![ The $\Delta$ dependence of the charge-excitation-gap ratio $R={\Delta E_{\rho,1}(U,\Delta,L)}/{\Delta E_{\rho,2}(U,\Delta,L)}$ for $L=10$-$16$ at $u=0.72$. The arrow shows the transition point $\Delta_\rho(U)$. The inset plots the $L$ dependence of $R$ at $\Delta_\rho(U)$ with a least-square-fitting line. []{data-label="FIG3"}](FIG3.eps){width="3.2in"}
![ The finite-size-scaling plots of the charge-excitation gap $\Delta E_{\rho,1}$ for systems of $L=14$-$18$ at $u=0.72$ and 0.80. We use the 2D-Ising critical exponent $\nu=1$. A dotted line (the slope 1) is given for the guide to eye. []{data-label="FIG4"}](FIG4.eps){width="3.2in"}
Furthermore, we shall investigate the critical behavior:[@Otsu02] According to the finite-size-scaling argument, we analyze the charge-excitation gap by using the following one-parameter scaling form: $$\Delta E_{\rho,1}(U,\Delta,L)=L^{-1}\Psi(L[\Delta-\Delta_\rho(U)]^\nu).
\label{eq-FSS}$$ Since $\Delta E_{\rho,1}\propto 1/\xi$ in the thermodynamic limit ($L/\xi\to\infty$), the scaling function is expected to asymptotically behave as $\Psi(x)\propto x$ for large $x$. On the other hand, the gap $\Delta E_{\rho,1}\propto 1/L$ on the critical point ($L/\xi\to0$) so that $\Psi(x)\simeq {\rm const}$ for $x\to0$.[@Barb83] Figure \[FIG4\] plots Eq. (\[eq-FSS\]) using the exponent of the 2D-Ising model $\nu=1$. Although due to the smallness of $L$ a scattering of the scaled data is visible especially near the transition point, the data of different system sizes are collapsed on the single curve, and its asymptotic behaviors agree with the expected ones. Therefore, we can check that, in the transition of the charge part, the deviation $\Delta-\Delta_\rho(U)$ plays a role of the thermal scaling variable on the 2D-Ising fixed point. Here, note that in the strong-coupling region the energy scale of the crossover behavior may be large enough to be detected even in the small-size systems. However, the finite-size-scaling nature may become obscure in the weak and intermediate couplings.
Lastly, we present the ground-state phase diagram. In order to determine it, the extrapolations of $\Delta_\nu(U,L)$ to the thermodynamic limit are carried out. For the spin part, it should be noted that Torio [*et al.*]{} evaluated the spin-gap transition line from the level crossing Eq. (\[eq\_SPNCR\]),[@Tori01] so here we perform the same calculations in order to complete the ground-state phase diagram. We employ the formula: $\Delta_{\sigma}(U,L)=\Delta_{\sigma}(U)+a
L^{-2} +b L^{-4}$, where $\Delta_{\sigma}(U)$, $a$ and $b$ are determined according to the least-square-fitting condition. Then, we extrapolated the data of $L=12$-$18$ as shown in Fig. \[FIG5\](a), where from bottom to top the data with fitting curves are given in the increasing order of $U$. Consequently, the spin-gap transition line $\Delta_{\sigma}(U)$ (open circles with a fitting curve) is given in Fig. \[FIG5\], where the reduced alternating potential parameter $\delta=\Delta/(\Delta+2)$ is used as the $y$ axis. On the other hand, for the extrapolation of $\Delta_\rho(U,L)$, we assume the following formula:[@Itzy89] $\Delta_{\rho}(U,L)=\Delta_{\rho}(U)+a L^{-3}$, and extrapolate the data of $L=10$-18 as shown in Fig. \[FIG5\](b). Consequently, Fig. \[FIG5\] shows that the critical line in the charge part (open squares with a fitting curve) does not coincide with the spin-gap transition line, i.e., $\Delta_{\sigma}(U)<\Delta_{\rho}(U)$, and that the 2D parameter space $\{(u,\delta)~|~0\le u,~\delta\le1\}$ is separated into the MI, BI, and SDI phases with SDW, CDW, and BCDW, respectively. Since the Hubbard gap provides a principal energy scale and a shape of the boundary is roughly determined so that the magnitude of the band gap becomes comparable to the scale, the $U$ dependence of the boundaries is expected to be weak in the small-$U$ region,[@Tsuc99; @Fabr99] which is in agreement with our observation. On the other hand, in order to clarify the behaviors in the large-$U$ region, we plot a magnification of the phase diagram around the $2\Delta=U$ line in Fig. \[FIG6\]. This shows that in the limit of $U\to\infty$ the boundaries do not merge to the line: More precisely, for $U=96$ we obtain $\Delta_{\rho}-U/2 \simeq -0.65$ and $\Delta_{\sigma}-U/2 \simeq -0.97$, respectively. In Ref. , adding to the spin part ($2\Delta_{\sigma}-U \simeq -1.91$ for $U$, $V\gg1$), they also reported $2\Delta_{\rho}-U \simeq -1.33$, which is close to our estimation. Consequently, we confirm that the intermediate SDI phase may survive in the large-$U$ limit, which is one of the nontrivial behaviors and is contrasted to the naive argument.
Here we shall perform a comparison with the previous DMRG results. As mentioned in Sec. \[sec\_INTRO\], while the DMRG calculations performed by several groups seem not to reach an agreement with respect to an existence of the SDI phase, it may be informative to provide a comparison with our result. Zhang [*et al.*]{} determined two-types of phase transition points $U_{c1}$ and $U_{c2}$ based on the structure factor of the BCDW order parameter;[@Zhan03] we plot their results in Fig. \[FIG5\] by using the filled squares and filled circles, respectively. This shows that their estimations of $U_{c1}$ agree well with our data $\Delta_\rho(U)$, although those of $U_{c2}$ considerably deviate from $\Delta_\sigma(U)$. Since the phase transition at $\Delta_\sigma(U)$ is the spin-gap transition, the logarithmic corrections to the power-law behaviors as well as the exponentially small magnitude of the spin gap generally make it difficult to determine the transition point. On one hand, as explained in the above, the LC method used here overcomes these difficulties in the determination of the transition points $\Delta_\sigma(U)$.
![ The ground-state phase diagram of the 1D Hubbard model with the alternating potential. The open circles (squares) with a fitting curve show the spin-gap (2D-Ising) transition line in the spin (charge) part. The stable regions of the MI, SDI, and BI phases are given in the 2D parameter space $(u,\delta)$ \[$u=U/(U+4)$ and $\delta=\Delta/(\Delta+2)$\]. Insets (a) and (b) show the extrapolations of the $L$-dependent transition points in the spin and the charge parts, respectively. For comparison, we also plot the DMRG calculation results given in Ref. by using the filled squares ($U_{c1}$ in their notation) and the filled circles ($U_{c2}$). []{data-label="FIG5"}](FIG5.eps){width="3.2in"}
![ The deviations of boundaries from the $2\Delta=U$ line, $\Delta_\nu(U,L)-U/2$. We use $u=U/(U+4)$ as the $x$ axis. The correspondence between marks and system sizes is given in the figure. Marks with solid (dotted) curves exhibit the deviations in the spin (charge) part. []{data-label="FIG6"}](FIG6.eps){width="3.2in"}
DISCUSSION and SUMMARY {#sec_DISCU}
======================
For the understanding of the phase diagram in the large-$U$ limit, let us see the perturbative treatment of Hamiltonian (\[eq\_HAMIL\]) under the condition of $U-2\Delta\gg1$. $\Delta_\sigma(U)$ may be related to the spin-gap transition point in the $S=\frac12$ $J_1$-$J_2$ model.[@Fabr99; @Kamp03] Therefore, using its numerical value[@Okam92] and perturbative expressions on $J_1$ and $J_2$,[@Naga86] we can approximately estimate $\Delta_\sigma(U)$ as a solution of the equation $J_2/J_1\simeq X/(1-4X)\simeq0.2411$, where $X=(1+4x^2-x^4)/U^2(1-x^2)^2$ and $x=2\Delta/U$. Then, we find a solution \[$\Delta'_\sigma(U)$\] to give a value $\Delta'_\sigma(U)-U/2\simeq-1.427$ in the limit. While, due to the lack of effects from the higher-order processes in the kinetic energy term, the approximate value deviates from the numerical estimation, this exhibits the following, i.e., the perturbative expansion becomes singular on the $2\Delta=U$ line so that the phase boundary deviates from the line. This singularity also exists in the perturbative calculations of the SDW and CDW state energies ($E_{\rm SDW}$ and $E_{\rm CDW}$). And then the direct transition line between these phases cannot be determined from the equation $E_{\rm SDW}=E_{\rm CDW}$, which is highly contrasted to the case of the extended Hubbard model (EHM) including the nearest-neighbor Coulomb interaction $\sum_j V
n_jn_{j+1}$.[@Dong94] Since the spin-charge coupling term with the dimerized spin part generates one of the relevant forces, $\Delta_\rho(U)$ should be affected by that of the spin part. Besides the present model, it is known that EHM possesses the coupling term $V\cos\sqrt8\phi_\rho\cos\sqrt8\phi_\sigma$ in its bosonized form,[@Voit92] and that the BCDW state with the locking points $\langle\sqrt8\phi_{\rho,\sigma}\rangle=0$ is stabilized around the $2V=U$ line in the weak- and intermediate-coupling region.[@NakaEX] The corrections to $g_\nu$ from higher-energy states stabilize it,[@Tsuc01] but the coupling term forces the boundaries to merge into the single first-order phase transition line between the SDW and CDW states in the strong-coupling region because it raises the BCDW state energy. However, in the present BCDW state, the locking point $\phi_0$ in Table \[TAB\_I\] may take a value so as not to bring about a large energy cost due to the coupling term Eq. (\[eq\_alter\]). Therefore, the existence of the SDI phase is not prohibited even in the strong-coupling limit in contrast to the EHM case. Of course, these arguments are qualitative and intuitive ones, so an effective theory in this limit is required for the precise description on the limiting behaviors.
![ Behavior of the ground-state expectation value of the twist operator $z_{\nu}$ ($\nu=\rho,\sigma$) near the $2\Delta=U$ line. The correspondence between marks and system sizes are given in the figure. []{data-label="FIG7"}](FIG7.eps){width="3.2in"}
![ Comparisons of the system-size dependences of the transition points obtained by the LC and PRG methods vs by the condition $z_{\nu}=0$. The fitting curves show the extrapolations of data to the thermodynamic limit. []{data-label="FIG8"}](FIG8.eps){width="3.2in"}
Finally, we comment on the Berry phase method.[@Rest95; @Resta98; @Alig99; @NV02; @Tori01] The Berry phases for the charge and the spin parts $\gamma_\nu$ are related to the ground-state expectation values of the twist operators as $\gamma_\nu={\rm Im}\log z_\nu$ where $$z_{\rho}=\langle U_{\uparrow}U_{\downarrow}\rangle,\quad
z_{\sigma}=\langle U_{\uparrow}U_{\downarrow}^{-1}\rangle,$$ and $U_{s}=\exp[(2\pi i/L)\sum_{j=1}^L j n_{j,s}]$.[@Resta98] Since $z_\nu$ is real at the half filling with zero-magnetic field, $\gamma_\nu$ (=0 or $\pi$) indicates the sign of $z_\nu$. On one hand, $z_\nu$ can be related to the bosonic field as $z_{\rho,\sigma}\propto\mp\langle\cos\sqrt{8}\phi_{\rho,\sigma}\rangle$, so that it includes the information of the locking points given in Table \[TAB\_I\].[@NV02] In Fig. \[FIG7\] we show behaviors of $z_\nu$ near the $2\Delta=U$ line for $U=16$ and find that with the increase of $\Delta$ both of these increase and change their sign. As shown in the lower panel of Fig. \[FIG8\], the condition $z_\sigma=0$ gives a close value to the result of the LC method, so it may provide a proper estimation of the spin-gap transition point $\Delta_\sigma$.[@Tori01; @NV02] On the other hand, the zero point of $z_\rho$ exhibits a deviation from the PRG result (see the upper panel of Fig. \[FIG8\]). Since $\phi_0$ continuously varies with $\Delta$, $z_\rho$ can take a finite value on the 2D-Ising transition point in the thermodynamic limit, which is highly contrasted to $z_\sigma$ on the spin-gap transition point. In fact, the size-dependent zero points are seemingly extrapolated to a value different from our PRG estimation, so that the condition $z_\rho=0$ does not specify the transition point. On the other hand, we also find in Fig. \[FIG7\] that there is a point $\Delta\simeq 7.3$ at which $z_\rho$ is almost independent of $L$. This crossing point is expected to be a good estimator for the 2D-Ising transition point in the charge part $\Delta_\rho$ because this is quite close to the PRG result even for small $L$. However, a theoretical explanation of this possibility is still open.
To summarize, we have investigated the ground-state phase diagram of the one-dimensional half-filled Hubbard model with the alternating potential, especially in order to verify the scenario given by Fabrizio, Gogolin, and Nersesyan, we have numerically treated the phase transitions observed in the spin and charge parts: We calculated the spin-gap transition points $\Delta_\sigma$ in the spin part by the level-crossing method (see also the argument for the spin-gap transition in Ref. ) and the two-dimensional Ising transition points $\Delta_\rho$ in the charge part by the phenomenological renormalization-group method. We confirmed that, adding to the Mott and band insulators, the “spontaneously dimerized insulator” accompanied by the long-range-ordered $2k_{\rm F}$ bond charge-density wave is stabilized as the intermediate phase for all $U>0$. Then we checked the SU(2)-symmetric Gaussian (2D-Ising) criticality of the spin (charge) part by treating the low-lying excitation levels in the finite-size systems, and, simultaneously, we performed the finite-size-scaling analysis of the charge-excitation gap to clarify the critical phenomena around $\Delta_\rho$. The comparison with the relating work was performed to check the reliability of our numerical results and to exhibit the efficiency of our approach.
After submission of this paper, we became aware of the work investigating the ground-state phase diagram and the universality of the transition in the charge part by the use of finite-size-scaling analysis of the DMRG calculation data.[@Manm03] They have found two transition points and succeeded to obtain $\nu=1$ in agreement with our conclusion, while the estimated exponent for the susceptibility of the BCDW order parameter shows a deviation from the theoretical value $\eta_1=1/4$, e.g., $\eta_1\simeq0.45$ at the point on the BI-SDI phase boundary $\Delta=10$ and $U_{c1}=21.385$ (in their notation). In this paper we have treated the elementary excitations in the TLL system specified by the discrete symmetries of the lattice Hamiltonian with the twisted boundary condition, whereas they have measured the BCDW order parameter, (i.e., a composite excitation of the spin and charge degrees of freedom) with the larger energy scale.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
One of the author (H.O.) is grateful to Y. Okabe for helpful discussions. M.N. thanks J. Voit for the collaboration in the early stage of the present work. M.N. is partly supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan through Grants-in-Aid No. 14740241. Main computations were performed using the facilities of Tokyo Metropolitan University, Yukawa Institute for Theoretical Physics, and the Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
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C. Itzykson and J.-M. Drouffe, [*Statistical Field Theory*]{}, (Cambridge University Press, New York, 1989). Vol. 1; T. Sakai and M. Takahashi, .
P.G.J. van Dongen, .
M. Tsuchiizu and A. Furusaki, ; M. Tsuchiizu and A. Furusaki, .
S.R. Manmana [[*et al.*]{}]{}, (eprint cond-mat/0307741).
|
---
abstract: 'Motivated by puzzling results of recent experiments, we re-examine the response of a weakly pinned two-dimensional Wigner crystal to a uniform AC electric field. We confirm that at some disorder and magnetic field dependent frequency $\omega_p$, an inhomogeneously broadened absorption line emerges. Although the line is conventionally broad in zero magnetic field, in strong fields it appears as a sharp resonance whose width is related to the density of states in the low-frequency tail of the zero-field phonon spectrum. This behavior originates due to the long-range Coulomb interactions.'
address:
- 'School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540'
- 'Physics Department, Princeton University, Princeton, New Jersey 08544'
author:
- 'M. M. Fogler'
- 'David A. Huse'
date:
title: 'Dynamical response of a weakly pinned two-dimensional Wigner crystal'
---
The earlier version of this paper is superseded by cond-mat/0004343.
|
---
abstract: 'The Keck Planet Imager and Characterizer (KPIC) is an upgrade to the Keck II adaptive optics system that includes an active fiber injection unit (FIU) for efficiently routing light from exoplanets to NIRSPEC, a high-resolution spectrograph. Towards the end of 2019, we will add a suite of new coronagraph modes as well as a high-order deformable mirror. One of these modes, operating in $K$-band (2.2$\mu m$), will be the first vortex fiber nuller to go on sky. Vortex Fiber Nulling (VFN) is a new interferometric method for suppressing starlight in order to spectroscopically characterize exoplanets at angular separations that are inaccessible with conventional coronagraph systems. A monochromatic starlight suppression of $6\times10^{-5}$ in 635 nm laser light has already been demonstrated on a VFN testbed in the lab. A polychromatic experiment is now underway and coupling efficiencies of $<5\times10^{-4}$ and $\sim5\%$ have been demonstrated for the star and planet respectively in 10% bandwidth light. Here we describe those experiments, the new KPIC VFN mode, and the expected performance of this mode using realistic parameters determined from on-sky tests done during the KPIC commissioning.'
author:
- |
Daniel Echeverri, Garreth Ruane, Nemanja Jovanovic, Thomas Hayama, Jacques-Robert Delorme, Jacklyn Pezzato, Charlotte Bond, Jason Wang, Dimitri Mawet J. Kent Wallace, Eugene Serabyn Department of Astronomy, California Institute of Technology, 1200 E. California Blvd.,\
Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,\
Pasadena, CA 91109, USA Institute for Astronomy, University of Hawaii, 640 North A’Ohoku Place, Hilo, HI 96720, USA
bibliography:
- 'Library.bib'
title: '**The vortex fiber nulling mode of the Keck Planet Imager and Characterizer (KPIC)**'
---
INTRODUCTION {#sec:intro}
============
The Keck Planet Imager and Characterizer (KPIC)[@Mawet2016_KPIC; @Mawet2017_KPIC; @Jovanovic2019SPIE] is an instrument designed to link the Keck adaptive optics (AO) system[@Wizinowich2000] to NIRSPEC[@Martin2018], a high-resolution near-infrared spectrograph. KPIC accomplishes this by coupling light from a point-like source into a single-mode fiber (SMF)[@Jovanovic2017], allowing for conventional stellar spectroscopy as well as direct spectroscopy of low mass companions, including giant exoplanets. In scenarios where the companion of interest is resolved with respect to the star (roughly speaking, when the angular separation is $>\lambda/D$, where $\lambda$ is the wavelength and $D$ is the telescope diameter), the starlight may be minimized at the position of the planet using a coronagraph and speckle nulling techniques[@Pezzato2019SPIE; @Mawet2017_HDCII]. However, in situations where the planet-star angular separation is $\sim \lambda/D$, fiber nulling interferometry[@Bracewell1978; @Haguenauer2006; @Serabyn2019PFN] may be a more effective method for reducing the amount of starlight entering the spectrograph. KPIC will have such an interferometric mode, based on the vortex fiber nulling (VFN) technique[@Ruane2018_VFN; @Echeverri2019_VFN; @Ruane2019SPIE], to enable spectroscopy of close-in companions. In this paper, we provide an overview of the VFN concept, an update on the laboratory demonstrations of VFN, and estimates of the on-sky performance of the KPIC VFN mode given the current AO performance.
VFN Concept {#sec:concept}
===========
VFN is an interferometric method for detecting and spectroscopically characterizing exoplanets at small angular separations that are inaccessible to conventional coronagraph instruments[@Ruane2018_VFN]. It leverages the modal selectivity of SMFs to reject starlight while efficiently coupling planet light that can then be fed into a spectrograph for analysis.
A vortex phase mask placed in the pupil plane[@Swartzlander2001] introduces an azimuthally increasing phase ramp pattern, shown in the left half of Fig. \[fig:MotherFig\]a, of the form $\exp(il\theta)$, where $\theta$ is the azimuthal coordinate and $l$ is an integer known as the charge which defines how many times the phase cycles around the beam. In theory, a VFN system can have the vortex mask located anywhere in the beam path. In the KPIC VFN mode, the vortex mask will be located in a pupil plane (see Fig. \[fig:MotherFig\]b). This results in the complex-valued point spread function (PSF) shown in the right half of Fig. \[fig:MotherFig\]a which is characterized by a “donut shape" in amplitude and a vortex phase structure that also varies as $\exp(il\theta)$.
![(a) The azimuthally varying phase pattern introduced by a charge $l=1$ vortex mask and the corresponding complex-valued PSF resulting when the vortex mask is placed in a pupil plane. (b) Diagram of a VFN system with the vortex mask in the pupil plane and SMF in the image plane. The mode of the fiber is denoted $\Psi(r)$. (c) Coupling efficiency, $\eta$, of a point source versus its angular separation from the optical axis, $\alpha$, for a charge $l=1$ (blue) and $l=2$ (orange) VFN system. The inset shows the coupling efficiency for all points in a field of view centered on the star/fiber.[]{data-label="fig:MotherFig"}](SPIE2019_MotherFig_SimpAx.jpg){width="\linewidth"}
The coupling efficiency, or fraction of light from a given source that gets into the SMF, can be computed as: $$\eta(\alpha)=\frac{\left|\int E(\mathbf{r};\alpha) \Psi(r)dA\right|^2}{\int \left| E(\mathbf{r};\alpha)\right|^2dA \int \left|\Psi(r)\right|^2dA} , \label{eqn:couplingeff}$$ where $E(\mathbf{r};\alpha)$ is the field at the entrance to the SMF, $\Psi(r)$ is the fundamental mode of the SMF, $\mathbf{r}=(r,\theta)$ are the coordinates in the fiber-tip plane, and $\alpha$ is the angular offset with respect to the optical axis (see Fig. \[fig:MotherFig\]b). To give a heuristic explanation of why the starlight is rejected by the fiber, we consider a simplified case where the system pupil is circular and unobscured and the post-vortex stellar field is composed of separate radial and azimuthal components, $E_s(\mathbf{r})=f_r(r)\exp(il\theta)$. Since the fiber mode is also radially symmetric, the coupling integral for the star is separable and the azimuthal term goes to zero for any $l\neq 0$. Thus, the complex stellar field is orthogonal to $\Psi(r)$ and does not couple into the fiber. An off-axis point source has the same PSF but is shifted with respect to the SMF mode such that the azimuthal term no longer vanishes and the field is not orthogonal to $\Psi(r)$. Thus, the off-axis planet light partially couples. For non-circular pupils, such as the Keck aperture, the computation is different but a similar orthogonality condition occurs.
Figure \[fig:MotherFig\]c shows the coupling efficiency of a point source as a function of the angular separation, $\alpha$, for the system shown in Fig. \[fig:MotherFig\]b. As expected, a complete rejection, or nulling condition, is obtained for the star since it is an on-axis point source, $\eta_s(\alpha=0)=0$. Meanwhile a peak coupling of 19% is achieved for an off-axis point source, such as a planet, at an angular separation of $\alpha=0.85\lambda/D$ with a charge $l=1$ vortex. Similarly, a peak coupling of 10% is obtained for $\alpha=1.3\lambda/D$ with a charge $l=2$ vortex. Following the conventional definition of inner-working-angle (IWA) as the separation where the planet throughput is $50\%$ of its peak value, a charge 1 VFN system has inner and outer-working angles of $0.4$ and $1.4\lambda/D$ respectively. The entire charge 1 VFN working region is thus within the typical IWA[@Guyon_2006] of conventional coronagraphs. Furthermore, the rotational symmetry of the coupling efficiency about the star allows for planets to be spectrally characterized even when the planet’s position angle as projected on sky is unknown, which is often the case for planets detected through radial velocity (RV) and transit techniques.
Ruane et al.[@Ruane2019SPIE] provide a more detailed explanation of the VFN concept, its benefits, and practical design considerations. Most importantly, VFN works well with Keck’s segmented aperture and requires few modifications to the optical design of KPIC.
Laboratory Demonstrations
=========================
Monochromatic Demonstration {#sec:monochromatic}
---------------------------
![(a) Schematic of the transmissive VFN testbed at Caltech. Stage 1 holds the fiber source which projects light onto L1, the collimating lens. An iris then sets the pupil diameter before passing the beam to the vortex mask in stage 2. A focusing lens images the beam onto stage 3 which holds the nulling fiber that is connected to a photodiode for coupling measurements. A retractable power meter, norm. PM, can be moved into the beam path to measure the power for normalization. (b) Picture of the Caltech VFN testbed.[]{data-label="fig:LabSetup"}](TestbedSetup_Norm.jpg){width="0.85\linewidth"}
Earlier this year, the VFN concept was demonstrated in monochromatic light for the first time in the lab[@Echeverri2019_VFN]. We were able to achieve stellar coupling fractions, or “null depths", of $6\times10^{-5}$ and an average peak planet coupling of $10\%$ using a monochromatic charge 1 vortex mask, a 635 nm laser source, and other commercial off-the-shelf optics (see Fig. \[fig:LabSetup\]). Furthermore, we predicted that the main reason for the reduced planet coupling relative to the theoretical $19\%$ for a charge 1 VFN system was an incorrect $F/\#$ in our setup. Thus, by correcting the $F\#$ and making a few minor modifications, we have since achieved an azimuthally averaged peak planet coupling of $16\%$ while still maintaining the same null depth (left plot of Fig. \[fig:LabResults\]). The inset 2D coupling map shows that the “donut" is rounder and more symmetric than before.
![(left) Azimuthally averaged radial coupling, $\eta$, with 635 nm laser light and a charge 1, monochromatic vortex mask. The inset is the two dimensional coupling map. (middle) Azimuthally averaged peak planet coupling, $\eta_p$, in broadband light at various bandwidths centered around $\lambda_0=790$ nm and with a charge 2 polychromatic vortex mask. (right) Same as middle, but showing the null depth or “stellar coupling", $\eta_s$.[]{data-label="fig:LabResults"}](FinalLabResults.jpg){width="\linewidth"}
We believe that the remaining $3\%$ missing from the expected planet coupling is primarily due to: a) throughput losses in the final focusing lens and fiber and b) errors in the calibration between our two power meters. In our experiments, we measure the normalization power (denominator in Eq. \[eqn:couplingeff\]) just before the final lens using a retractable power meter, norm. PM, as shown in Fig. \[fig:LabSetup\]. We assume that the throughput losses in the lens and fiber match the manufacturers’ specifications but we have not measured these losses precisely to confirm that they are correct. As such, this may provide a marginal deviation in coupling efficiency. The larger deviation could arise from the calibration between our two power meters. The normalization power is measured with a Thorlabs S120C detector head but the coupled power (numerator in Eq. \[eqn:couplingeff\]) is measured on a separate fiber-coupled photodiode, , at the output of the nulling SMF. We carefully calibrated these two detectors relative to each other but there is some uncertainty in the accuracy of the calibration across the multiple gain settings needed for these measurements. As such, we believe that a more accurate coupling fraction can be obtained by modifying the system so that both the normalization and the coupled power can be measured on the same detector as explained in Sec \[sec:testbedRedesign\].
Polychromatic Demonstration {#sec:polychromatic}
---------------------------
With some minor modifications, we were able to use the same testbed shown in Fig. \[fig:LabSetup\] to demonstrate the VFN concept in polychromatic light. We replaced the 635 nm laser with a supercontinuum white light source (NKT Photonics SuperK EXTREME) and tunable filter (NKT Photonics SuperK VARIA) which allows us to select the bandwidth and central wavelength within the visible spectrum. We also replaced the monochromatic charge 1 vector vortex mask with an analogous polychromatic charge 2 mask, manufactured by Beam Co.[@Tabiryan2017], that nominally operates from $420\text{-}870$ nm. We found the wavelength that produced the best null with this mask by scanning through central wavelengths with 3 nm bandwidth and tuning the vortex and fiber positions.
The optimal wavelength was 790 nm, where we achieved a null depth of $2\times10^{-4}$ and an azimuthally averaged peak planet coupling of $\sim5\%$ with a 3 nm bandwidth. We then gradually increased the bandwidth without changing anything else in the system and measured the broadband coupling efficiencies shown in Fig. \[fig:LabResults\]. At 10% bandwidth (defined as $\Delta\lambda/\lambda_0$), we achieved a broadband null of $4.2\times10^{-4}$ and an azimuthally averaged peak planet coupling of $\sim4.5\%$. Thus, there was very little degradation in performance with increasing bandwidth. The main limitation on the maximum bandwidth was the upper wavelength limit of 840 nm from the VARIA filter. With a longer-wavelength source, we would likely achieve larger bandwidths with similar performance. Furthermore, the slight decrease in peak planet coupling at the larger bandwidths is likely due to the fact that our transmissive system uses a singlet focusing lens with an inherently wavelength-dependent focal length. As such, the planet coupling at the edges of the band is reduced by defocus.
The peak planet coupling of 5% is half of the ideal theoretical coupling for a charge 2 vortex. These coupling losses may be due to a few factors including that: a) we are using SM600 fibers at wavelengths beyond their specified range, b) the anti-reflection (AR) coating on the final lens, L2, is sub-optimal at wavelengths beyond 700 nm, and c) the pupil diameter is currently set to produce the ideal $F/\#$ for 635 nm light, not 790 nm. Even with this sub-optimal configuration, we believe this experiment demonstrates that VFN has the potential to provide broadband nulling over a typical astronomical band ($\sim$20%).
Planned Testbed Upgrades {#sec:testbedRedesign}
------------------------
To improve the broadband performance and capabilities of the VFN testbed at Caltech, we have redesigned the system and are planning to implement the following upgrades in coming months:
1. Replace the transmissive lenses with reflective off-axis parabolas (OAPs).
2. Modify the source stage to use a pinhole and add space for polarizers.
3. Replace the current fiber positioning actuators with larger-range, higher-precision actuators.
4. Modify the fiber injection stage to support two fibers: the nulling SMF and a multi-mode fiber.
5. Use a fiber coupler to combine the two fibers.
6. Use anti-reflection (AR) coated fibers.
The use of OAPs will minimize chromatic aberrations and especially remove the chromatic focal shift such that the planet and star coupling efficiencies will no longer be affected at the edges of the band when working with large bandwidths. This will also allow us to switch easily between central wavelengths which will be helpful as we transition from our current visible experiments to planned $K$-band $(2.2~\mu m)$ tests in the future. Using a pinhole ensures that the source remains unresolved which is particularly useful in the charge 1 VFN case since it is much more sensitive to the angular size of the source[@Ruane2018_VFN]. Furthermore, the new design creates space before the pinhole for polarizers to be placed without reducing the wavefront quality of the system. Filtering for a single polarization state will allow us to further investigate polarization-dependent aberrations in the vortex mask and reduce stellar leakage due to imperfect retardance.
The new fiber positioning actuators (PI Q-545.240) will provide closed loop accuracy down to $\sim6$ nm. This will allow us to measure the coupling versus position more precisely while ensuring that we have the accuracy needed to center the fiber on the optimal null location. These actuators also have a 25 mm range which will allow us to easily switch between two fibers in the focal plane: the nulling SMF and a 400 micron core multi-mode fiber (MMF). The MMF will couple $>99.9\%$ of the light incident on the fiber plane for normalization purposes. Additionally, by using fibers with the same AR coating, we’ll implicitly account for Fresnel reflections in the normalization as well. Using a fiber coupler to combine the MMF and SMF allows us to use the same photodiode for the coupling and normalization measurements which removes the need to cross calibrate two different power meters. These upgrades combined will provide more accurate coupling efficiency, $\eta$, measurements.
VFN with KPIC {#sec:KPIC}
=============
The KPIC VFN Mode {#KPICVFNMode}
-----------------
![(a) Schematic of the KPIC VFN mode. Light arrives from the telescope after passing through the facility AO system. The near-infrared pyramid wavefront sensor (PyWFS) and high-order deformable mirror (DM) further correct the wavefront before allowing the beam to pass through the vortex mask and then the atmospheric dispersion compensator (ADC). The tip-tilt mirror (TTM) centers the star PSF on the fiber which feeds NIRSPEC, the high-resolution near-infrared spectrograph. A dichroic reflects $J$- and $H$-bands to a tracking camera, which provides simultaneous imaging for PSF tracking, calibration, and control algorithms. Light in the science channel ($K$-band) transmits through the dichroic and is routed to NIRSPEC via the SMF. $L$-band is unused in the initial VFN configuration. (b) Nominal KPIC observation mode for direct exoplanet spectroscopy with the apodizer in the pupil and the planet aligned to the fiber. (c) KPIC VFN mode with the vortex mask in the pupil and the star aligned to the fiber.[]{data-label="fig:KPIC_VFNMode"}](KPIC_VFNModeFull.jpg){width="\linewidth"}
Given the simplicity of this approach and the recent progress on laboratory demonstrations, we are preparing to add a VFN mode to the KPIC instrument as part of an upcoming upgrade. This VFN mode shares many modules with the other KPIC observation modes but there are a few key differences. As shown in Fig. \[fig:KPIC\_VFNMode\]a, the VFN mode uses the near-infrared pyramid wavefront sensor[@Bond2018_PyWFS] (PyWFS) and high-order deformable mirror to perform an additional wavefront control loop after the facility AO system. The corrected beam then passes through the vortex mask and atmospheric dispersion compensator (ADC). A tip-tilt mirror (TTM) aligns the stellar PSF with the science SMF that feeds NIRSPEC (see Fig. \[fig:KPIC\_VFNMode\]c) to null the starlight. Feedback for the TTM control loop is provided primarily by the tracking camera which images the PSF just before the final focusing optics, but further feedback can also be obtained at a slower cadence using the slit-viewing camera of NIRSPEC.
Thus, the VFN mode (Fig. \[fig:KPIC\_VFNMode\]c) is slightly different from the direct exoplanet spectroscopy mode of KPIC (Fig. \[fig:KPIC\_VFNMode\]b), which a) uses an optional apodizer instead of a vortex and b) aligns the planet with the fiber. The direct spectroscopy mode is better suited for characterization of known exoplanets at larger separations from their host-stars ($>\lambda/D$), while the VFN mode is better for blind or targeted surveys and characterization of close-in companions ($\sim\lambda/D$).
In order to enable the VFN mode, a charge 2 $K$-band vector vortex mask will be installed in the pupil mask stage alongside the apodizer, as shown in Fig. \[fig:KPIC\_CoronagraphModule\]a. Pezzato et al.[@Pezzato2019SPIE] describe this module and the custom-designed apodization mask it carries in detail. A charge 2 vortex mask was chosen for VFN based on the predicted and early on-sky performance of KPIC. Although a charge 1 vortex yields higher planet throughput at smaller angular separations which can significantly decrease the integration time needed to observe an exoplanet, this improved planet sensitivity comes at the cost of increased sensitivity to tip/tilt errors. For example, to achieve a null depth of $\eta_{s}=10^{-4}$, a charge 1 VFN system requires less than $0.01\lambda/D$ RMS tip-tilt jitter whereas a charge 2 needs $0.1\lambda/D$ for a similar null depth[@Ruane2019SPIE]. While a charge 2 vortex takes a hit in planet coupling, it relaxes tip/tilt requirements, which will be useful during the early stages of VFN development. As more on-sky measurements of the AO performance are made, we will reconsider whether a charge 1 or 2 vortex is optimal given the current system performance.
![(a) Model of the pupil mask stage for KPIC. (b) Charge 2 $K$-band vector vortex masks being considered for deployment in KPIC as viewed in cross-polarization. (c) The PSF produced by the vortex masks in (b) when placed in a pupil plane, which takes on the expected donut shape. (d) Same as (c), but on a logarithmic scale.[]{data-label="fig:KPIC_CoronagraphModule"}](KPIC_CoronagraphStage.jpg){width="\linewidth"}
We chose to start with $K$-band $(2.2\mu m)$ operation first for similar reasons; the wavefront errors scale with wavelength. Nevertheless, we have considered the possibility of including $H$- $(1.65\mu m)$ or $J$- $(1.25\mu m)$ band operation in the future and have left a clear path to implementing this capability if the AO performance allows for it. Longer wavelengths are also possible, but the performance may be limited by thermal background.
Given these design considerations, we have started testing charge 2 $K$-band vortex masks in the lab (see Fig. \[fig:KPIC\_CoronagraphModule\]b). We measured the transmission of these masks at $2\mu m$ to be $>99\%$. We also put these vortex masks in the pupil plane of a simple optical system to image their PSF. The resulting PSFs (Fig. \[fig:KPIC\_CoronagraphModule\]c-d) show the expected donut pattern. We plan to further validate these masks with polychromatic coupling measurements on the upgraded VFN testbed as well as on a dedicated KPIC testbed at Caltech[@Pezzato2019SPIE].
Predicted On-Sky VFN Performance {#sec:VFNSimulations}
--------------------------------
To predict the performance of the KPIC VFN mode, we have started simulating the system assuming wavefront errors based on PyWFS measurements made during KPIC on-sky engineering runs. The PyWFS provides the residual wavefront error sampled at 1 kHz. Although the PyWFS beam path has optics that are non-common with the VFN beam path, we assume that the measurements represent the wavefront just before the vortex mask. In practice, image sharpening routines will be run on the tracking camera during the daytime before any KPIC observation run to minimize the non-common path aberrations. The PyWFS also provides the residual tip/tilt error at 1 kHz which we can feed into the simulator as well. For now, these simulations assume that the TTM is not being used for fast control so we apply the tip/tilt residuals from the PyWFS directly. Additionally, the simulator also accounts for the predicted residual atmospheric dispersion left over by the ADC.
For this paper, we used PyWFS data obtained on June 17^th^, 2019 during an on-sky engineering run. Figure \[fig:RawPyWFSData\] shows the residual wavefront errors measured by the PyWFS. Figure \[fig:RawPyWFSData\]a is a sample of the wavefront residuals while Fig. \[fig:RawPyWFSData\]b has the tip/tilt residuals in milliarcseconds (mas) for the full 60 seconds of data at 1 kHz. The average seeing that night was about 0.6 arcseconds. The average RMS wavefront residuals in the minute-long sample were 150 nm while the tip/tilt residuals were 2.6 and 2.5 mas RMS respectively. The spatial wavefront sampling shown in Fig. \[fig:RawPyWFSData\]a is sufficient for simulating the VFN performance since the VFN is fairly insensitive to high-frequency aberrations[@Ruane2018_VFN].
![(a) Sample of the PyWFS wavefront residuals shown as projected onto the low-order facility DM. The average RMS wavefront residual over the 60 seconds of data was 150 nm. (b) Tip/Tilt residuals as reported by the PyWFS. The RMS tip and tilt residuals for the 1 minute sample are 2.6 and 2.5 mas respectively.[]{data-label="fig:RawPyWFSData"}](PyWFS_RawData_Hist.jpg){width="0.9\linewidth"}
In the simulator, we decompose the real wavefront data into Zernike coefficients and then reconstruct the wavefront as projected onto the real Keck pupil. Figure \[fig:SimulatorResults\] shows the final frame, or time-step, of the simulator for a charge 2 (Fig. \[fig:Charge2Simulation\]) and charge 1 (Fig. \[fig:Charge1Simulation\]) VFN case. As such, the reconstructed wavefront at this final time step is shown in the upper left plots. We then add in the tip/tilt residuals from Fig. \[fig:RawPyWFSData\]b at the given time sample as well as the predicted chromatic dispersion left over from the ADC to get the net pupil phase as a function of wavelength. We apply the vortex phase assuming an ideal, achromatic charge 1 or charge 2 vortex mask and calculate the resulting PSF at five sample wavelengths across the band. The upper middle plot of Figs. \[fig:SimulatorResults\]a,b shows the broadband PSF, which would be imaged on the tracking camera.
To get the coupling efficiencies across the band, we compute Eq. \[eqn:couplingeff\] for every point in the field at each sample wavelength. The resulting 2D coupling map is shown in the upper right plot of Figs. \[fig:SimulatorResults\]a,b for the central operating wavelength of $2.2\mu m$. We calculate the predicted planet coupling, $\eta_p$, as the average for all points between $0.8\text{-}1.0\lambda/D$ for the charge 1 case and $1.3\text{-}1.5\lambda/D$ for charge 2 to account for uncertainties in the planet location. The region used in this average is shown between the two red circles superimposed on the 2D coupling map and encompasses the separation at which the theoretical peak coupling occurs for each vortex charge. The resulting planet coupling, plotted against time, is shown in the lower left plot of Figs. \[fig:SimulatorResults\]a,b. The time-averaged planet coupling is shown at the bottom left of this plot. Under these assumptions, we find that the predicted time-averaged planet coupling is 8% for a charge 2 vortex and just over 14% for charge 1.
In order to compute the star coupling, $\eta_s$, we must choose where to place the SMF in our simulations. Due to the tip/tilt residuals, the PSF moves around with respect to the SMF much faster than we can track and compensate for with the current TTM control loop. This means that taking the null point in the coupling map at each frame would be an unfair representation of the actual on-sky performance since we would be assuming that we can align the PSF with the SMF core infinitely fast. We therefore take the average of all the coupling maps and find the optimal null location in this time-averaged map. We then place our simulated fiber at that location and compute the coupling efficiency there at each time sample. This is representative of what we expect from a realistic TTM control loop which will tend to average out the tip/tilt residuals. The resulting star coupling is shown in the lower middle plot of Figs. \[fig:SimulatorResults\]a,b. The time-averaged null depth is reported in the upper left corner of this plot. Given the wavefront residuals used in this simulation as well as the predicted ADC residuals, we get an average null depth of 0.6% $(6\times10^{-3})$ for the charge 2 case and 1.3% $(1.3\times10^{-2})$ for charge 1. The final, lower right, plot in Figs. \[fig:SimulatorResults\]a,b shows the instantaneous coupling efficiency for the star and planet at each of the 5 sample wavelengths across our $K$-band simulation. This assumes a flat spectrum for both the star and planet.
Thus, Fig. \[fig:Charge2Simulation\] represents the predicted performance for the planned KPIC Charge 2 VFN mode while Fig. \[fig:Charge1Simulation\] shows a possible charge 1 case for comparison. As expected, the planet coupling at the peak planet location increases to 14% with the charge 1 vortex mask but the null depth also degrades to 1.3%. The tradeoff is whether the decrease in null depth is worth the access to closer companions. The images shown in Fig. \[fig:SimulatorResults\] are stills of the final frame in the simulation. The video version of these figures, showing the instantaneous wavefront, PSF, and coupling, is available in the online copy of these proceedings.
The results of these simulations are promising and indicate that the current PyWFS performance is sufficient for obtaining $<10^{-2}$ nulls while coupling 8% of the planet light with a charge 2 vortex as planned. We can expect that this performance will improve further once the high-order DM is integrated into the KPIC system. However, these simulations are preliminary and there are other effects that may impact the VFN performance including realistic polarization aberrations, on-sky ADC residuals, and non-common path aberrations. We are also working towards simulating the characterization capabilities of the KPIC VFN mode by injecting simulated planet atmospheric spectra, accounting for the planet-star contrast ratios, applying the throughput losses in the rest of the system, and attempting to extract molecules from the resulting signal[@Wang2017].
![(Video online) KPIC VFN performance simulations given the on-sky performance of the PyWFS as well as the predicted ADC residuals with an ideal vortex mask of (a) charge 2 and (b) charge 1. The upper left plot in each is the reconstructed PyWFS residuals. The upper middle plot is the system PSF while the upper right is the corresponding coupling efficiency for all points within a $6\times6\lambda/D$ field of view. The lower left plot shows the average coupling for a planet within the region bound by the two red circles in the coupling map. The lower middle plot is the star coupling. The lower right plot is the planet and star coupling at each of the 5 sample wavelengths across the $K$-band.[]{data-label="fig:SimulatorResults"}](Sims_Charge2.png "fig:"){width="\linewidth"} \[fig:Charge2Simulation\]
![(Video online) KPIC VFN performance simulations given the on-sky performance of the PyWFS as well as the predicted ADC residuals with an ideal vortex mask of (a) charge 2 and (b) charge 1. The upper left plot in each is the reconstructed PyWFS residuals. The upper middle plot is the system PSF while the upper right is the corresponding coupling efficiency for all points within a $6\times6\lambda/D$ field of view. The lower left plot shows the average coupling for a planet within the region bound by the two red circles in the coupling map. The lower middle plot is the star coupling. The lower right plot is the planet and star coupling at each of the 5 sample wavelengths across the $K$-band.[]{data-label="fig:SimulatorResults"}](Sims_Charge1.png "fig:"){width="\linewidth"} \[fig:Charge1Simulation\]
Summary and Future Outlook {#sec:Conclusion}
==========================
VFN is a promising new concept that will soon be tested on-sky with the KPIC instrument. The concept has been demonstrated in the lab with a visible wavelength, monochromatic null depth of $6\times10^{-5}$ and planet coupling of 16% with a charge 1 vortex. We have also demonstrated a polychromatic null of $4.2\times10^{-4}$ and planet coupling of $4.5\%$ in 10% bandwidth visible light with a charge 2 vortex mask. We have laid out a path for upgrading the Caltech VFN testbed in order to further improve the VFN performance as well as move to longer wavelength demonstrations in preparation for the deployment of a KPIC VFN mode. Using measurements taken from the KPIC pyramid wavefront sensor during an on-sky engineering night, we have simulated the predicted performance of the VFN mode at first light.
Moving forward, we plan to continue developing the VFN concept on the upgraded testbed at Caltech. Some of the planned experiments include switching to longer wavelengths and larger bandwidths as well as testing scalar vortex designs. We also plan on installing a low-order DM into the VFN testbed to start developing wavefront control algorithms that are tailored to match VFN requirements.
For on-sky operations of a VFN mode on KPIC, we will need to develop automated algorithms and procedures for finding and maintaining a deep null. These are already being developed on the VFN testbed and will eventually be moved to the telescope system along with the wavefront control algorithms. Once the VFN architecture is integrated, we will start commissioning the VFN mode following a similar fashion to what was done for the direct planet spectroscopy mode. Thus, we will target bright binaries first to confirm that we can null one and extract the spectral signatures of the other on the NIRSPEC spectrograph. As the acquisition, tracking, and extraction algorithms improve, we will move towards known, RV-detected, young giant planets and eventually transition to blind detection surveys around promising M-dwarf stars.
VFN may enable us to collect and analyze the reflected spectra of exoplanets that are inaccessible to conventional coronagraphic techniques; i.e. at separations $<1\lambda/D$. These are mainly young giant planets of which there is a large population detected by RV and transit techniques. Thus, a VFN mode on a large ground-based telescope (e.g. Keck or TMT) or on a space-based telescope (e.g. the LUVOIR mission concept[@Bolcar2018_LUVOIR]) would be complementary to proposed coronagraphic instruments and would increase the scientific yield with minimal modifications to the system.
Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA).
|
---
abstract: 'We prove homological mirror symmetry for Milnor fibers of simple singularities, which are among the log Fano cases of [@1806.04345 Conjecture 1.5]. The proof is based on a relation between matrix factorizations and Calabi–Yau completions. As an application, we give an explicit computation of the symplectic cohomology group of the Milnor fiber of a simple singularity in all dimensions.'
address:
- ' Department of Mathematics King’s College London Strand London WC2R 2LS'
- ' Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan.'
author:
- Yanki Lekili
- Kazushi Ueda
bibliography:
- 'bibs.bib'
title: Homological mirror symmetry for Milnor fibers of simple singularities
---
Introduction
============
A *simple singularity* is an isolated hypersurface singularity of modality zero. Arnold classified such singularities; up to right equivalence, they are given by one of the following: $$\begin{aligned}
\label{eq:simple}
\begin{split}
A_\ell &\colon x_1^{\ell+1} + x_2^2 + \cdots + x_{n+1}^2 = 0, \\
D_\ell &\colon x_1^{\ell-1} + x_1 x_2^2 + x_3^2+ \cdots + x_{n+1}^2 = 0, \\
E_6 &\colon x_1^4 + x_2^3 + x_3^2 + \cdots + x_{n+1}^2 = 0, \\
E_7 &\colon x_1^3 + x_1 x_2^3 + x_3^2 + \cdots + x_{n+1}^2 = 0, \\
E_8 &\colon x_1^5 + x_2^3 + x_3^2 + \cdots + x_{n+1}^2 = 0.
\end{split}\end{aligned}$$ In the case $n=2$, simple surface singularities have many other characterizations, such as Kleinian singularities, rational double points, or canonical singularities, to name a few.
Let $\wv$ be one of these defining polynomials, which we think of as a holomorphic function on $\bC^{n+1}$, and equip $\wv^{-1}(1)$ with the Liouville structure induced from the standard one on $\bC^{n+1}$. This is the Liouville completion of the *Milnor fiber*, which is the Liouville domain obtained by intersecting $\wv^{-1}(1)$ with a ball. Let $\cW \lb \wv^{-1}(1) \rb$ denote the idempotent-complete derived wrapped Fukaya category of $\wv^{-1}(1)$.
For $n \geq 2$, since $\wv^{-1}(1)$ is not a log Calabi–Yau manifold but a log Fano manifold, its mirror is not a manifold but a *Landau–Ginzburg model*, by which we mean a pair of a stack and a section of a line bundle on it. One way to obtain a Landau–Ginzburg mirror of a log Fano manifold is to first remove a divisor to make it log Calabi–Yau, then find its mirror, which is another log Calabi–Yau manifold, and finally add a potential to this mirror [@MR2386535; @MR2537081]. This produces a Landau–Ginzburg mirror whose underlying manifold is of the same dimension as the original manifold. When the singularity is toric (i.e., a simple surface singularity of type A), there is a standard choice for the divisor to remove, and the resulting mirror is the Landau–Ginzburg model consisting of a complement of a toric divisor in the minimal resolution of the singularity of the same type and a monomial function on it (see e.g. [@MR3502098 Section 9.2]). The choice of the divisor is not unique in general, and there are multiple mirrors for a given Milnor fiber.
In this paper, we consider an alternative mirror of the Milnor fiber of a simple singularity based on transposition of invertible polynomials introduced in [@MR1214325; @MR1310310]. A weighted homogeneous polynomial $
\w \in \bC[x_1,\ldots,x_{n+1}]
$ with an isolated critical point at the origin is *invertible* if there is an integer matrix $
A = (a_{ij})_{i, j=1}^{n+1}
$ with non-zero determinant such that $$\begin{aligned}
\w = \sum_{i=1}^{n+1} \prod_{j=1}^{n+1} x_j^{a_{ij}}.
$$ The *transpose* of $\w$ is defined as $$\begin{aligned}
\wv = \sum_{i=1}^{n+1} \prod_{j=1}^{n+1} x_j^{a_{ji}},\end{aligned}$$ whose exponent matrix $\Av$ is the transpose matrix of $A$. The group $$\begin{aligned}
\Gamma_{\w} \coloneqq
\lc (t_0, t_1, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^{a_{1,1}} \cdots t_{n+1}^{a_{1,{n+1}}}
= \cdots
= t_1^{a_{{n+1},1}} \cdots t_{n+1}^{a_{{n+1},{n+1}}}
= t_0 t_1 \cdots t_{n+1}
\rc \end{aligned}$$ acts naturally on $
\bA^{n+2} \coloneqq \Spec \bC[x_0,\ldots,x_{n+1}].
$ Let $
\mf \lb \bA^{n+2}, \Gamma_{\w}, \w + x_0 \cdots x_{n+1} \rb
$ denote the idempotent completion of the dg category of $\Gamma_{\w}$-equivariant coherent matrix factorizations of $\w + x_0 \cdots x_{n+1}$ on $\bA^{n+2}$ in the sense of [@MR3366002]. below is given in [@1806.04345 Conjecture 1.5]:
\[cj:hms\] For any invertible polynomial $\w$, one has a quasi-equivalence $$\begin{aligned}
\label{eq:hms}
\mf \lb \bA^{n+2}, \Gamma_\w, \w+x_0 \cdots x_{n+1} \rb
\simeq \cW \lb \wv^{-1}(1) \rb.\end{aligned}$$
In other words, the Landau–Ginzburg model $
\lb \ld \bA^{n+2}/\Gamma_{\w} \rd, \w + x_0 \cdots x_{n+1} \rb
$ is mirror to the Liouville manifold $\wv^{-1}(1)$. The main result of this paper is the following:
\[th:main\] holds for $n \ge 2$ and $\wv$ one of the defining polynomials of simple singularities appearing in .
For $n=1$, was recently studied by [@2003.01106] who proved an equivalence of the full subcategories consisting of objects with finite-dimensional morphism spaces.
This paper is organized as follows: In , we collect basic definitions and results on Calabi–Yau completions and trivial extension algebras. In , we recall the description of the wrapped Fukaya category of the Milnor fiber of a simple singularity for $n \ge 2$ in terms of the $n$-Calabi–Yau completion of a Dynkin quiver of the corresponding type. In , we show that $\mf(\bA^{n+2}, \Gamma, \w)$ is the $n$-Calabi–Yau completion of $\mf(\bA^{n+1}, \Gamma_{\w}, \w)$ under mild conditions on $\Gamma$ and $\w$. When $\wv$ is one of the defining polynomials of simple singularities appearing in , $\mf(\bA^{n+2}, \Gamma_{\w}, \w+x_0 \cdots x_{n+1})$ is quasi-equivalent to $\mf(\bA^{n+2}, \Gamma_{\w}, \w)$. Since $\mf(\bA^{n+1}, \Gamma_{\w}, \w)$ is quasi-equivalent to the derived category of representations of the Dynkin quiver of the corresponding type, is proved. As an application, we explicitly compute the symplectic cohomology of the Milnor fiber of a simple singularity in all dimensions in .
*Acknowledgment*: Y. L. is partially supported by the Royal Society (URF) and the NSF grant DMS-1509141. K. U. is partially supported by Grant-in-Aid for Scientific Research (15KT0105, 16K13743, 16H03930).
Calabi–Yau completions and trivial extension algebras {#sc:quiver}
=====================================================
The *$n$-Calabi–Yau completion* (or the *derived $n$-preprojective algebra*) of a dg algebra $
\scrA
$ is defined in [@MR2795754 Section 4.1] as the tensor algebra $$\begin{aligned}
\Pi_{n} (\scrA)
&\coloneqq T_{\scrA}(\theta)
\coloneqq \scrA \oplus \theta \oplus \theta \otimes_\scrA \theta \oplus \cdots,\end{aligned}$$ where the $\scrA$-bimodule $
\theta
\coloneqq \Theta[n-1]
$ is a shift of the inverse dualizing complex $
\Theta
\coloneqq \hom_{\scrA^{\mathrm{e}}} (\scrA, \scrA^{\mathrm{e}}).
$
The *Ginzburg dg algebra* $
\scrG_Q^{n}
$ of a quiver $Q$ (without potential) is a model of the $n$-Calabi–Yau completion $\Pi_{n} \lb A_Q \rb$ of the path algebra $A_Q$, defined in [@MR2795754 Section 6.2] after [@0612139] as the path algebra of the graded quiver $\overline{Q}$ with same vertices as $Q$ and arrows consisting of
- the original arrows $g \in Q_1$ in degree $1$,
- the opposite arrows $g^*$ for each arrow $g \in Q_1$ in degree $1-n$, and
- loops $h_v$ at each vertex $v \in Q_0$ in degree $1-n$,
equipped with the differential $d$ given by $$\begin{aligned}
dg = dg^* = 0 \ \mbox{ and } \ dh = \sum_{g\in Q_1} g^* g - g g^*\end{aligned}$$ where $h = \sum_{v\in Q_0} h_v$.
The *degree $n$ trivial extension algebra* of a finite-dimensional algebra $A$ is defined as $
A \oplus A^\dual[-n]
$ equipped with the multiplication $
(a, f) \cdot (b, g) = (ab, ag+fb),
$ where $
A^\dual
$ is the dual of $A$ as a vector space.
The degree $n$ trivial extension algebra $
B_Q^{n}
$ of the path algebra $A_Q$ of a Dynkin quiver $Q$ is the (derived) Koszul dual of $\scrG^{n}_Q$ in the sense that $$\begin{aligned}
\label{eq:Koszul_duality}
\hom_{\scrG_Q^{n}} \lb \bfk_{\scrG}, \bfk_{\scrG} \rb
\simeq B_Q^{n},
\qquad
\hom_{\lb B_Q^{n} \rb^{\mathrm{op}}} \lb \bfk_B, \bfk_B \rb
\simeq \lb \scrG_Q^{n} \rb^{\mathrm{op}},\end{aligned}$$ where $
\bfk_{\scrG} \coloneqq \bigoplus_{v \in Q_0} S_v
$ is the direct sum of simple $\scrG_Q^{n}$-modules $S_v$ associated with vertices $v \in Q_0$, and similarly for $\bfk_B$. The Koszul duality implies an isomorphism $$\begin{aligned}
\label{eq:hh}
\HH^* \lb \scrG_Q^{n} \rb
\cong \HH^* \lb B_{Q}^{n} \rb\end{aligned}$$ of Hochschild cohomologies (cf. e.g. [@MR3941473 Theorem 3.4]).
Wrapped Fukaya category of the Milnor fiber of simple singularity {#sc:wrap}
=================================================================
Let $\wv$ be one of the defining polynomials of a simple singularity and $M^{n} = \wv^{-1}(1)$ be the Milnor fiber, which we view as a Weinstein manifold where the Weinstein structure is induced by restriction from the ambient $\bC^{n+1}$. It is well known that this Weinstein manifold is symplectomorphic (in fact, Weinstein homotopic) to plumbing $X_Q$ of cotangent bundles of spheres $T^*S^{n}$ according to the Dynkin diagram $Q$ corresponding to the simple singularity. One way to see this is to verify it directly for $n=1$, and then use the fact that in higher dimensions the Milnor fiber is obtained by stabilization — increasing the dimension corresponds to suspension of the Lefschetz fibration [@MR2651908]. See also [@MR2786590] for an explicit construction of a symplectic structure on plumbings. This stabilization point of view also enables one to describe $M$ via Legendrian surgery. Namely $M$ is obtained by attaching critical handles to a Legendrian link $\Lambda_Q^{n-1}$ on $\partial \mathbb{D}^{n}$ whose components are unknotted Legendrian spheres $S^{n-1}$ which are clasped together (as in Hopf link) according to the Dynkin diagram $Q$. The direct sum of co-cores to the critical handles (i.e., cotangent fibers away from the plumbing region) form a generating object of the wrapped Fukaya category by the main theorem in [@1712.09126], and the surgery formula of [@MR2916289; @1906.07228] allows one to explicitly compute the endomorphism algebra of this generator as the Chekanov–Eliasberg algebra $\CE^*(\Lambda_Q^{n-1})$.
This Chekanov–Eliashberg algebra was computed directly in the case $n=2$ in the paper [@MR3692968] and the resulting dg algebra was shown to be quasi-isomorphic to the derived multiplicative preprojective algebra of the corresponding Dynkin type. Moreover, working over $\bC$, it was shown that the derived multiplicative preprojective algebra of Dynkin type $Q$ is quasi-isomorphic to the Ginzburg algebra $\mathscr{G}^2_Q$, also known as the derived (additive) preprojective algebra of Dynkin type $Q$.
For $n \ge 3$, one can do a direct computation in an analogous way, but we can also deduce this by the Koszul duality result given in [@1701.01284 Theorem 58] which shows that $\CE^*(\Lambda_Q^{n-1})$ is the (derived) Koszul dual of the endomorphism algebra of the union of the core spheres of the plumbing. Notice that for $n \ge 3$, $\wv$ is suspended at least twice, thus the formality of the endomorphism algebra of vanishing cycles in the compact Fukaya category of $\wv^{-1}(1)$ follows automatically by a result of Seidel [@MR2651908]. Putting it all together, we conclude that $\CE^*(\Lambda_Q^{n-1})$ is Koszul dual to the degree $n$ trivial extension algebra $B_Q^{n}$ of the path algebra $A_Q$ of a Dynkin quiver of the corresponding type (see also [@MR3977875] for another example).
As a result of these computations, for $n \ge 2$ we have a quasi-isomorphism $$\begin{aligned}
\label{eq:iso}
\CE^* \lb \Lambda_Q^{n-1} \rb \simeq \scrG^{n}_Q\end{aligned}$$ over $\bC$, which implies a quasi-equivalence $$\begin{aligned}
\label{eq:W_Pi}
\cW(\wv^{-1}(1)) \simeq \per \Pi_{n}(A_Q)\end{aligned}$$ between the wrapped Fukaya category of $\wv^{-1}(1)$ and the dg derived category of perfect modules over $\Pi_{n}(A_Q)$.
Note from [@MR1882336 Proposition 3.4] that $A_Q$ is derived equivalent to the Fukaya–Seidel category $\cF(\wv)$ of the LG-model $
\wv \colon \bC^{n+1} \to \bC.
$ Thus shows that $\cW \lb \wv^{-1}(1) \rb$ is the Calabi–Yau completion of $\cF(\wv)$ for $n \ge 2$. Although this relationship between $\cF(\wv)$ and $\cW \lb \wv^{-1}(1) \rb$ is not true in general, we expect it to hold when $\wv$ is a double suspension of an invertible polynomial whose Milnor fiber is a log Fano manifold.
The isomorphism remains true for $n \ge 3$ over an arbitrary commutative ring but for $n=2$ we have to require that $2$ is invertible for type $D_\ell, E_6,E_7,E_8$, $3$ is invertible for type $E_6,E_7, E_8$, and 5 is invertible for type $E_8$. Otherwise, $\CE^*(\Lambda_Q)$ is quasi-isomorphic to the derived multiplicative preprojective algebra (see [@MR4033516]) which is not quasi-isomorphic to the derived (additive) preprojective algebra $\Pi_{n}(A_Q)$.
Matrix factorizations and Calabi–Yau completions {#sc:CY-completion}
================================================
Let $\Gamma$ be a subgroup of $(\Gm)^{n+1}$ acting diagonally on $
\bA^{n+1} \coloneqq \Spec \bC[x_1,\ldots,x_{n+1}].
$ Assume that $\Gamma$ is a finite extension of the multiplicative group $\Gm$, so that the group $
\Char(\Gamma)
\coloneqq \Hom(\Gamma, \Gm)
$ of characters of $\Gamma$ is a finite extension of $\bZ$. The coordinate ring $\bC[x_1,\ldots,x_{n+1}]$ has a $\Char(\Gamma)$-grading coming from the $\Gamma$-action on $\bA^{n+1}$, and we set $\chi_i \coloneqq \deg x_i$ for $i \in \{ 1, \ldots, n+1 \}$. Let $
\w \in \bC[x_1,\ldots,x_{n+1}]_\chi
\coloneqq \lb \bC[x_1,\ldots,x_{n+1}] \otimes \chi \rb^\Gamma
$ be a homogeneous element of degree $
\chi
\in
\Char(\Gamma).
$ Asume that $\w$ has an isolated critical point at the origin, so that the structure sheaf $\cO_0$ of the origin split-generates $
\mf \lb \bA^{n+1}, \w \rb
$ by [@MR2776613 Proposition A.2] (see also [@MR2735755; @MR2824483]). Let $
R \subset \Char(\Gamma)
$ be a set of representatives of the group $
\Char(\Gamma)/(\chi),
$ which we assume to be finite. Then $
\cE \coloneqq \bigoplus_{\rho \in R} \cO(\rho)
$ generates $
\mf \lb \bA^{n+1}, \Gamma, \w \rb,
$ since the autoequivalence $
M \mapsto M(\chi)
$ of $
\mf \lb \bA^{n+1}, \Gamma, \w \rb
$ shifting the $\Gamma$-weight by $\chi$ is isomorphic to the functor $M \mapsto M[2]$ shifting the cohomological grading by 2.
The $n$-Calabi–Yau completion of the dg Yoneda algebra $
\scrA \coloneqq \hom(\cE, \cE)
$ is given by $$\begin{aligned}
\Pi_{n} (\scrA)
\coloneqq \scrA \oplus \theta \oplus \theta \otimes_\scrA \theta \oplus \cdots
\simeq \bigoplus_{i=0}^\infty \hom(\cE, \theta^i(\cE)),\end{aligned}$$ where we abuse notation and use the same symbol for an autoequivalence and its graph bimodule. Note that the autoequivalence $\theta$ is isomorphic to a shift of the inverse Serre functor $\bS^{-1}$; $$\begin{aligned}
\label{eq:theta_S}
\theta \simeq \bS^{-1}[n-1].\end{aligned}$$
Now, as in [@1806.04345 Section 2], we introduce another variable $x_0$ of degree $
\chi_0
\coloneqq \chi - (\chi_1 + \cdots + \chi_{n+1}),
$ and consider the polynomial ring $
\bC[x_0,x_1,\ldots,x_{n+1}]
$ in $n+2$ variables, which naturally contains $
\bC[x_1,\ldots,x_{n+1}]
$ as a subring. One can show, e.g., by taking the trivial $G$, $H$, and $v$ in [@MR3270588 Lemma 3.52], that $$\begin{aligned}
\label{eq:kunneth}
\mf(\bA^{n+2},\w) \simeq \coh \bA^1_{x_0} \otimes \mf(\bA^{n+1},\w).\end{aligned}$$ As shown in [@MR3063907 Corollary 2.5], graded Auslander–Reiten duality [@MR915178] implies that $$\begin{aligned}
\label{eq:mf_Serre}
\bS \coloneqq (\chi_0)[n-1]\end{aligned}$$ is a Serre functor on $\mf \lb \bA^{n+1}, \Gamma, \w\rb$. It follows from and that $$\begin{aligned}
\label{eq:theta_chi}
\theta \simeq (-\chi_0).\end{aligned}$$
Let $\cF$ be the generator of $\mf \lb \bA^{n+2}, \Gamma, \w \rb$ obtained from the tensor product of the generator $\cE$ of $
\mf \lb \bA^{n+1}, \Gamma, \w \rb
$ and the generator $\bC[x_0]$ of $\coh \bA^1$. If we write both of the forgetful functors $
\mf \lb \bA^{n+1}, \Gamma, \w \rb \to \mf \lb \bA^{n+1}, \w \rb
$ and $
\mf \lb \bA^{n+2}, \Gamma, \w \rb \to \mf \lb \bA^{n+2}, \w \rb
$ as $\overline{(\bullet)}$, then one has $$\begin{aligned}
\label{eq:non-equiv}
\hom \lb \overline{\cF},\overline{\cF} \rb
\simeq \hom \lb \overline{\cE}, \overline{\cE} \rb \otimes \bC[x_0]
\simeq \bigoplus_{\rho \in \Char(\Gamma)} \hom(\cE, \cE(\rho)) \otimes \bC[x_0].\end{aligned}$$ Since $\deg (x_0) = \chi_0$, by taking the $\Gamma$-invariant part of and using , one obtains $$\begin{aligned}
\hom(\cF,\cF)
\simeq \bigoplus_{i=0}^\infty \hom(\cE, \cE(-i \chi_0))
\simeq \bigoplus_{i=0}^\infty \hom(\cE, \theta^i(\cE))
\simeq \Pi_{n}(\scrA),\end{aligned}$$ which shows that $\mf \lb \bA^{n+2}, \Gamma, \w \rb$ is the $n$-Calabi–Yau completion of $\mf \lb \bA^{n+1}, \Gamma, \w \rb$.
When $\wv$ is one of the defining polynomials of simple singularities appearing in , it follows from [@0506347; @MR2803848; @MR3030671; @1903.01351] and the Knörrer periodicity that there exists a generator $\cG$ of $
\mf \lb \bA^{n+1}, \Gamma_\w, \w \rb
$ whose dg Yoneda algebra is quasi-isomorphic to the path algebra $A_Q$ of a Dynkin quiver of the corresponding type, so that $$\begin{aligned}
\label{eq:mf_Pi}
\mf \lb \bA^{n+2}, \Gamma_\w, \w \rb \simeq \per \Pi_{n}(A_Q).\end{aligned}$$
When $n$ is greater than one, the polynomial $\w+x_0 \cdots x_{n+1}$ considered as an element of $\bC[\![x_0]\!][\![x_1,\ldots,x_{n+1}]\!]$ (i.e., a formal one-parameter deformation of a formal germ of $\w$) is right equivalent to $\w$ by a formal coordinate change (i.e., there exists $
\varphi
\in \Aut_{\bC[\![x_0]\!]} \bC[\![x_0]\!][\![x_1,\ldots,x_{n+1}]\!]
$ such that $\varphi^*(\w+x_0 \cdots x_{n+1}) = \w$) since the degree of $x_1 \cdots x_{n+1}$ is greater than that of any element in the Jacobi ring $$\begin{aligned}
\Jac_{\w} \coloneqq \bC[x_1,\ldots,x_{n+1}] / (\partial_{x_1} \w, \ldots, \partial_{x_{n+1}} \w)\end{aligned}$$ of $\w$ (see e.g. [@MR777682 Section 12.6]). Moreover, one can choose $\varphi$ to be $\Gamma_\w$-equivariant, so that $
\varphi
\in \Aut_{\bC[x_0]} \bC[x_0][\![x_1,\ldots,x_{n+1}]\!]
$ (in fact, for any $i \in \{ 1, \ldots, n+1 \}$, the coefficient $a_{i,m_1,\ldots,m_{n+1}}(x_0)$ of the expansion $
\varphi^* (x_i)
= \sum_{m_1, \ldots, m_{n+1}=0}^\infty
a_{i,m_1,\ldots,m_{n+1}}(x_0) x_1^{m_1} \cdots x_{n+1}^{m_{n+1}}
$ is a monomial in $x_0$). This implies $$\begin{aligned}
\label{eq:completion}
\mf \lb \bA^{n+2}, \Gamma_\w, \w+x_0 \cdots x_{n+1} \rb
\simeq \mf \lb \bA^{n+2}, \Gamma_\w, \w \rb\end{aligned}$$ by [@MR2735755 Theorem 2.10], and is proved.
Symplectic cohomology of the Milnor fiber of simple singularity {#sc:SH}
===============================================================
Symplectic cohomology as Hochschild cohomology {#sc:SH_HH}
----------------------------------------------
The closed-open map of any Weinstein manifold is an isomorphism because of [@MR3121862 Theorem 1.1] and the fact, implied by [@1712.09126 Theorem 1.4] which builds on [@MR3121862; @1712.00225], that any Weinstein manifold is non-degenerate in the sense of [@MR3121862 Definition 1.1]; $$\begin{aligned}
\label{eq:SH_HH}
\mathrm{SH}^*(M) \simto \HH^*(\cW(M)).\end{aligned}$$ The combination of , , , and the derived Morita invariance of Hochschild cohomology shows that the symplectic cohomology of the Milnor fiber $\wv^{-1}(1)$ of a simple singularity is isomorphic to $
\HH^*(\mf(\bA^{n+2}, \Gamma_\w, \w)).
$ As we discuss below, the latter has an explicit description which allows us to compute $\SH^*(\wv^{-1}(1))$ for the Milnor fibers of all simple singularities. Previous partial results in this direction based on different techniques were obtained in [@MR3692968; @MR3483060; @MR3489066], which are in agreement with our computations.
Hochshschild cohomology via matrix factorizations
-------------------------------------------------
We use the same notation as in , and set $$\begin{aligned}
V \coloneqq \bC x_0 \oplus \bC x_1 \oplus \cdots \oplus \bC x_{n+1}.\end{aligned}$$ Then [@MR2824483; @MR3084707; @MR3108698; @MR3270588] (cf. also [@1806.04345 Theorem 3.1]) shows that $
\HH^t \lb \mf \lb \bA^{n+2}, \Gamma, \w \rb \rb
$ is isomorphic to $$\begin{gathered}
\label{eq:HHmf}
\lb \bigoplus_{\substack{\gamma \in \ker \chi, \ l \geq 0 \\ t - \dim N_\gamma = 2u }}
H^{-2l}(d \w_\gamma) \otimes \chi^{\otimes (u+l)} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual \right. \\
\left. \oplus \bigoplus_{\substack{\gamma \in \ker \chi, \ l \geq 0 \\ t - \dim N_\gamma = 2u+1}}
H^{-2l-1}(d \w_\gamma) \otimes \chi^{\otimes (u+l+1)} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\vee \rb^\Gamma.\end{gathered}$$ Here $H^i(d \w_\gamma)$ is the $i$-th cohomology of the Koszul complex $$\begin{aligned}
\label{eq:Koszul}
C^*(d \w_\gamma) \coloneqq \lc
\cdots
\to \Lambda^2 V_\gamma^\dual \otimes \chi^{\otimes (-2)} \otimes S_\gamma
\to V_\gamma^\dual \otimes \chi^\dual \otimes S_\gamma
\to S_\gamma \rc,\end{aligned}$$ where the rightmost term $S_\gamma$ sits in cohomological degree 0, and the differential is the contraction with $$\begin{aligned}
d \w_\gamma \in \lb V_\gamma \otimes \chi \otimes S_\gamma \rb^\Gamma.\end{aligned}$$ The vector space $V_\gamma$ is the subspace of $\gamma$-invariant elements in $V$, $S_\gamma$ is the symmetric algebra of $V_\gamma$, $\w_\gamma$ is the restriction of $\w$ to $\Spec S_\gamma$, and $N_\gamma$ is the complement of $V_\gamma$ in $V$ so that $V \cong V_\gamma \oplus N_\gamma$ as a $\Gamma$-module.
If $\w_\gamma$ has an isolated critical point at the origin, then the cohomology of is concentrated in degree 0, so that only the summand $$\begin{aligned}
\label{eq:l=0}
\lb \Jac_{\w_\gamma} \otimes \chi^{\otimes u} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual \rb^\Gamma\end{aligned}$$ with $l = 0$ in contributes to $\HH^{2 u + \dim N_\gamma}$.
If $V_\gamma$ contains $\bC x_0$, then the Koszul complex $C^*(d \w_\gamma)$ is isomorphic to the tensor product of $C^*(d \w_\gamma')$ and the complex $
\lc \bfk x_0^\dual \otimes \chi^\dual \otimes \bfk[x_0] \to \bfk[x_0] \rc
$ concentrated in cohomological degree $[-1,0]$ with the zero differential, where $\w_\gamma'$ is the restriction of $\w$ to the complement $V_\gamma'$ of $\bC x_0$ in $V_\gamma$. If $\w_\gamma'$ has an isolated critical point at the origin, then $C^*(d \w_\gamma')$ is quasi-isomorphic to $\Jac_{\w_\gamma'}$ concentrated at the origin, so that only the summands $$\begin{aligned}
\label{eq:HH0_0}
\lb \Jac_{\w_\gamma'}
\otimes \bC[x_0]
\otimes \chi^{\otimes u}
\otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual \rb^\Gamma\end{aligned}$$ and $$\begin{aligned}
\label{eq:HH0_1}
\lb \bC x_0^\dual \otimes \Jac_{\w_\gamma'}
\otimes \bfk[x_0]
\otimes \chi^{\otimes u}
\otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual \rb^\Gamma\end{aligned}$$ with $l=0$ in contribute to $\HH^{2u+\dim N_\gamma}$ and $\HH^{2u+\dim N_\gamma+1}$ respectively.
Bigrading on Hochschild cohomology
----------------------------------
The Hochschild cohomology of a graded algebra $B$ (with no differential) has a bigrading such that $$\begin{aligned}
\HH^{r+s} \lb B \rb^s
\coloneqq \Ext_{B^{\mathrm{op}} \otimes B}^r \lb B, B[s] \rb.\end{aligned}$$
When $B$ is the trivial extension algebra $B^{n}$ of a finite-dimensional algebra $A$, by introducing a $\Gm$-action on $B^{n}$ such that $A$ has weight $0$ and $A^\dual[-n]$ has weight $n$, the $s$-grading on $\HH^* \lb B^{n} \rb$ can be described as the weight of the induced $\Gm$-action.
For any positive integer $m$, the underlying ungraded algebra of the trivial extension algebras $B^{mn}$ is isomorphic to $B^{n}$, and only the cohomological gradings are different; that of the former is $m$ times that of the latter. It follows that one has an isomorphism $$\begin{aligned}
\HH^{r+ms} \lb B^{mn} \rb^{ms}
\cong \HH^{r+s} \lb B^{n} \rb^s\end{aligned}$$ of vector spaces for any positive integer $m$ such that the parities of $n$ and $mn$ are the same (note that the signs in the Hochschild complex depend on the parity of the cohomological grading).
When $Q$ is a Dynkin quiver, one can transport the $\Gm$-action on $B_Q^{n}$ to $\scrG_Q^{n}$ through the Koszul duality , so that $g$ for $g \in Q_1$ has weight $0$, $g^*$ for $g \in Q_1$ has weight $-n$, and $h_v$ for $v \in Q_0$ has weight $-n$. This makes the isomorphism $\Gm$-equivariant, so that the $\Gm$-weights on both sides agree.
Since $\w$ does not depend on $x_0$, the $\Gm$-action on $\bA^{n+2}$ such that the weight of $x_i$ is $-n$ for $i=0$ and $0$ for $i \in \{ 1, \ldots, n+1 \}$ keeps $\w$ invariant. This induces a $\Gm$-action on $\mf(\bA^{n+2}, \Gamma, \w)$, and hence on $B_Q^{n}$, whose weight is $0$ on $A_Q$ and $n$ on $A_Q^\dual[-n]$ just as in [@1806.04345]. This allows us to compute the $s$-grading on $\HH^* \lb B_Q^{n} \rb$ as the $\Gm$-weight on . This $\Gm$-action is mirror to the one introduced in [@MR2929070] and studied further for type A Milnor fibers in [@MR3033519].
Type $A_\ell$
-------------
Consider the case $$\begin{aligned}
\w = x_1^{\ell+1} + x_2^2 + \cdots + x_{n+1}^2 \in \bC[x_0, x_1,\ldots,x_{n+1}]\end{aligned}$$ with $$\begin{aligned}
\Gamma
&= \Gamma_\w
\coloneqq \lc \gamma = (t_0, t_1, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^{\ell+1} = t_2^2 = \cdots = t_{n+1}^2 = t_0 t_1 \cdots t_{n+1} \rc,\end{aligned}$$ so that $
\chi(\gamma) = t_1^{\ell+1},
$ $
\ker \chi \cong \bmu_{\ell+1} \times \lb \bmu_2 \rb^{n},
$ and $\Char(\Gamma)$ is generated by $\chi$ and $\chi_i = \deg x_i$ for $i \in \{ 0, \ldots, n+1 \}$ with relations $$\begin{aligned}
\chi = (\ell+1) \chi_1 = 2 \chi_2 = \cdots = 2 \chi_{n+1}.\end{aligned}$$
###
For any $\gamma \in \ker \chi$, one has $$\begin{aligned}
\Jac_{\w_\gamma} =
\begin{cases}
\bC[x_0] \otimes \bC[x_1]/(x_1^\ell) & \bC x_0 \oplus \bC x_1 \subset V_\gamma, \\
\bC[x_0] & \bC x_0 \subset V_\gamma \text{ and } \bC x_1 \not \subset V_\gamma, \\
\bC[x_1]/(x_1^\ell) & \bC x_0 \not \subset V_\gamma \text{ and } \bC x_1 \subset V_\gamma, \\
\bC & \text{otherwise}.
\end{cases}\end{aligned}$$ If we write an element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ as $$\begin{aligned}
x_0^{k_0} x_1^{k_1}
\otimes x_{j_1}^\dual \wedge x_{j_2}^\dual \wedge \ldots \wedge x_{j_s}^\dual,\end{aligned}$$ where $k_0 = 0$ if $\bC x_0 \not \subset V_\gamma$ and $k_1 = 0$ if $\bC x_1 \not \subset V_\gamma$, then its degree is given by $$\begin{aligned}
k_0 \chi_0 + k_1 \chi_1 - \chi_{j_1} - \cdots - \chi_{j_s},\end{aligned}$$ which can be proportional to $\chi$ only if $V_\gamma$ is either $V$, $\bC x_0 \oplus \bC x_1$, $\bC x_0$, or $0$.
###
One has $
V_\gamma = V
$ if and only if $\gamma$ is the identity element. The degree of $
x_0^{k_0} x_1^{k_1} \in \Jac_\w
$ is $$\begin{aligned}
k_0 \chi - (k_0 - k_1) \chi_1 - k_0 \chi_2 - \cdots - k_0 \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is even and $\ell+1$ divides $k_0 - k_1$. Such an element can be written as $$\begin{aligned}
\bsa_{k,m} \coloneqq x_0^{k+m(\ell+1)} x_1^k,\end{aligned}$$ where $
k \in \{ 0, \ldots, \ell-1 \}
$ and $
m \in \bN
$ satisfies
- if $\ell$ is even, then the parities of $k$ and $m$ agree, and
- if $\ell$ is odd, then $k$ is even.
Since $$\begin{aligned}
\deg \lb x_0^{k+m(\ell+1)} x_1^k \rb
&= (k+m(\ell+1)) \chi - m \chi - \frac{1}{2} (k+m(\ell+1))n \chi \\
&= \lb (k+m \ell) - \frac{1}{2} (k+m(\ell+1))n \rb \chi,\end{aligned}$$ the element $x_0^{k+m(\ell+1)} x_1^k$ for such $(k,m)$ contributes $\bC((k+m(\ell+1))n)$ to $\HH^t$ for $t = 2(k+m \ell) - (k+m(\ell+1))n$. Similarly, for each such $(k,m)$, the element $$\begin{aligned}
\bsalpha_{k,m} \coloneqq x_0^\dual \otimes x_0^{k+m(\ell+1)+1} x_1^k
\in \bC x_0^\dual \otimes \Jac_{\w}
$$ contributes $\bC((k+m(\ell+1))n)$ to $\HH^{t+1}$ for $t = 2(k+m \ell) - (k+m(\ell+1))n$.
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_1
$ if and and only if $n$ is even and $
\gamma = (1, 1, -1, \ldots, -1).
$ The degree of $$\begin{aligned}
x_0^{k_0} x_1^{k_1} \otimes x_2^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi + (k_1 - k_0) \chi_1 - (k_0+1) \chi_2 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is odd and $\ell+1$ divides $k_1 - k_0$. Such an element can be written as $$\begin{aligned}
\bsa_{k,m} \coloneqq x_0^{k+m(\ell+1)} x_1^k \otimes x_2^\dual \wedge \cdots \wedge x_{n+1}^\dual,\end{aligned}$$ where $
k \in \{ 0, \ldots, \ell-1 \}
$ and $
m \in \bN
$ satisfies
- if $\ell$ is even, then the parities of $k$ and $m$ differ, and
- if $\ell$ is odd, then $k$ is odd.
Since the degree of this element is $$\begin{aligned}
\lb (k+m \ell) - \frac{1}{2} (k+m(\ell+1)+1)n \rb \chi,\end{aligned}$$ each such $(k,m)$ contributes $\bC((k+m(\ell+1))n)$ to $\HH^t$ for $$\begin{aligned}
t
&= 2 \lb (k+m \ell) - \frac{1}{2}(k+m(\ell+1)+1)n \rb + \dim N_\gamma \\
&= 2 (k+m \ell) - (k+m(\ell+1))n.\end{aligned}$$ Similarly, for each such $(k,m)$, there is an element $\bsalpha_{k,m}$ contributing $\bC((k+m(\ell+1))n)$ to $\HH^{t+1}$ for $t = 2(k+m \ell) - (k+m(\ell+1))n$.
###
One has $
V_\gamma = \bC x_0
$ if and only if both $\ell$ and $n$ are odd and $
\gamma = (1, -1, \ldots, -1).
$ The degree of $$\begin{aligned}
x_0^{k_0} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi - (k_0+1) \chi_1 - (k_0+1) \chi_2 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $\ell+1$ divides $k_0+1$. Such an element can be written as $$\begin{aligned}
\bsb_m \coloneqq x_0^{m(\ell+1)-1} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual\end{aligned}$$ for $
m \in \bN \setminus \{ 0 \}.
$ Since the degree of this element is $$\begin{aligned}
\lb (m \ell-1) - \frac{1}{2} m(\ell+1)n \rb \chi,\end{aligned}$$ each such element contributes $\bC((m(\ell+1)-1)n)$ to $\HH^t$ for $$\begin{aligned}
t
&= 2 \lb (m \ell-1) - \frac{1}{2}m(\ell+1)n \rb + \dim N_\gamma \\
&= (2 m \ell+1) - (m(\ell+1)-1)n.\end{aligned}$$ Similarly, for each $m \in \bN$, the element $$\begin{aligned}
\bsbeta_m \coloneqq x_0^\dual \otimes x_0^{m(\ell+1)} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \bC x_0^\dual \otimes \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ contributes $\bC(m(\ell+1)-1)n)$ to $\HH^{t+1}$ for $
t = (2 m \ell-1) - (m(\ell+1)-1)n.
$
###
For $
\gamma = (t_0, \ldots, t_{n+1}) \in \ker \chi,
$ one has $V_\gamma = 0$ if and only if $t_i \ne 1$ for all $i \in \{ 0, \ldots, n+1 \}$. This is the case if and only if $
t_2 = \cdots = t_{n+1} = -1,
$ $
t_1 \in \bsmu_{\ell+1} \setminus \{ 1 \},
$ and $$\begin{aligned}
\label{eq:t0_A}
t_0 = (-1)^{n} t_1^{-1} \ne 1.\end{aligned}$$ If $n$ is odd, then holds if and only if $t_1 \ne -1$, so that the number of such $\gamma$ is $\ell$ if $\ell$ is even, and $\ell-1$ if $\ell$ is odd. If $n$ is even, then always holds, and the number of such $\gamma$ is $\ell$. Each such $\gamma$ contributes $\bC(-n)$ to $\HH^{n}$.
###
To sum up, the Hochschild cohomology group has a basis consisting of the following elements:
- $\bsa_{k,m}$ of degree $
2(k+m \ell) - (k+m (\ell+1))n
$ and weight $
-(k+m(\ell+1))n,
$ and $\bsalpha_{k,m}$ of degree $
2(k+m \ell) - (k+m (\ell+1))n+1
$ and weight $
-(k+m(\ell+1))n,
$ where $(k,m)$ runs over
- $\{ 0, \ldots, \ell-1 \} \times \bN$ if $n$ is even, and
- the subset of $\{ 0, \ldots, \ell-1 \} \times \bN$ consisting of $(k,m)$ such that
- the parities of $k$ and $m$ agree if $n$ is odd and $\ell$ is even,
- $k$ is even if both $n$ and $\ell$ are odd, and
- if both $n$ and $\ell$ are odd, then
- $\bsb_m$ of degree $
2m \ell-1-(m(\ell+1)-1)n
$ and weight $
-(m(\ell+1)-1)n
$ for $m \in \bN \setminus \{ 0 \}$, and
- $\bsbeta_{m}$ of degree $
2 m \ell-(m(\ell+1)-1)n
$ and weight $
-(m(\ell+1)-1)n
$ for $m \in \bN$.
- $\bss_h$ of degree $n$ and weight $n$, where $h$ runs over
- $\{ 1, 2, \ldots, \ell-1 \}$ if both $\ell$ and $n$ are odd, and
- $\{ 1, 2, \ldots, \ell \}$ otherwise,
###
As an example, consider the case $\ell=1$. The Hochschild cohomology group in this case is spanned by
- $\bsa_{0,m}$ for $m \in \bN$ of degree $-2m(n-1)$ and weight $-2mn$,
- $\bsalpha_{0,m}$ for $m \in \bN$ of degree $-2m(n-1)+1$ and weight $-2mn$,
and, if $n$ is odd, in addition to the above,
- $\bsb_m$ for $m \in \bN \setminus \{ 0 \}$ of degree $-(2m-1)(n-1)$ and weight $-(2m-1)n$,
- $\bsbeta_m$ for $m \in \bN$ of degree $-(2m-1)(n-1)+1$ and weight $-(2m-1)n$,
and, if $n$ is even, in addition to the above,
- $\bss_1$ of degree $n$ and weight $n$.
This is consistent with the isomorphism $$\begin{aligned}
\SH^*(T^* S^{n})
\cong H_{n-*}(\scrL S^{n}),\end{aligned}$$ which is a special case of the isomorphism between the symplectic cohomology of the cotangent bundle and the homology of the free loop space [@1805.01316 Theorem 3.1] (see e.g. [@MR2039760 Theorem 2] for the homology of the free loop space of spheres).
Another example is the case when $n=2$ and $\ell$ is arbitrary. In this case, $\SH^*(\wv^{-1}(1))$ was computed in [@MR3692968] as a bigraded ring. This is compatible with the computation given here.
Type $D_\ell$ {#sc:D}
-------------
Consider the case $$\begin{aligned}
\w = x_1^{2\ell-2} + x_2^2 + \cdots + x_{n+1}^2 \in \bC[x_0, x_1,\ldots,x_{n+1}]\end{aligned}$$ with the non-maximal group $$\begin{aligned}
\Gamma
= \lc \gamma = (t_0, t_1, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^{\ell-1} t_2 = t_2^2 = \cdots = t_{n+1}^2 = t_0 t_1 \cdots t_{n+1} \rc,\end{aligned}$$ so that $
\ker \chi \cong \bmu_{2\ell-2} \times \lb \bmu_2 \rb^{n-1}
$ and $\Char(\Gamma)$ is generated by $\chi$ and $\chi_i = \deg x_i$ for $i \in \{ 0, \ldots, n+1 \}$ with relations $$\begin{aligned}
\label{eq:D_rel}
\chi = (\ell-1) \chi_1 + \chi_2 = 2 \chi_2 = \cdots = 2 \chi_{n+1} = \chi_0 + \cdots + \chi_{n+1}.\end{aligned}$$ By a change of coordinates, this is equivalent to $$\begin{aligned}
\w' = x_1^{\ell-1} x_2 + x_2^2 + \cdots + x_{n+1}^2\end{aligned}$$ with $
\Gamma = \Gamma_{\w'},
$ whose Berglund–Hübsch transpose $$\begin{aligned}
\wv' = x_1^{\ell-1} + x_1 x_2^2 + x_3^2 + \cdots + x_{n+1}^2\end{aligned}$$ defines the $D_\ell$-singularity. The relations imply $$\begin{aligned}
\chi_2 &= \chi - (\ell-1) \chi_1, \\
\chi &= (2\ell-2) \chi_1, \\
\chi_0 &= \chi - \chi_1 - \cdots - \chi_n \\
&= (\ell-2) \chi_1 - \chi_3 - \cdots - \chi_{n+1}.\end{aligned}$$
###
For any $\gamma \in \ker \chi$, one has $$\begin{aligned}
\Jac_{\w_\gamma} =
\begin{cases}
\bC[x_0] \otimes \bC[x_1]/(x_1^{2\ell-3}) & \bC x_0 \oplus \bC x_1 \subset V_\gamma, \\
\bC[x_0] & \bC x_0 \subset V_\gamma \text{ and } \bC x_1 \not \subset V_\gamma, \\
\bC[x_1]/(x_1^{2\ell-3}) & \bC x_0 \not \subset V_\gamma \text{ and } \bC x_1 \subset V_\gamma, \\
\bC & \text{otherwise}.
\end{cases}\end{aligned}$$ If we write an element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ as $$\begin{aligned}
x_0^{k_0} x_1^{k_1}
\otimes x_{j_1}^\dual \wedge x_{j_2}^\dual \wedge \ldots \wedge x_{j_s}^\dual,\end{aligned}$$ where $k_0 = 0$ if $\bC x_0 \not \subset V_\gamma$ and $k_1 = 0$ if $\bC x_1 \not \subset V_\gamma$, then its degree is given by $$\begin{aligned}
k_0 \chi_0 + k_1 \chi_1 - \chi_{j_1} - \cdots - \chi_{j_s},\end{aligned}$$ which can be proportional to $\chi$ only if
- $
V_\gamma \cap \lb \bC x_3 \oplus \cdots \oplus \bC x_{n+1} \rb
$ is either $
\bC x_3 \oplus \cdots \oplus \bC x_{n+1}
$ or $0$.
We will assume this condition for the rest of .
###
Since $t_0 = (t_1 \cdots t_{n+1})^{-1} = \pm t_1^{-1}$, one has $t_0=1$ only if $t_1 = \pm 1$. If $t_1 = 1$, then $t_2 = 1$, and one has $t_0 = 1$ if and only if $(t_3)^{n-1} = 1$. If $t_1=-1$, then $t_2 = (-1)^{\ell-1}$, and one has $t_0 = 1$ if and only if $(-1)^{\ell} t_3^{n-1} = 1$. It follows that
- $V_\gamma$ contains $\bC x_0$ if and only if
- $\gamma = (1,\ldots,1)$, where $V_\gamma = V$,
- $\gamma = (1,1,1,-1,\ldots,-1)$ with odd $n$, where $V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2$,
- $\gamma = (1,-1,-1,1,\ldots,1)$ with even $\ell$, where $V_\gamma = \bC x_0 \oplus \bC x_3 \oplus \cdots \oplus \bC x_{n+1}$,
- $\gamma = (1,-1,-1,-1,\ldots,-1)$ with even $\ell$ and odd $n$, where $V_\gamma = \bC x_0$,
- $\gamma = (1,-1,1,-1,\ldots,-1)$ with odd $\ell$ and even $n$, where $V_\gamma = \bC x_0 \oplus \bC x_2$.
###
The smallest positive integer $k$ such that the degree of $x_0^k$ is proportional to $\chi$ is $2 \ell-2$. One has $$\begin{aligned}
\deg x_0^{2\ell-2}
&= (2\ell-2)(\chi-\chi_1-\cdots-\chi_{n+1}) \\
&= \lb (2\ell-3)-(\ell-1) n \rb \chi.\end{aligned}$$
###
One has $
V_\gamma = V
$ if and only if $\gamma$ is the identity element. The degree of $
x_0^{k_0} x_1^{k_1} \in \Jac_\w
$ is $$\begin{aligned}
k_0 \chi - (k_0-k_1) \chi_1 - k_0 \chi_2 - \cdots - k_0 \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is even and $2\ell-2$ divides $k_0 - k_1$. Such an element can be written as $$\begin{aligned}
\bsa_{k,m} \coloneqq x_0^{2k+(2\ell-2)m} x_1^{2k}\end{aligned}$$ for $
(k,m) \in \{ 0, \ldots, \ell-2 \} \times \bN
$ which contributes $
\bC((2k+(2\ell-2)m)n)
$ to $\HH^t$ for $
t = 4k+(4\ell-6)m - (2k+(2\ell-2)m)n
$ since $$\begin{aligned}
\deg x_0^{2k} x_1^{2k}
&= (2k - kn) \chi.\end{aligned}$$ Similarly, for each $(k,m) \in \{ 0, \ldots, \ell-2 \} \times \bN$, there is an element $\bsalpha_{k,m}$ contributing $
\bC((2k+(2\ell-2)m)n)
$ to $\HH^t$ for $
t = 4k+1+(4\ell-6)m - (2k+(2\ell-2)m)n.
$
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2
$ if and only if $\gamma = (1,1,1,-1,\ldots,-1)$ and $n$ is odd. The degree of $
x_0^{k_0} x_1^{k_1} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ is $$\begin{aligned}
k_0 \chi - (k_0 - k_1) \chi_1 - k_0 \chi_2 - (k_0+1) \chi_3 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is odd and $2\ell-2$ divides $k_0 - k_1-(\ell-1)$. Such an element can be written as $$\begin{aligned}
\bsb_{k,m} \coloneqq
x_0^{k+\ell-1+(2\ell-2)m} x_1^{k} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual\end{aligned}$$ for $$\begin{aligned}
(k,m) \in \lc (k,m) \in \{ 0, \ldots, 2 \ell-4 \} \times \bZ \relmid
k+\ell \text{ is even and } k+\ell-1+m(2\ell-2) \ge 0 \rc.\end{aligned}$$ It contributes $
\bC((k+\ell-1+(2\ell-2)m)n)
$ to $\HH^t$ for $$\begin{aligned}
t
&= 2 \deg \lb x_0^{k+\ell-1+(2\ell-2)m} x_1^k \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb + \dim N_\gamma \\
&= 2k+2\ell-3+(4\ell-6)m-(k+\ell-1-(2\ell-2)m)n,\end{aligned}$$ since $$\begin{aligned}
&\deg \lb x_0^{k+\ell-1} x_1^k \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb \\
&\qquad= (k+\ell-1) \chi - (\ell-1) \chi_1 - (k+\ell-1) \chi_2 - (k+\ell) \chi_3 - \cdots - (k+\ell) \chi_{n+1} \\
&\qquad = \lb k+\ell-1 -\frac{1}{2}(k+\ell)n \rb \chi.\end{aligned}$$ Similarly, for each $$\begin{aligned}
(k,m) \in \lc (k,m) \in \{ 0, \ldots, 2 \ell-4 \} \times \bZ \relmid
k+\ell \text{ is even and } k+\ell+m(2\ell-2) \ge 0 \rc,\end{aligned}$$ the element $$\begin{aligned}
\bsbeta_{k,m} \coloneqq
x_0^\dual \otimes x_0^{k+\ell+(2\ell-2)m} x_1^{k} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \lb (\bC x_0)^\dual \otimes \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual \rb^\Gamma\end{aligned}$$ contributes $
\bC(k+\ell-1+(2\ell-2)m)
$ to $\HH^t$ for $$\begin{aligned}
t = 2k+2\ell-2+(4\ell-6)m-(k+\ell-1-(2\ell-2)m)n.\end{aligned}$$
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_3 \oplus \cdots \oplus \bC x_{n+1}
$ if and only if $\gamma = (1,-1,-1,1,\ldots,1)$ with even $\ell$. An element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ whose degree is proportional to $\chi$ can be written as $$\begin{aligned}
\bsc_m \coloneqq x_0^{\ell-2 + (2\ell-2)m} \otimes x_1^\dual \wedge x_2^\dual\end{aligned}$$ for $
m \in \bN,
$ which contributes $\bC((\ell-2+(2\ell-2)m)n)$ to $\HH^t$ for $$\begin{aligned}
t
&= 2 \deg \lb x_0^{\ell-2} \otimes x_1^\dual \wedge x_2^\dual \rb + \dim N_\gamma \\
&= 2 \ell-4+(4\ell-6)m-(\ell-2+(2\ell-2)m)n\end{aligned}$$ since $$\begin{aligned}
\deg \lb x_0^{\ell-2} \otimes x_1^\dual \wedge x_2^\dual \rb
&= \lb \ell-3 - \frac{1}{2} (\ell-2)n \rb \chi.\end{aligned}$$ Similarly, for each $m \in \bN$, there is an element $\bsgamma_m$ contributing $\bC((\ell-2+(2\ell-2)m)n)$ to $\HH^t$ for $$\begin{aligned}
t
&= 2\ell-3+(4\ell-6)m-(\ell-2+(2\ell-2)m)n.\end{aligned}$$
###
One has $
V_\gamma = \bC x_0
$ if and only if $\ell$ is even, $n$ is odd, and $
\gamma = (1, -1, \ldots, -1) \in \ker \chi.
$ The degree of $$\begin{aligned}
x_0^{k_0} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi - (k_0+1) \chi_1 - (k_0+1) \chi_2 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $2\ell-2$ divides $k_0+1$. Such an element can be written as $$\begin{aligned}
\bsd_m \coloneqq
x_0^{-1+m(2\ell-2)} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual\end{aligned}$$ for $
m \in \bN \setminus \{ 0 \}.
$ Since $$\begin{aligned}
\deg \lb x_0^{-1} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb
= -\chi,\end{aligned}$$ each such element contributes $\bC((-1+(2\ell-2)m)n)$ to $\HH^t$ for $$\begin{aligned}
t
&= 2 \deg \lb x_0^{-1+(2\ell-2)m} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb
+ \dim N_\gamma \\
&= -1+(4\ell-6)m - (-1+(2\ell-2)m)n.\end{aligned}$$ Similarly, for each $m \in \bN$, the element $$\begin{aligned}
\bsdelta_m \coloneqq
x_0^\dual \otimes x_0^{m(2\ell-2)} \otimes x_1^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \bC x_0^\dual \otimes \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ contributes $\bC((-1+(2\ell-2)m)n)$ to $\HH^t$ for $$\begin{aligned}
t
&= (4\ell-6)m - (-1+(2\ell-2)m)n.\end{aligned}$$
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_2
$ if and only if $\ell$ is odd, $n$ is even, and $
\gamma = (1, -1, 1, -1, \ldots, -1).
$ The degree of $$\begin{aligned}
x_0^{k_0} \otimes x_1^\dual \wedge x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi_0 - \chi_1 - \chi_3 - \cdots - \chi_{n+1}
&= (k_0 (\ell-2) - 1) \chi_1 - (k_0+1) \chi_3 -\cdots- (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is odd and $2\ell-2$ divides $k_0(\ell-2)-1$. Such an element can be written as $$\begin{aligned}
\bse_m \coloneqq x_0^{\ell-2+(2\ell-2)m}
\otimes x_1^\dual \wedge x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual\end{aligned}$$ for $
m \in \bN,
$ which contributes $
\bC((\ell-2+(2\ell-2)m)n)
$ to $\HH^t$ for $$\begin{aligned}
t
&= 2\ell-4+(4\ell-6)m-(\ell-2+(2\ell-2)m)n\end{aligned}$$ since $$\begin{aligned}
\deg \lb x_0^{\ell-2}
\otimes x_1^\dual \wedge x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb
&= \frac{1}{2} \lb 2\ell-4-(\ell-1)n \rb \chi.\end{aligned}$$ Similarly, for each $
m \in \bN,
$ there is an element $\bsepsilon_m$ contributing $
\bC((\ell-2+(2\ell-2)m)n)
$ to $\HH^t$ for $$\begin{aligned}
t &= 2\ell-3+(4\ell-6)m-(\ell-2+(2\ell-2)m)n.\end{aligned}$$
###
Set $
\zeta \coloneqq \exp \lb 2 \pi \sqrt{-1}/(2\ell-2) \rb.
$ For a given $\gamma = (t_0, \ldots, t_{n+1}) \in \ker \chi$, we write $t_1 = \zeta^p$ for $p \in \{ 0, \ldots, 2 \ell-3 \}$. Then one has $t_2 = (-1)^p$, so that
- $V_\gamma$ contains $\bC x_1$ if and only if $p=0$, and
- $V_\gamma$ contains $\bC x_2$ if and only if $p$ is even.
###
If $\bC x_0 \not \subset V_\gamma$ and $\bC x_1 \subset V_\gamma$, then one has $\gamma = (-1,1,1,-1,\ldots,-1)$ for even $n$ and $V_\gamma = \bC x_1 \oplus \bC x_2$. The element $$\begin{aligned}
x_1^{\ell-2} \otimes x_0^\dual \wedge x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual\end{aligned}$$ has degree $$\begin{aligned}
(\ell-2) \chi_1 - \chi_0 - \chi_3 - \cdots - \chi_{n+1}
&= 0,\end{aligned}$$ so that it contributes $\bC(-n)$ to $\HH^t$ for $
t = \dim N_\gamma = n.
$
###
An element $\gamma \in \ker \chi$ with $V_\gamma = \bC x_2$, $\bC x_2 \oplus \cdots \oplus \bC x_{n+1}$, or $\bC x_3 \oplus \cdots \oplus \bC x_{n+1}$ does not contribute to $\HH^*$, since $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ contains a unique element, whose degree is not proportional to $\chi$.
###
One has $V_\gamma = 0$ if and only if $
t_3 = \cdots = t_{n+1} = -1,
$ $
t_1 = \zeta^{2m+1}
$ for $m \in \{ 0, \ldots, \ell-2 \}$, and $$\begin{aligned}
\label{eq:t0_D}
t_0 = (-1)^{n} \zeta^{-2m-1} \ne 1.\end{aligned}$$ The number of such $\gamma$ is $\ell-2$ if $\ell$ is even and $n$ is odd, and $\ell-1$ otherwise. Each such $\gamma$ contributes $\bC(-n)$ to $\HH^{n}$.
###
To sum up, the Hochschild cohomology group has a basis consisting of the following elements:
- $\bsa_{k,m}$ of degree $
4k+(4\ell-6)m-(2k+(2\ell-2)m)n
$ and weight $
- (2k+(2\ell-2)m)n
$ for $
(k,m) \in \{ 0, \ldots, \ell-2 \} \times \bN,
$
- $\bsalpha_{k,m}$ of degree $
4k+1+(4\ell-6)m-(2k+(2\ell-2)m)n
$ and weight $
- (2k+(2\ell-2)m)n
$ for $
(k,m) \in \{ 0, \ldots, \ell-2 \} \times \bN,
$
- if $n$ is odd, $\bsb_{k,m}$ of degree $
2k+2\ell-3+(4\ell-6)m-(k+\ell-1-(2\ell-2)m)n
$ and weight $
-(k+\ell-1+(2\ell-2)m)n
$ for $
\lc (k,m) \in \{ 0, \ldots, 2 \ell-4 \} \times \bZ \relmid
k+\ell \text{ is even and } k+\ell-1+m(2\ell-2) \ge 0 \rc,
$
- if $n$ is odd, $\bsbeta_{k,m}$ of degree $
2k+2\ell-2+(4\ell-6)m-(k+\ell-1-(2\ell-2)m)n
$ and weight $
-(k+\ell-1-(2\ell-2)m)n
$ for $
\lc (k,m) \in \{ 0, \ldots, 2 \ell-4 \} \times \bZ \relmid
k+\ell \text{ is even and } k+\ell+m(2\ell-2) \ge 0 \rc,
$
- if $\ell$ is even, $\bsc_m$ of degree $
2 \ell-4+(4\ell-6)m-(\ell-2+(2\ell-2)m)n
$ and weight $
-(\ell-2+(2\ell-2)m)n
$ for $
m \in \bN,
$
- if $\ell$ is even, $\bsgamma_m$ of degree $
2 \ell-3+(4\ell-6)m-(\ell-2+(2\ell-2)m)n
$ and weight $
-(\ell-2+(2\ell-2)m)n
$ for $
m \in \bN,
$
- if $\ell$ is even and $n$ is odd, $\bsd_m$ of degree $
-1+(4\ell-6)m - (-1+(2\ell-2)m)n
$ and weight $
- (-1+(2\ell-2)m)n
$ for $m \in \bN \setminus \{ 0 \}$,
- if $\ell$ is even and $n$ is odd, $\bsdelta_m$ of degree $
(4\ell-6)m - (-1+(2\ell-2)m)n
$ and weight $
- (-1+(2\ell-2)m)n
$ for $m \in \bN$,
- if $\ell$ is odd and $n$ is even, $\bse_m$ of degree $
2\ell-4+(4\ell-6)m-(\ell-2+(2\ell-2)m)n
$ and weight $
-(\ell-2+(2\ell-2)m)n
$ for $m \in \bN$,
- if $\ell$ is odd and $n$ is even, $\bsepsilon_m$ of degree $
2\ell-3+(4\ell-6)m-(\ell-2+(2\ell-2)m)n
$ and weight $
-(\ell-2+(2\ell-2)m)n
$ for $m \in \bN$, and
- $\bss_h$ of degree $n$ and weight $n$, where $h$ runs over a set consisting of
- $\ell-2$ elements if $\ell$ is even and $n$ is odd,
- $\ell-1$ elements if both $\ell$ and $n$ are odd, and
- $\ell$ elements otherwise.
Type $E_6$ {#sc:E6}
----------
Consider the case $$\begin{aligned}
\w = x_1^4 + x_2^3 + x_3^2 + \cdots + x_{n+1}^2 \in \bC[x_0, x_1,\ldots,x_{n+1}]\end{aligned}$$ with $$\begin{aligned}
\Gamma
= \Gamma_\w
\coloneqq \lc \gamma = (t_0, t_1, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^4 = t_2^3 = t_3^2 = \cdots = t_{n+1}^2 = t_0 t_1 \cdots t_{n+1} \rc,\end{aligned}$$ so that $
\ker \chi \cong \bmu_4 \times \bmu_3 \times \lb \bmu_2 \rb^{n-1}
$ and $\Char(\Gamma)$ is generated by $\chi$ and $\chi_i = \deg x_i$ for $i \in \{ 0, \ldots, n+1 \}$ with relations $$\begin{aligned}
\chi = 4 \chi_1 = 3 \chi_2 = 2 \chi_3 = \cdots = 2 \chi_{n+1} = \chi_0 + \cdots + \chi_{n+1}.\end{aligned}$$
###
For any $\gamma \in \ker \chi$, one has $$\begin{aligned}
\Jac_{\w_\gamma} \cong
\begin{cases}
\bC[x_0] & \bC x_0 \subset V_\gamma \\
\bC & \bC x_0 \not \subset V_\gamma
\end{cases}
\otimes
\begin{cases}
\bC[x_1]/(x_1^3) & \bC x_1 \subset V_\gamma \\
\bC & \bC x_1 \not \subset V_\gamma
\end{cases}
\otimes
\begin{cases}
\bC[x_2]/(x_2^2) & \bC x_2 \subset V_\gamma \\
\bC & \bC x_2 \not \subset V_\gamma.
\end{cases}\end{aligned}$$ If we write an element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ as $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2}
\otimes x_{j_1}^\dual \wedge x_{j_2}^\dual \wedge \ldots \wedge x_{j_s}^\dual,\end{aligned}$$ where $k_i = 0$ if $\bC x_i \not \subset V_\gamma$ for $i = 0, 1, 2$, then its degree is given by $$\begin{aligned}
k_0 \chi_0 + k_1 \chi_1 + k_2 \chi_2 - \chi_{j_1} - \cdots - \chi_{j_s},\end{aligned}$$ which can be proportional to $\chi$ only if $
V_\gamma \cap \lb \bC x_3 \oplus \cdots \bC x_{n+1} \rb
$ is either $\bC x_3 \oplus \cdots \oplus \bC x_{n+1}$ or $0$. We will assume this condition for the rest of .
###
Since $t_0=1$ implies $t_2 = 1$ and $t_1 = \pm 1$, one has the following:
- $V_\gamma$ contains $\bC x_0$ if and only if either
- $\gamma = (1, \ldots, 1)$, where $V_\gamma = V$,
- $\gamma = (1,1,1,-1,\ldots,-1)$ with odd $n$, where $V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2$,
- $\gamma = (1,-1,1,-1,\ldots,-1)$ with even $n$, where $V_\gamma = \bC x_0 \oplus \bC x_2$.
###
One has $
V_\gamma = V
$ if and only if $\gamma$ is the identity element. The degree of $
x_0^{k_0} x_1^{k_1} x_2^{k_2} \in \Jac_\w
$ is $$\begin{aligned}
k_0 \chi - (k_0 - k_1) \chi_1 - (k_0 - k_2) \chi_2 - k_0 \chi_3 - \cdots - k_0 \chi_n,\end{aligned}$$ which is proportional to $\chi$ if and only if 4 divides $k_0-k_1$, 3 divides $k_0 - k_2$, if $n=1$ and, in addition $k_0$ is even, if $n>1$.
Thus, for $n=1$, we must have $$\begin{aligned}
\label{eq:e6a}
5k_0 + 3k_1 + 4k_2 = 12m \end{aligned}$$ for $m \in \bN$, in which case the constant of proportionality is $$\begin{aligned}
t/2 = m\end{aligned}$$ For each $m \in \bN$ such that $5\nmid m$, the equation (\[eq:e6a\]) has a unique solution with $(k_1,k_2) \in \{0,1,2\} \times \{0,1\}$ and if $5 \mid m$, then there are precisely two contributions with $(k_1,k_2)= (0,0)$ and $(k_1,k_2)=(2,1)$ such that $(k_1,k_2,m) \in \{0,1,2\} \times \{0,1\} \times \mathbb{N}$ except if $m=0$, then only $(k_1,k_2) = (0,0)$ contributes. Each such $(k_1,k_2,m)$ contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
For $n>1$, in addition equation (\[eq:e6a\]), we must have that $k_0$ is even. This forces $k_1 \ne 1$, and the possible $(k_0, k_1, k_2)$ and $
t = 2 \deg(x_0^{k_0} x_1^{k_1} x_2^{k_2})/\chi
$ are given by $$\begin{aligned}
\label{eq:kE6}
\begin{array}{ccc}
\toprule
(k_1, k_2) & k_0 & t \\
\midrule
(0,0) & 12m & 22m-12mn \\
(0,1) & 4+12m & 8+22m-(4+12m)n \\
(2,0) & 6+12m & 12+22m-(6+12m)n \\
(2,1) & 10+12m & 20+22m-(10+12m)n \\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$. Each $(k_0, k_1, k_2)$ from contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2
$ if and and only if $n$ is odd and $
\gamma = (1, 1, 1, -1, \ldots, -1) \in \ker \chi.
$ The degree of $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi - (k_0 - k_1) \chi_1 - (k_0 - k_2 ) \chi_2 - (k_0+1) \chi_3 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $k_0$ is odd, $4$ divides $k_0 - k_1$, and $3$ divides $k_0 - k_2$. This forces $k_1 = 1$ and the possible $(k_0, k_1, k_2)$ and $$\begin{aligned}
t = 2 \deg(x_0^{k_0} x_1^{k_1} x_2^{k_2} \otimes x_3^\dual \wedge \cdots \wedge x_n^\dual)/\chi + \dim N_\gamma\end{aligned}$$ are given by $$\begin{aligned}
\label{eq:kE6_2}
\begin{array}{ccc}
\toprule
(k_1, k_2) & k_0 & t \\
\midrule
(1,0) & 9+12m & 17+22m-(9+12m)n \\
(1,1) & 1+12m & 1+22m-(1+12m)n \\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$. Each $(k_0, k_1, k_2)$ from contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
One has $
V_\gamma = \bC x_0 \oplus \bC x_2
$ if and only if $n$ is even and $
\gamma = (1, -1, 1, -1, \ldots, -1) \in \ker \chi.
$ The degree of $$\begin{aligned}
x_0^{k_0} x_2^{k_2} \otimes x_1^\dual \wedge x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
k_0 \chi - (k_0+1) \chi_1 - (k_0 - k_2 ) \chi_2 - (k_0+1) \chi_3 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $4$ divides $k_0+1$ and $3$ divides $k_0 - k_2$. The possible $(k_0, k_2)$ and $$\begin{aligned}
t = 2 \deg(x_0^{k_0} x_2^{k_2} \otimes x_1^\dual \wedge x_3^\dual \wedge \cdots \wedge x_n^\dual)/\chi + \dim N_\gamma\end{aligned}$$ are given by $$\begin{aligned}
\label{eq:kE6_3}
\begin{array}{ccc}
\toprule
k_2 & k_0 & t \\
\midrule
0 & 3+12m & 6+22m-(3+12m)n \\
1 & 7+12m & 14+22m-(7+12m)n \\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$. Each $(k_0, k_2)$ from contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
If $V_\gamma = \bC x_1$, then one has $$\begin{aligned}
\deg \lb x_1^{k_1} \otimes x_0^\dual \wedge x_2^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb
&= - \chi_0 + k_1 \chi_1 - \chi_2 - \cdots - \chi_{n+1} \\
&= - \chi + (k_1+1) \chi_1,\end{aligned}$$ which is not proportional to $\chi$ for any $k_1 \in \{ 0, 1, 2 \}$. Similarly, $\gamma$ with $\bC x_0 \not \subset V_\gamma$ and $V_\gamma \ne 0$ does not contribute to $\HH^*$.
###
One has $V_\gamma = 0$ if and only if $
t_1 \in \lb \bsmu_4 \setminus \{ 1 \} \rb,
$ $
t_2 \in \lb \bmu_3 \setminus \{ 1 \} \rb,
$ and $
t_3 = \cdots, t_{n+1} = -1,
$ since $t_2 \ne 1$ implies $
t_0 = (-1)^{n-1} t_1^{-1} t_2^{-1} \ne 1.
$ There are six such $\gamma$, and each of them contributes $\bC(-n)$ to $\HH^{n}$.
Type $E_7$ {#sc:E7}
----------
Consider the case $$\begin{aligned}
\w = x_1^3 x_2 + x_2^3 + x_3^2 + \cdots + x_{n+1}^2 \in \bC[x_0, x_1,\ldots,x_{n+1}]\end{aligned}$$ with $$\begin{aligned}
\Gamma
= \Gamma_\w
\coloneqq \lc \gamma = (t_0, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^3 t_2 = t_2^3 = t_3^2 = \cdots = t_{n+1}^2 = t_0 \cdots t_{n+1} \rc,\end{aligned}$$ so that $
\ker \chi \cong \bmu_9 \times \lb \bmu_2 \rb^{n-1}
$ and $\Char(\Gamma)$ is generated by $\chi$ and $\chi_i = \deg x_i$ for $i \in \{ 0, \ldots, n+1 \}$ with relations $$\begin{aligned}
\chi = 3 \chi_1 + \chi_2 = 3 \chi_2 = 2 \chi_3 = \cdots = 2 \chi_{n+1} = \chi_0 + \cdots + \chi_{n+1}.\end{aligned}$$ These relations imply $$\begin{aligned}
\chi_2 &= \chi - 3 \chi_1, \\
9 \chi_1 &= 2 \chi, \\
\chi_0
&= \chi - \chi_1 - \cdots - \chi_{n+1} \\
&= 2 \chi_1 - \chi_3 - \cdots - \chi_{n+1}.\end{aligned}$$
###
For any $\gamma \in \ker \chi$, the intersection $
V_\gamma \cap \lb \bC x_1 \oplus \bC x_2 \rb
$ can be either $\bC x_1 \oplus \bC x_2$, $\bC x_2$, or $0$, where $\Jac_{\w_\gamma'}$ is isomorphic to $
\bC[x_1,x_2]/(3 x_1^2 x_2, x_1^3 + 3 x_2^2),
$ $
\bC[x_2]/(3 x_2^2),
$ or $
\bC
$ respectively. A basis of $
\bC[x_1,x_2]/(3 x_1^2 x_2, x_1^3 + 3 x_2^2)
$ is given by $\{ 1, x_1, x_1^2, x_1^3, x_1^4, x_2, x_1 x_2 \}$.
If we write an element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ as $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2}
\otimes x_{j_1}^\dual \wedge x_{j_2}^\dual \wedge \ldots \wedge x_{j_s}^\dual,\end{aligned}$$ then its degree is given by $$\begin{aligned}
k_0 \chi_0 + k_1 \chi_1 + k_2 \chi_2 - \chi_{j_1} - \cdots - \chi_{j_s},\end{aligned}$$ which can be proportional to $\chi$ only if $
V \cap \lb \bC x_3 \oplus \cdots \oplus \bC x_{n+1} \rb
$ is either $
\bC x_3 \oplus \cdots \oplus \bC x_{n+1}
$ or $0$. We assume this condition for the rest of .
###
Since $t_0 = 1$ implies $t_1=t_2=1$, one has $\bC x_0 \subset V_\gamma$ if and only if either $V_\gamma = V$ or $V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2$.
###
One has $
V_\gamma = V
$ if and only if $\gamma$ is the identity element. The degree of $
x_0^{k_0} x_1^{k_1} x_2^{k_2} \in \Jac_\w
$ is $$\begin{aligned}
& k_0 (2 \chi_1 - \chi_3 - \cdots - \chi_{n+1}) + k_1 \chi_1 + k_2 (\chi - 3 \chi_1) \\
&\qquad =
k_2 \chi + (2 k_0 + k_1 - 3 k_2) \chi_1 - k_0 \chi_3 - \cdots - k_0 \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if, 9 divides $2k_0 + k_1- 3k_2$ if $n=1$ and in addition $k_0$ is even, if $n>1$. Thus, for $n=1$, the constant of proportionality is $$\begin{aligned}
t/2 = k_2 + \frac{2}{9}(2k_0+k_1-3k_2) \end{aligned}$$ The possible $(k_0, k_1, k_2)$ and $
t = 2 \deg (x_0^{k_0} x_1^{k_1} x_2^{k_2})/\chi
$ are given by $$\begin{aligned}
\begin{array}{ccc}
\toprule
(k_1, k_2) & k_0 & t \\
\midrule
(0,0) & 9m & 8m \\
(1,0) & 4+9m & 4+8m \\
(2,0) & 8+9m & 8+8m \\
(3,0) & 3+9m & 4+8m \\
(4,0) & 7+9m & 8+8m \\
(0,1) & 6+9m & 5+8m \\
(1,1) & 1+9m & 1+8m \\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$.
For $n>1$, the constant of proportionality is $$\begin{aligned}
t/2= k_2 + \frac{2}{9}(2 k_0 + k_1 - 3 k_2) - \frac{1}{2} k_0 (n-1).\end{aligned}$$ The possible $(k_0, k_1, k_2)$ and $
t = 2 \deg (x_0^{k_0} x_1^{k_1} x_2^{k_2})/\chi
$ are given by $$\begin{aligned}
\label{eq:kE7}
\begin{array}{ccc}
\toprule
(k_1, k_2) & k_0 & t \\
\midrule
(0,0) & 18m & 34m-18mn \\
(1,0) & 4+18m & 8+34m-(4+18m)n \\
(2,0) & 8+18m & 16+34m-(8+18m)n \\
(3,0) & 12+18m & 24+34m-(12+18m)n \\
(4,0) & 16+18m & 32+34m-(16+18m)n \\
(0,1) & 6+18m & 12+34m-(6+18m)n \\
(1,1) & 10+18m & 20+34m-(10+18m)n \\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$. Each $(k_0, k_1, k_2)$ from contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
For $n>1$, in addition, one has $
V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2
$ if and only if $n$ is odd and $\gamma = (1,1,1,-1,\ldots,-1)$. The degree of $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is given by $$\begin{aligned}
& k_0 (2 \chi_1 - \chi_3 - \cdots - \chi_n) + k_1 \chi_1 + k_2 (\chi - 3 \chi_1)
- \chi_3 - \cdots - \chi_{n+1} \\
&\qquad =
k_2 \chi + (2 k_0 + k_1 - 3 k_2) \chi_1 - (k_0+1) \chi_3 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if 9 divides $2 k_0 + k_1 - 3 k_2$ and $k_0$ is odd, in which case the constant of proportionality is $$\begin{aligned}
k_2 + \frac{2}{9}(2 k_0 + k_1 - 3 k_2) - \frac{1}{2} (k_0+1) (n-1).\end{aligned}$$ The possible $(k_0, k_1,k_2)$ and $$\begin{aligned}
t = 2 \deg \lb x_0^{k_0} x_1^{k_1} x_2^{k_2} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual \rb / \chi + \dim N_\gamma\end{aligned}$$ are given by $$\begin{aligned}
\label{eq:kE7_2}
\begin{array}{ccc}
\toprule
(k_1, k_2) & k_0 & t \\
\midrule
(0,0) & 9+18m & 18+34m-(9+18m)n\\
(1,0) & 13+18m & 26+34m-(13+18m)n\\
(2,0) & 17+18m & 24+34m-(17+18m)n\\
(3,0) & 3+18m & 6+34m-(3+18m)n\\
(4,0) & 7+18m & 14+34m-(7+18m)n\\
(0,1) & 15+18m & 30+34m-(15+18m)n\\
(1,1) & 1+18m & 2+34m-(1+18m)n\\
\bottomrule
\end{array}\end{aligned}$$ for $m \in \bN$. Each $(k_0, k_1, k_2)$ from contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
In addition, for the case $(k_1,k_2)= (2,0)$, the element $
x_0^\dual \otimes x_1^2 \otimes
x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
$ corresponding to $m=-1$ in has degree 0, and contributes $\bC(-n)$ to $\HH^n$.
###
For any $
t_1 \in \bmu_9
$ satisfying $
t_2 \coloneqq t_1^{-3} \ne 1,
$ the element $
\gamma = ((-1)^{n-1} t_1^2, t_1, t_1^{-3}, -1, \ldots, -1)
\in \ker \chi
$ satisfies $V_\gamma = 0$. There are six such elements, each of which contributes $\bC(-n)$ to $\HH^{n}$.
Type $E_8$ {#sc:E8}
----------
Consider the case $$\begin{aligned}
\w = x_1^5 + x_2^3 + x_3^2 + \cdots + x_{n+1}^2 \in \bC[x_0, x_1,\ldots,x_{n+1}]\end{aligned}$$ with $$\begin{aligned}
\Gamma
= \Gamma_\w
\coloneqq \lc \gamma = (t_0, \ldots, t_{n+1}) \in (\Gm)^{n+2} \relmid
t_1^5 = t_2^3 = t_3^2 = \cdots = t_{n+1}^2 = t_0 \cdots t_{n+1} \rc,\end{aligned}$$ so that $
\ker \chi \cong \bmu_5 \times \bmu_3 \times \lb \bmu_2 \rb^{n-1}
$ and $\Char(\Gamma)$ is generated by $\chi$ and $\chi_i = \deg x_i$ for $i \in \{ 0, \ldots, n+1 \}$ with relations $$\begin{aligned}
\chi = 5 \chi_1 = 3 \chi_2 = 2 \chi_3 = \cdots = 2 \chi_{n+1} = \chi_0 + \cdots + \chi_{n+1}.\end{aligned}$$
###
If we write an element of $
\Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual
$ as $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2}
\otimes x_{j_1}^\dual \wedge x_{j_2}^\dual \wedge \ldots \wedge x_{j_s}^\dual,\end{aligned}$$ then its degree is given by $$\begin{aligned}
k_0 \chi_0 + k_1 \chi_1 + k_2 \chi_2 - \chi_{j_1} - \cdots - \chi_{j_s},\end{aligned}$$ which can be proportional to $\chi$ only if $
V \cap \lb \bC x_3 \oplus \cdots \oplus \bC x_{n+1} \rb
$ is either $
\bC x_3 \oplus \cdots \oplus \bC x_{n+1}
$ or $0$. We assume this condition for the rest of .
###
Since $t_0 = 1$ implies $t_1=t_2=1$, one has $\bC x_0 \subset V_\gamma$ if and only if either $V_\gamma = V$ or $V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2$.
###
One has $
V_\gamma = V
$ if and only if $\gamma$ is the identity element. The degree of $
x_0^{k_0} x_1^{k_1} x_2^{k_2} \in \Jac_\w
$ is $$\begin{aligned}
k_0 \chi - (k_0-k_1) \chi_1 - (k_0-k_2) \chi_2 - k_0 \chi_3 - \cdots - k_0 \chi_{n+1},\end{aligned}$$
which is proportional to $\chi$ if and only if $$\begin{aligned}
k_0 &\equiv k_1 \mod 5, \\
k_0 &\equiv k_2 \mod 3, \end{aligned}$$ if $n=1$ and, in addition to the above, $$\begin{aligned}
k_0 &\equiv 0 \mod 2,\end{aligned}$$ if $n>1$.
Thus, for $n=1$, we must have $$\begin{aligned}
\label{eq:e8a}
7k_0 + 3 k_1 + 5k_2 = 15 m \end{aligned}$$ for $m \in \bN$, in which case the constant of proportionality is $$\begin{aligned}
t/2 = m \end{aligned}$$ For each $m \in \mathbb{N}$ such that $7 \nmid m$, the equation (\[eq:e8a\]) has a unique solution with $(k_1,k_2) \in \{0,1,2,3\} \times \{0,1\}$ and if $7 \mid m$, then there are precisely two contributions with $(k_1,k_2)=(0,0)$ and $(k_1,k_2)= (3,1)$ such that $(k_1, k_2, m) \in \{ 0, 1, 2, 3 \} \times \{ 0, 1 \} \times \bN$ except if $m=0$, then only $(k_1,k_2)=(0,0)$ contributes.
For $n>1$, we must have $$\begin{aligned}
\label{eq:e8b} 7k_0 + 3 k_1 + 5 k_2 = 15m \end{aligned}$$ for $m \in \bN$, and in addition $k_0$ must be in $2\bN$. Thus, we can re-write (\[eq:e8b\]) as $$\begin{aligned}
\label{eq:e8b2} k_0 = 6k_1 + 10k_2+ 30m'\end{aligned}$$ with $m' = k_0/2 - m$. The constant of proportionality is $$\begin{aligned}
t/2 = \frac{(2-n)k_0}{2} - m' = 6k_1+10k_2 +29m' - (3k_1+5k_2 +15m')n \end{aligned}$$ Each $(k_1,k_2,m') \in \{0,1,2,3\} \times \{0,1\} \times \bN$ contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
If $n>1$, in addition, one has $
V_\gamma = \bC x_0 \oplus \bC x_1 \oplus \bC x_2
$ if and only if $n$ is odd and $\gamma = (1,1,1,-1,\ldots,-1)$. The degree of $$\begin{aligned}
x_0^{k_0} x_1^{k_1} x_2^{k_2} \otimes x_3^\dual \wedge \cdots \wedge x_{n+1}^\dual
\in \Jac_{\w_\gamma} \otimes \Lambda^{\dim N_\gamma} N_\gamma^\dual\end{aligned}$$ is $$\begin{aligned}
k_0 \chi - (k_0-k_1) \chi_1 - (k_0-k_2) \chi_2 - (k_0+1) \chi_3 - \cdots - (k_0+1) \chi_{n+1},\end{aligned}$$ which is proportional to $\chi$ if and only if $$\begin{aligned}
\label{eq:e8c}
14k_0 + 6k_1 + 10k_2 - 15(n-1) = 30 m
$$ for $m \in \bZ$ and in addition we must have $k_0$ odd. Thus, again we can rewrite (\[eq:e8c\]) as $$\begin{aligned}
\label{eq:e8c2}
k_0 = 15 + 6 k_1 + 10 k_2 + 30m' \end{aligned}$$ where $m' = (k_0 -n)/2 -m$. The constant of proportionality is $$\begin{aligned}
(t- \dim N_\gamma)/2 = m - \frac{(n-1)k_0}{2} = 15 + 6k_1 + 10k_2 + 29m' - (8+3k_1+5k_2+15m)n
$$ Each $(k_1,k_2,m') \in \{0,1,2,3\} \times \{0,1\} \times \bZ$ such that $$15 + 6k_1+ 10 k_2 + 30m ' \geq 0$$ contributes $
\bC(k_0n)
$ to $\HH^t$ and $\HH^{t+1}$.
###
For any $
(t_1, t_2) \in \lb \bmu_5 \setminus \{ 1 \} \rb \times \lb \bmu_3 \setminus \{ 1 \} \rb,
$ the element $
\gamma = ((-1)^{n-1} (t_1 t_2)^{-1}, t_1, t_2, -1, \ldots, -1)
$ satisfies $V_\gamma = 0$. There are eight such elements, each of which contributes $\bC(-n)$ to $\HH^{n}$.
|
---
abstract: 'We study some global aspects of the bifurcation of an equivariant family of volume-contracting vector fields on the three-dimensional sphere. When part of the symmetry is broken, the vector fields exhibit Bykov cycles. Close to the symmetry, we investigate the mechanism of the emergence of heteroclinic tangencies coexisting with transverse connections. We find persistent suspended horseshoes accompanied by attracting periodic trajectories with long periods.'
address: |
Centro de Matemática da Universidade do Porto\
and Faculdade de Ciências da Universidade do Porto\
Rua do Campo Alegre 687, 4169–007 Porto, Portugal
author:
- 'Isabel S. Labouriau'
- 'Alexandre A. P. Rodrigues'
title: Partial symmetry breaking and heteroclinic tangencies
---
[^1]
Introduction
============
Heteroclinic cycles and networks associated to equilibria, periodic solutions and chaotic sets may be responsible for intermittent dynamics in nonlinear systems. Heteroclinic cycles may also be seen as the skeleton for the understanding of complicated switching between physical states – see Field [@Field], Golubitsky and Stewart [@GS] and Melbourne *et al* [@MPR].
The homoclinic cycle associated to a saddle-focus [@Homburg] provides one of the main examples for the occurrence of chaos involving suspended hyperbolic horseshoes and strange attractors; the complexity of the dynamics near these cycles has been discovered by the pionner L. P. Shilnikov [@Shilnikov63; @Shilnikov65; @Shilnikov67A]. The simplest heteroclinic cycles between two saddle-foci of different Morse indices where one heteroclinic connection is structurally stable and the other is not have been first studied by Bykov [@Bykov] and are thus called *Bykov cycles*. Recently there has been a renewal of interest of this type of heteroclinic bifurcation in different contexts – see [@Homburg; @LR; @Rodrigues3] and references therein. We also refer Lamb *et al* [@Lamb2005] who have studied Bykov cycles in the context of reversible systems.
Explicit examples of vector fields for which such cycles may be found are reported in Aguiar *et al* [@ACL2] and Rodrigues and Labouriau [@LR2]. These examples start with a differential equation with symmetry, $\dot{x}=f_0(x)$ whose flow has a globally attracting three-dimensional sphere, containing an asymptotically stable heteroclinic network with two saddle-foci. When part of the symmetry is destroyed by a small non-equivariant perturbation, it may be shown by the Melnikov method that the two-dimensional invariant manifolds intersect transversely. When some symmetry remains, the connection of the one-dimensonal manifolds is preserved, giving rise to Bykov cycles forming a network.
The main goal of this article is to describe and characterize the transition from the dynamics of the flow of the fully symmetric system $\dot{x}=f_0(x)$ and the perturbed system $\dot{x}=f_\lambda(x)$, for small $\lambda \neq 0$. For $\dot{x}=f_0(x)$, there is a heteroclinic network $\Sigma^0$ whose basin of attraction has positive Lebesgue measure. When $\lambda\neq 0$, the intersection of the invariant manifolds is transverse giving rise to a network $\Sigma^\star$ that cannot be removed by any small smooth perturbation. The transverse intersection implies that the set of all trajectories that lie for all time in a small neighbourhood of $\Sigma^\star$ has a locally-maximal hyperbolic set, admiting a complete description in terms of symbolic dynamics [@Shilnikov67]. Labouriau and Rodrigues [@LR] proved that for the perturbed system $\dot{x}=f_\lambda(x)$, the flow contains a Bykov network $\Sigma^\star$ and uniformly hyperbolic horseshoes accumulating on it.
Suppose the fully symmetric network $\Sigma^0$ is asymptotically stable and let $V^0$ be a neighbourhood of $\Sigma^0$ whose closure is compact and positively flow-invariant; hence it contains the $\omega$–limit sets of all its trajectories. The union of these limit sets is a maximal invariant set in $V^0$. For $\dot{x}=f_0(x)$, this union is simply the network $\Sigma_0$. For symmetry-breaking perturbations of $f_0$ it contains, but does not coincide with, a nonwandering set $\Omega_\lambda(\Sigma^\star)$ of trajectories that remain close to $\Sigma^\star$, the suspension of horseshoes accumulating on $\Sigma^\star$. The goal of this article is to investigate the larger limit set that contains nontrivial hyperbolic subsets and attracting limit cycles with long periods in $V^0$. This is what Gonchenko *et al* [@Gonchenko2007] call a *strange attractor*: an attracting limit set containing nontrivial hyperbolic subsets as well as attracting periodic solutions of extremely long periods.
When $\lambda \rightarrow 0$, the horseshoes in $\Omega_\lambda(\Sigma^\star)$ lose hyperbolicity giving rise to *heteroclinic tangencies* with infinitely many sinks nearby. A classical problem in this context is the study of heteroclinic bifurcations that lead to the birth of stable periodic sinks – see Afraimovich and Shilnikov [@Afraimovich83] and Newhouse [@Newhouse74; @Newhouse79]. When we deal with a heteroclinic tangency of the invariant manifolds, the description of all solutions that lie near the cycle for all time becomes more difficult. The problem of a *complete description* is unsolvable: the source of the difficulty is that arbitrarily small perturbations of any differential equation with a quadratic homo/heteroclinic tangency (the simplest situation) may lead to the creation of new tangencies of higher order, and to the birth of degenerate periodic orbits — Gonchenko [@Gonchenko2012].
Large-scale invariant sets of planar Poincaré maps vary discontinuously in size under small perturbations. Global bifurcations of observable sets, such as the emergence of attractors or metamorphoses of their basin boundaries, are easily detected numerically and regularly described. However, in the example described in [@LR2], the global bifurcation from a neighbourhood of $\Sigma^\star$ to $V^0$ is still a big mistery. The present paper contributes to a better understanding of the transition between uniform hyperbolicity (Smale horseshoes with infinitely many slabs) and the emergence of heteroclinic tangencies in a dissipative system close the symmetry.
Framework of the paper {#framework-of-the-paper .unnumbered}
----------------------
This paper is organised as follows. In section \[object\] we state our main result and review some of our recent results related to the object of study, after some basic definitions given in section \[preliminaries\]. The coordinates and other notation used in the rest of the article are presented in section \[localdyn\], where we also obtain a geometrical description of the way the flow transforms a curve of initial conditions lying across the stable manifold of an equilibrium. In section \[hyperbolicity\], we investigate the limit set that contains nontrivial hyperbolic subsets and we explain how the horseshoes in $\Omega_\lambda(\Sigma^\star)$ lose hyperbolicity, as $\lambda \rightarrow 0$. The first obstacle towards hyperbolicity is the emergence of tangencies and the existence of thick suspended Cantor sets near the network. In section \[sec tangency\], we prove that there is a sequence of parameter values $\lambda_i$ accumulating on $0$ such that the flow of $f_{\lambda_i}$ has heteroclinic tangencies and thus infinitely many attracting periodic trajectories. We include in section \[Conclusion\] a short conclusion about the results.
Preliminaries
=============
Let $f$ be a $C^1$ vector field on ${{\rm\bf R}}^{n}$ with flow given by the unique solution $x(t)=\varphi(t,x_{0})\in {{\rm\bf R}}^{n}$ of $\dot{x}=f(x)$ and $ x(0)=x_{0}$. Given two equilibria $p$ and $q$, an $m$-dimensional *heteroclinic connection* from $p$ to $q$, denoted $[p\rightarrow q]$, is an $m$-dimensional connected flow-invariant manifold contained in $W^{u}(p)\cap W^{s}(q)$. There may be more than one trajectory connecting $p$ and $q$.
Let $\mathcal{S=}\{p_{j}:j\in \{1,\ldots,k\}\}$ be a finite ordered set of mutually disjoint invariant saddles. Following Field [@Field], we say that there is a [*heteroclinic cycle* ]{}associated to $\mathcal{S}$ if $$\forall j\in \{1,\ldots,k\},W^{u}(p_{j})\cap W^{s}(p_{j+1})\neq
\emptyset \pmod k .$$ Sometimes we refer to the equilibria defining the heteroclinic cycle as *nodes*. A *heteroclinic network* is a finite connected union of heteroclinic cycles. Throughout this article, all nodes will be hyperbolic; the dimension of the local unstable manifold of an equilibria $p$ will be called the *Morse index* of $p$.
In a three-dimensional manifold, a *Bykov cycle* is a heteroclinic cycle associated to two hyperbolic saddle-foci with different Morse indices, in which the one-dimensional manifolds coincide and the two-dimensional invariant manifolds have a transverse intersection. It arises as a bifurcation of codimension 2 and it is also called by *$T$–point*.
Let $\mu$ denote a measure on a smooth manifold $M$ locally equivalent to the Lebesgue measure on charts. Given $x \in M$, let $$\omega(x) = \bigcap_{T > 0} \overline{\{\varphi(t,x) : t \ge T\}}$$ denote the $\omega$-limit set of the solution through $x$. If $X\subset M$ is a compact and flow-invariant subset, we let $\mathcal{B}(X) = \{x \in M : \omega(x) \subset X\}$ denote the *basin of attraction* of $X$. A compact invariant subset $X$ of $M$ is a *Milnor attractor* if $\mu(\mathcal{B}(X)) > 0$ and for any proper compact invariant subset $Y$ of $X$, $\mu(\mathcal{B}(X)\smallsetminus \mathcal{B}(Y)) > 0$.
The object of study {#object}
===================
The organising centre
---------------------
The starting point of the analysis is a differential equation $\dot{x}=f_0(x)$ on the unit sphere ${{\rm\bf S}}^3 =\{X=(x_1,x_2,x_3,x_4) \in {{\rm\bf R}}^4: ||X||=1\}$ where $f_0: {{\rm\bf S}}^3 \rightarrow \mathbf{T}{{\rm\bf S}}^3$ is a $C^1$ vector field with the following properties:
1. \[P1\] The vector field $f_0$ is equivariant under the action of $ {{\rm\bf Z}}_2 \oplus {{\rm\bf Z}}_2$ on ${{\rm\bf S}}^3$ induced by the two linear maps on ${{\rm\bf R}}^4$: $$\gamma_1(x_1,x_2,x_3,x_4)=(-x_1,-x_2,x_3,x_4) \qquad \text{and} \qquad \gamma_2(x_1,x_2,x_3,x_4)=(x_1,x_2,-x_3,x_4).$$
2. \[P2\] The set $Fix( {{\rm\bf Z}}_2 \oplus {{\rm\bf Z}}_2)=\{x \in {{\rm\bf S}}^3:\gamma_1 x=\gamma_2 x = x \}$ consists of two equilibria ${{\rm\bf v}}=(0,0,0,1)$ and ${{\rm\bf w}}=(0,0,0,-1)$ that are hyperbolic saddle-foci, where:
- the eigenvalues of $df_0({{\rm\bf v}})$ are $-C_{{{\rm\bf v}}} \pm \alpha_{{{\rm\bf v}}}i$ and $E_{{{\rm\bf v}}}$ with $\alpha_{{{\rm\bf v}}} \neq 0$, $C_{{{\rm\bf v}}}>E_{{{\rm\bf v}}}>0$
- the eigenvalues of $df_0({{\rm\bf w}})$ are $E_{{{\rm\bf w}}} \pm \alpha_{{{\rm\bf w}}} i$ and $-C_{{{\rm\bf w}}}$ with $\alpha_{{{\rm\bf w}}} \neq 0$, $C_{{{\rm\bf w}}}>E_{{{\rm\bf w}}}>0$.
3. \[P3\] The flow-invariant circle $Fix({\langle\gamma_{1}\rangle})=\{x \in {{\rm\bf S}}^3:\gamma_1 x = x \}$ consists of the two equilibria ${{\rm\bf v}}$ and ${{\rm\bf w}}$, a source and a sink, respectively, and two heteroclinic trajectories from ${{\rm\bf v}}$ to ${{\rm\bf w}}$ that we denote by $[{{\rm\bf v}}\rightarrow {{\rm\bf w}}]$.
4. \[P4\] The $f_0$-invariant sphere $Fix({\langle\gamma_{2}\rangle})=\{x \in {{\rm\bf S}}^3:\gamma_2 x = x \}$ consists of the two equilibria ${{\rm\bf v}}$ and ${{\rm\bf w}}$, and a two-dimensional heteroclinic connection from ${{\rm\bf w}}$ to ${{\rm\bf v}}$. Together with the connections in (P\[P3\]) this forms a heteroclinic network that we denote by $\Sigma^0$.
5. \[P6\] For sufficiently small open neighbourhoods $V$ and $W$ of ${{\rm\bf v}}$ and ${{\rm\bf w}}$, respectively, given any trajectory $\varphi$ going once from $V$ to $W$, if one joins the starting point of $\varphi$ in $\partial V$ to the end point in $\partial W$ by a line segment, one obtains a closed curve that is linked to $\Sigma^0$ (figure \[orientations\]).
![There are two different possibilities for the geometry of the flow around a Bykov cycle depending on the direction trajectories turn around the heteroclinic connection $[{{\rm\bf v}}\rightarrow {{\rm\bf w}}]$. We assume here that all trajectories turn in the same direction near ${{\rm\bf v}}$ and near ${{\rm\bf w}}$ as drawn on the left (a). When the endpoints of the trajectory are joined, the closed curve is linked to the cycle. The case where trajectories turn in opposite directions around the connection is shown in (b). In this case, when the endpoints of the trajectory are joined, the closed curve may not be linked to the cycle.[]{data-label="orientations"}](OurNetworkSameOrientLink.eps){height="4.6cm"}
Condition (P\[P6\]) means that the curve $\varphi$ and the cycle $\Sigma^0$ cannot be separated by an isotopy. This property is persistent under perturbations: if it holds for the organising centre $f_0$, then it is still valid for vector fields near it, as long as the heteroclinic connection remains. An explicit example of a family of differential equations where this assumption is valid is constructed in Rodrigues and Labouriau [@LR2].
The heteroclinic network of the organising centre
-------------------------------------------------
The heteroclinic connections in the network $\Sigma^0$ are contained in fixed point subspaces satisfying the hypothesis (H1) of Krupa and Melbourne [@KM1]. Since the inequality $C_{{\rm\bf v}}C_{{\rm\bf w}}>E_{{\rm\bf v}}E_{{\rm\bf w}}$ holds, the Krupa and Melbourne stability criterion [@KM1] may be applied to $\Sigma^0$ and we have:
\[propNetworkIstStable\] Under conditions (P\[P1\])–(P\[P4\]) the heteroclinic network $\Sigma^0$ is asymptotically stable.
In particular, we obtain:
The basin of attraction of the heteroclinic network $\Sigma^0$ whose vector field satisfies (P\[P1\])–(P\[P4\]) has positive Lebesgue measure.
Proposition \[propNetworkIstStable\] implies that there exists an open neighbourhood $V^0$ of the network $\Sigma^0$ such that every trajectory starting in $V^0$ remains in it for all positive time and is forward asymptotic to the network. The neighbourhood may be taken to have its boundary transverse to the vector field $f_0$.
The fixed point hyperplane defined by $Fix({\langle\gamma_{2}\rangle})=\{(x_1, x_2, x_3, x_4) \in {{\rm\bf S}}^3: x_3=0\}$ divides ${{\rm\bf S}}^3$ in two flow-invariant connected components, preventing jumps between the two cycles in $\Sigma^0$. Due to the ${\langle\gamma_{2}\rangle}$–symmetry, trajectories whose initial condition lies outside the invariant subspaces will approach one of the cycles in positive time. Successive visits to both cycles require breaking this symmetry.
Bykov cycles
------------
Since ${{\rm\bf v}}$ and ${{\rm\bf w}}$ are hyperbolic equilibria, then any vector field close to it in the $C^1$ topology still has two equilibria ${{\rm\bf v}}$ and ${{\rm\bf w}}$ with eigenvalues satisfying (P\[P2\]). The dimensions of the local stable and unstable manifolds of ${{\rm\bf v}}$ and ${{\rm\bf w}}$ do not change. If we retain the symmetry $\gamma_1$, then the one-dimensional connection of (P\[P3\]) remains, as it takes place in the flow-invariant circle $Fix({\langle\gamma_{1}\rangle})$, but generically when the symmetry $\gamma_2$ is broken, the two dimensional heteroclinic connection is destroyed, since the fixed point subset $Fix(\textbf{Z}_2({\langle\gamma_{2}\rangle}))$ is no longer flow-invariant. Generically, the invariant two-dimensional manifolds meet transversely at two trajectories, for small ${\langle\gamma_{1}\rangle}$-equivariant perturbations of the vector field. We denote these heteroclinic connections by $[{{\rm\bf w}}\rightarrow {{\rm\bf v}}]$ This gives rise to a network $\Sigma^*$ consisting of four copies of the simplest heteroclinic cycle between two saddle-foci of different Morse indices, where one heteroclinic connection is structurally stable and the other is not. This cycle is called a *Bykov cycle*. Note that the networks $\Sigma^0$ and $\Sigma^*$ are not of the same nature.
Transversality ensures that the neighbourhood $V^0$ is still positively invariant for vector fields $C^1$ close to $f_0$ and contains the network $\Sigma^*$. Since the closure of $V^0$ is compact and positively invariant it contains the $\omega$-limit sets of all its trajectories. The union of these limit sets is a maximal invariant set in $V^0$. For $f_0$, this is the cycle $\Sigma^0$, by Proposition \[propNetworkIstStable\], whereas for symmetry-breaking perturbations of $f_0$ it contains $\Sigma^*$ but does not coincide with it. From now on, our aim is to obtain information on this invariant set.
A systematic study of the dynamics in a neighbourhood of the Bykov cycles in $\Sigma^*$ was carried out in Aguiar [*et al*]{} [@ACL; @NONLINEARITY], Labouriau and Rodrigues [@LR] and Rodrigues [@Rodrigues3]; we proceed to review these local results. In the next section we will discuss some global aspects of the dynamics.
Given a Bykov cycle $\Gamma$ involving ${{\rm\bf v}}$ and ${{\rm\bf w}}$, let $U_{{\rm\bf v}}$ and $U_{{\rm\bf w}}\subset V^0$ be disjoint neighbourhoods of these points. Let $S_p$ and $S_q$ be local cross-sections of $f_\lambda$ at two points $p$ and $q$ in the connections $[{{\rm\bf v}}\rightarrow{{\rm\bf w}}]$ and $[{{\rm\bf w}}\rightarrow{{\rm\bf v}}]$, respectively, with $p, q\not\in U_{{\rm\bf v}}\cup U_{{\rm\bf w}}$. Saturating the cross-sections by the flow, one obtains two flow-invariant tubes joining $U_{{\rm\bf v}}$ and $U_{{\rm\bf w}}$ containing the connections in their interior. We call the union of these tubes, $U_{{\rm\bf v}}$ and $U_{{\rm\bf w}}$ a *tubular neighbourhood* $V^\Gamma$ of the Bykov cycle.
With these conventions we have:
\[teorema T-point switching\] If a vector field $f_0$ satisfies (P\[P1\])–(P\[P6\]), then the following properties are satisfied generically by vector fields in an open neighbourhood of $f_0$ in the space of ${\langle\gamma_{1}\rangle}$–equivariant vector fields of class $C^1$ on ${{\rm\bf S}}^3$:
1. \[item0\] there is a heteroclinic network $\Sigma^*$ consisting of four Bykov cycles involving two equilibria ${{\rm\bf v}}$ and ${{\rm\bf w}}$, two heteroclinic connections $[{{\rm\bf v}}\rightarrow{{\rm\bf w}}]$ and two heteroclinic connections $[{{\rm\bf w}}\rightarrow{{\rm\bf v}}]$;
2. \[item1\] the only heteroclinic connections from ${{\rm\bf v}}$ to ${{\rm\bf w}}$ are those in the Bykov cycles and there are no homoclinic connections;
3. \[item6\] any tubular neighbourhood $V^\Gamma$ of a Bykov cycle $\Gamma$ in $\Sigma^*$ contains points not lying on $\Gamma$ whose trajectories remain in $V^\Gamma$ for all time;
4. \[item4\] any tubular neighbourhood $V^\Gamma$ of a Bykov cycle $\Gamma$ in $\Sigma^*$ contains at least one $n$-pulse heteroclinic connection $[{{\rm\bf w}}\to{{\rm\bf v}}]$;
5. \[item5\] given a cross-section $S_q\subset V^\Gamma$ at a point $q$ in $[{{\rm\bf w}}\rightarrow{{\rm\bf v}}]$, there exist sets of points such that the dynamics of the first return to $S_q$ is uniformly hyperbolic and conjugate to a full shift over a finite number of symbols. These sets accumulate on the cycle.
Notice that assertion of Theorem \[teorema T-point switching\] implies the existence of a bigger network: beyond the original transverse connections $[{{\rm\bf w}}\rightarrow{{\rm\bf v}}]$, there exist infinitely many subsidiary heteroclinic connections turning around the original Bykov cycle. We will restrict our study to one cycle.
It is a folklore result that a hyperbolic invariant set of a $C^2$–diffeomorphism has zero Lebesgue measure – see Bowen [@Bowen75]. However, since the authors of [@LR] worked in the $C^1$ category, this chain of horseshoes might have positive Lebesgue measure as the “fat Bowen horseshoe” described in [@Bowen]. Rodrigues [@Rodrigues3] proved that this is not the case:
\[zero measure\] Let ${V}^\Gamma$ be a tubular neighbourhood of one of the Bykov cycles $\Gamma$ of Theorem \[teorema T-point switching\]. Then in any cross-section $S_q\subset V^\Gamma$ at a point $q$ in $[{{\rm\bf w}}\rightarrow{{\rm\bf v}}]$ the set of initial conditions in $S_q \cap V^\Gamma$ that do not leave $V^\Gamma$ for all time has zero Lebesgue measure.
It follows from Theorem \[zero measure\] that the shift dynamics does not trap most trajectories in the neighbourhood of the cycle. In particular, the cycle cannot be Lyapunov stable (and therefore cannot be asymptotically stable).
One astonishing property of the heteroclinic network $\Sigma^*$ is the possibility of shadowing $\Sigma^*$ by the property called *switching*: any infinite sequence of pseudo-orbits defined by admissible heteroclinic connections can be shadowed, as we proceed to define.
A *path* on $\Sigma^* $ is an infinite sequence $s^k=(c_{j})_{j\in {{\rm\bf N}}}$ of heteroclinic connections $c_{j}=[A_{j}\rightarrow B_{j}]$ in $\Sigma^* $ such that $A_{j}, B_{j} \in \{{{\rm\bf v}}, {{\rm\bf w}}\}$ and $B_{j}=A_{j+1}$. Let $U_{{\rm\bf v}}, U_{{\rm\bf w}}\subset V^0$ be neighbourhoods of ${{\rm\bf v}}$ and ${{\rm\bf w}}$. For each heteroclinic connection in $\Sigma^*$, consider a point $p$ on it and a small neighbourhood $U_p\subset V^0$ of $p$. We assume that all these neighbourhoods are pairwise disjoint.
The trajectory $\varphi(t,q)$ of $\dot x=f_\lambda(x)$, *follows* the path $s^k=(c_{j})_{j\in {{\rm\bf N}}}$ within a given set of neighbourhoods as above, if there exist two monotonically increasing sequences of times $(t_{i})_{i\in {{\rm\bf N}}}$ and $(z_{i})_{i\in {{\rm\bf N}}}$ such that for all $i \in {{\rm\bf N}}$, we have $t_{i}<z_{i}<t_{i+1}$ and:
- $\varphi(t,q)\subset V^0$ for all $t\ge 0$;
- $\varphi(t_{i},q) \in U_{A_{i}}$ and $\varphi (z_{i},q)\in U_{p}$, $p\in c_i$ and
- for all $t\in (z_{i},z_{i+1})$, $\varphi (t,q)$ does not visit the neighbourhood of any other node except $A_{i+1}$.
There is *switching* near $\Sigma^*$ if for each path there is a trajectory that follows it within every set of neighbourhoods as above.
\[teorema switching\] There is switching on the network $\Sigma^*$.
The solutions that realise switching lie for all positive time in the union of tubular neighbourhoods $V^\Gamma$ of all cycles $\Gamma\subset\Sigma^*$. Hence we may adapt the proof of Theorem \[zero measure\] to obtain:
The switching of Theorem \[teorema switching\] is realised by a set of initial conditions with zero Lebesgue measure.
Non-hyperbolic dynamics
-----------------------
When the symmetry $\gamma_2$ is broken, the dynamics changes dramatically, as can be seen in the next result:
\[teorema tangency\] If a vector field $f_0$ satisfies (P\[P1\])–(P\[P6\]), then in any open neighbourhood of $f_0$ in the space of ${\langle\gamma_{1}\rangle}$-equivariant vector fields of class $C^1$ on ${{\rm\bf S}}^3$, there are vector fields $f_*$ whose flow has a heteroclinic tangency between $W^u({{\rm\bf w}})$ and $W^s({{\rm\bf v}})$ in $V^0$.
Theorem \[teorema tangency\] is proved in section \[sec tangency\]. The vector fields $f_*$ will be obtained in a generic one-parameter unfolding $f_\lambda$ of $f_0$, for which we will find a sequence of $\lambda_i$ converging to zero such that the flow of $f_{\lambda_i}$ has the required property. When $\lambda_i \rightarrow 0$, these tangencies accumulate on the transverse connections. Persistent tangencies in a dissipative diffeomorphism are related to the coexistence of infinitely many sinks and sources [@Newhouse74; @Newhouse79]. Moreover, any parametrised family of diffeomorphisms going through a heteroclinic tangency associated to a dissipative cycle must contain a sequence of Hénon-like families [@Colli]. Hence, for $\lambda \approx 0$, return maps to appropriate domains close to the tangency are conjugate to Hénon-like maps and thus:
\[attractors\] Suppose a vector field $f_0$ satisfies (P\[P1\])–(P\[P6\]). Then in the space of ${\langle\gamma_{1}\rangle}$-equivariant vector fields of class $C^3$ on ${{\rm\bf S}}^3$, there is a set $\mathcal{C}$ of vector fields accumulating on $f_0$ such that all $f_\lambda \in \mathcal{C}$ possess infinitely many strange (coexisting) attractors in $V^0$ .
In Ovsyannikov and Shilnikov [@OS], it is shown that there are small perturbations of $f_0$ with a periodic solution as close as desired to the cycle whose stable and unstable manifolds are tangent. By Colli [@Colli], the result follows. Note that although the statement in [@Colli] asks for $C^\infty$ perturbations, in [@BonattiEcia] the coexistence of strange attractors only requires $C^3$.
Local dynamics near the network {#localdyn}
===============================
In order to obtain results on ${\langle\gamma_{1}\rangle}$–equivariant vector fields $C^1$ close to $f_0$ we study the bifurcation of a generic one-parameter family of differential equations $\dot{x}=f_\lambda(x)$ on the unit sphere ${{\rm\bf S}}^3 =\{X=(x_1,x_2,x_3,x_4) \in {{\rm\bf R}}^4: ||X||=1\}$. The unfolding $f_\lambda: {{\rm\bf S}}^3 \rightarrow \mathbf{T}{{\rm\bf S}}^3$ of $f_0$ is a family of $C^1$ vector fields with the following properties:
1. \[PartialSymmetry\] For each $\lambda$ the vector field $f_\lambda$ is ${\langle\gamma_{1}\rangle}$–equivariant.
2. \[P4.5\] There are two equilibria ${{\rm\bf v}}$ and ${{\rm\bf w}}$ satisfying (P\[P2\]) and (P\[P3\]).
3. \[P5\] For $\lambda \neq 0$, the local two-dimensional manifolds $W^u({{\rm\bf w}})$ and $W^s({{\rm\bf v}})$ intersect transversely at two trajectories. Together with the connections in (P\[P3\]) this forms a Bykov heteroclinic network that we denote by $\Sigma^\lambda$.
In order to describe the dynamics around the Bykov cycles, we start by introducing local coordinates near the saddle-foci ${{\rm\bf v}}$ and ${{\rm\bf w}}$ and we define some terminology that will be used in the rest of the paper. Since by assumption (P\[P2\]) we have $C_{{\rm\bf v}}\neq E_{{\rm\bf v}}$ and $C_{{\rm\bf w}}\neq E_{{\rm\bf w}}$, then by Samovol’s Theorem [@Samovol], the vector field $f_\lambda$ is $C^1$–conjugate to its linear part around each saddle-focus — see also Homburg and Sandstede [@HS] (section 3.1). In cylindrical coordinates $(\rho ,\theta ,z)$ the linearization at ${{\rm\bf v}}$ takes the form $$\dot{\rho}=-C_{{{\rm\bf v}}}\rho \qquad \dot{\theta}=1 \qquad \dot{z}=E_{{{\rm\bf v}}}z$$ and around ${{\rm\bf w}}$ it is given by: $$\dot{\rho}=E_{{{\rm\bf w}}}\rho \qquad \dot{\theta}= 1 \qquad \dot{z}=-C_{{{\rm\bf w}}}z .$$
![Cylindrical neighbourhoods of the saddle-foci ${{\rm\bf w}}$ (a) and ${{\rm\bf v}}$ (b). []{data-label="neigh_vw"}](neigh_vw){height="6cm"}
We consider cylindrical neighbourhoods $V$ and $W$ in ${{{\rm\bf S}}}^3$ of ${{\rm\bf v}}$ and ${{\rm\bf w}}$, respectively, of radius $\varepsilon>0$ and height $2\varepsilon$ — see figure \[neigh\_vw\]. After a linear rescaling of the local variables, we may take $\varepsilon=1$. Their boundaries consist of three components: the cylinder wall parametrized by $x\in {{\rm\bf R}}\pmod{2\pi}$ and $|y|\leq 1$ with the usual cover $ (x,y)\mapsto (1 ,x,y)=(\rho ,\theta ,z)$ and two discs (top and bottom of the cylinder). We take polar coverings of these discs $(r,\varphi )\mapsto (r,\varphi , \pm 1)=(\rho ,\theta ,z)$ whith $0\leq r\leq 1$ and $\varphi \in {{\rm\bf R}}\pmod{2\pi}$ and use the following terminology, as in figure \[neigh\_vw\]:
- $In({{\rm\bf v}})$, the cylinder wall of $V$, consists of points that go inside $V$ in positive time;
- $Out({{\rm\bf v}})$, the top and bottom of $V$, consists of points that go inside $V$ in negative time;
- $In({{\rm\bf w}})$, the top and bottom of $W$, consists of points that go inside $W$ in positive time;
- $Out({{\rm\bf w}})$, the cylinder wall of $W$, consists of points that go inside $W$ in negative time.
The flow is transverse to these sets and moreover the boundaries of $V$ and of $W$ may be written as the closures of the disjoint unions $In({{\rm\bf v}}) \cup Out ({{\rm\bf v}})$ and $In({{\rm\bf w}}) \cup Out ({{\rm\bf w}})$, respectively. The trajectories of all points $(x,y)$ in $In({{\rm\bf v}}) \backslash W^s({{\rm\bf v}})$, leave $V$ at $Out({{\rm\bf v}})$ at $$\Phi_{{{\rm\bf v}}}(x,y)=\left(|y|^{\delta_{{\rm\bf v}}},-\frac{\ln |y|}{E_{{\rm\bf v}}}+x\right)=(r,\phi)
\qquad \mbox{where}\qquad
\delta_{{\rm\bf v}}=\frac{C_{{{\rm\bf v}}}}{E_{{{\rm\bf v}}}} > 1 \ .
\label{local_v}$$ Similarly, points $(r,\phi)$ in $In({{\rm\bf w}}) \backslash W^s({{\rm\bf w}})$, leave $W$ at $Out({{\rm\bf w}})$ at $$\Phi_{{{\rm\bf w}}}(r,\varphi )=\left(-\frac{\ln r}{E_{{\rm\bf w}}}+\varphi,r^{\delta_{{\rm\bf w}}}\right)=(x,y)
\qquad \mbox{where}\qquad
\delta_{{\rm\bf w}}=\frac{C_{{{\rm\bf w}}}}{E_{{{\rm\bf w}}}} >1 \ .
\label{local_w}$$
We will denote by $W^u_{loc}({{\rm\bf v}})$ the portion of $W^u({{\rm\bf v}})$ that goes from ${{\rm\bf v}}$ up to $In({{\rm\bf w}})$ not intersecting the interior of $W$. Similarly, $W^s_{loc}({{\rm\bf v}})$ is the portion of $W^s({{\rm\bf v}})$ outside $W$ that goes directly from $Out({{\rm\bf w}})$ into ${{\rm\bf v}}$, and $W^u_{loc}({{\rm\bf w}})$ and $W^s_{loc}({{\rm\bf w}})$ connect ${{\rm\bf w}}$ to $\partial V$, not intersecting the interior of $V$.
The flow sends points in $Out({{\rm\bf v}})$ near $W^u_{loc}({{\rm\bf v}})$ into $In({{\rm\bf w}})$ along the connection $[{{\rm\bf v}}\rightarrow{{\rm\bf w}}]$. We will assume that this map $\Psi_{{{\rm\bf v}}\rightarrow {{\rm\bf w}}}$ is the identity, this is compatible with hypothesis (P\[P6\]); nevertheless all the results follow if $\Psi_{{{\rm\bf v}}\rightarrow {{\rm\bf w}}}$ is either a uniform contraction or a uniform expansion. We make the convention that one of the connections $[{{\rm\bf v}}\rightarrow{{\rm\bf w}}]$ links points with $y>0$ in $V$ to points with $y>0$ in $W$. There is also a well defined transition map $ \Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}:Out({{\rm\bf w}})\longrightarrow In({{\rm\bf v}})$ that will be discussed later.
By (P\[P5\]), the manifolds $W^u({{\rm\bf w}})$ and $W^s({{\rm\bf v}})$ intersect transversely for $\lambda \neq 0$. For $\lambda$ close to zero, we are assuming that $W^s({{\rm\bf v}})$ intersects the wall $Out({{\rm\bf w}})$ of the cylinder $W$ on a closed curve represented in figure \[elipse\] by an ellipse — this is the expected unfolding from the coincidence of the invariant manifolds of the equilibria.
![For $\lambda$ close to zero, we are assuming that $W^s({{\rm\bf v}})$ intersects the wall $Out({{\rm\bf w}})$ of the cylinder $W$ on a closed curve – this is the expected unfolding from the coincidence of the invariant manifolds of the equilibria.[]{data-label="elipse"}](elipse1){height="12cm"}
From the geometrical behaviour of the local transition maps (\[local\_v\]) and (\[local\_w\]), we need some definitions: a *segment* $\beta $ on $In({{\rm\bf v}})$ is a smooth regular parametrized curve $\beta :[0,1)\rightarrow In({{\rm\bf v}})$ that meets $W^{s}_{loc}({{\rm\bf v}})$ transversely at the point $\beta (1)$ only and such that, writing $\beta (s)=(x(s),y(s))$, both $x$ and $y$ are monotonic functions of $s$ – see figure \[Structures\] (a).
A *spiral* on a disc $D$ around a point $p\in D$ is a curve $\alpha :[0,1)\rightarrow D$ satisfying ${\displaystyle}\lim_{s\to 1^-}\alpha (s)=p$ and such that if $\alpha (s)=(r(s),\theta(s))$ is its expressions in polar coordinates around $p$ then the maps $r$ and $\theta$ are monotonic, and $\lim_{s\to 1^-}|\theta(s)|=+\infty$.
Consider a cylinder $C$ parametrized by a covering $(\theta,h )\in {{\rm\bf R}}\times[a,b]$, with $a<b\in{{\rm\bf R}}$ where $\theta $ is periodic. A *helix* on the cylinder $C$ *accumulating on the circle* $h=h_{0}$ is a curve $\gamma :[0,1)\rightarrow C$ such that its coordinates $(\theta (s),h(s))$ satisfy $ \lim_{s\to 1^-}h(s)=h_{0}$, $\lim_{s\to 1^-}|\theta (s)|=+\infty$ and the maps $\theta$ and $h$ are monotonic.
Using these definitions and the expressions and for $\Phi_{{{\rm\bf v}}}$ and $\Phi_{{{\rm\bf w}}}$ we get:
\[Structures\] When (P\[P6\]) holds, then a segment on $In({{\rm\bf v}})$ is mapped by $\Phi _{{{\rm\bf v}}}$ into a spiral on $Out({{\rm\bf v}})$ around $W^u_{loc}({{\rm\bf v}})\cap Out({{\rm\bf v}}) $. This spiral is mapped by $ \Psi_{{{\rm\bf v}}\rightarrow {{\rm\bf w}}}$ into another spiral around $W^s_{loc}({{\rm\bf w}})\cap In({{\rm\bf v}})$, which is mapped by $\Phi _{{{\rm\bf w}}}$ into a helix on $Out({{\rm\bf w}})$ accumulating on the circle $Out({{\rm\bf w}}) \cap W^{u}({{\rm\bf w}})$.
Hyperbolicity
=============
In this section we show that the hyperbolicity of Theorem \[teorema T-point switching\] only holds when we restrict our attention to trajectories that remain near the cycles in the network. The construction also indicates how the geometrical content of Theorems \[teorema T-point switching\], \[zero measure\] and \[teorema switching\] is obtained.
Let $(P_{{\rm\bf w}}^1,0)$ and $(P_{{\rm\bf w}}^2,0)$ with $0<P_{{\rm\bf w}}^1<P_{{\rm\bf w}}^2<2\pi$ be the coordinates of the two points in $W^u_{loc}({{\rm\bf w}})\cap W^s_{loc}({{\rm\bf v}})\cap Out({{\rm\bf w}})$ where the connections $[{{\rm\bf w}}\rightarrow {{\rm\bf v}}]$ meet $Out({{\rm\bf w}})$, as in figure \[elipse\]. Analogously, let $(P_{{\rm\bf v}}^1,0)$ and $(P_{{\rm\bf v}}^2,0)$ be the coordinates of the two corresponding points in $W^u_{loc}({{\rm\bf w}})\cap W^s_{loc}({{\rm\bf v}})\cap In({{\rm\bf v}})$ where $[{{\rm\bf w}}\rightarrow {{\rm\bf v}}]$ meets $In({{\rm\bf v}})$, with the convention that $(P_{{\rm\bf w}}^j,0)$ and $(P_{{\rm\bf v}}^j,0)$ are on the same trajectory for $j=1,2$.
For small $\lambda>0$ we may write $W^s({{\rm\bf v}})\cap Out({{\rm\bf w}})$ as the graph of a smooth function $y=g(x)$, with $g(P_{{\rm\bf w}}^j)=0$, $j=1,2$. Similarly, $W^u({{\rm\bf w}})\cap In({{\rm\bf v}})$ is the graph of a smooth function $y=h(x)$, with $g(P_{{\rm\bf v}}^j)=0$, $j=1,2$. For definiteness, we number the points in the connections to have $g^\prime(P_{{\rm\bf w}}^1)>0$. Hence $g^\prime(P_{{\rm\bf w}}^2)<0$, and $h^\prime(P_{{\rm\bf v}}^1)<0$, $h^\prime(P_{{\rm\bf v}}^2)>0$.
\[propLines\] For the first hit map $\eta= \Phi _{{{\rm\bf w}}}\circ \Psi_{{{\rm\bf v}}\rightarrow {{\rm\bf w}}}\circ \Phi _{{{\rm\bf v}}}:In({{\rm\bf v}}){\longrightarrow}Out({{\rm\bf w}})$, we have:
1. \[horizontal\] any horizontal line segment $[a,b]\times\{y_0\}\subset In({{\rm\bf v}})$ is mapped by $\eta$ into a horizontal line segment $[c,d]\times \{y_0^\delta\}\subset Out({{\rm\bf w}})$, with $\delta>1$;
2. \[vertical\] any vertical line segment $\{x_0\}\times[0,y_0]\subset In({{\rm\bf v}})$ is mapped by $\eta$ into a helix accumulating on the circle $Out({{\rm\bf w}})\cap W^u_{loc}({{\rm\bf w}})$;
3. \[intervals\] given $x_0$, there are positive constants $a<b\in{{\rm\bf R}}$ and a sequence of intervals\
$\mathcal{I}_n= \{x_0\} \times [e^{-2n\pi/K}e^a, e^{-2n\pi/K}e^b] $ such that $\eta\left(\mathcal{I}_n\right)$ crosses $W^s_{loc}({{\rm\bf v}})\cap Out({{\rm\bf w}})$ twice transversely;
4. \[disjoint\] if $K>1$ then the intervals $\mathcal{I}_n$ are disjoint.
Assertion \[horizontal\]) is immediate from the expression of $\eta=\Phi_{{{\rm\bf w}}} \circ \Psi_{{{\rm\bf v}}\rightarrow {{\rm\bf w}}} \circ \Phi_{{{\rm\bf v}}}$ in coordinates: $$\eta(x,y)=\left(x-K \ln y , y^{\delta} \right)
\qquad\mbox{where}\qquad
K= \frac{C_{{\rm\bf v}}+E_{{\rm\bf w}}}{E_{{\rm\bf v}}E_{{\rm\bf w}}} > 0
\qquad\mbox{and}\qquad
\delta=\delta_{{\rm\bf v}}\delta_{{\rm\bf w}}>1$$ and assertion \[vertical\]) follows from Proposition \[Structures\].
For \[intervals\]) let $y_*$ be the maximum value of $g(x)$ and let $m\in{{\rm\bf Z}}$ such that $x_0-P_{{\rm\bf w}}^2\le \frac{K}{\delta}\ln y_*+2m\pi$. We take $a=(x_0-P_{{\rm\bf w}}^1-2m\pi)/K$ and $b=(x_0-P_{{\rm\bf w}}^2-2m\pi)/K$. Then $a<b$, the second coordinate of $\eta(x_0,e^b)$ is less than $y_*$, its first coordinate is $P_{{\rm\bf w}}^2+2m\pi$, and that of $\eta(x_0,e^a)$ is $P_{{\rm\bf w}}^1+2m\pi$. Hence the curve $\eta(\mathcal{I}_0)$ goes across the graph of $g(x)$ that corresponds to $W^s({{\rm\bf v}})$ as in figure \[figPropLines\]. The first coordinates of the end points of $\eta(\mathcal{I}_n)$ for the other intervals are $P_{{\rm\bf w}}^1+2(m+n)\pi$ and $P_{{\rm\bf w}}^2+2(m+n)\pi$ and their second coordinates are also less than $y_*$, so each curve $\eta(\mathcal{I}_n)$ also crosses the graph of $g(x)$ transversely.
If $K>1$, and since $P_{{\rm\bf w}}^1-P_{{\rm\bf w}}^2<2\pi<2K\pi$, then $x_0-P_{{\rm\bf w}}^2-2m\pi-2K\pi<x_0-P_{{\rm\bf w}}^1-2m\pi$ and hence $b-2(n+1)\pi/K<a-2n\pi/K$, implying that $\mathcal{I}_n\cap\mathcal{I}_{n+1}=\emptyset$.
![Proof of \[intervals\]) in Proposition \[propLines\]: the transition map $\eta$ sends the intervals $\mathcal{I}_n$ into curves that cross $W^s_{loc}({{\rm\bf v}})$ transversely. []{data-label="figPropLines"}](proof_10.eps){height="7cm"}
We are interested in the images of rectangles in $In({{\rm\bf v}})$ under iteration by the first return map to $In({{\rm\bf v}})\backslash W^s_{loc}({{\rm\bf v}})$ given by $ \zeta= \Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}} \circ \eta$.
Consider a rectangle $R_{{\rm\bf v}}=[P_{{\rm\bf v}}^1-\tau,P_{{\rm\bf v}}^1+\tau]\times[\tilde{y},\hat{y}]\subset In({{\rm\bf v}})$ with $\tau>0$ small and $0<\tilde{y}<\hat{y}<1$. This is mapped by $\eta$ into a strip whose boundary consists of two horizontal line segments and two pieces of helices. A calculation similar to that in Proposition \[propLines\] shows that there are many choices of $\tilde{y}$ and $\hat{y}$ for which the strip crosses $W^s_{loc}({{\rm\bf v}})$ transversely near $P_{{\rm\bf w}}^1$. This strip is then mapped by $\Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}$ into $In(v)$ crossing $W^s_{loc}({{\rm\bf v}})$ transversely. The final strip $\zeta(R_{{\rm\bf v}})=\Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}\circ\eta(R_{{\rm\bf v}})$ remains close to $W^u_{loc}({{\rm\bf w}})$. Hence the effect of $\zeta$ is to stretch $R_{{\rm\bf v}}$ in the vertical direction and map it with the stretched direction approximately parallel to $W^u_{loc}({{\rm\bf w}})$. Since $R_{{\rm\bf v}}$ has been chosen to contain a piece of $W^u_{loc}({{\rm\bf w}})$, then $\zeta(R_{{\rm\bf v}})$ will cross $R_{{\rm\bf v}}$. Repeating this for successive disjoint intervals $[\tilde{y},\hat{y}]$ gives rise to horseshoes.
This is the idea of the proof of the local results of Theorem \[teorema T-point switching\]: in a neighbourhood of the connection point $(P_{{\rm\bf v}}^1,0)$, one finds infinitely many disjoint rectangles, each one containing a Cantor set of points whose orbits under $\zeta$ remain in the Cantor set, and hence return to a neighbourhood of $(P_{{\rm\bf v}}^1,0)$ for all future iterations. In the gaps between these rectangles one finds another set of disjoint rectangles that are first mapped by $\zeta$ into a neighbourhood of the other connection point $(P_{{\rm\bf v}}^2,0)$. Repeating the construction near the second connection one obtains the switching of Theorem \[teorema switching\].
It can also be shown that the first return map $\zeta$ is uniformly hyperbolic at the points in the rectangle $R_{{\rm\bf v}}$, with the choices of $\tilde{y},\hat{y}$ above. This means that at each of these points there is a well defined contraction direction and this is the main tool in the proof of Theorem \[zero measure\].
Since all this takes place inside a positively invariant neighbourhood $V^0$, it would be natural to try to extend this reasoning to larger rectangles in $In({{\rm\bf v}})$. We finish this section explaining why this fails.
Consider a rectangle $R_{{\rm\bf v}}$ as above, containing points of the Cantor set. Since $\eta$ expands vertical lines, the local unstable manifolds of these points forms a lamination on $R_{{\rm\bf v}}$ whose sheets are approximately vertical. Now, take a larger rectangle $\widehat{R}_{{\rm\bf v}}$ with $y\in[\tilde{y},\bar{y}]$, $\bar{y}>\hat{y}$, so as to have the maximum of the curve $y=g(x)$ lying in $\eta(\widehat{R}_{{\rm\bf v}})$, and increasing $\tau$ if necessary. The sheets of the lamination still follow the vertical direction in the enlarged rectangle, and their image by $\eta$ is approximately a helix on $Out({{\rm\bf w}})$. Changing the value of the bifurcation parameter $\lambda$ moves the graph $y=g(x)$ (the stable manifold of ${{\rm\bf v}}$) but does not affect the map $\eta$. Hence, by varying $\lambda$ we can get a sheet of the lamination tangent to $y=g(x)$, say, $W^u(x_0,y_0)$ for some point $(x_0,y_0)$ in the Cantor set as in figure \[lamina\]. This breaks the hyperbolicity, since it means that $W^u(x_0, y_0)$ is tangent to $W^s({{\rm\bf v}})$. This phenomenon has been studied by Gonchenko [*et al.*]{} [@Gonchenko2007] — it corresponds to a decrease in topological entropy as in figure \[transition1\]. As the images of the rectangles move down, each time one of them crosses a rectangle a sequence of saddle-node bifurcations starts, together with a period-doubling cascade, as on the right hand side of figure \[transition1\].
A more rigorous construction will be made in the next section.
![Near the point of maximum height of $W^s_{loc}({{\rm\bf v}})\cap Out({{\rm\bf w}})$ the unstable manifold of some point is tangent to $W^s({{\rm\bf v}})$. By varying $\lambda$ this point may be taken to be in the Cantor set of points that remain near the cycle. []{data-label="lamina"}](lamination){height="5cm"}
![When $\lambda$ decreases, the Cantor set of points of the horseshoes that remain near the cycle is losing topological entropy, as the set loses hyperbolicity. This happens when the unstable manifold of some point in the Cantor set is tangent to $W^s({{\rm\bf v}})$.[]{data-label="transition1"}](transition1){height="8.7cm"}
Heteroclinic Tangency {#sec tangency}
=====================
In this section we show how to find values of the bifurcation parameter $\lambda$ for which $W^u({{\rm\bf w}})$ is tangent to $W^s({{\rm\bf v}})$.
With the notation of section \[hyperbolicity\], the two points $(P_{{\rm\bf w}}^1,0)$ and $(P_{{\rm\bf w}}^2,0)$ divide the closed curve $y=g(x)$ where $W_{loc}^s({{\rm\bf v}})$ intersects $Out({{\rm\bf w}})$ in two components, corresponding to different signs of the second coordinate. With the conventions of section \[hyperbolicity\], we get $g(x)>0$ for $x\in\left(P_{{\rm\bf w}}^1,P_{{\rm\bf w}}^2\right)$. Then the region in $Out({{\rm\bf w}})$ delimited by $W_{loc}^s({{\rm\bf v}})$ and $W_{loc}^u({{\rm\bf w}})$ between $P_{{\rm\bf w}}^1$ and $P_{{\rm\bf w}}^2$ gets mapped by $ \Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}$ into the $y<0$ component of $In({{\rm\bf v}})$, while all other points in $Out({{\rm\bf w}})$ with positive second coordinates, are mapped ito the $y>0$ component of $In({{\rm\bf v}})$ as in figure \[Psiwv\]. The maximum value of the coordinate $y$ for the curve $W_{loc}^s({{\rm\bf v}})\cap Out({{\rm\bf w}})$ is of the order of $\lambda$, attained at some point $(x,y)\approx (x_m,\lambda)$ with $P_{{\rm\bf w}}^1<x_m<P_{{\rm\bf w}}^2\pmod{2\pi}$.
Consider now the closed curve where $W_{loc}^u({{\rm\bf w}})$ intersects $In({{\rm\bf v}})$. For small values of $\lambda$ this is approximately an ellipse, crossing $W_{loc}^s({{\rm\bf v}})$ at the two points $(P_{{\rm\bf v}}^1,0)$ and $(P_{{\rm\bf v}}^2,0)$, see figure \[elipse\]. With the conventions of section \[hyperbolicity\], this is the graph $y=h(x)$ with $h(x)>0$ for $x\in\left(P_{{\rm\bf v}}^2,P_{{\rm\bf v}}^1\right)$. In particular, the portion of this curve that lies in the upper half of $In({{\rm\bf v}})$, parametrised by $(x,y)$, $y>0$, may be written as the union of two segments $\sigma_1$ and $\sigma_2$ that meet at the point where the coordinate $y$ attains its maximum value on the curve. Without loss of generality, let $(x_*,\lambda)$ be the coordinates in $In({{\rm\bf v}})$ of this point, whith $P_{{\rm\bf v}}^2<x_*<P_{{\rm\bf v}}^1\pmod{2\pi}$.
![The transition map $ \Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}$ sends the shaded area in $Out({{\rm\bf w}})$, between $W_{loc}^s({{\rm\bf v}})$ and $W_{loc}^u({{\rm\bf w}})$ and with positive second coordinate, into the shaded area in $In({{\rm\bf v}})$ with negative second coordinate. Conventions: black line stands for $W_{loc}^u({{\rm\bf w}})$, grey line is $W_{loc}^s({{\rm\bf v}})$. The rest of the upper part of $Out({{\rm\bf w}})$ is mapped into the upper part of $In({{\rm\bf v}})$.[]{data-label="Psiwv"}](nova){height="4.5cm"}
By Proposition \[Structures\], the image of each segment $\sigma_j$ by $\eta$ is a helix on $Out({{\rm\bf w}})$ accumulating on $W_{loc}^u({{\rm\bf w}})$. Hence, the curve $\eta(\sigma_1\cup \sigma_2)$ is a double helix. The projection of this curve into $W_{loc}^u({{\rm\bf w}})$ is regular at all points, except for a fold at $\eta(x_*,\lambda)=(x_*-K\ln\lambda, \lambda^\delta)=(x(\lambda),y(\lambda))$, as in figure \[homoclinic1\]. As $\lambda$ decreases to zero, the first coordinate $x(\lambda)$ of $\eta(x_*,\lambda)$ tends to infinity, hence the point $\eta(x_*,\lambda)$ makes infinitely many turns around the cylinder $Out({{\rm\bf w}})$.
On the other hand, $y(\lambda)= \lambda^\delta$ with $\delta>1$, so $y(\lambda)$ decreases to zero faster than $\lambda$, the maximum height of the curve $W_{loc}^s({{\rm\bf v}})\cap Out({{\rm\bf w}})$. Therefore, given any small $\lambda_0>0$, there exists a positive $\lambda_1<\lambda_0$ such that $x(\lambda_1)=x_m$ and moreover $\eta(x_*,\lambda_1)=(x(\lambda_1),y(\lambda_1))$ lies in the region in $Out({{\rm\bf w}})$ between $W_{loc}^s({{\rm\bf v}})$ and $W_{loc}^u({{\rm\bf w}})$ that gets mapped into the lower part of $In({{\rm\bf v}})$. For $\lambda=\lambda_1$, the points on the curve $W_{loc}^u({{\rm\bf w}})\cap In({{\rm\bf v}})$ close to $(x_*,\lambda_1)$ are also mapped by $\eta$ into the lower half of $Out({{\rm\bf w}})$.
Furthermore, there exists a positive $\lambda_2<\lambda_1$ such that $P_{{\rm\bf w}}^2<x(\lambda_2)<P_{{\rm\bf w}}^1\pmod{2\pi}$ and hence $\eta(x_*,\lambda_2)=(x(\lambda_2),y(\lambda_2))$ is mapped by $ \Psi_{{{\rm\bf w}}\rightarrow {{\rm\bf v}}}$ into the upper part of $In({{\rm\bf v}})$. Again, for $\lambda=\lambda_2$, points on the curve $W_{loc}^u({{\rm\bf w}})\cap In({{\rm\bf v}})$ close to $(x_*,\lambda)$ return to the upper part of $Out({{\rm\bf w}})$.
Therefore, for some $\lambda_*$, with $\lambda_2<\lambda_*<\lambda_1$, the image of the curve $W_{loc}^u({{\rm\bf w}})\cap In({{\rm\bf v}})$ by the first return map to $In({{\rm\bf v}})$ is tangent to $W_{loc}^s({{\rm\bf v}})\cap In({{\rm\bf v}})$, given in local coordinates by $y=0$. This completes the proof of Theorem \[teorema tangency\] — given any $\lambda_0>0$ we have found a positive $\lambda_*<\lambda_0$ such that $W^u({{\rm\bf w}})$ is tangent to $W^w({{\rm\bf v}})$ for $f_{\lambda_*}$.
Conclusion {#Conclusion}
==========
For the present study, we have used the symmetry $\gamma_1$ and its flow-invariant fixed-point subspace to ensure the persistence of the connections $[{{\rm\bf v}}\rightarrow {{\rm\bf w}}]$ of one-dimensional manifolds. The symmetry is not essential for our exposition but its existence makes it more natural. In particular, the networks we describe are persistent within the class of differential equations with the prescribed symmetry.
As a global structure, the transition from in the dynamics from $\dot{x}=f_0(x)$ to $\dot{x}=f_\lambda(x)$, $\lambda \approx 0$ is intriguing and has not always attracted appropriate attention. There is a neighbourhood $V^0$ with positive Lebesgue measure that is positively invariant for the flow of $\dot{x}=f_\lambda(x)$. For $\lambda=0$ all trajectories approach the network $\Sigma^0$. For $\lambda > 0$ and for a sufficiently small tubular neighbourhood $V^\Gamma \subset V^0 $ of any of the Bykov cycles in $\Sigma^\lambda$, almost all trajectories might return to $V^\Gamma$ but they do not necessarily remain there for all future time. Trajectories that remain in $V^\Gamma$ for all future time form an infinite set of suspended horseshoes with zero Lebesgue measure.
For a fixed $\lambda>0$, when we take a larger tubular neighbourhood $V^\Gamma$ of a Bykov cycle $\Gamma$, the suspended horseshoes lose hyperbolicity. While small symmetry-breaking terms generically destroy the attracting cycle $\Sigma^0$, there will still be an attractor lying close to the original cycle. This is the main point of section \[hyperbolicity\]: when local invariant manifolds are extended, they develop tangencies, which explain the attractivity. The existence of primary heteroclinic tangencies is proved in section \[sec tangency\].
Heteroclinic tangencies give rise to attracting periodic trajectories of large periods and small basins of attraction, appearing in large (possibly infinite) numbers. Heteroclinic tangencies also create new tangencies near them in phase space and for nearby parameter values. We know very little about the geometry of these strange attractors, we also do not know the size and the shape of their basins of attraction.
When $\lambda \rightarrow 0$, the infinite number of periodic sinks lying close to the network of Bykov cycles will approach the ghost of $\Sigma^0$. For the parameter values where we observe heteroclinic tangencies, each Bykov cycle possesses infinitely many sinks whose basins of attractions have positive three-dimensional Lesbesgue measure. The attractors must lie near $\overline{W^s({{\rm\bf v}}) \cup W^u({{\rm\bf w}})}$ and they collapse into $\Sigma^0$ as $\lambda \rightarrow 0$. A lot more needs to be done before the subject is well understood.
![In any open neighbourhood of $f_0$ in the space of ${\langle\gamma_{1}\rangle}$–equivariant vector fields of class $C^1$ on ${{\rm\bf S}}^3$, there is a sequence of vector fields $f_{\lambda_i}$ accumulating of $f_0$ whose flow has a heteroclinic tangency between $W^u({{\rm\bf w}})$ and $W^s({{\rm\bf v}})$.[]{data-label="homoclinic1"}](homoclinic1){height="4.5cm"}
**Acknowledgements:** The authors would like to thank Maria Carvalho for helpful discussions.
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[^1]: CMUP is supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the project PEst-C/MAT/UI0144/2011. A.A.P. Rodrigues was supported by the grant SFRH/BPD/84709/2012 of FCT
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---
abstract: 'We compute endomorphism algebras of Kuga-Satake varieties associated to $K3$ surfaces.'
author:
- Evgeny Mayanskiy
bibliography:
- 'EndomorphismAlgebrasKugaSatake.bib'
nocite: '[@*]'
title: 'Endomorphism algebras of Kuga-Satake varieties.'
---
Preliminary remarks.
====================
Let $V$ be a $\mathbb Q$-lattice of transcendental cycles on a $K3$ surface $X$, $\phi\colon V{\otimes}_{\mathbb Q} V\rightarrow \mathbb Q$ the polarization of the weight $2$ Hodge structure on $V$, $E=End_{Hdg}(V)$, $\Phi\colon V{\otimes}_{E} V\rightarrow E$ the hermitian or bilinear form constructed in [@Zarhin], $\phi=tr \circ \Phi$.\
Let $C^{}(V)$ be the Clifford algebra of the quadratic space $(V,\phi)$ over $\mathbb Q$, $C^{+}(V)$ the even Clifford algebra and $KS(X)$ the Kuga-Satake variety of $X$. Here we define $KS(X)$ from the weight $2$ Hodge structure on the lattice of transcendental cycles $V$ rather than on the whole lattice of primitive cycles $H^2(X,\mathbb Q)_{prim}$. In particular, the Kuga-Satake variety defined here is isogenous to a power of the Kuga-Satake variety defined using the whole lattice of primitive cycles (see [@KugaSatake], [@Morrison], §4).\
We want to compute the endomorphism algebra $End(KS(X))_{\mathbb Q}=End_{Hdg}(C^{+}(V))$.\
Let $Z(\Phi)$ be the $\mathbb Q$-algebraic group $Res_{E/\mathbb Q}(SO(V,\Phi))$, if $E$ is a totally real field, or $Res_{E_0/\mathbb Q}(U(V,\Phi))$, if $E=E_0(\theta)$ is a CM-field (with the totally real subfield $E_0$). Recall, that according to [@Zarhin], $Z(\Phi)$ is the Hodge group of the Hodge structure on $V$.\
Let $CSpin(\phi)\colon = \{ g\in C^{+}(V)^{*} \; \mid \; gVg^{-1}\subset V \}$. Consider the vector representation $\rho \colon CSpin(\phi)\rightarrow GL(V)$, $g\mapsto (v\mapsto gvg^{-1})$ and the spin representation $\sigma\colon CSpin(\phi)\rightarrow GL(C^{+}(V))$, $g\mapsto (x\mapsto gx)$. Let $ZSpin(\Phi)\colon =\{ g\in CSpin(\phi) \;\mid\; {\rho}(g)\in Z(\Phi) \}=$ ${\rho}^{-1}(Z(\Phi))\subset CSpin(\phi)$. Note that ${\rho}(ZSpin(\Phi))=Z(\Phi)$.\
[**Lemma 1.**]{} [*The Mumford-Tate group of the weight $1$ Hodge structure on $C^{+}(V)$ is the preimage with respect to $\rho$ of the Mumford-Tate group of the weight $2$ Hodge structure on $V$.*]{}\
[*Proof:*]{} The same as Proposition 6.3 in [@vanGeemen]. If $h_X\colon S^{1}\rightarrow GL(V)$ and $h_{KS(X)}\colon S^{1}\rightarrow GL(C^{+}(V))$ denote the corresponding Hodge structures, then $h_X={\rho}\circ {\sigma}^{-1}\circ h_{KS(X)}$ (as shown in [@vanGeemen]). [*QED*]{}\
[**Corollary.**]{} [*$End(KS(X))_{\mathbb Q}\cong End_{ZSpin(\Phi)}(C^{+}(V))$, where $ZSpin(\Phi)$ acts on $C^{+}(V)$ via the spin representation ${\sigma}{\mid}_{ZSpin(\Phi)}$.*]{}\
So, if $C^{+}(V)=\bigoplus_{j} T_{j}^{\oplus m_j}$ is the decomposition of ${\sigma}{\mid}_{ZSpin(\Phi)}$ into a direct sum of irreducible (mutually non-isomorphic) representations $T_j$, then $End(KS(X))_{\mathbb Q}\cong \prod_{j} Mat_{m_j\times m_j}(D_j)$ as $\mathbb Q$-algebras, where $D_j=End_{CSpin(\Phi)}(T_j)$.\
Let us assume that $m=dim_EV\geq 3$, if $E=E_0$ is totally real, and $m=dim_EV \geq 2$, if $E=E_0(\theta)$ is a CM-field. In the totally real case condition $m\geq 3$ is automatically satisfied for any $K3$ surface $X$ (see [@Mayanskiy] and [@vanGeemen2]). In what follows we will often denote the field of rational numbers $\mathbb Q$ by $k$ and $E_0$ by $L$. Our approach is not invariant in the sense that we choose a basis in $V$ which diagonalizes $\Phi$ right from the start (see Section 2).\
Consider the epimorphism $\pi\colon CSpin(\phi)\rightarrow SO(\phi)$ of algebraic groups over $\mathbb Q$ (induced by the vector representation $\rho$ above) with fiber $ker(\pi)={\mathbb G}_m\subset CSpin(\phi)$ and its restriction ${\pi}_0\colon Spin(\phi)\rightarrow SO(\phi)$ to the subgroup $Spin(\phi)\subset CSpin(\phi)$. Then ${\pi}_0$ is a double etale covering [@BConrad].\
The argument above shows that the Hodge group $Hdg$ of the Kuga-Satake structure on $C^{+}(V)$ satisfies inclusions: $$Hdg\subset ({{\pi}_0}^{-1}(Z(\Phi)))^{0} \cdot {\mathbb G}_m \; \; \mbox{and} \; \; \; ({{\pi}_0}^{-1}(Z(\Phi)))^{0} \subset Hdg$$ (hereafter for an algebraic group $G$ we let $G^{0}$ denote the connected component of the identity and $Lie(G)$ the Lie algebra of $G$).\
Hence the $\mathbb Q$-algebra $$End_{Hdg}(C^{+}(V))=End_{({{\pi}_0}^{-1}(Z(\Phi)))^{0}}(C^{+}(V))=End_{Lie({{\pi}_0}^{-1}(Z(\Phi)))}(C^{+}(V))=End_{Lie(Z(\Phi))}(C^{+}(V)).$$
Let ${\mathfrak{g}}=Lie(Z(\Phi))$. Then ${\mathfrak{g}}=Res_{E/k}({\mathfrak{so}}(\Phi))$, if $E$ is totally real, or ${\mathfrak{g}}=Res_{E_0/k}({\mathfrak{u}}(\Phi))$, if $E=E_0(\theta)$ is a CM-field (${\theta}^2\in E_0$), where $k=\mathbb Q$.\
Hence what we are looking for is the algebra of intertwining operators $End_{{\mathfrak{g}}}(C^{+}(V))$ of the $\mathbb Q$-linear representation of the Lie algebra ${\mathfrak{g}}$ over $\mathbb Q$ induced by the spin representation of ${\mathfrak{so}}(\phi)$ in $C^{+}(V)$ via the inclusion of Lie algebras ${\mathfrak{g}}\subset {\mathfrak{so}}(\phi)$ corresponding to the inclusion of the $\mathbb Q$-algebraic groups $Z(\Phi)\subset SO(\phi)$ above.\
The problem of computing endomorphism algebras of Kuga-Satake varieties was addressed earlier by Bert van Geemen in papers [@vanGeemen1] and [@vanGeemen2]. In particular, in [@vanGeemen1] he considered the case of the CM-field, which is quadratic over $\mathbb Q$ and in [@vanGeemen2] he considered the case of the totally real field, computed the endomorphism algebra in several special cases and made some general remarks. A different computation of the endomorphism algebra of the Kuga-Satake variety in the totally real case was done by Ulrich Schlickewei [@Schlickewei].\
Our solution uses the same ideas as (some of the ideas) in papers [@vanGeemen1] and [@vanGeemen2]. We compute the decomposition of the restriction to ${\mathfrak{g}}$ of the spin representation of ${\mathfrak{so}}(\phi)$ into irreducible subrepresentations over a splitting field of ${\mathfrak{g}}$, and then apply Galois descent.\
Our main result is Theorem 1 in Section 4 complemented by the computation of primary representations (which are the multiples of irreducible representations $T_j$ above) and division algebras (which are the endomorphism algebras of $T_j$) in subsequent sections. In this text a ’primary representation’ means a multiple of an irreducible representation. Some general observations regarding representations over arbitrary fields are collected in the Section 2. In Section 3 we introduce Galois-invariant Cartan subalgebras. In Section 4 we compute decompositions of representations over a splitting field. In Section 5 we construct primary representations over $\mathbb Q$ whose irreducible components appear in Theorem 1. In Section 6 we compute the division algebras which are the endomorphism algebras of those irreducible components. Section 7 is devoted to examples.\
Some remarks on Galois theory of representations.
=================================================
Let $F/k=\mathbb Q$ be a finite Galois extension, ${\mathfrak{g}}={\mathfrak{c}}\oplus {{\mathfrak{g}}}'$ be a reductive Lie algebra over $k$, ${\mathfrak{c}}\subset {\mathfrak{g}}$ be its center and ${{\mathfrak{g}}}'\subset {\mathfrak{g}}$ be its derived subalgebra. Let $S=Gal(F/k)$ and ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_k F$ be a Galois-invariant (i.e. such that $g({\mathfrak{h}})={\mathfrak{h}}$ for any $g\in S$) splitting Cartan subalgebra. Let $B$ be a basis of the root system $R$ of $({\mathfrak{g}}\otimes_k F,{\mathfrak{h}})$. In what follows we assume that all the representations of ${\mathfrak{g}}$ we are dealing with are finite-dimensional and can be integrated to representations of a reductive algebraic group with Lie algebra ${\mathfrak{g}}$ (in order to guarantee their complete reducibility).\
Let $\rho\colon {\mathfrak{g}}\rightarrow End_k(W)$ be a representation of ${\mathfrak{g}}$ over $k$ and $W\otimes_k F=\oplus_{\alpha} V_{\alpha}$ its decomposition into irreducible subrepresentations over $F$. Let ${\rho}_{\alpha}={\rho}{\mid}_{V_{\alpha}}$ be an irreducible representation of ${\mathfrak{g}}\otimes_k F$ with primitive element $v_{\alpha}\in W\otimes_k F$ with highest weight ${\omega}_{\alpha}\in Hom_F({\mathfrak{h}},F)$ (with resprect to $B$). Then for any $g\in S$, ${\rho}_{\alpha}^{g}\colon =\rho {\mid}_{g(V_{\alpha})}$ is an irreducible representation of ${\mathfrak{g}}\otimes_k F$ with primitive element $g(v_{\alpha})\in W\otimes_k F$ with highest weight $g\circ {{\omega}_{\alpha}}\circ g^{-1}\in Hom_F({\mathfrak{h}},F)$ with respect to the basis $g\circ B\circ g^{-1}$ of $R$. Since the Weyl group ${\mathcal W}_{R}$ of $R$ acts simply transitively on the set of bases of $R$, for any $g\in S$ there exists unique $w(g)\in {\mathcal W}_R$ such that $g\circ B\circ g^{-1}=w(g)(B)$. Hence ${\rho}_{\alpha}^{g}$ is an irreducible representation of ${\mathfrak{g}}\otimes_k F$ with primitive element $g(v_{\alpha})\in W\otimes_k F$ with highest weight ${\omega}_{\alpha}^{g}\colon = w(g)^{-1}(g\circ {\omega}_{\alpha}\circ g^{-1})\in Hom_F({\mathfrak{h}},F)$ (with respect to $B$).\
[**Lemma 3.**]{} [*Suppose that ${\rho}_1\colon {\mathfrak{g}}\rightarrow End_k(W_1)$ and ${\rho}_2\colon {\mathfrak{g}}\rightarrow End_k(W_2)$ are two irreducible representations of ${\mathfrak{g}}$ over $k$, $V_{\alpha}\subset W_1\otimes_k F$ and $V_{\beta}\subset W_2\otimes_k F$ are two irreducible subrepresentations of ${\mathfrak{g}}\otimes_k F$ over $F$. Then $W_1\cong W_2$ as ${\mathfrak{g}}$-modules over $k$, if and only if there exist ${\sigma}, {\tau}\in S$ such that $({\rho}_1 {\mid}_{V_{\alpha}})^{\sigma}\cong ({\rho}_2 {\mid}_{V_{\beta}})^{\tau}$ as ${\mathfrak{g}}\otimes_k F$-modules over $F$.*]{}\
[*Proof:*]{} Schur’s lemma. [*QED*]{}\
[**Corollary.**]{} [*If ${\rho}_{\alpha}\colon {\mathfrak{g}}\otimes_k F \rightarrow End_F(V_{\alpha})$ is an irreducible representation of ${\mathfrak{g}}\otimes_k F$ over $F$, then there exists at most one irreducible representation $\rho\colon {\mathfrak{g}}\rightarrow End_k(W)$ of ${\mathfrak{g}}$ over $k$, such that ${\rho}_{\alpha}$ is a subrepresentation of $\rho \otimes_k F$.*]{}\
Using the notation of the remark preceeding Lemma 3, let $W=\oplus_{\gamma} W_{\gamma}$ be a decomposition of $\rho$ into irreducible subrepresentations over $k$. Then for any $\gamma$ such that $V_{\alpha}\subset W_{\gamma}\otimes_k F$, by Galois descent we have: $$\bigoplus_{{\gamma}'\colon W_{{\gamma}'}\cong W_{{\gamma}}\; \mbox{as}\; {\mathfrak{g}}\mbox{-modules} } W_{{\gamma}'}=\left( \bigoplus_{{\alpha}' \colon {\rho}_{\alpha}^{\tau}\cong {\rho}_{{\alpha}'}^{\sigma}\; \mbox{for some} \; \tau, \sigma \in S} V_{{\alpha}'} \right)^{S}.$$
Hence $dim_k \left( \bigoplus_{{\gamma}'\colon W_{{\gamma}'}\cong W_{{\gamma}} \;\mbox{as}\; {\mathfrak{g}}\mbox{-modules} } W_{{\gamma}'} \right)=$ $dim_F \left( \bigoplus_{{\alpha}'\colon {\rho}_{\alpha}^{\tau}\cong {\rho}_{{\alpha}'}^{\sigma}\; \mbox{for some}\; \tau, \sigma \in S } V_{{\alpha}'} \right)=$ $\sum_{{\alpha}'\colon \exists \tau, \sigma \in S \colon {\omega}_{\alpha}^{\tau}={\omega}_{{\alpha}'}^{\sigma}} dim_F (V_{{\alpha}'})=$ $m_{\alpha}\cdot dim_k(W_{\gamma})$, where $m_{\alpha}$ is the multiplicity of $W_{\gamma}$ in the decomposition above.\
So, if $W_{{\gamma}_1}, ..., W_{{\gamma}_p}$ are pairwise nonisomorphic (as ${\mathfrak{g}}$-modules) irreducible ${\mathfrak{g}}$-submodules of $W$ over $k$ (with the corresponding ${\mathfrak{g}}\otimes_k F$-submodules $V_{{\alpha}_i}\subset W\otimes_k F$) appearing in the decomposition above, then $W=\oplus_{i} W_{{\gamma}_i}^{\oplus m_{{\alpha}_i}}$ and $$End_{{\mathfrak{g}}}(W)\cong \prod_i Mat_{m_{{\alpha}_i}\times m_{{\alpha}_i}}(D_i) \; \; \mbox{as}\; k-\mbox{algebras},$$ where $D_i=End_{{\mathfrak{g}}}(W_{{\gamma}_i})$, $W_{{\gamma}_i}$ is the unique irreducible ${\mathfrak{g}}$-module over $k$ such that $W_{{\gamma}_i}\otimes_k F$ contains $V_{{\alpha}_i}$ as a ${\mathfrak{g}}\otimes_k F$-submodule over $F$ and $m_{{\alpha}_i}= \left( \sum_{{\alpha}'\colon \exists \sigma \in S \colon {\omega}_{{\alpha}'}={\omega}_{{\alpha}_i}^{\sigma}} dim_F(V_{{\alpha}'}) \right) / dim_k(W_{{\gamma}_i})$. We can also write: $$m_{{\alpha}_i}=\frac{dim_F(V_{{\alpha}_i}) \cdot \sum_{\sigma \in S} mult({\omega}_{{\alpha}_i}^{\sigma}) }{n_{{\omega}_{{\alpha}_i}} \cdot dim_k(W_{{\gamma}_i})},$$ where $milt(\omega)$ is the multiplicity of the irreducible representation of ${\mathfrak{g}}\otimes_k \mathbb C$ with highest weight $\omega$ (relative to the chosen ${\mathfrak{h}}$ and $B$) in $W\otimes_k \mathbb C$ and $n_{\omega}$ is the stabilizer of $\omega$ under the action of the Galois group $S=Gal(F/k)$ on weights. Note that $\{ {\omega}_{{\alpha}_i} \}$ is a set of representatives of the orbits of the action of $S$ on the set of highest weights of irreducible representations of ${\mathfrak{g}}\otimes_k \mathbb C$ appearing as irreducible components of $W\otimes_k \mathbb C$.\
This reduces the study of $End_{{\mathfrak{g}}}(W)$ to the study of the (uniquely determined) ($k=\mathbb Q$)-forms of irreducible ${\mathfrak{g}}\otimes_k \mathbb C$-submodules of $W\otimes_k \mathbb C$ (i.e. $D_i=End_{{\mathfrak{g}}}(W_{{\gamma}_i})$ and $dim_k(W_{{\gamma}_i})$) and the description of the Galois action (of the finite group $Gal(F/k)$) on the weights of ${\mathfrak{g}}\otimes_k \mathbb C$ over $\mathbb C$.\
Description of the Galois action, Cartan subalgebras and bases of the root systems.
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According to Section 2, we need to specify a splitting field $F$ of ${\mathfrak{g}}$ (which should be a Galois extension of $k$), a Galois-invariant splitting Cartan subalgebra ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_{k} F$ (i.e. ${\mathfrak{h}}$ should be $Gal(F/k)$-stable) and a basis $B$ of the root system $R$ of the split reductive Lie algebra $({\mathfrak{g}}\otimes_{k} F, {\mathfrak{h}})$.\
Let us assume that $\Phi=d_1\cdot X_1^2+...+d_m\cdot X_m^2$ (if $E=E_0=L$ is totally real) or $\Phi=d_1\cdot X_1\bar{X_1}+...+d_m\cdot X_m\bar{X_m}$ (if $E=E_0(\theta)$, ${\theta}^2\in E_0=L$ is a CM-field), where $d_i\in L$ for any $i$. In other words, we reduce the Hermitian (or quadratic) form $\Phi$ to a diagonal form, i.e. choose an orthogonal (with respect to $\Phi$) basis of $V$ such that $X_i$ are the corresponding coordinates.\
Let $k=\mathbb Q$ and $F / k$ be a finite Galois extension such that $F$ contains $L$, $\sqrt{d_i}$ for any $i$, $\sqrt{-1}$ and $\theta$ (if $E=E_0(\theta)$ is a CM-field, ${\theta}^2\in E_0$).\
Let $r=[L\colon k]$ and ${\sigma}_1,..., {\sigma}_r \colon L\hookrightarrow F$ be the list of all field embeddings of $L$ into $F$.\
Case of the totally real field.
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Let us consider first the case ${\mathfrak{g}}=Res_{L/k}({\mathfrak{so}}(\Phi))\subset {\mathfrak{so}}(\phi)$ (i.e. $E=E_0$ is totally real). We will denote by $E_{i,j}$ a matrix with all entries equal to $0$ except for the entry $(i,j)$ which is equal to $1$.\
Let ${{\mathfrak{h}}}_0=Span_L(A_1,...,A_l)$, where $l=[\frac{m}{2}]$ and $A_i=d_{m-i+1}\cdot E_{m-i+1,i}-d_i\cdot E_{i,m-i+1}$, $1\leq i\leq l$. Let ${{\mathfrak{h}}}_i={{\mathfrak{h}}}_0\otimes_{L,{\sigma}_i} F\subset {\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ and ${\mathfrak{h}}={{\mathfrak{h}}}_1\times ...\times {{\mathfrak{h}}}_r\subset \oplus_{i=1}^{r}({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)\cong Res_{L/k}({\mathfrak{so}}(\Phi))\otimes_{k} F=g \otimes_{k} F$. Then ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_{k} F$ is a splitting Cartan subalgebra.\
Note that over $F$ we have $\Phi=d_1\cdot X_1^2+...+d_m\cdot X_m^2=\sum_{i=1}^{l}Y_i\cdot Y_{-i}+{\epsilon} {Y_0}^2$, where $\epsilon =0$, if $m$ is even, $\epsilon =1$, if $m$ is odd, $Y_i=\sqrt{d_i}\cdot X_i+\sqrt{-d_{m-i+1}}\cdot X_{m-i+1}$, $Y_{-i}=\sqrt{d_i}\cdot X_i-\sqrt{-d_{m-i+1}}\cdot X_{m-i+1}$ and $Y_0=\sqrt{d_{l+1}}\cdot X_{l+1}$.\
This implies that for any $i, j$ we have $A_j\otimes_{L,{\sigma}_i} 1={\Gamma}_j\cdot H_j$, where ${\Gamma}_j=-\sqrt{{\sigma}_i(d_j)}\cdot \sqrt{-{\sigma}_i(d_{m-j+1})}\in F$ ($1\leq j\leq l$) (in future we will be writing $d_j$ instead of ${\sigma}_i(d_j)$) and $H_j=E_{j,j}-E_{-j,-j}$ (using notation form [@Bourbaki], §13). Hence for any $i$ subalgebra ${{\mathfrak{h}}}_i\subset {\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ is the same splitting Cartan subalgebra as in [@Bourbaki], §13. By construction ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_{k} F$ is Galois-invariant.\
Let $R_0$ be the root system of type $B_l$, if $m=2l+1$ (respectively, of type $D_l$, if $m=2l$) from [@Bourbaki], §13, i.e. $R_0=\{\pm {\epsilon}_p, \pm {\epsilon}_p \pm {\epsilon}_q\}$ (respectively, $R_0=\{\pm {\epsilon}_p \pm {\epsilon}_q\}$) with basis $B_0=\{ {\epsilon}_1-{\epsilon}_2, {\epsilon}_2-{\epsilon}_3,..., {\epsilon}_{l-1}-{\epsilon}_l, {\epsilon}_l \}$ (respectively, $B_0=\{ {\epsilon}_1-{\epsilon}_2, {\epsilon}_2-{\epsilon}_3,..., {\epsilon}_{l-1}-{\epsilon}_l, {\epsilon}_{l-1}+{\epsilon}_l \}$) (using notation from [@Bourbaki], §13).\
Then for any $i$ the root system of $(so(\Phi)\otimes_{L,{\sigma}_i} F, h_0\otimes_{L,{\sigma}_i} F)$ is $R_i=\{\pm {\epsilon}_p\otimes_{L,{\sigma}_i} {\Gamma}_p, \pm {\epsilon}_p\otimes_{L,{\sigma}_i} {\Gamma}_p \pm {\epsilon}_q\otimes_{L,{\sigma}_i} {\Gamma}_q\}$ with basis $$B_i=\{ {\epsilon}_1\otimes_{L,{\sigma}_i} {\Gamma}_1-{\epsilon}_2\otimes_{L,{\sigma}_i} {\Gamma}_2, {\epsilon}_2\otimes_{L,{\sigma}_i} {\Gamma}_2-{\epsilon}_3\otimes_{L,{\sigma}_i} {\Gamma}_3,..., {\epsilon}_{l-1}\otimes_{L,{\sigma}_i} {\Gamma}_{l-1}-{\epsilon}_l\otimes_{L,{\sigma}_i} {\Gamma}_l, {\epsilon}_l\otimes_{L,{\sigma}_i} {\Gamma}_l \}$$ (respectively, $R_i=\{\pm {\epsilon}_p\otimes_{L,{\sigma}_i} {\Gamma}_p \pm {\epsilon}_q\otimes_{L,{\sigma}_i} {\Gamma}_q\}$ with basis $$\begin{gathered}
B_i=\{ {\epsilon}_1\otimes_{L,{\sigma}_i} {\Gamma}_1-{\epsilon}_2\otimes_{L,{\sigma}_i} {\Gamma}_2, {\epsilon}_2\otimes_{L,{\sigma}_i} {\Gamma}_2-{\epsilon}_3\otimes_{L,{\sigma}_i} {\Gamma}_3,..., {\epsilon}_{l-1}\otimes_{L,{\sigma}_i} {\Gamma}_{l-1}-{\epsilon}_l\otimes_{L,{\sigma}_i} {\Gamma}_l,\\
{\epsilon}_{l-1}\otimes_{L,{\sigma}_i} {\Gamma}_{l-1}+ {\epsilon}_l\otimes_{L,{\sigma}_i} {\Gamma}_l \}).\end{gathered}$$ Then $R=R_1\sqcup ... \sqcup R_r$ is the root system of $({\mathfrak{g}}\otimes_k F, {\mathfrak{h}})$ and as a basis we can take $B=B_1\sqcup ... \sqcup B_r\subset R$.\
The action of the Galois group $S=Gal(F/k)$ on weights reduces to its action by permutation on factors of $R_1\times ... \times R_r$ (or on the left cosets $Gal(F/k)/Gal(F/L)$) and to switching signes in front of various ${\Gamma}_p$.\
Note the isomorphism of root systems $R\cong R_0\sqcup ... \sqcup R_0$ ($r$ factors) under which basis $B$ is identified with $B_0\sqcup ... \sqcup B_0$ ($r$ factors).\
Let $w_p\in {\mathcal W}_{R_0}$ (where ${\mathcal W}_{R}$ denotes the Weyl group of a root system $R$) be the element of the Weyl group such that $w_p(B_0)={\sigma}_p(B_0)$, where ${\sigma}_p$ is a linear transformation of the $\mathbb Q$-vector space generated by the roots of $R_0$ which switches the sign in front of ${\epsilon}_p$ and does not change other ${\epsilon}_q$’s. Then in the notation of Section 2 for any $g\in S$, ${\omega}_{\alpha}^{g}=(\prod_{p\in P_1(g)} {w_p}^{-1})\sqcup ... \sqcup (\prod_{p\in P_r(g)} {w_p}^{-1}) (g\circ {\omega}_{\alpha} \circ g^{-1})\in Hom_F({\mathfrak{h}},F)$, where $P_i(g)=\{ p \; \mid \; g^{-1}({\epsilon}_p \otimes_{L,{\sigma}_i} {\Gamma}_p)=-{\epsilon}_p \otimes_{L,g^{-1}\circ {\sigma}_i} {\Gamma}_p \}$.\
Case of the CM-field.
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Now let us consider the case ${\mathfrak{g}}=Res_{L/k}({\mathfrak{u}}(\Phi))\subset {\mathfrak{so}}(\phi)$ (i.e. $E=E_0(\theta)$ is a CM-field, ${\theta}^2\in E_0=L$).\
Let ${{\mathfrak{h}}}_0=Span_L(A_1,...,A_m)$, where $A_i=\theta\cdot E_{i,i}$, ${{\mathfrak{h}}}_i={{\mathfrak{h}}}_0\otimes_{L,{\sigma}_i} F \subset {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m, F)$ and ${\mathfrak{h}}={{\mathfrak{h}}}_1\times ... \times {{\mathfrak{h}}}_r\subset \oplus_{i=1}^{r} ({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F)\cong {\mathfrak{gl}}(m,F)^{\otimes r}\cong Res_{L/k}({\mathfrak{u}}(\Phi))\otimes_k F={\mathfrak{g}}\otimes_k F$. Then ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_k F$ is a splitting Cartan subalgebra.\
Note that over $F$ we have $\Phi=d_1\cdot X_1\bar{X_1}+...+d_m\cdot X_m\bar{X_m}=Y_1\bar{Y_1}+...+Y_m\bar{Y_m}$, where $Y_i=\sqrt{d_i}\cdot X_i$. Hence for any $i, j$ we have $A_j\otimes_{L,{\sigma}_i} 1=\theta \cdot E_{j,j}$ (more precisely we have to write $\sqrt{{{\sigma}_i}({\theta}^2)}$ instead of $\theta$ here) in ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)$ and so for any $i$ subalgebra ${{\mathfrak{h}}}_i\subset {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)$ is the same splitting Cartan subalgebra as in [@Bourbaki], §13. By construction ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_k F$ is Galois-invariant.\
Let $R_0$ be the root system of type $A_{m-1}$ (for the reductive Lie algebra ${\mathfrak{gl}}(m)={\mathfrak{c}}\oplus {\mathfrak{sl}}(m)$, where ${\mathfrak{c}}\subset {\mathfrak{gl}}(m)$ is the center), i.e. $R_0=\{ {\epsilon}_p-{\epsilon}_q \}_{p\neq q}$ with basis $B_0=\{ {\epsilon}_1-{\epsilon}_2, ... ,{\epsilon}_{m-1}-{\epsilon}_m \}$.\
Then for any $i$ the root system of $({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F, {{\mathfrak{h}}}_i)\cong ({\mathfrak{gl}}(m),$ diagonal matrices $)$ is $R_i=\{ {\epsilon}_p\otimes_{L,{\sigma}_i} \theta-{\epsilon}_q\otimes_{L,{\sigma}_i} \theta \}$ with basis $B_i=\{ {\epsilon}_1\otimes_{L,{\sigma}_i} \theta-{\epsilon}_2\otimes_{L,{\sigma}_i} \theta, ... ,{\epsilon}_{m-1}\otimes_{L,{\sigma}_i} \theta-{\epsilon}_m\otimes_{L,{\sigma}_i} \theta \}$. Then $R=R_1\sqcup ... \sqcup R_r$ is the root system of $({\mathfrak{g}}\otimes_k F, {\mathfrak{h}})$ and as a basis we can take $B=B_1\sqcup ... \sqcup B_r \subset R$.\
The action of the Galois group $S=Gal(F/k)$ on weights reduces to its action by permutation on factors of $R_1\sqcup ... \sqcup R_r$ (or on the left cosets $Gal(F/k)/Gal(F/L)$) and to multiplication of various $\theta$ by $-1$.\
Note the isomorphism of root systems $R\cong R_0\sqcup ... \sqcup R_0$ ($r$ factors) under which basis $B$ is identified with $B_0\sqcup ... \sqcup B_0$ ($r$ factors).\
Let $w_0 \in {\mathcal W}_{R_0}$ be such that $w_0(B_0)=-B_0$. Then in the notation of Section 2 for any $g\in S$, ${\omega}_{\alpha}^{g}={(w_0)^{-P_1(g)}}\sqcup ... \sqcup {(w_0)^{-P_r(g)}} (g\circ {\omega}_{\alpha} \circ g^{-1})\in Hom_F({\mathfrak{h}},F)$, where $P_i(g)=1$, if $g^{-1}({{\epsilon}_p}\otimes_{L,{\sigma}_i} \theta)=- {{\epsilon}_p}\otimes_{L,g^{-1}\circ{\sigma}_i} \theta$ and $P_i(g)=0$ otherwise (the action of $w_0$ is extended to the center of ${\mathfrak{gl}}(m,F)$ as multiplication by $-1$).\
Decomposition of the restriction of the spin representation over a splitting field.
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In order to apply the general statements of Section 2, we need to decompose the $F$-linear extension of the restriction of the spin representation of ${\mathfrak{so}}(\phi)$ in $C^{+}(V)$ to ${\mathfrak{g}}\subset {\mathfrak{so}}(\phi)$ over $F$. For this we need to describe the embedding of Cartan subalgebras induced by the embedding of Lie algebras ${\mathfrak{g}}\otimes_k F\subset {\mathfrak{so}}(\phi)\otimes_k F$.\
[**Lemma 2.**]{} [*If $E$ is totally real, then the Lie algebra homomorphism $\oplus_{i=1}^{r}{\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F \subset {\mathfrak{so}}(\oplus_{i=1}^{r}(\Phi \otimes_{L, {\sigma}_i} F))={\mathfrak{so}}(\phi)\otimes_k F$ sends $(M_1,...,M_r)$ to $diag(M_1,...,M_r)$.\
If $E=E_0(\theta)$ is a CM-field (and ${\theta}^2\in E_0=L$ as usual), then the Lie algebra homomorphism $\oplus_{i=1}^{r} {\mathfrak{gl}}(m,F)\cong \oplus_{i=1}^{r}{\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F \subset {\mathfrak{so}}(\oplus_{i=1}^{r}( (\Phi \otimes_{E, {\sigma}_i} F)\oplus (\Phi \otimes_{E, \bar{\sigma}_i} F) ))={\mathfrak{so}}(\phi)\otimes_k F$ (where in the last formula ${\sigma}_i$ and $\bar{\sigma}_i$ denote the two extensions of ${\sigma}_i$ to an embedding of $E/k$ into $F/k$) sends $(M_1,...,M_r)$ to $diag(M_1, -\Phi\cdot {M_1}^T\cdot {\Phi}^{-1},...,M_r, -\Phi\cdot {M_r}^T\cdot {\Phi}^{-1})$.*]{}\
[*Proof:*]{} One should notice that $Res_{L/k}$ on vector spaces over $L$ is the forgetful functor to the vector spaces over $k$. Hence on the $Res_{L/k}({\mathfrak{so}}(\Phi))$ (respectively, $Res_{L/k}({\mathfrak{u}}(\Phi))$), which is the Galois-invariant subspace of the source, our homomorphisms have exactly the form needed. Extending scalars to $F$ gives the result. See also Proposition 3.8 in [@vanGeemen1] and [@vanGeemen2]. [*QED*]{}\
Case of the totally real field.
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Let $E=L$ be a totally real field.\
For any $i=1,...,r$, $j=1,...,l$ (where $l=\left[ \frac{m}{2} \right]$) let $\hat{H}_{j}^{i}={{\sigma}_i(d_{m-j+1})}\cdot E_{m-j+1+m(i-1),j+m(i-1)}-{{\sigma}_i(d_j)}\cdot E_{j+m(i-1),m-j+1+m(i-1)}\in {\mathfrak{so}}(\phi)\otimes_k F$. Then $\hat{H}_{j}^{i}$ are linearly independent elements of the splitting Cartan subalgebra $\hat{{\mathfrak{h}}}\subset {\mathfrak{so}}(\phi)\otimes_k F$ described in [@Bourbaki], §13. They form a basis of $\hat{{\mathfrak{h}}}$, if $m$ is even or $r=1$. If $m$ is odd and $r\geq 2$, then $\hat{H}_{j}^{i}$ together with $\hat{H}_{l+1}^{1},...,\hat{H}_{l+1}^{[\frac{r}{2}]}$ form a basis of $\hat{{\mathfrak{h}}}$, if we take $\hat{H}_{l+1}^{i}={{\sigma}_{r-i+1}(d_{l+1})}\cdot E_{(l+1)(r-i+1),(l+1)i}-{{\sigma}_{i}(d_{l+1})}\cdot E_{(l+1)i,(l+1)(r-i+1)}$, $1\leq i\leq [\frac{r}{2}]$.\
Let us denote by $\{ \hat{\epsilon}_{j}^{i} \}$ the corresponding dual basis of $\hat{{\mathfrak{h}}}^{*}=Hom_F(\hat{{\mathfrak{h}}},F)$. Its elements differ from the elements of the corresponding basis of the dual Cartan subalgebra considered in [@Bourbaki], §13 by scalar factors of the form $-\sqrt{{{\sigma}_i}(d_j)}\cdot \sqrt{-{{\sigma}_i}(d_{m-j+1})}$.\
Lemma 2 above implies that the restriction of $\hat{\epsilon}_{l+1}^{i}$ to the Cartan subalgebra ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_k F$ is zero, while for any $j\leq l$ the restriction of $\hat{\epsilon}_{j}^{i}$ to ${\mathfrak{h}}$ is the corresponding element of the dual basis of ${{\mathfrak{h}}}^{*}$ of the basis $\{ A_j\otimes_{L,{\sigma}_i} 1 \; \mid \; 1\leq i\leq r, \; 1\leq j\leq l \}$ of ${\mathfrak{h}}$.\
If $m\cdot r=dim_k(V)\geq 5$, then according to [@Bourbaki], §13 the weights of the spin representation of ${\mathfrak{so}}(\phi)\otimes_k F$ in $C^{+}(V)\otimes_k F$ ($V$ is considered as a vector space over $k$) are $\frac{1}{2}\sum_{i,j} {\hat{\epsilon}}_{j}^{i} - \sum_{(i,j)\in I} {\hat{\epsilon}}_{j}^{i}$, where $I$ runs over the subsets of the set of parameters $i$ and $j$ (i.e. $I\subset \{ (i,j) \; \mid \; 1\leq j\leq l\; \mbox{and}\; 1\leq i\leq r \; \mbox{or (if }\; m\; \mbox{is odd and }\; r\geq 2)\; j=l+1\;\mbox{ and }\; 1\leq i\leq [\frac{r}{2}] \}$) and each weight has multiplicity $\frac{dim_k(C^{+}(V))}{2^{[{mr}/{2}]}}=2^{mr-1-[\frac{mr}{2}]}$.\
As it was remarked in [@vanGeemen2], Lemma 5.5, this implies (if $m\geq 5$) that the restrictions of these weights to $h\subset \hat{h}$ are exactly the weights of the exterior tensor product of the spin representations of ${\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ in $C^{+}(V)\otimes_{L,{\sigma}_i} F$ ($V$ is considered as a vector space over $L$), $1\leq i\leq r$, taken with multiplicity $\frac{2^{mr-1-[mr/2]}}{(2^{m-1-l})^r}=2^{r-1}$, if $m$ is even, or with multiplicity $\frac{2^{mr-1-[mr/2]}}{(2^{m-1-l})^r}\cdot 2^{[\frac{r}{2}]}=2^{r-1}$, if $m$ is odd.\
[**Corollary 1.**]{} [*If $E=E_0=L$ is totally real, then the restriction of the spin representation $\rho\colon {\mathfrak{so}}(\phi)\otimes_k F\rightarrow End_F(C^{+}(V\otimes_k F))$ to ${\mathfrak{g}}\otimes_k F=\oplus_{i=1}^{r}({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)\subset {\mathfrak{so}}(\phi)\otimes_k F$ is the exterior tensor product $\Gamma\cdot ({\rho}_1 \boxtimes ... \boxtimes {\rho}_r)$ of spin representations ${\rho}_i \colon {\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F \rightarrow End_F(C^{+}(V \otimes_{L,{\sigma}_i} F))$ with multiplicity $\Gamma = 2^{r-1}$.*]{}\
Case of the CM-field.
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Let $E=E_0(\theta), {\theta}^2\in E_0=L$ be a CM-field.\
For any $i=1,...,r$, $j=1,...,m$ let $\hat{H}_{j}^{i}=E_{j+2m(i-1),j+2m(i-1)}- E_{j+m+2m(i-1),j+m+2m(i-1)}\in {\mathfrak{so}}(\phi)\otimes_k F$. Then $\hat{H}_{j}^{i}$ form a basis of the splitting Cartan subalgebra $\hat{{\mathfrak{h}}}\subset {\mathfrak{so}}(\phi)\otimes_k F$ described in [@Bourbaki], §13. Let us denote by $\{ \hat{\epsilon}_{j}^{i} \}$ the corresponding dual basis of $\hat{{\mathfrak{h}}}^{*}=Hom_F(\hat{{\mathfrak{h}}},F)$. This is the same Cartan subalgebra and the same basis as considered in [@Bourbaki], §13.\
Lemma 2 above implies that the restriction of $\hat{\epsilon}_{j}^{i}$ to the Cartan subalgebra ${\mathfrak{h}}\subset {\mathfrak{g}}\otimes_k F$ is the element $(0,...,\hat{\epsilon}_j,...,0)$ (with $0$ outside of the $i$-th spot) of the Cartan subalgebra (consisting of diagonal matrices) of ${\mathfrak{gl}}(m,F)^{\oplus r}$, where $\hat{\epsilon}_j\cong E_{j,j}\in {\mathfrak{gl}}(m,F)$ is the $j$-th element of the dual basis of the Cartan subalgebra of ${\mathfrak{gl}}(m,F)$ considered in [@Bourbaki], §13.\
If $m\cdot r=\frac{1}{2}\cdot dim_k(V)\geq 3$, then according to [@Bourbaki], §13 the weights of the spin representation of ${\mathfrak{so}}(\phi)\otimes_k F$ in $C^{+}(V)\otimes_k F$ ($V$ is considered as a vector space over $k$) are $\frac{1}{2}\sum_{i,j} {\hat{\epsilon}}_{j}^{i} - \sum_{(i,j)\in I} {\hat{\epsilon}}_{j}^{i}$. Here $I$ runs over the subsets of $[1,...,r]\times [1,...,m]$. Each weight has multiplicity $\frac{dim_k(C^{+}(V))}{2^{mr}}=2^{mr-1}$ ([@Bourbaki], §13).\
Suppose $m\geq 2$. Then the restrictions of these weights to $h\subset \hat{h}$ are exactly the weights of the exterior tensor product of the exterior algebra representations of ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)$ in ${\wedge}_{E}^{*}(V)\otimes_{E,{\sigma}_i} F$ ($V$ is considered as a vector space over $E$) twisted by $D^{-1/2}$, $1\leq i\leq r$. Here $D^{c}$, $c\in\mathbb Q$ denotes the representation of ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)={\mathfrak{c}}\oplus {\mathfrak{sl}}(m,F)$ in ${\wedge}_{E}^{m}(V)\otimes_{E,{\sigma}_i} F\cong {\wedge}_{F}^{m}(V\otimes_{E,{\sigma}_i} F)$ such that ${\mathfrak{sl}}(m,F)$ acts trivially, while $1\in F\cong {\mathfrak{c}}$ acts as $c\cdot Id$. In other words, $D^{c}\colon {\mathfrak{gl}}(m,F)\rightarrow End_F({\wedge}_{E}^{m}(V)\otimes_{E,{\sigma}_i} F)$, $M \mapsto c\cdot Tr(M)\cdot Id$.\
Indeed, for any $i$, $\sum_{j}\hat{\epsilon}_{j}^{i}$ restricts to $0$ to the Cartan subalgebra of the semi-simple part ${\mathfrak{sl}}(m,F)\subset{\mathfrak{gl}}(m,F)\cong {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F$ and to $m\cdot Id_F$ to the center $F\cong {\mathfrak{c}}\subset {\mathfrak{gl}}(m,F)\cong {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F$.\
The exterior tensor product above has multiplicity $\Gamma = 2^{mr-1}$. Indeed, $dim_F(C^{+}(V)\otimes_k F)=2^{2mr-1}$ and $dim_F({\wedge}_{E}^{*}(V)\otimes_{E,{\sigma}_i} F)=2^m$. Hence the dimention of the exterior tensor product is $(dim_F({\wedge}_{E}^{*}(V)\otimes_{E,{\sigma}_i} F))^r=2^{mr}$ and so the multiplicity is $2^{2mr-1}/2^{mr}=2^{mr-1}$.\
[**Corollary 2.**]{} [*If $E=E_0(\theta), {\theta}^2\in E_0=L$ is a CM-field, then the restriction of the spin representation $\rho\colon {\mathfrak{so}}(\phi)\otimes_k F\rightarrow End_F(C^{+}(V\otimes_k F))$ to ${\mathfrak{g}}\otimes_k F=\oplus_{i=1}^{r}({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F)\cong {{\mathfrak{gl}}(m,F)}^{\oplus r}\subset {\mathfrak{so}}(\phi)\otimes_k F$ is the exterior tensor product $\Gamma\cdot ({\rho}_1 \boxtimes ... \boxtimes {\rho}_r)$ of exterior algebra representations ${\rho}_i \colon {\mathfrak{gl}}(m,F) \rightarrow End_F({\wedge}_{F}^{*}(V\otimes_{E,{\sigma}_i} F)\otimes_F F )$ twisted by one-dimensional representations $D^{-1/2}\colon {\mathfrak{gl}}(m,F)\rightarrow End_F(F)\cong F$, $M\mapsto (-\frac{1}{2m})\cdot Tr(M)$ with multiplicity $\Gamma = 2^{mr-1}$.*]{}\
[**Remark.**]{} ${\rho}_i$ is a double-valued ’spin’ representation of $GL(m,F)$.\
From these Corollaries one can deduce the highest weights of irreducible subrepresentations over $F$ of the restriction to ${\mathfrak{g}}\otimes_k F \subset {\mathfrak{so}}(\phi)\otimes_k F$ of the spin representation $\rho\colon {\mathfrak{so}}(\phi) \rightarrow End_k(C^{+}(V))$. Then one can use the description of the Galois action of $S=Gal(F/k)$ on weights of ${\mathfrak{g}}\otimes_k F$ given above in order to break down the highest weights into orbits $\{ S\cdot {\omega}_1,...,S\cdot {\omega}_t \}$. Let us denote the dimension of the irreducible representation of ${\mathfrak{g}}\otimes_k F$ with highest weight ${\omega}_i$ by $d_i$. Let $\hat{\rho}_i \colon {\mathfrak{g}}\rightarrow End_k(W_i)$ be the (unique) irreducible representation of ${\mathfrak{g}}$ over $k$ such that $W_i\otimes_k F$ contains the irreducible representation of ${\mathfrak{g}}\otimes_k F$ with highest weight ${\omega}_i$ as a $({\mathfrak{g}}\otimes_k F)$-submodule. Then our analysis in Section 2 implies:\
[**Theorem 1.**]{} [*$$End(KS(X))_{\mathbb Q} \cong End_{{\mathfrak{g}}}(W)\cong \prod_i Mat_{m_i\times m_i}(D_i) \; \; \mbox{as}\; {\mathbb Q}-\mbox{algebras},$$ where $D_i=End_{{\mathfrak{g}}}(W_{i})$, $m_i=(d_i/ dim_k(W_i))\cdot \sum_{\omega\in S\cdot {\omega}_i} mult(\omega)$ and $mult(\omega)$ is the multiplicity of the irreducible subrepresentation of the representation of ${\mathfrak{g}}\otimes_k F$ on $C^{+}(V\otimes_k F)$ with highest weight $\omega$*]{}.\
[**Remark.** ]{} In the analysis above we assumed that $m=dim_EV\geq 5$ (if $E$ is totally real) or $m\geq 2$ (if $E$ is a CM-field and $r=[E:k]/2\geq 2$) or $m\geq 3$ (if $E$ is a CM-field and $r=[E:k]/2=1$). In the case of small $m$ Lie algebras we consider ’degenerate’ and requre a separate consideration.\
$\mathbb Q$-forms of spin representations.
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Let us describe more explicitely $\mathbb Q$-forms $W_i$ above or at least the corresponding primary representations. We will use corestriction of algebraic structures, as in [@vanGeemen2], §6 and (in the case of totally real fields) representation spaces which we are going to construct in the following subsection.\
Galois-invariant sums of ideals of Clifford algebra.
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Let $k=\mathbb Q$, $E=L$ be a totally real number field, $r=[L:k]$. Let $\Phi=d_1\cdot X_1^2+...+d_m\cdot X_m^2$ with respect to basis $\{ e_1,...,e_m \}$ of $V$, $m=dim_LV$. Let $F/k$ be a finite Galois extension containing $L$, $\sqrt{-1}$ and $\sqrt{d_i}$ for all $i$. Let ${\sigma}_1,...,{\sigma}_r\colon L\hookrightarrow F$ be all the field embeddings over $k$.\
Let $f_i=\frac{1}{\sqrt{d_i}}\cdot e_i+\frac{1}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1}$, $f_{-i}=\frac{1}{\sqrt{d_i}}\cdot e_i-\frac{1}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1}$, $1\leq i\leq l=[\frac{m}{2}]$ and $f_0=\frac{1}{\sqrt{d_{l+1}}}\cdot e_{l+1}$. Then $\{ f_i, \; f_{-i}\; \mid \; 1\leq i\leq l \}$ (if $m$ is even) or $\{ f_0,\; f_i, \; f_{-i}\; \mid \; 1\leq i\leq l \}$ (if $m$ is odd) is a basis of $V\otimes_{L,{\sigma}_i} F$, where we denote ${{\sigma}_i}(d_j)$ by $d_j$. With respect to this basis $\Phi=2\sum_{i=1}^{l}Y_i\cdot Y_{-i}+{\epsilon}Y_0^2$, where ${\epsilon}=(1-(-1)^m)/2$.\
### Even dimension.
Assume that $m$ is even. Let $f_{{\alpha}_1,...,{\alpha}_l}^i=f_{{\alpha}_1\cdot 1}\cdot...\cdot f_{{\alpha}_l\cdot l}\in C(V\otimes_{L,{\sigma}_i} F)$ for various ${\alpha}_i\in \{ \pm 1 \}$ and $I_{{\alpha}_1,...,{\alpha}_l}^{i}=C(V\otimes_{L,{\sigma}_i} F)\cdot f_{{\alpha}_1,...,{\alpha}_l}^i$, $1\leq i\leq r$. $I_{{\alpha}_1,...,{\alpha}_l}^{i}$ are left ideals of the Clifford algebra $C(V\otimes_{L,{\sigma}_i} F)$ viewed as $F$-vector subspaces.\
Consider the direct sum of $F$-vector spaces $$\tilde{C}(V\otimes_{L,{\sigma}_i} F)=\tilde{C}(V)\otimes_{L,{\sigma}_i} F=\bigoplus_{{\alpha}_1,...,{\alpha}_l\in \{ \pm 1 \}} I_{{\alpha}_1,...,{\alpha}_l}^{i}.$$
Note that $g(f_i)\in \{ \pm f_i, \pm f_{-i} \}$ for any $i$ and $g\in S$. Hence the Galois group $S=Gal(F/k)$ acts on $\tilde{C}(V\otimes_{L,{\sigma}_i} F)$ (by sending an element of the summand $I_{{\alpha}_1,...,{\alpha}_l}$ to its image under the action of $S$ on $C(V\otimes_{L,{\sigma}} F)$ viewed as an element of the summand $I_{{\beta}_1,...,{\beta}_l}$, where $f_{{\beta}_1,...,{\beta}_l}$ is upto a scalar factor the image of $f_{{\alpha}_1,...,{\alpha}_l}$).\
It follows from the construction that $F$-vector subspaces $\oplus_{i=1}^{r} I_{{\alpha}_1^i,...,{\alpha}_l^i}^{i}\subset \oplus_{i=1}^r \tilde{C}(V)\otimes_{L,{\sigma}_i} F$ for various choices of ${\alpha}_j^i\in \{ \pm 1 \}$ are permuted among themselves under the action of the Galois group $S=Gal(F/k)$.\
[**Remark.**]{} For any ${\alpha}_1,...,{\alpha}_l$ the left ideal $I_{{\alpha}_1,...,{\alpha}_l}^{i}\subset C(V\otimes_{L,{\sigma}_i} F)$ is an $({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F )$-subrepresentation of the spin representation, which is either irreducible (if $m$ is odd) or is the sum of two irreducible and non-isomorphic (semi-spin) representations [@Chevalley], [@FultonHarris]. In the latter case, let us write $I_{{\alpha}_1,...,{\alpha}_l}^{i}=I_{{\alpha}_1,...,{\alpha}_l}^{i, +}\oplus I_{{\alpha}_1,...,{\alpha}_l}^{i, -}$ for the corresponding (unique) decomposition.\
### Odd dimension.
Assume that $m$ is odd. Let $f_{{\alpha}_1,...,{\alpha}_l,\gamma}^i=f_{{\alpha}_1\cdot 1}\cdot...\cdot f_{{\alpha}_l\cdot l} \cdot (1+\gamma\cdot f_0) \in C(V\otimes_{L,{\sigma}_i} F)$ for various ${\alpha}_i,\gamma \in \{ \pm 1 \}$ and $I_{{\alpha}_1,...,{\alpha}_l,\gamma}^{i}=C(V\otimes_{L,{\sigma}_i} F)\cdot f_{{\alpha}_1,...,{\alpha}_l,\gamma}^i$, $1\leq i\leq r$. $I_{{\alpha}_1,...,{\alpha}_l,\gamma}^{i}$ are left ideals of the Clifford algebra $C(V\otimes_{L,{\sigma}_i} F)$ viewed as $F$-vector subspaces.\
Consider the direct sum of $F$-vector spaces $$\tilde{C}(V\otimes_{L,{\sigma}_i} F)=\tilde{C}(V)\otimes_{L,{\sigma}_i} F=\bigoplus_{{\alpha}_1,...,{\alpha}_l,\gamma \in \{ \pm 1 \}} I_{{\alpha}_1,...,{\alpha}_l,\gamma}^{i}.$$
Note that $g(1+\gamma \cdot f_0)=(1\pm\gamma \cdot f_0)$ for any $g\in S$. Hence the Galois group $S=Gal(F/k)$ acts on $\tilde{C}(V\otimes_{L,{\sigma}_i} F)$ (by sending an element of the summand $I_{{\alpha}_1,...,{\alpha}_l,\gamma}$ to its image under the action of $S$ on $C(V\otimes_{L,{\sigma}} F)$ viewed as an element of the summand $I_{{\beta}_1,...,{\beta}_l,{\gamma}'}$, where $f_{{\beta}_1,...,{\beta}_l,{\gamma}'}$ is upto a scalar factor the image of $f_{{\alpha}_1,...,{\alpha}_l,\gamma}$).\
It follows from the construction that $F$-vector subspaces $\oplus_{i=1}^{r} I_{{\alpha}_1^i,...,{\alpha}_l^i,{\gamma}^i}^{i}\subset \oplus_{i=1}^r \tilde{C}(V)\otimes_{L,{\sigma}_i} F$ for various choices of ${\alpha}_j^i, {\gamma}^i\in \{ \pm 1 \}$ are permuted among themselves under the action of the Galois group $S=Gal(F/k)$.\
[**Remark.**]{} For any ${\alpha}_1,...,{\alpha}_l,\gamma$ the left ideal $I_{{\alpha}_1,...,{\alpha}_l,\gamma}^{i}\subset C(V\otimes_{L,{\sigma}_i} F)$ is an irreducible $({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F )$-subrepresentation of the spin representation (since $m$ is odd by assumption) [@Chevalley], [@FultonHarris].\
We will use $\tilde{C}(V\otimes_{L,{\sigma}_i} F)$ as representation spaces of $({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F )$ (the direct sum of its representations on the left ideals of the Clifford algebra) in order to construct primary $\mathbb Q$-forms of spin representations.\
Case of the totally real field and odd dimension.
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Let $E=E_0=L$ be totally real and $m=dim_LV$ odd. Let ${\Sigma}_i\subset C^{+}(V\otimes_{L,{\sigma}_i} F)$, $1\leq i\leq r$ be the irreducible subrepresentation of the spin representation of ${\mathfrak{so}}(\Phi) \otimes_{L,{\sigma}_i} F$. Then ${\Sigma}_1 \otimes_F ... \otimes_F {\Sigma}_r$ is an irreducible representation of $\oplus_{i=1}^{r}({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)={\mathfrak{g}}\otimes_k F$.\
Let $\tilde{C}(V\otimes_{L,{\sigma}_i} F)=\oplus_p S_{p}^{i}$ be a decomposition into irreducible components of the representation of ${\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ considered above. Let ${\Omega}'$ be the finite set of $F$-vector subspaces of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ (or of $C(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F C(V\otimes_{L,{\sigma}_r} F)$) of the form ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$ for various $p_1,...,p_r$. These subspaces are irreducible subrepresentations of the exterior tensor product of spin representations as a representation of $\oplus_{i=1}^{r} ({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)$.\
Galois group $S=Gal(F/k)$ acts on ${\Omega}'$. Take any element ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$ of ${\Omega}'$. Let $U\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ be the sum of the elements of ${\Omega}'$ (as subspaces of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$) lying in the $S$-orbit of ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$. Then $U\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ is an $S$-submodule.\
Since the actions of ${\mathfrak{g}}\subset {\mathfrak{g}}\otimes_k F$ and $S=Gal(F/k)$ commute, by Galois descent $$\left( U \right)^S\cong \left( ({\Sigma}_1 \otimes_F ... \otimes_F {\Sigma}_r)^{\oplus n_0} \right)^{S}$$ is a primary representation of ${\mathfrak{g}}$ over $k$ of dimension $n_0\cdot 2^{l\cdot r}$, which contains ${\Sigma}_1 \otimes_F ... \otimes_F {\Sigma}_r$ after extending scalars to $F$.\
Multiplicity $n_0$ is the length of the $S$-orbit in ${\Omega}'$ of the chosen element ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$ of ${\Omega}'$.\
[**Remark.**]{} We will use notation introduced above. Consider the action of $S=Gal(F/k)$ on $2^{l+1}$ elements (or more precisely on the lines generated by them) $f_{{\beta}_1,...,{\beta}_l,\gamma}$ of $C(V\otimes_L F)$ for various ${{\beta}_1,...,{\beta}_l,\gamma}$ by sign changes in front of $\sqrt{d_i}$’s and $\sqrt{-d_{m-i+1}}$’s in the definition of $f_i$ in terms of $e_j$ (see notation above). Then (if we choose all $S_{p_i}$ to be the same) $$n_0=\frac{\mbox{order of }\; S=Gal(F/k) }{\mbox{order of the stabilizer of}\; f_{1,...,1,1}}.$$
Case of the totally real field and even dimension.
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Let $E=E_0=L$ be a totally real field and $m=dim_LV$ even. Let ${\Sigma}_i^{+}, {\Sigma}_i^{-} \subset C^{+}(V\otimes_{L,{\sigma}_i} F)$, $1\leq i\leq r$ be irreducible (semi-spin) subrepresentations of the spin representation of ${\mathfrak{so}}(\Phi) \otimes_{L,{\sigma}_i} F$.\
Consider the finite set $\Omega$ of $F$-vector spaces of the form ${\Sigma}_1^{{\alpha}_1} \otimes_F ... \otimes_F {\Sigma}_r^{{\alpha}_r}$ for various ${\alpha}_i \in \{ +,- \}$. They are exactly the irreducible components of the exterior tensor product of spin representations ${\Sigma}_i={\Sigma}_i^{+}\oplus {\Sigma}_i^{-}\subset C^{+}(V\otimes_{L,{\sigma}_i} F)$ of $\oplus_{i=1}^{r} ({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)$ (see [@Bourbaki], §13, [@Chevalley], [@FultonHarris]). They are also the isomorphism classes of simple $\oplus_{i=1}^{r} ({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)$-submodules of $C(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F C(V\otimes_{L,{\sigma}_r} F)$. Let $\tilde{C}(V\otimes_{L,{\sigma}_i} F)=\oplus_p S_{p}^{i}$ be a decomposition into irreducible components of the representation of ${\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ considered above. Let ${\Omega}'$ be the finite set of $F$-vector subspaces of $C(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F C(V\otimes_{L,{\sigma}_r} F)$ (or of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$) of the form ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$ for various $p_1,...,p_r$. These subspaces are irreducible subrepresentations of the exterior tensor product of spin representations as a representation of $\oplus_{i=1}^{r} ({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)$.\
Galois group $S=Gal(F/k)$ acts naturally on both $\Omega$ and ${\Omega}'$. Let ${\Omega}_1,...,{\Omega}_u$ be the orbits of $S$ on $\Omega$. For any $i$ choose $({\alpha}_1,...,{\alpha}_r)\in {\Omega}_i$ and define $U_i\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ to be the sum of the elements of ${\Omega}'$ (as subspaces of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$) lying in the $S$-orbit of any ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$, which is isomorphic to ${\Sigma}_1^{{\alpha}_1} \otimes_F ... \otimes_F {\Sigma}_r^{{\alpha}_r}$ as an $\oplus_{i=1}^{r} ({\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F)$-module.\
Then $U_i\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ is an $S$-submodule and $$\left( U_i \right)^S\cong \left( \bigoplus_{({\alpha}_1,...,{\alpha}_r)\in {\Omega}_i} ({\Sigma}_1^{{\alpha}_1} \otimes_F ... \otimes_F {\Sigma}_r^{{\alpha}_r})^{\oplus n_{{\alpha}_1,...,{\alpha}_r}} \right)^{S}$$ is a primary representation of ${\mathfrak{g}}$ over $k$ of dimension $\sum_{({\alpha}_1,...,{\alpha}_r)\in {\Omega}_i} n_{{\alpha}_1,...,{\alpha}_r} \cdot 2^{r\cdot (l-1)}$. These representations $\left( U_i \right)^S$, $1\leq i \leq u$ contain all representations of ${\mathfrak{g}}\otimes_k F$ of the form ${\Sigma}_1^{{\alpha}_1} \otimes_F ... \otimes_F {\Sigma}_r^{{\alpha}_r}$ after extending scalars to $F$.\
Multiplicities $n_{{\alpha}_1,...,{\alpha}_r}$ can be computed as follows: $$n_{{\alpha}_1,...,{\alpha}_r}=\frac{\mbox{order of the stabilizer of}\; ({\alpha}_1,...,{\alpha}_r)\in\Omega}{\mbox{order of the stabilizer of}\; ({p}_1,...,{p}_r)\in{\Omega}'}.$$
[**Remark.**]{} We will use notation introduced above. Consider the action of $S=Gal(F/k)$ on $2^l$ elements (or more precisely on the lines generated by them) $f_{{\beta}_1,...,{\beta}_l}$ of $C(V\otimes_L F)$ for various ${{\beta}_1,...,{\beta}_l}$ by sign changes in front of $\sqrt{d_i}$’s and $\sqrt{-d_{m-i+1}}$’s in the definition of $f_i$ in terms of $e_j$ (see notation above). Then (if we choose all $S_{p_i}$ to be the same) $$(\mbox{stabilizer of}\; (p_1,...,p_r)\in {\Omega}')=(\mbox{stabilizer of}\; ({\alpha}_1,...,{\alpha}_r)\in \Omega )\cap (\mbox{stabilizer of}\; f_{1,...,1}).$$
[**Remark.**]{} Instead of $\tilde{C}(V\otimes_{L} F)$ one can also consider the Clifford algebra $C(V\otimes_{L} F)$ (or its even part ${C}^{+}(V\otimes_{L} F)$). Then the corestriction of $C(V)$ (or of $C^{+}(V)$) (with $V$ viewed as a vector space over $L$) from $L$ to $k=\mathbb Q$ (or Galois-fixed subspaces of sums (inside of tensor products of $C(V)\otimes_L F$) of tensor products of $({\mathfrak{g}}\otimes_k F)$-invariant $F$-vector subspaces (or ideals used above) of $C(V)\otimes_L F$, which form a single Galois orbit) would be a representation of ${\mathfrak{g}}$ over $\mathbb Q=k$, whose extension of scalars to $F$ contains all the irreducible representations (and only them) of ${\mathfrak{g}}\otimes_k F$ over $F$ which we need. In particular, in the case of odd $m$ it would be another primary representation of ${\mathfrak{g}}$ over $k$.\
Case of the CM-field.
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Let $E=E_0(\theta), {\theta}^2\in E_0 =L$ be a CM-field.\
Note that the tautological representation of ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F$ in $V\otimes_{L,{\sigma}_i} F$ splits into the direct sum of two representations of ${\mathfrak{gl}}(m,F)\cong{\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F$: $$V\otimes_{L,{\sigma}_i} F=(V\otimes_{E,{\sigma}_i} F)\oplus (V\otimes_{E,\bar{{\sigma}_i}} F),$$ where ${\sigma}_i$ and $\bar{{\sigma}_i}$ are the two extensions of ${\sigma}_i \colon E_0\rightarrow F$ to embeddings $E\rightarrow F$.\
Since the exterior power representations ${\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)$ and ${\wedge}_F^{p}(V\otimes_{E,{{{\sigma}_i}}} F)$ of ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)$ are identified by the Lie algebra automorphism ${\mathfrak{gl}}(m,F)\rightarrow {\mathfrak{gl}}(m, F)$, $M\mapsto -\Phi\cdot M^{T}\cdot {\Phi}^{-1}$, we have isomorphisms $${\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)\rightarrow {\wedge}_F^{m-p}(V\otimes_{E,{{{\sigma}_i}}} F)\otimes_F D^{-1}$$ and hence also isomorphisms $${\tau}_p\colon {\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)\otimes_F (E\otimes_{E,\bar{{\sigma}_i}} F)\rightarrow {\wedge}_F^{m-p}(V\otimes_{E,{{{\sigma}_i}}} F)\otimes_F D^{-1/2}, \;1\leq p\leq m$$ of representations of ${\mathfrak{gl}}(m,F)\cong {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F$.\
Let ${\wedge}_{i}^{j}\subset {\wedge}_{F}^{*}(V \otimes_{E,{\sigma}_i} F)\otimes_F F$, $1\leq i \leq r$, $1\leq j\leq m$ be the irreducible representation of ${\mathfrak{gl}}(m,F)$ on the $F$-vector space ${\wedge}_{F}^{j}(V \otimes_{E,{\sigma}_i} F)$ twisted by $D^{-1/2}$. We define an $E_0$-linear representation $D^{c}$, $c\in\mathbb Q$ of ${\mathfrak{u}}(\Phi)$ in the $E_0$-vector space $E$ in exactly the same way as for ${\mathfrak{gl}}(m,F)$ above, i.e. by taking the trace of a matrix and multiplying it by $\frac{c}{m}$.\
Consider the finite set $\Omega$ of $F$-vector spaces of the form ${\wedge}_1^{{j}_1} \otimes_F ... \otimes_F {\wedge}_r^{{j}_r}$ for various ${j}_i \in \{ 1,...,m \}$. They are exactly the isomorphism classes of irreducible subrepresentations of the exterior tensor product of (twisted by $D^{-1/2}$ and extended to $F$) exterior algebra representations ${\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_i} F)\otimes_F (E\otimes_{L,{\sigma}_i} F)$ of $\oplus_{i=1}^{r} ({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F)\cong {\mathfrak{gl}}(m,F)^{\oplus r}$.\
Let ${\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_i} F)\otimes_F (E\otimes_{L,{\sigma}_i} F)=\oplus_p S_{p}^{i}$ be the decomposition into irreducible components of the representation of ${\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {\mathfrak{gl}}(m,F)$ obtained from the decompositions $E\otimes_{L,{\sigma}_i} F=(E\otimes_{E,{\sigma}_i} F)\oplus (E\otimes_{E,\bar{{\sigma}_i}} F)\cong D^{-1/2}\oplus D^{1/2} \cong F\oplus F$ and $V\otimes_{L,{\sigma}_i} F=(V\otimes_{E,{\sigma}_i} F)\oplus (V\otimes_{E,\bar{{\sigma}_i}} F)$ above.\
Let ${\Omega}'$ be the finite set of $F$-vector subspaces of $({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$ of the form ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$ for various $p_1,...,p_r$. These subspaces are irreducible subrepresentations of the exterior tensor product of exterior algebra representations as a representation of $\oplus_{i=1}^{r} ({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F)$.\
Galois group $S=Gal(F/k)$ acts on $\Omega$ by permuting factors in tensor products. It also acts on ${\Omega}'$. Let ${\Omega}_1,...,{\Omega}_u$ be the orbits of $S$ on $\Omega$. For any $i$ choose $({j}_1,...,{j}_r)\in {\Omega}_i$ and define $U_i\subset ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$ to be the sum of the elements of ${\Omega}'$ (as subspaces of $({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$) lying in the $S$-orbit of any ${S}_{p_1}^{1} \otimes_F ... \otimes_F {S}_{p_r}^{r}$, which is isomorphic to ${\wedge}_1^{{j}_1} \otimes_F ... \otimes_F {\wedge}_r^{{j}_r}$ as a $\oplus_{i=1}^{r} ({\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F)\cong {{\mathfrak{gl}}(m,F)}^{\oplus r}$-module.\
Then $U_i\subset ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$ is an $S$-submodule and $$\left( U_i \right)^S\cong \left( \bigoplus_{({j}_1,...,{j}_r)\in {\Omega}_i} ({\wedge}_1^{{j}_1} \otimes_F ... \otimes_F {\wedge}_r^{{j}_r})^{\oplus n_{j_1,...,j_r}} \right)^{S}$$ is a primary representation of ${\mathfrak{g}}$ over $k$ of dimension $\sum_{({j}_1,...,{j}_r)\in {\Omega}_i} n_{j_1,...,j_r}\cdot {\binom{m}{j_1}} \cdot ... \cdot {\binom{m}{j_r}}$. These representations $\left( U_i \right)^S$, $1\leq i \leq u$ contain all representations of ${\mathfrak{g}}\otimes_k F$ of the form ${\wedge}_1^{{j}_1} \otimes_F ... \otimes_F {\wedge}_r^{{j}_r}$ after extending scalars to $F$.\
The reason why nontrivial multiplicities may appear is exactly the doubling $V\otimes_{L,{\sigma}_i} F=(V\otimes_{E,{\sigma}_i} F)\oplus (V\otimes_{E,\bar{{\sigma}_i}} F)$ described above. Hence one can compute multiplicities $n_{j_1,...,j_r}$ as follows. Consider the finite set ${\Omega}''$ of $r$-tuples of signs $+$ and $-$, i.e. ${\Omega}''=\{ ({\alpha}_1,...,{\alpha}_r)\;\mid\; {\alpha}_i=\pm \}$. Note that the $i$-th sign corresponds to the $i$-th embedding ${\sigma}_i\colon L\rightarrow F$ over $k$. Consider the action of $S=Gal(F/k)$ on ${\Omega}''$ such that $g\in S$ acts on entries of $r$-tuples by the same permutations as on the set of left cosets $S/\tilde{H}$ (where $\tilde{H}=\{ g\in S \;\mid \; g\circ {\sigma}_1={\sigma}_1 \}$) and $g$ changes the sign in the $i$-th entry to the opposit sign (in the $j$-th entry, where ${\sigma}_j=g\circ {\sigma}_i$) if and only if $g(\theta)=-\theta$. Then $$n_{j_1,...,j_r}=\frac{\mbox{order of the stabilizer of}\; ({j}_1,...,{j}_r)\in\Omega}{\mbox{order of the intersection of stabilizers of}\; (+,...,+)\in{\Omega}''\;\mbox{ and of }\; ({j}_1,...,{j}_r)\in \Omega }.$$
This gives a description of some multiples of $(k=\mathbb Q)$-linear irreducible representations $W_i$ of ${\mathfrak{g}}$ mentioned in the Theorem above (as well as formulas for their dimensions - some multiples of $dim_k(W_i)$) in terms of the Galois action.\
Cohomology classes of division algebras.
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In this section we compute division algebras $D_i$ as elements of the Brauer group $Br(F/C_j)\cong H^2(Gal(F/C_j), F^{*})$ as well as their centers $C_j$.\
Case of the totally real field and odd dimension.
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Let $E=E_0=L$ be totally real and $m=dim_EV$ odd. We saw above how to construct a primary representation $W=U^S$ of ${\mathfrak{g}}$ over $k=\mathbb Q$, which contains irreducible representation ${\rho}^0 \boxtimes ... \boxtimes {\rho}^0$ (the exterior tensor product of irreducible spin representations) of ${\mathfrak{g}}\otimes_k F\cong \oplus_{i=1}^{r} {\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ after extending scalars to $F$. This means that $W\cong W_0^{\oplus \mu}$, where $W_0$ is an irreducible representation of ${\mathfrak{g}}$ over $k$ and $W_0\otimes_k F\cong \frac{dim_kW}{\mu \cdot (dim_F({\rho}^0))^r}\cdot {\rho}^0 \boxtimes ... \boxtimes {\rho}^0$. Since we are interested only in the endomorphism algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ which is a central division algebra over $k$ split over $F$, we can describe it by computing the Galois cohomology invariant of the central simple algebra $A=End_{{\mathfrak{g}}}(W)\cong Mat_{\mu \times \mu}(D_0)$, i.e. its Brauer invariant in $Br(F/k)\cong H^2(S,F^{*})$, where $S=Gal(F/k)$. Then $\mu=\frac{deg(A)}{deg(D_0)}=\frac{n_0}{deg(D_0)}$.\
We will use the same notation as above with the following exceptions: $$f_{{\alpha}_1,...,{\alpha}_l,\gamma} = (1+\gamma \cdot f_0)\cdot f_{{\alpha}_1\cdot 1}\cdot ... \cdot f_{{\alpha}_l\cdot l},$$
$$f_{{\alpha}\cdot i}=\left(e_i+\alpha \cdot \frac{\sqrt{d_i}}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1}\right).$$ Some parts of our construction (in particular, the construction of the generators of endomorphism algebras) may be viewed as a generalization of some constructions of van Geemen [@vanGeemen1], §3.\
Consider $F$-linear homomorphisms $$r_{(({\alpha}_i),\gamma),(({\beta}_i),\tilde{\gamma})} \colon \tilde{C}(V\otimes_L F)\rightarrow I_{{\beta}_1,...,{\beta}_l,\tilde{\gamma}}, \;\xi \mapsto {\tau}^{{\delta}(\gamma,\tilde{\gamma})} (\xi\cdot R_{(({\alpha}_i),\gamma),({\beta}_i)}),$$ where $\tau \colon {C}(V\otimes_L F)\rightarrow {C}(V\otimes_L F)$ is the algebra homomorphism induced by multiplication by $(-1)$ on $V$, ${\delta}(\gamma,\tilde{\gamma}) = 1$, if $\gamma\neq \tilde{\gamma}\cdot (-1)^{P(\alpha,\beta)}$ (where $P(\alpha,\beta)=card \{ i \;\mid\; {\alpha}_i\neq {\beta}_i \}$) and $0$ otherwise, and $$R_{(({\alpha}_i),\gamma),({\beta}_i)}=\frac{(-1)^{c(\alpha,\beta)}}{\prod_{i\colon {\alpha}_i= {\beta}_i} \Phi (f_i,f_{-i})} \cdot \prod_{i\colon {\alpha}_i= {\beta}_i} (f_{-{\alpha}_i \cdot i}\cdot f_{{\alpha}_i \cdot i}) \cdot \prod_{i\colon {\alpha}_i\neq {\beta}_i} f_{{\beta}_i \cdot i},$$ where $c(\alpha,\beta)$ is the number of transpositions of factors needed to transform the product $\prod_i f_{{\alpha}_i \cdot i}\cdot \prod_{i\colon {\alpha}_i\neq {\beta}_i} f_{{\beta}_i \cdot i}$ into the product $q\cdot \prod_i f_{{\beta}_i \cdot i}$ with some coefficient $q\in C(V\otimes_L F)$. Then $r_{(({\alpha}_i),\gamma),(({\beta}_i),\tilde{\gamma})}$ is nonzero only on the factor $I_{{\alpha}_1,...,{\alpha}_l,{\gamma}}$ of $\tilde{C}(V\otimes_L F)$ and induces an isomorphism $I_{{\alpha}_1,...,{\alpha}_l,{\gamma}} \rightarrow I_{{\beta}_1,...,{\beta}_l,\tilde{\gamma}}$ which commutes with the action of ${\mathfrak{so}}(\Phi)\otimes_L F$.\
In order to simplify notation we will denote index $(({\alpha}_i),\gamma)$ by $\alpha$.\
One can choose coefficients ${\lambda}_{\alpha,\beta}\in F^{*}$ such that under an isomorphism of $F$-algebras $Mat(F)\cong End_{{\mathfrak{so}}(\Phi)\otimes_L F}(\tilde{C}(V\otimes_L F))$ matrices of the form $E_{ij}$ (in the notation of [@Bourbaki], §13) correspond to endomorphisms ${\lambda}_{\alpha,\beta}\cdot r_{\alpha,\beta}$. In order to do this, one can choose and fix index ${\alpha}^0=(({\alpha}_i^0),{\gamma}^0)$ and take $${\lambda}_{{\alpha}^0,\beta}=1, \; {\lambda}_{\beta,{\alpha}^0}=(-1)^{P({\alpha}^0,\beta)\cdot {\delta}({\gamma}^0,\tilde{\gamma})+P({\alpha}^0,\beta)\cdot (P({\alpha}^0,\beta)-1)/2}\cdot \prod_{i\colon {\alpha}_i^0 \neq {\beta}_i}\frac{1}{\Phi(f_i,f_{-i})}$$ and $${\lambda}_{\alpha,\beta}={\lambda}_{\alpha,{\alpha}^0}\cdot (-1)^{e(\alpha,\beta)+ {\delta}(\gamma,\tilde{\gamma})\cdot (l+P(\alpha,\beta)) + {\delta}(\gamma,{\gamma}^0)\cdot (l+P(\alpha,{\alpha}^0)) + {\delta}({\gamma}^0,\tilde{\gamma})\cdot (l+P({\alpha}^0,\beta)) }\cdot \prod_{i\colon {\alpha}_i = {\beta}_i \neq {\alpha}_i^0 } \Phi(f_i,f_{-i}),$$ where ${\alpha}=(({\alpha}_i),{\gamma})$, ${\beta}=(({\beta}_i),\tilde{\gamma})$, $e(\alpha,\beta)$ is the number of transpositions of factors needed in order to transform the product $\prod_{i\colon {\alpha}_i^0 \neq {\alpha}_i} f_{{\alpha}_i\cdot i}\cdot \prod_{i\colon {\alpha}_i^0 \neq {\beta}_i} f_{-{\beta}_i\cdot i}$ into the product $\prod_{i\colon {\alpha}_i \neq {\beta}_i} f_{{\alpha}_i\cdot i}\cdot \prod_{i\colon {\alpha}_i= {\beta}_i \neq {\alpha}_i^0} (f_{{\beta}_i\cdot i} \cdot f_{-{\beta}_i\cdot i} )$. Note that in this construction ${\lambda}_{\alpha,\beta}\in L^{*}$.\
Then we construct endomorphisms $$\begin{gathered}
r_{({\alpha}^i),({\beta}^i)}=r_{{\alpha}^1,{\beta}^1}^{1}\circ ... \circ r_{{\alpha}^r,{\beta}^r}^{r}\colon \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\rightarrow \\
\rightarrow I_{{\beta}^1}^1\otimes_F ... \otimes_F I_{{\beta}^r}^r\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\end{gathered}$$ which commute with ${\mathfrak{g}}\otimes_k F$, where ${\alpha}^p=(({\alpha}_{1}^{p},...,{\alpha}_{l}^{p}),{\gamma}^p)$, ${\beta}^p=(({\beta}_{1}^{p},...,{\beta}_{l}^{p}),{\tilde{\gamma}}^p)$ and $$\begin{gathered}
r_{{\alpha}^p,{\beta}^p}^{p}=1\otimes_F ... \otimes_F (r_{{\alpha}^p,{\beta}^p}) \otimes_F ... \otimes_F 1\colon \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\rightarrow \\
\rightarrow \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\end{gathered}$$ (with $1$ outside of the $p$-th spot).\
As in [@vanGeemen1], Proposition 3.6 $F$-algebra $End_{{\mathfrak{g}}\otimes_k F}(W\otimes_k F)=A\otimes_k F$ is generated by elements $r_{({\alpha}^i),({\beta}^i)}$ (more precisely, by those of them which correspond to the summands of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ included in $W\otimes_k F=U\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$) or by elements $r_{{\alpha},{\beta}}^{p}$, while $k$-algebra $A=End_{{\mathfrak{g}}}(W)=(A\otimes_k F)^S$ is generated by elements $r_{{\alpha},{\beta}}^{p,q}=\sum_{g \in S} g(e_q)\cdot g\circ r_{{\alpha},{\beta}}^{p}$, where $\{ e_q \}$ is a basis of $F/k$.\
Let us denote by $(c_{q,g})$ the inverse matrix of the matrix $(g(e_q))$. Then $r_{\alpha,\beta}^p=\sum_{q} c_{q,Id}\cdot r_{\alpha,\beta}^{p,q}$ and for any $g\in S=Gal(F/k)$ if we denote by ${\phi}_g\colon A\otimes_k F\rightarrow A\otimes_k F$ the conjugation by $g\colon a\otimes f\mapsto a\otimes g(f)$, then $${\phi}_g(r_{({\alpha}^i),({\beta}^i)})=g\circ r_{({\alpha}^i),({\beta}^i)}= r_{g({\alpha}^i),g({\beta}^i)},$$ where the action of $S$ on upper indices $i$ (which number embeddings ${\sigma}_i\colon L\hookrightarrow F$) coincides with its action on left cosets $S/ \tilde{H}$, where $\tilde{H}=\{ g\in S \; \mid\; g {\mid}_{{\sigma}_1(L)}=Id_{{\sigma}_1(L)} \}$ and the action of $g\in S$ on indices $\alpha=(({\alpha}_1,...,{\alpha}_l),\gamma)$ is given by the rule $g(\alpha)=((c_1(g)\cdot{\alpha}_1,...,c_l(g)\cdot{\alpha}_l),c_0(g)\cdot \gamma)$, where $c_i(g)\in \{ \pm 1 \}$ and $g(f_{{\alpha}_1,...,{\alpha}_l,\gamma})=f_{c_1(g)\cdot{\alpha}_1,...,c_l(g)\cdot{\alpha}_l,c_0(g)\cdot \gamma}$.\
Hence the matrix of $m(g)\in GL(W\otimes_k F)$ is such that $$m(g)\cdot E_{i,j}\cdot {m(g)}^{-1}={\phi}_g (E_{i,j})=\left( \prod_{i=1}^r \frac{g({\lambda}_{{\alpha}^i,{\beta}^i})}{{\lambda}_{g({\alpha}^i),g({\beta}^i)}} \right) \cdot E_{g(i),g(j)},$$ where $E_{i,j}$ denotes a matrix from $Mat(F)\cong End_{{\mathfrak{g}}\otimes_k F} (W\otimes_k F)$ corresponding to $r_{({\alpha}^i),({\beta}^i)}$, i.e. upto a scalar multiple conjugation by $m(g)$ acts on matrices as the (same) permutation of columns and rows induced by $g$ on indices $(({\alpha}_{j}^{i}),{\gamma}^i)$.\
Then the element of $H^2(S,F^{*})$ corresponding to the central division algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ is the class of a $2$-cocycle $\lambda\colon S\times S\rightarrow F^{*}\cong F^{*}\cdot Id\subset Mat(F)$, $(g_1,g_2)\mapsto m(g_1g_2)\cdot (g_1 (m(g_2)))^{-1} \cdot m(g_1)^{-1}$ [@Kuznetsov], [@Jacobson].\
Case of the totally real field and even dimension.
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Let $E=E_0=L$ be totally real and $m=dim_EV$ even. We saw above how to construct a primary representation $W=(U_i)^S$ of ${\mathfrak{g}}$ over $k=\mathbb Q$, which contains irreducible representation ${\rho}^{{\alpha}_1} \boxtimes ... \boxtimes {\rho}^{{\alpha}_r}$ (the exterior tensor product of irreducible semi-spin representations) of ${\mathfrak{g}}\otimes_k F\cong \oplus_{i=1}^{r} {\mathfrak{so}}(\Phi)\otimes_{L,{\sigma}_i} F$ after extending scalars to $F$ (as well as its Galois conjugates). This means that $W\cong W_0^{\oplus \mu}$, where $W_0$ is an irreducible representation of ${\mathfrak{g}}$ over $k$, $W\otimes_k F\cong \oplus_i W_i$ and $W_i\cong \frac{dim_F{W_i}}{(dim_F({\rho}^{{\alpha}_1}))^r}\cdot {\rho}^{{{\alpha}_1}'} \boxtimes ... \boxtimes {\rho}^{{{\alpha}_r}'}$ are the isotypical components (over $F$). Since we are interested only in the endomorphism algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ which is a division algebra over $k$ (and over its center $C$) split over $F$, we can describe it by computing the Galois cohomology invariant of the central simple algebra $A=End_{{\mathfrak{g}}}(W)\cong Mat_{\mu \times \mu}(D_0)$ (over $C$), i.e. its Brauer invariant in $Br(F/C)\cong H^2(S',F^{*})$, where $S'=Gal(F/C)$. Then $\mu=\frac{deg(A)}{deg(D_0)}=\frac{n_{{\alpha}_1,...,{\alpha}_r}}{deg(D_0)}$.\
We will use the same notation as above with the following exceptions: $$f_{{\alpha}_1,...,{\alpha}_l} = f_{{\alpha}_1\cdot 1}\cdot ... \cdot f_{{\alpha}_l\cdot l},$$
$$f_{{\alpha}\cdot i}=\left(e_i+\alpha \cdot \frac{\sqrt{d_i}}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1}\right).$$ Some parts of our construction (in particular, the construction of the generators of endomorphism algebras) may be viewed as a generalization of some constructions of van Geemen [@vanGeemen1], §3.\
Consider $F$-linear homomorphisms $$r_{({\alpha}_i),({\beta}_i)} \colon \tilde{C}(V\otimes_L F)\rightarrow I_{{\beta}_1,...,{\beta}_l}, \; \xi \mapsto \xi\cdot R_{({\alpha}_i),({\beta}_i)},$$ where $P(\alpha,\beta)=card \{ i \;\mid\; {\alpha}_i\neq {\beta}_i \}$ and $$R_{({\alpha}_i),({\beta}_i)}=\frac{(-1)^{c(\alpha,\beta)}}{\prod_{i\colon {\alpha}_i= {\beta}_i} \Phi (f_i,f_{-i})} \cdot \prod_{i\colon {\alpha}_i= {\beta}_i} (f_{-{\alpha}_i \cdot i}\cdot f_{{\alpha}_i \cdot i}) \cdot \prod_{i\colon {\alpha}_i\neq {\beta}_i} f_{{\beta}_i \cdot i},$$ where $c(\alpha,\beta)$ is the number of transpositions of factors needed to transform the product $\prod_i f_{{\alpha}_i \cdot i}\cdot \prod_{i\colon {\alpha}_i\neq {\beta}_i} f_{{\beta}_i \cdot i}$ into the product $q\cdot \prod_i f_{{\beta}_i \cdot i}$ with some coefficient $q\in C(V\otimes_L F)$. Then $r_{({\alpha}_i),({\beta}_i)}$ is nonzero only on the factor $I_{{\alpha}_1,...,{\alpha}_l}$ of $\tilde{C}(V\otimes_L F)$ and induces an isomorphism $I_{{\alpha}_1,...,{\alpha}_l} \rightarrow I_{{\beta}_1,...,{\beta}_l}$ which commutes with action of ${\mathfrak{so}}(\Phi)\otimes_L F$. Without mentioning this explicitely, we will be restricting all our endomorphisms to the factors of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ contributing to an isotypical component $W_i\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$.\
In order to simplify notation we will denote index $({\alpha}_i)$ by $\alpha$.\
One can choose coefficients ${\lambda}_{\alpha,\beta}\in F^{*}$ such that under an isomorphism of $F$-algebras $Mat(F)\cong End_{{\mathfrak{so}}(\Phi)\otimes_L F}(W_i)$ (note that $W_i\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ and see the remark above) matrices of the form $E_{ij}$ correspond to endomorphisms ${\lambda}_{\alpha,\beta}\cdot r_{\alpha,\beta}$. In order to do this, one can choose and fix index ${\alpha}^0=({\alpha}_i^0)$ and take $${\lambda}_{{\alpha}^0,\beta}=1, \; {\lambda}_{\beta,{\alpha}^0}=(-1)^{P({\alpha}^0,\beta)\cdot (P({\alpha}^0,\beta)-1)/2}\cdot \prod_{i\colon {\alpha}_i^0 \neq {\beta}_i}\frac{1}{\Phi(f_i,f_{-i})}$$ and $${\lambda}_{\alpha,\beta}={\lambda}_{\alpha,{\alpha}^0}\cdot (-1)^{e(\alpha,\beta)}\cdot \prod_{i\colon {\alpha}_i = {\beta}_i \neq {\alpha}_i^0 } \Phi(f_i,f_{-i}),$$ where ${\alpha}=({\alpha}_i)$, ${\beta}=({\beta}_i)$, $e(\alpha,\beta)$ is the number of transpositions of factors needed in order to transform the product $\prod_{i\colon {\alpha}_i^0 \neq {\alpha}_i} f_{{\alpha}_i\cdot i}\cdot \prod_{i\colon {\alpha}_i^0 \neq {\beta}_i} f_{-{\beta}_i\cdot i}$ into the product $\prod_{i\colon {\alpha}_i \neq {\beta}_i} f_{{\alpha}_i\cdot i}\cdot \prod_{i\colon {\alpha}_i= {\beta}_i \neq {\alpha}_i^0} (f_{{\beta}_i\cdot i} \cdot f_{-{\beta}_i\cdot i} )$. Note that in this construction ${\lambda}_{\alpha,\beta}\in L^{*}$.\
Then we construct endomorphisms $$\begin{gathered}
r_{({\alpha}^i),({\beta}^i)}=r_{{\alpha}^1,{\beta}^1}^{1}\circ ... \circ r_{{\alpha}^r,{\beta}^r}^{r}\colon \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\rightarrow \\
\rightarrow I_{{\beta}^1}^1\otimes_F ... \otimes_F I_{{\beta}^r}^r\subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\end{gathered}$$ which commute with ${\mathfrak{g}}\otimes_k F$, where ${\alpha}^p=({\alpha}_{1}^{p},...,{\alpha}_{l}^{p})$, ${\beta}^p=({\beta}_{1}^{p},...,{\beta}_{l}^{p})$ and $$\begin{gathered}
r_{{\alpha}^p,{\beta}^p}^{p}=1\otimes_F ... \otimes_F (r_{{\alpha}^p,{\beta}^p}) \otimes_F ... \otimes_F 1\colon \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\rightarrow \\
\rightarrow \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)\end{gathered}$$ (with $1$ outside of the $p$-th spot).\
As in [@vanGeemen1], Proposition 3.6 $F$-algebra $End_{{\mathfrak{g}}\otimes_k F}(W\otimes_k F)=A\otimes_k F$ is generated by elements $r_{({\alpha}^i),({\beta}^i)}$ (more precisely, by those of them which correspond to the summands of $\tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$ included in various isotypical components $W_{i'}\otimes_k F\subset U_i \subset \tilde{C}(V\otimes_{L,{\sigma}_1} F)\otimes_F ... \otimes_F \tilde{C}(V\otimes_{L,{\sigma}_r} F)$) or by elements $r_{{\alpha},{\beta}}^{p}$, while $k$-algebra $A=End_{{\mathfrak{g}}}(W)=(A\otimes_k F)^S$ is generated by elements $r_{{\alpha},{\beta}}^{p,q}=\sum_{g \in S} g(e_q)\cdot g\circ r_{{\alpha},{\beta}}^{p}$, where $\{ e_q \}$ is a basis of $F/k$.\
The center $C$ of $A$ (and of $D_0$) consists of Galois averages (as above) of $F$-linear combinations of sums $C_{{i}'}=\sum_{({\alpha}^{j})\in I_{{i}'}} (\prod_{i=1}^{r} {{\sigma}_i}({\lambda}_{{\alpha}^i,{\alpha}^i})) \cdot r_{({\alpha}^j),({\alpha}^j)}$ (over the sets $I_{{i}'}$ of indices ${\alpha}^{j}$ corresponding to irreducible subrepresentations over $F$ of $W\otimes_k F$ contained in various isotypical components $W_{i'}$). Each of the coefficients of these $F$-linear combinations gives a field embedding $C\rightarrow F$ over $k=\mathbb Q$. Note that $A\otimes_k F \cong \prod A\otimes_C F$, where the product is taken over these embeddings (which are numbered by the isotypical components $W_{{i}'}$ of $W\otimes_k F$ over $F$) and $A\otimes_C F\cong End_{{\mathfrak{g}}\otimes_k F}(W_{{i}'})$. Moreover, the projection $A\otimes_k F \rightarrow A\otimes_C F$ is given by annihilating endomorphisms between irreducible subrepresentations of isotypical components $W_{{i}''}$ different from $W_{{i}'}$. More explicitely the subfield $C\subset F$ under the embedding corresponding to an isotypical component $W_{{i}'}$ is the fixed subfield of the subgroup $S'\subset S$ consisting of those $g\in S$ which preserve the isotypical component: $g(W_{{i}'})=W_{{i}'}$. Let us choose one such embedding $C\rightarrow F$ (which corresponds to a choice of an isotypical component $W_{{i}'}$ of $W\otimes_k F$).\
Let us denote by $(c_{q,g})$ the inverse matrix of the matrix $(g(e_q))$. Then $r_{\alpha,\beta}^p=\sum_{q} c_{q,Id}\cdot r_{\alpha,\beta}^{p,q}$ and for any $g\in S'=Gal(F/C)\subset S=Gal(F/k)$ if we denote by ${\phi}_g\colon A\otimes_C F\rightarrow A\otimes_C F$ the conjugation by $g\colon a\otimes f\mapsto a\otimes g(f)$, then $${\phi}_g(r_{({\alpha}^i),({\beta}^i)})=g\circ r_{({\alpha}^i),({\beta}^i)}= r_{g({\alpha}^i),g({\beta}^i)},$$ where the action of $S'\subset S$ on upper indices $i$ (which number embeddings ${\sigma}_i\colon L\hookrightarrow F$) coincides with its action on left cosets $S/ \tilde{H}$, where $\tilde{H}=\{ g\in S \; \mid\; g {\mid}_{{\sigma}_1(L)}=Id_{{\sigma}_1(L)} \}$ and the action of $g\in S'\subset S$ on indices $\alpha=({\alpha}_1,...,{\alpha}_l)$ is given by the rule $g(\alpha)=(c_1(g)\cdot{\alpha}_1,...,c_l(g)\cdot{\alpha}_l)$, where $c_i(g)\in \{ \pm 1 \}$ and $g(f_{{\alpha}_1,...,{\alpha}_l})=f_{c_1(g)\cdot{\alpha}_1,...,c_l(g)\cdot{\alpha}_l}$.\
Hence the matrix of $m(g)\in GL(W_{{i}'})$ is such that $$m(g)\cdot E_{i,j}\cdot {m(g)}^{-1}={\phi}_g (E_{i,j})=\left( \prod_{i=1}^r \frac{g({\lambda}_{{\alpha}^i,{\beta}^i})}{{\lambda}_{g({\alpha}^i),g({\beta}^i)}} \right) \cdot E_{g(i),g(j)},$$ where $E_{i,j}$ denotes a matrix from $Mat(F)\cong End_{{\mathfrak{g}}\otimes_k F} (W_{{i}'})$ corresponding to $r_{({\alpha}^i),({\beta}^i)}$, i.e. upto a scalar multiple conjugation by $m(g)$ acts on matrices as the (same) permutation of columns and rows induced by $g$ on indices $({\alpha}_{j}^{i})$.\
Then the element of $H^2(S',F^{*})$ corresponding to the central division algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ (over $C$) is the class of a $2$-cocycle $\lambda\colon S'\times S'\rightarrow F^{*}\cong F^{*}\cdot Id\subset Mat(F)$, $(g_1,g_2)\mapsto m(g_1g_2)\cdot (g_1 (m(g_2)))^{-1} \cdot m(g_1)^{-1}$ [@Kuznetsov], [@Jacobson].\
Case of the CM-field.
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Let $E=E_0(\theta), {\theta}^2\in E_0=L$ be a CM-field and $m=dim_EV$. We saw above how to construct a primary representation $W=(U_i)^S$ of ${\mathfrak{g}}$ over $k=\mathbb Q$, which contains the irreducible representation ${\rho}_{j_1}^{{\alpha}_1} \boxtimes ... \boxtimes {\rho}_{j_r}^{{\alpha}_r}$ of ${\mathfrak{g}}\otimes_k F\cong \oplus_{i=1}^{r} {\mathfrak{u}}(\Phi)\otimes_{L,{\sigma}_i} F\cong {{\mathfrak{gl}}(m,F)}^{\oplus r}$ after extending scalars to $F$ (as well as its Galois conjugates). Here ${\alpha}_{i}\in \{ \pm \}$, $1\leq j_i\leq m$ and $${\rho}_{j_i}^{{\alpha}_i}\colon {\mathfrak{gl}}(m,F)\rightarrow End_{F}({\wedge}_{F}^{j_i}(V\otimes_{E,{\alpha_i\cdot {\sigma}}} F)\otimes_F F)$$ is the exterior product representation twisted by $D^{{\alpha}_i/2}$, where $\pm {\sigma}\colon E\rightarrow F$ are the two embeddings extending ${\sigma}\colon L\rightarrow F$. This means that $W\cong W_0^{\oplus \mu}$, where $W_0$ is an irreducible representation of ${\mathfrak{g}}$ over $k$, $W\otimes_k F\cong \oplus_i W_i$ and $W_i\cong \frac{dim_F{W_i}}{dim_F({\rho}_{j_1}^{{\alpha}_1})\cdot ... \cdot dim_F({\rho}_{j_r}^{{\alpha}_r})}\cdot {\rho}_{j_1}^{{{\alpha}_1}'} \boxtimes ... \boxtimes {\rho}_{j_r}^{{{\alpha}_r}'}$ are the isotypical components (over $F$). Since we are interested only in the endomorphism algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ which is a division algebra over $k$ (and over its center $C$) split over $F$, we can describe it by computing the Galois cohomology invariant of the central simple algebra $A=End_{{\mathfrak{g}}}(W)\cong Mat_{\mu \times \mu}(D_0)$ (over $C$), i.e. its Brauer invariant in $Br(F/C)\cong H^2(S',F^{*})$, where $S'=Gal(F/C)$. Then $\mu=\frac{deg(A)}{deg(D_0)}=\frac{n_{{j}_1,...,{j}_r}}{deg(D_0)}$.\
Our computation is analogous to the case of a totally real field considered above.\
Consider $F$-linear homomorphisms $$r_{{\alpha},{\beta}} \colon {\wedge}_{F}^{*}(V\otimes_{E,{\alpha\cdot {\sigma}}} F)\otimes_F F\rightarrow {\wedge}_{F}^{*}(V\otimes_{E,{\beta\cdot {\sigma}}} F)\otimes_F F, \; \xi\mapsto ({\tau}_{*})^{P(\alpha,\beta)} (\xi),$$ where $P(-1,+1)=1$, $P(+1,-1)=-1$, $P(\alpha,\alpha)=0$ and ${\tau}_{*}=\oplus_p {\tau}_p$ is the direct sum of isomorphisms of ${\mathfrak{gl}}(m,F)$-modules $${\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)\otimes_F (E\otimes_{E,\bar{{\sigma}_i}} F)\rightarrow {\wedge}_F^{m-p}(V\otimes_{E,{{{\sigma}_i}}} F)\otimes_F D^{-1/2}$$ introduced above. Then $r_{{\alpha},{\beta}}$ induces an isomorphism $${\wedge}_{F}^{j_i}(V\otimes_{E,{\alpha\cdot {\sigma}}} F)\otimes_F F\rightarrow {\wedge}_{F}^{{j_i}'}(V\otimes_{E,{\beta\cdot {\sigma}}} F)\otimes_F F$$ which commutes with the action of ${\mathfrak{u}}(\Phi)\otimes_L F\cong {\mathfrak{gl}}(m,F)$. Without mentioning this explicitely, we will be restricting all our endomorphisms to the factors of $({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$ contributing to an isotypical component $W_i\subset ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$.\
Then we construct endomorphisms $$\begin{gathered}
r_{({\alpha}^i),({\beta}^i)}=r_{{\alpha}^1,{\beta}^1}^{1}\circ ... \circ r_{{\alpha}^r,{\beta}^r}^{r}\colon ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F \ldots \\
\ldots \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))\rightarrow \\
\rightarrow ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F)),\end{gathered}$$ which commute with ${\mathfrak{g}}\otimes_k F$, where $({\alpha}^i)=({\alpha}^{1},...,{\alpha}^{r})$, $({\beta}^i)=({\beta}^{1},...,{\beta}^{r})$ and $$\begin{gathered}
r_{{\alpha}^p,{\beta}^p}^{p}=1\otimes_F ... \otimes_F (r_{{\alpha}^p,{\beta}^p})\otimes_F ... \otimes_F 1\colon ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F \ldots \\
\ldots \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))\rightarrow \\
\rightarrow ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))\end{gathered}$$ (with $1$ outside of the $p$-th spot).\
As in the case of a totally real field $E$, $F$-algebra $End_{{\mathfrak{g}}\otimes_k F}(W\otimes_k F)=A\otimes_k F$ is generated by elements $r_{({\alpha}^i),({\beta}^i)}$ (more precisely, by those of them which correspond to the summands of $({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$ included in various isotypical components $W_{i'}\otimes_k F\subset U_i \subset ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_1} F)\otimes_F (E\otimes_{L,{\sigma}_1} F))\otimes_F ... \otimes_F ({\wedge}_{F}^{*}(V \otimes_{L,{\sigma}_r} F)\otimes_F (E\otimes_{L,{\sigma}_r} F))$) or by elements $r_{{\alpha},{\beta}}^{p}$, while $k$-algebra $A=End_{{\mathfrak{g}}}(W)=(A\otimes_k F)^S$ is generated by elements $r_{{\alpha},{\beta}}^{p,q}=\sum_{g \in S} g(e_q)\cdot g\circ r_{{\alpha},{\beta}}^{p}$, where $\{ e_q \}$ is a basis of $F/k$.\
The center $C$ of $A$ (and of $D_0$) can be computed exactly as in the case of a totally real field. In particular, field embeddings $C\rightarrow F$ correspond to the isotypical components $W_{{i}'}$ of $W\otimes_k F$ over $F$, $A\otimes_C F\cong End_{{\mathfrak{g}}\otimes_k F}(W_{{i}'})$, the projection $A\otimes_k F\cong \prod A\otimes_C F \rightarrow A\otimes_C F$ is given by annihilating endomorphisms between irreducible subrepresentations of isotypical components $W_{{i}''}$ different from $W_{{i}'}$ and the subfield $C\subset F$ under the embedding corresponding to an isotypical component $W_{{i}'}$ is the fixed subfield of the subgroup $S'\subset S$ consisting of those $g\in S$ which preserve the isotypical component: $g(W_{{i}'})=W_{{i}'}$. Let us choose one such embedding $C\rightarrow F$ (which corresponds to a choice of an isotypical component $W_{{i}'}$ of $W\otimes_k F$).\
Let us denote by $(c_{q,g})$ the inverse matrix of the matrix $(g(e_q))$. Then $r_{\alpha,\beta}^p=\sum_{q} c_{q,Id}\cdot r_{\alpha,\beta}^{p,q}$ and for any $g\in S'=Gal(F/C)\subset S=Gal(F/k)$ if we denote by ${\phi}_g\colon A\otimes_C F\rightarrow A\otimes_C F$ the conjugation by $g\colon a\otimes f\mapsto a\otimes g(f)$, then $${\phi}_g(r_{({\alpha}^i),({\beta}^i)})=g\circ r_{({\alpha}^i),({\beta}^i)}=\left( \prod_k {{\lambda}_{{\alpha}^k,{\beta}^k}}(g) \right)\cdot r_{g({\alpha}^i),g({\beta}^i)},$$ where the action of $S'\subset S$ on upper indices $i$ (which number embeddings ${\sigma}_i\colon L\hookrightarrow F$) coincides with its action on the left cosets $S/ \tilde{H}$, where $\tilde{H}=\{ g\in S \; \mid\; g {\mid}_{{\sigma}_1(L)}=Id_{{\sigma}_1(L)} \}$ and moreover $g\in S'\subset S$ multiplies the $i$-th index ${{\alpha}}^i$ in the $r$-tuple $({\alpha}^i)=({\alpha}^1,...,{\alpha}^r)$ by $g(\theta)/\theta=\pm 1$.\
Here ${{\lambda}_{{\alpha}^k,{\beta}^k}}(g)\in F^{*}$ are suitable constants. In order to compute them, note that isomorphisms $$\begin{gathered}
{\tau}_p\colon {\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)\otimes_F (E\otimes_{E,\bar{{\sigma}_i}} F)\rightarrow {\wedge}_F^{p}(V\otimes_{E,{\bar{{\sigma}_i}}} F)\otimes_F (E\otimes_{E,\bar{{\sigma}_i}} F)\cong \\
\cong {\wedge}_F^{p}((V\otimes_{E,{{{\sigma}_i}}} F)^{*})\otimes_F (E\otimes_{E,{{\sigma}_i}} F)^{*} \rightarrow {\wedge}_F^{m-p}(V\otimes_{E,{{{\sigma}_i}}} F)\otimes_F (E\otimes_{E,{{\sigma}_i}} F)\end{gathered}$$ (where the first arrow is the isomorphism determined by the matrix of ${\Phi}^{-1}$) are defined over $E$. If we assume that the isomorphism ${\wedge}_E^{p}(V)^{*} \rightarrow {\wedge}_E^{m-p}(V)\otimes_E E$ is defined via the pairing $${\wedge}_E^{p}(V) \otimes_E {\wedge}_E^{m-p}(V) \rightarrow {\wedge}_E^{m}(V)\cong E, \; x\otimes y\mapsto x\wedge y,$$ then we find that ${{\lambda}_{{\alpha}^k,{\beta}^k}}(g)=1$, if $g(\theta)=\theta$ or ${\alpha}^k={\beta}^k$ and ${{\lambda}_{{\alpha}^k,{\beta}^k}}(g)=(-1)^{p(m-p)}\cdot (g({\sigma}_k(disc(\Phi))))^{-P({\alpha}^k,{\beta}^k)}$ otherwise.\
Hence the matrix of $m(g)\in GL(W_{{i}'})$ is such that $$m(g)\cdot E_{i,j}\cdot {m(g)}^{-1}={\phi}_g (E_{i,j})= \left( \prod_k {{\lambda}_{{\alpha}^k,{\beta}^k}}(g) \right) E_{g(i),g(j)},$$ where $E_{i,j}$ denotes a matrix from $Mat(F)\cong End_{{\mathfrak{g}}\otimes_k F} (W_{{i}'})$ corresponding to $r_{({\alpha}^i),({\beta}^i)}$, i.e. conjugation by $m(g)$ acts on matrices upto a constant as the (same) permutation of columns and rows induced by $g$ on indices $({\alpha}^{i})$.\
Then the element of $H^2(S',F^{*})$ corresponding to the central division algebra $D_0=End_{{\mathfrak{g}}}(W_0)$ (over $C$) is the class of a $2$-cocycle $\lambda\colon S'\times S'\rightarrow F^{*}\cong F^{*}\cdot Id\subset Mat(F)$, $(g_1,g_2)\mapsto m(g_1g_2)\cdot (g_1 (m(g_2)))^{-1} \cdot m(g_1)^{-1}$ [@Kuznetsov], [@Jacobson].\
Example.
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Let $k=\mathbb Q$, $r=3$ and $5\leq m\leq 6$. Let $\rho < 0$ be the negative root of the cubic polynomial $f(t)=t^3-3t+1$. Then $\frac{1}{1-\rho}$ and $1-\frac{1}{\rho}$ are the other two roots of $f(t)$ and $E=L=k(\rho)$ is a totally real cyclic cubic Galois number field [@Kim].\
Let $\Phi=-\rho\cdot X_1^2-\rho\cdot X_2^2-X_3^2-...-X_m^2$. Then by [@Mayanskiy] there is a $K3$ surface $X$ such that $End_{Hdg}(V)\cong E$ (where $V$ is the $\mathbb Q$-lattice of transcendental cycles on $X$), $dim_{E}V=m$ and $\Phi\colon V\otimes_E V\rightarrow E$ is the quadratic form constructed in [@Zarhin].\
Let $F=k\left( \sqrt{\rho},\sqrt{\frac{1}{1-\rho}},\sqrt{1-\frac{1}{\rho}}\right)$ be our choice of a splitting field. Note that $L\subset F$ and $\sqrt{-1}=\sqrt{\rho}\cdot\sqrt{\frac{1}{1-\rho}}\cdot\sqrt{1-\frac{1}{\rho}}\in F$. Then $$S=Gal(F/k)\cong ({\mathbb Z}/2{\mathbb Z})^{\oplus 3}\rtimes {\mathbb Z}/3{\mathbb Z}$$ is a nonabelian extension of ${\mathbb Z}/3{\mathbb Z}\cong Gal(L/k)$ with generator $g$ by $({\mathbb Z}/2{\mathbb Z})^{\oplus 3}$ with generators $h_1,h_2,h_3$, where $g$ acts on the generators $(h_1,h_2,h_3)$ by the permutation $(123)$. We also denote by $g$ the element of $S$ such that $g(\sqrt{\rho})=\sqrt{\frac{1}{1-\rho}}$, $g\left(\sqrt{\frac{1}{1-\rho}}\right)=\sqrt{1-\frac{1}{\rho}}$, $g\left(\sqrt{1-\frac{1}{\rho}}\right)=\sqrt{\rho}$. We assume that each generator $h_i$, $1\leq i\leq 3$ multiplies by $-1$ the $i$-th square root among $\sqrt{\rho}, \sqrt{\frac{1}{1-\rho}}, \sqrt{1-\frac{1}{\rho}}$ and does not change the others and that $h_i {\mid}_{L}=Id$.\
There are $3$ field embeddings $L\hookrightarrow F$: ${\sigma}_1=Id$, ${\sigma}_2=g{\mid}_L$ and ${\sigma}_3=g^2{\mid}_L$. Then $\sqrt{{{\sigma}_1}(d_1)}=\sqrt{{{\sigma}_1}(d_2)}=\sqrt{-1}\cdot \sqrt{\rho}$, $\sqrt{{{\sigma}_2}(d_1)}=\sqrt{{{\sigma}_2}(d_2)}=\sqrt{-1}\cdot \sqrt{\frac{1}{1-\rho}}$, $\sqrt{{{\sigma}_3}(d_1)}=\sqrt{{{\sigma}_3}(d_2)}=\sqrt{-1}\cdot \sqrt{1-\frac{1}{\rho}}$, $\sqrt{{{\sigma}_1}(d_3)}=\sqrt{{{\sigma}_2}(d_3)}=\sqrt{{{\sigma}_3}(d_3)}=\sqrt{-1}$, $\sqrt{-{{\sigma}_i}(d_{m-j+1})}=1$ for any $i=1,2,3$, $1\leq j\leq l=[\frac{m}{2}]$. Hence ${\otimes_{L,{\sigma}_1}} {\Gamma}_1={\otimes_{L,{\sigma}_1}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{\rho}$, ${\otimes_{L,{\sigma}_2}} {\Gamma}_1={\otimes_{L,{\sigma}_2}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{\frac{1}{1-\rho}}$, ${\otimes_{L,{\sigma}_3}} {\Gamma}_1={\otimes_{L,{\sigma}_3}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{1-\frac{1}{\rho}}$, ${\otimes_{L,{\sigma}_i}} {\Gamma}_3=\sqrt{-1}$ for all $i$ (if $m=6$).\
(1) Let us consider first the case $m=5$. The root system is of type $B_2$: $R_0=\{ \pm {\epsilon}_p,\; \pm {\epsilon}_p \pm {\epsilon}_q \; \mid \; p,q=1,2 \}$ with basis $B_0=\{ {\epsilon}_1-{\epsilon}_2, {\epsilon}_2 \}$. Hence $B_i=\{ {\epsilon}_1 {\otimes_{L,{\sigma}_i} {\Gamma}_1}-{\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2}, {\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2} \}$, $1\leq i \leq 3$. The restriction of the spin representation of ${\mathfrak{so}}(\phi)\otimes_k F$ in $C^{+}(V\otimes_k F)$ to ${\mathfrak{g}}\otimes_k F=Res_{L/k}({\mathfrak{so}}(\Phi))\otimes_k F$ is isomorphic over $F$ to $2^8$ copies of the exterior tensor product ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$ of the irreducible spin representation of ${\mathfrak{so}}(\Phi)\otimes_L F$. Hence over $k=\mathbb Q$ the restriction of the spin representation of ${\mathfrak{so}}(\phi)$ in $C^{+}(V)$ to ${\mathfrak{g}}=Res_{L/k}({\mathfrak{so}}(\Phi))\subset {\mathfrak{so}}(\phi)$ is one single irreducible representation with multiplicity $\mu$ which splits over $F$ into $\frac{2^8}{\mu}$ copies of ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$: $C^{+}(V)\cong U^{\oplus \mu}$.\
In order to estimate $\frac{2^8}{\mu}$ (which divides $n_0$), let us consider $$f_{1,...,1,1}=f_1\cdot ... \cdot f_l \cdot (1+f_0)=q\cdot \prod_{i=1}^{l} \left( e_i+\frac{\sqrt{d_i}}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1} \right)\cdot \left( 1+\frac{1}{\sqrt{d_{l+1}}}\cdot e_{l+1} \right)$$ (we use notation as above), where $q\in F$ is such that ${\sigma}(q)=\pm q$ for any ${\sigma}\in S=Gal(F/k)$. In our case $$f_{1,...,1,1}=q\cdot (e_1+\sqrt{-1}\cdot \sqrt{\rho}\cdot e_5)\cdot (e_2+\sqrt{-1}\cdot \sqrt{\rho}\cdot e_4)\cdot (1-\sqrt{-1}\cdot e_3).$$ Hence the stabilizer of (the line in $C(V\otimes_L F)$ generated by) $f_{1,...,1,1}$ consists of the elements $g^k$, i.e. has order $3$. Since $Gal(F/k)$ has $24$ elements total, we find that $n_0=8$. Hence either $\frac{2^8}{\mu}=1$ or $\frac{2^8}{\mu}=2$ or $\frac{2^8}{\mu}=4$ or $\frac{2^8}{\mu}=8$. In the first case, ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$ is already defined over $\mathbb Q$ and $\mu=2^8$, while in the other cases $\mu=2^7, \mu=2^6$ and $\mu=2^5$ respectively.\
Hence in this case $End(KS(X))_{\mathbb Q}\cong Mat_{\mu\times \mu}(D)$, where $D=End_{{\mathfrak{g}}}(U)$ is a division algebra. Let us check that $D\cong \mathbb Q$.\
Let us compute the cohomological invariant of $D$. In our case $$W\otimes_k F=V_{(1,1,1)}\oplus V_{(1,1',1')}\oplus V_{(1',1,1')}\oplus V_{(1',1',1)}\oplus V_{(2',2,2)}\oplus V_{(2,2',2)}\oplus V_{(2,2,2')}\oplus V_{(2',2',2')},$$ where $V_{(p_1,p_2,p_3)}=S_{p_1}^1\otimes_F S_{p_2}^2\otimes_F S_{p_3}^3$ in the notation of Section 5.2 and the values $1,1',2,2'$ of $p_i$ correspond to the indices $({\alpha}_1,{\alpha}_2,\gamma)$ of ideals $I_{{\alpha}_1,{\alpha}_2,\gamma}$ as follows: $1=(+++)$, $1'=(--+)$, $2=(---)$, $2'=(++-)$.\
Let us denote $\bar{1}=2$, $\bar{1'}=2'$, $\bar{2}=1$, $\bar{2'}=1'$ and $\tilde{1}=2'$, $\tilde{1'}=2$, $\tilde{2}=1'$, $\tilde{2'}=1$. Then $g(V_{(p_1,p_2,p_3)})=V_{(p_3,p_1,p_2)}$ and $h_i(V_{(p_1,p_2,p_3)})= V_{(q_1,q_2,q_3)}$, where $q_i=\tilde{p_i}$ and $q_j=\bar{p_j}$ for $j\neq i$.\
Let us denote $a=(1,1',1')$, $b=(1',1,1')$, $c=(1',1',1)$, $d=(1,1,1)$, $p=(2',2,2)$, $q=(2,2',2)$, $r=(2,2,2')$, $s=(2',2',2')$. Then using formulas from Section 6 we can choose coefficients ${\lambda}_{\alpha,\beta}=\prod_{i=1}^r {\lambda}_{{\alpha}^i,{\beta}^i}\in F^{*}$ as follows:
- ${\lambda}_{\alpha,\beta}=1$ for $(\alpha,\beta)\in \{ (d,-), (s,-), (a,a), (b,b), (c,c), (p,p), (q,q), (r,r) \}$,
- ${\lambda}_{\alpha,\beta}=1$ for $(\alpha,\beta)\in \{ (b,q), (a,p), (c,r), (q,b), (p,a), (r,c) \}$,
- ${\lambda}_{\alpha,\beta}=c_1$ for $(\alpha,\beta)\in \{ (b,a), (b,p), (c,a), (c,p), (q,a), (q,p), (r,a), (r,p) \}$,
- ${\lambda}_{\alpha,\beta}=c_2$ for $(\alpha,\beta)\in \{ (a,b), (a,q), (c,b), (c,q), (p,b), (p,q), (r,b), (r,q) \}$,
- ${\lambda}_{\alpha,\beta}=c_3$ for $(\alpha,\beta)\in \{ (b,c), (b,r), (a,c), (a,r), (p,c), (p,r), (q,c), (q,r) \}$,
- ${\lambda}_{\alpha,\beta}=c_1c_2$ for $(\alpha,\beta)\in \{ (c,d), (c,s), (r,d), (r,s) \}$,
- ${\lambda}_{\alpha,\beta}=c_1c_3$ for $(\alpha,\beta)\in \{ (b,d), (b,s), (q,d), (q,s) \}$,
- ${\lambda}_{\alpha,\beta}=c_2c_3$ for $(\alpha,\beta)\in \{ (a,d), (a,s), (p,d), (p,s) \}$.
Here we denoted $c_i={\sigma}_i\left(\frac{-1}{\Phi(f_1,f_{-1})\cdot \Phi(f_2,f_{-2})}\right)={\sigma}_i\left(\frac{-1}{4{\rho}^2}\right)$.\
Then in the formulas in Section 6 we can take:
- $m(g)= \begin{pmatrix} G & 0\\ 0 & G \end{pmatrix}$ is an $8\times 8$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcspqr)$,
- $m(h_1)= \begin{pmatrix} 0 & X_1^{-1}\\ \frac{1}{c_2c_3}\cdot X_1 & 0 \end{pmatrix}$ is an $8\times 8$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcpsrq)$,
- $m(h_2)= \begin{pmatrix} 0 & X_2^{-1}\\ \frac{1}{c_1c_3}\cdot X_2 & 0 \end{pmatrix}$ is an $8\times 8$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcqrsp)$,
- $m(h_3)= \begin{pmatrix} 0 & X_3^{-1}\\ \frac{1}{c_1c_2}\cdot X_3 & 0 \end{pmatrix}$ is an $8\times 8$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcrqps)$,
- $m(g^k\cdot h_1^{a_1} h_2^{a_2} h_3^{a_3})=m(g)^k\cdot g^k \left( m(h_1)^{a_1}\cdot m(h_2)^{a_2}\cdot m(h_3)^{a_3} \right)$, where $0\leq a_i\leq 1$, $k\geq 0$.
Here we denoted $G=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}$, $X_1=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & c_2c_3 & 0 & 0\\ 0 & 0 & c_3 & 0\\ 0 & 0 & 0 & c_2 \end{pmatrix}$, $X_2=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & c_3 & 0 & 0\\ 0 & 0 & c_1c_3 & 0\\ 0 & 0 & 0 & c_1 \end{pmatrix}$ and $X_3=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & c_2 & 0 & 0\\ 0 & 0 & c_1 & 0\\ 0 & 0 & 0 & c_1c_2 \end{pmatrix}$.\
Note that $m(h_i)\cdot m(h_j)=m(h_j)\cdot m(h_i)$, $m(h_i)^2=\frac{c_i}{c_1c_2c_3}$, $m(g)^3=1$ and $m(gh_ig^{-1})=m(g)\cdot g(m(h_i))\cdot m(g)^{-1}$.\
This implies that the class of $D$ in $H^2(S,F^{*})$ is represented by the $2$-cocycle $\lambda \colon S\times S\rightarrow F^{*}$ such that ${\lambda}(h_1^{a_1}h_2^{a_2}h_3^{a_3},h_1^{b_1}h_2^{b_2}h_3^{b_3})=(c_2c_3)^{x_1}\cdot (c_1c_3)^{x_2}\cdot (c_1c_2)^{x_3}$ and ${\lambda}(g^kh,g^lh')=g^{k+l}({\lambda}(g^{-l}hg^l,h'))$, where $0\leq a_i\leq 1$, $0\leq b_i\leq 1$, $x_i=1$ if $a_i=b_i=1$ and $0$ otherwise, and $h, h'$ are elements of the subgroup $({\mathbb Z}/2{\mathbb Z})^{\oplus 3}\subset S$ generated by $h_1, h_2, h_3$.\
Since $c_ic_j=\left( \frac{1}{4\cdot {\sigma}_i(\rho){\sigma}_j(\rho)} \right)^2$ is a square in $L^{*}$, we conclude that $\lambda$ is a coboundary. Namely, the required morphism $c\colon S\rightarrow F^{*}$ (whose coboundary is $\lambda$) can be defined as follows: $$c(g^k\cdot h_1^{a_1} h_2^{a_2} h_3^{a_3})=g^k\left( (\sqrt{c_2c_3})^{a_1}\cdot (\sqrt{c_1c_3})^{a_2}\cdot (\sqrt{c_1c_2})^{a_3} \right),$$ where $0\leq a_i\leq 1$, $k\geq 0$. Note that $c(gh_ig^{-1})=g(c(h_i))$. So, the class of $D$ in $H^2(S,F^{*})$ vanishes. Hence $D\cong \mathbb Q$.\
So, in this example $End(KS(X))_{\mathbb Q}\cong Mat_{256\times 256}(\mathbb Q)$.\
(2) Now let us consider the case $m=6$. The root system is of type $D_3$: $R_0=\{ \pm {\epsilon}_p \pm {\epsilon}_q \; \mid \; p,q=1,2,3 \}$ with basis $B_0=\{ {\epsilon}_1-{\epsilon}_2, {\epsilon}_2-{\epsilon}_3, {\epsilon}_2+{\epsilon}_3 \}$. Hence $B_i=\{ {\epsilon}_1 {\otimes_{L,{\sigma}_i} {\Gamma}_1}-{\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2}, {\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2}-{\epsilon}_3 {\otimes_{L,{\sigma}_i} {\Gamma}_3}, {\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2}+{\epsilon}_3 {\otimes_{L,{\sigma}_i} {\Gamma}_3} \}$, $1\leq i \leq 3$, and the Weyl group is generated by sign inversions in front of two of ${\epsilon}_1, {\epsilon}_2, {\epsilon}_3$ and by all possible permutations of ${\epsilon}_1, {\epsilon}_2, {\epsilon}_3$.\
The restriction of the spin representation of ${\mathfrak{so}}(\phi)\otimes_k F$ in $C^{+}(V\otimes_k F)$ to ${\mathfrak{g}}\otimes_k F=Res_{L/k}({\mathfrak{so}}(\Phi))\otimes_k F$ is isomorphic over $F$ to the sum of the exterior tensor products of semi-spin representations (in all possible combinations) each with multiplicity $2^8$: $C^{+}(V\otimes_k F)\cong \bigoplus_{{\alpha}_1,{\alpha}_2,{\alpha}_3 \in \{ \pm \} } 2^8\cdot ({\rho}^{{\alpha}_1} \boxtimes {\rho}^{{\alpha}_2} \boxtimes {\rho}^{{\alpha}_3})$. Hence the set $\Omega$ of highest weights consists of the elements ${\omega}_{{\alpha}_1,{\alpha}_2,{\alpha}_3}=\frac{1}{2}\cdot \sum_{i=1}^{3}( {\epsilon}_1 {\otimes_{L,{\sigma}_i} {\Gamma}_1}+ {\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2} + {\alpha}_i\cdot {\epsilon}_3 {\otimes_{L,{\sigma}_i} {\Gamma}_3})$ for various ${\alpha}_i\in \{ \pm 1 \}$.\
Note that $g({\omega}_{{\alpha}_1,{\alpha}_2,{\alpha}_3})={\omega}_{{\alpha}_3,{\alpha}_1,{\alpha}_2}$ and $h_i({\omega}_{{\alpha}_1,{\alpha}_2,{\alpha}_3})={\omega}_{-{\alpha}_1,-{\alpha}_2,-{\alpha}_3}$. So, $\Omega={\Omega}_1\cup {\Omega}_2$ has two $S$-orbits: ${\Omega}_1=\{ {\omega}_{+,+,+},{\omega}_{-,-,-} \}$ and ${\Omega}_2=\{ {\omega}_{+,+,-},{\omega}_{+,-,+},{\omega}_{-,+,+},{\omega}_{-,-,+}, {\omega}_{-,+,-} , {\omega}_{+,-,-} \}$.\
Hence over $k=\mathbb Q$ we have: $C^{+}(V)\cong U^{\oplus \mu} \oplus V^{\oplus \nu}$ as ${\mathfrak{g}}$-modules, where $U$ and $V$ are not isomorphic as representations of ${\mathfrak{g}}=Res_{L/k}({\mathfrak{so}}(\Phi))$. $U\otimes_k F$ splits into $\frac{2^8}{\mu}$ copies of ${\rho}^{+} \boxtimes {\rho}^{+} \boxtimes {\rho}^{+}$ and $\frac{2^8}{\mu}$ copies of ${\rho}^{-} \boxtimes {\rho}^{-} \boxtimes {\rho}^{-}$, while $V\otimes_k F$ splits into $\frac{2^8}{\nu}$ copies of ${\rho}^{{\alpha}_1} \boxtimes {\rho}^{{\alpha}_2} \boxtimes {\rho}^{{\alpha}_3}$ with other ${{\alpha}_i}$’s.\
In order to estimate multiplicities $\mu$ and $\nu$, let us consider $$f_{1,...,1}=f_1\cdot ... \cdot f_l=q\cdot \prod_{i=1}^{l} \left( e_i+\frac{\sqrt{d_i}}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1} \right)$$ (we use notation introduced above), where $q\in F$ is such that ${\sigma}(q)=\pm q$ for any ${\sigma}\in S=Gal(F/k)$. In our case $$f_{1,...,1}=q\cdot (e_1+\sqrt{-1}\cdot \sqrt{\rho}\cdot e_6)\cdot (e_2+\sqrt{-1}\cdot \sqrt{\rho}\cdot e_5)\cdot (e_3+\sqrt{-1} \cdot e_4).$$ Hence the stabilizer of (the line in $C(V\otimes_L F)$ generated by) $f_{1,...,1}$ consists of the elements $g^k$. Since the stabilizer of ${\omega}_{+,+,+}\in\Omega$ as a subgroup of $S$ is generated by elements $g,h_1h_2,h_1h_3,h_2h_3$, we conclude that $n_{+,+,+}=4$. Since the stabilizer of ${\omega}_{+,+,-}\in\Omega$ has $4$ elements: $Id$ and $h_1h_2, h_1h_3, h_2h_3$, we conclude that $n_{+,+,-}=4$ as well. The same computation as in the case $m=5$ above shows that $\frac{2^8}{\mu}=\frac{2^8}{\nu}=1$, i.e. $\mu=\nu=256$, and the division algebras $D_1=End_{{\mathfrak{g}}}(U)$ and $D_2=End_{{\mathfrak{g}}}(V)$ are fields, i.e. coincide with their centers.\
According to Section 6.2, the center $C_1$ of $D_1$ is the subfield of $F$ fixed by the stabilizer of ${\omega}_{+,+,+}\in\Omega$, i.e. $D_1=C_1\cong k(\sqrt{-1})$. Similarly, the center $C_2$ of $D_2$ is the subfield of $F$ fixed by the stabilizer of ${\omega}_{+,+,-}\in\Omega$, i.e. $D_2=C_2\cong k(\sqrt{-1},\rho)$.\
So, in this example $End(KS(X))_{\mathbb Q}\cong Mat_{256\times 256}(\mathbb Q (\sqrt{-1})) \times Mat_{256\times 256}(\mathbb Q (\sqrt{-1},\rho) )$.\
\(3) Let us modify the first example above. Consider the same number $\rho$ and the same totally real cubic field $E=L=k(\rho)$, but a different quadratic form $$\Phi=-(a+\rho)\cdot X_1^2-(a+\rho)\cdot X_2^2-X_3^2-X_4^2-X_5^2,$$ where $a$ is a fixed rational number between $0$ and $-\rho$: $0<a<-\rho$. As above, these quadratic form and totally real field correspond to a $K3$ surface $X$ ([@Mayanskiy]). Assume that $1+3a-a^3>0$ is not a square of a rational number.\
Let $F=k\left(\sqrt{a+\rho},\sqrt{a+\frac{1}{1-\rho}},\sqrt{a+1-\frac{1}{\rho}},\sqrt{-1}\right)$ be our choice of a splitting field. Note that $L\subset F$ and $\sqrt{a+\rho}\cdot \sqrt{a+\frac{1}{1-\rho}}\cdot \sqrt{a+1-\frac{1}{\rho}}=\sqrt{-1-3a+a^3}$. Then $$S=Gal(F/k)\cong {\mathbb Z}/2{\mathbb Z}\oplus G,$$ where $G$ is the group isomorphic to the Galois group of the splitting field from the first example above, i.e. $G$ is a noncommutative group extension of ${\mathbb Z}/3{\mathbb Z}\cong Gal(L/k)$ by $({\mathbb Z}/2{\mathbb Z})^{\oplus 3}$. Let $g$ be a generator of ${\mathbb Z}/3{\mathbb Z}$ such that $g(\sqrt{a+\rho})=\sqrt{a+\frac{1}{1-\rho}}$, $g\left(\sqrt{a+\frac{1}{1-\rho}}\right)=\sqrt{a+1-\frac{1}{\rho}}$, $g\left(\sqrt{a+1-\frac{1}{\rho}}\right)=\sqrt{a+\rho}$, $g(\sqrt{-1})=\sqrt{-1}$. Let $h_1,h_2,h_3$ be the generators of $({\mathbb Z}/2{\mathbb Z})^{\oplus 3}$ and $h_0$ be the generator of the first factor ${\mathbb Z}/2{\mathbb Z}$ in $S$ above such that each $h_i$, $0\leq i\leq 3$ multiplies by $-1$ the $i$-th square root among $\sqrt{-1}, \sqrt{a+\rho}, \sqrt{a+\frac{1}{1-\rho}}, \sqrt{a+1-\frac{1}{\rho}}$ and does not change the others. We also assume that $h_i {\mid}_{L}=Id$, $0\leq i\leq 3$.\
There are $3$ field embeddings $L\hookrightarrow F$: ${\sigma}_1=Id$, ${\sigma}_2=g{\mid}_L$ and ${\sigma}_3=g^2{\mid}_L$. Then $\sqrt{{{\sigma}_1}(d_1)}=\sqrt{{{\sigma}_1}(d_2)}=\sqrt{-1}\cdot \sqrt{a+\rho}$, $\sqrt{{{\sigma}_2}(d_1)}=\sqrt{{{\sigma}_2}(d_2)}=\sqrt{-1}\cdot \sqrt{a+\frac{1}{1-\rho}}$, $\sqrt{{{\sigma}_3}(d_1)}=\sqrt{{{\sigma}_3}(d_2)}=\sqrt{-1}\cdot \sqrt{a+1-\frac{1}{\rho}}$, $\sqrt{{{\sigma}_1}(d_3)}=\sqrt{{{\sigma}_2}(d_3)}=\sqrt{{{\sigma}_3}(d_3)}=\sqrt{-1}$, $\sqrt{-{{\sigma}_i}(d_4)}=\sqrt{-{{\sigma}_i}(d_5)}=1$ for any $i=1,2,3$. Hence ${\otimes_{L,{\sigma}_1}} {\Gamma}_1={\otimes_{L,{\sigma}_1}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{a+\rho}$, ${\otimes_{L,{\sigma}_2}} {\Gamma}_1={\otimes_{L,{\sigma}_2}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{a+\frac{1}{1-\rho}}$, ${\otimes_{L,{\sigma}_3}} {\Gamma}_1={\otimes_{L,{\sigma}_3}} {\Gamma}_2=\sqrt{-1}\cdot \sqrt{a+1-\frac{1}{\rho}}$.\
As in the first example above, the root system is of type $B_2$: $R_0=\{ \pm {\epsilon}_p,\; \pm {\epsilon}_p \pm {\epsilon}_q \; \mid \; p,q=1,2 \}$ with basis $B_0=\{ {\epsilon}_1-{\epsilon}_2, {\epsilon}_2 \}$. Hence $B_i=\{ {\epsilon}_1 {\otimes_{L,{\sigma}_i} {\Gamma}_1}-{\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2}, {\epsilon}_2 {\otimes_{L,{\sigma}_i} {\Gamma}_2} \}$, $1\leq i \leq 3$. The restriction of the spin representation of ${\mathfrak{so}}(\phi)\otimes_k F$ in $C^{+}(V\otimes_k F)$ to ${\mathfrak{g}}\otimes_k F=Res_{L/k}({\mathfrak{so}}(\Phi))\otimes_k F$ is isomorphic over $F$ to $2^8$ copies of the exterior tensor product ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$ of the irreducible spin representation of ${\mathfrak{so}}(\Phi)\otimes_L F$. Hence over $k=\mathbb Q$ the restriction of the spin representation of ${\mathfrak{so}}(\phi)$ in $C^{+}(V)$ to ${\mathfrak{g}}=Res_{L/k}({\mathfrak{so}}(\Phi))\subset {\mathfrak{so}}(\phi)$ is one single irreducible representation with multiplicity $\mu$ which splits over $F$ into $\frac{2^8}{\mu}$ copies of ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$: $C^{+}(V)\cong U^{\oplus \mu}$.\
In order to estimate $\frac{2^8}{\mu}$ (which divides $n_0$), let us consider $$f_{1,...,1,1}=f_1\cdot ... \cdot f_l \cdot (1+f_0)=q\cdot \prod_{i=1}^{l} \left( e_i+\frac{\sqrt{d_i}}{\sqrt{-d_{m-i+1}}}\cdot e_{m-i+1} \right)\cdot \left( 1+\frac{1}{\sqrt{d_{l+1}}}\cdot e_{l+1} \right)$$ (we use notation as above), where $q\in F$ is such that ${\sigma}(q)=\pm q$ for any ${\sigma}\in S=Gal(F/k)$. In our case $$f_{1,...,1,1}=q\cdot (e_1+\sqrt{-1}\cdot \sqrt{a+\rho}\cdot e_5)\cdot (e_2+\sqrt{-1}\cdot \sqrt{a+\rho}\cdot e_4)\cdot (1-\sqrt{-1}\cdot e_3).$$ Hence the stabilizer of (the line in $C(V\otimes_L F)$ generated by) $f_{1,...,1,1}$ consists of the elements $g^k$, i.e. has order $3$. Since $Gal(F/k)$ has $48$ elements total, we find that $n_0=16$. Hence either $\frac{2^8}{\mu}=1$ or $\frac{2^8}{\mu}=2$ or $\frac{2^8}{\mu}=4$ or $\frac{2^8}{\mu}=8$ or $\frac{2^8}{\mu}=16$. In the first case, ${\rho}^0 \boxtimes {\rho}^0 \boxtimes {\rho}^0$ is already defined over $\mathbb Q$ and $\mu=2^8$, while in the other cases $\mu=2^7, \mu=2^6$, $\mu=2^5$ and $\mu=2^4$ respectively.\
Hence in this case $End(KS(X))_{\mathbb Q}\cong Mat_{\mu\times \mu}(D)$, where $D=End_{{\mathfrak{g}}}(U)$ is a division algebra. Let us compute the cohomological invariant of $D$. In our case $$\begin{gathered}
W\otimes_k F=V_{(1,1,1)}\oplus V_{(1',1,1)}\oplus V_{(1,1',1)}\oplus V_{(1,1,1')}\oplus V_{(1',1',1')}\oplus V_{(1,1',1')}\oplus V_{(1',1,1')}\oplus V_{(1',1',1)}\oplus \\
\oplus V_{(2,2,2)}\oplus V_{(2',2,2)}\oplus V_{(2,2',2)}\oplus V_{(2,2,2')}\oplus V_{(2',2',2')}\oplus V_{(2,2',2')}\oplus V_{(2',2,2')}\oplus V_{(2',2',2)},\end{gathered}$$ where $V_{(p_1,p_2,p_3)}=S_{p_1}^1\otimes_F S_{p_2}^2\otimes_F S_{p_3}^3$ in the notation of Section 5.2 and the values $1,1',2,2'$ of $p_i$ correspond to the indices $({\alpha}_1,{\alpha}_2,\gamma)$ of ideals $I_{{\alpha}_1,{\alpha}_2,\gamma}$ as follows: $1=(+++)$, $1'=(--+)$, $2=(++-)$, $2'=(---)$.\
Let us denote $\bar{1}=1'$, $\bar{1'}=1$, $\bar{2}=2'$, $\bar{2'}=2$ and $\tilde{1}=2'$, $\tilde{1'}=2$, $\tilde{2}=1'$, $\tilde{2'}=1$. Then $g(V_{(p_1,p_2,p_3)})=V_{(p_3,p_1,p_2)}$, $h_0(V_{(p_1,p_2,p_3)})= V_{(\tilde{p_1},\tilde{p_2},\tilde{p_3})}$ and $h_i(V_{(p_1,p_2,p_3)})= V_{(q_1,q_2,q_3)}$, $1\leq i\leq 3$, where $q_i=\bar{p_i}$ and $q_j={p_j}$ for $j\neq i$.\
Let us denote $a=(1',1,1)$, $b=(1,1',1)$, $c=(1,1,1')$, $d=(1,1,1)$, $a'=(1,1',1')$, $b'=(1',1,1')$, $c'=(1',1',1)$, $d'=(1',1',1')$, $p=(2',2,2)$, $q=(2,2',2)$, $r=(2,2,2')$, $s=(2,2,2)$, $p'=(2,2',2')$, $q'=(2',2,2')$, $r'=(2',2',2)$, $s'=(2',2',2')$. Consider the set of indices $T=\{ d,a,b,c,d',a',b',c',s,p,q,r,s',p',q',r' \}$ and the morphism $t\colon T\rightarrow T, s\mapsto d, p\mapsto a, q\mapsto b, r\mapsto c, s'\mapsto d', p'\mapsto a', q'\mapsto b', r'\mapsto c'$ and $x\mapsto x$ for all other $x\in T$.\
Then using formulas from Section 6 we can choose coefficients ${\lambda}_{\alpha,\beta}=\prod_{i=1}^r {\lambda}_{{\alpha}^i,{\beta}^i}\in F^{*}$ as follows:
- ${\lambda}_{d,x}={\lambda}_{s,x}={\lambda}_{x,x}=1$, ${\lambda}_{d',d}=c_1c_2c_3$ and ${\lambda}_{x,y}={\lambda}_{t(x),t(y)}$ for any $x,y\in T$,
- ${\lambda}_{\alpha,\beta}=1$ for $(\alpha,\beta)\in \{ (a,d'), (a,b'), (a,c'), (b,d'), (b,a'), (b,c') \}$,
- ${\lambda}_{\alpha,\beta}=1$ for $(\alpha,\beta)\in \{ (c,d'), (c,b'), (c,a'), (a',d'), (b',d'), (c',d') \}$,
- ${\lambda}_{\alpha,\beta}=c_1$ for $(\alpha,\beta)\in \{ (a,a'), (a,d), (a,b), (a,c), (d',a'), (b',c), (b',a'), (c',b), (c',a') \}$,
- ${\lambda}_{\alpha,\beta}=c_2$ for $(\alpha,\beta)\in \{ (b,b'), (b,d), (b,a), (b,c), (d',b'), (a',c), (a',b'), (c',a), (c',b') \}$,
- ${\lambda}_{\alpha,\beta}=c_3$ for $(\alpha,\beta)\in \{ (c,c'), (c,d), (c,b), (c,a), (d',c'), (a',b), (a',c'), (b',a), (b',c') \}$,
- ${\lambda}_{\alpha,\beta}=c_1c_2$ for $(\alpha,\beta)\in \{ (d',c), (c',c), (c',d) \}$,
- ${\lambda}_{\alpha,\beta}=c_1c_3$ for $(\alpha,\beta)\in \{ (d',b), (b',b), (b',d) \}$,
- ${\lambda}_{\alpha,\beta}=c_2c_3$ for $(\alpha,\beta)\in \{ (d',a), (a',a), (a',d) \}$.
Here we denoted $c_i={\sigma}_i\left(\frac{-1}{\Phi(f_1,f_{-1})\cdot \Phi(f_2,f_{-2})}\right)={\sigma}_i\left(\frac{-1}{4(a+{\rho})^2}\right)$.\
Then in the formulas in Section 6 we can take:
- $m(g)= \begin{pmatrix} G & 0 & 0 & 0\\ 0 & G & 0 & 0\\ 0 & 0 & G & 0\\ 0 & 0 & 0 & G \end{pmatrix}$ is a $16\times 16$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcd'a'b'c'spqrs'p'q'r')$,
- $m(h_1)= \begin{pmatrix} 0 & 1 & 0 & 0\\ \frac{1}{c_1}\cdot 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & \frac{1}{c_1}\cdot 1 & 0 \end{pmatrix}$ is a $16\times 16$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (da'bcad'c'b'sp'qrps'r'q')$,
- $m(h_2)= \begin{pmatrix} 0 & 1 & 0 & 0\\ \frac{1}{c_2}\cdot 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & \frac{1}{c_2}\cdot 1 & 0 \end{pmatrix}$ is a $16\times 16$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dab'cbc'd'a'spq'rqr's'p')$,
- $m(h_3)= \begin{pmatrix} 0 & 1 & 0 & 0\\ \frac{1}{c_3}\cdot 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & \frac{1}{c_3}\cdot 1 & 0 \end{pmatrix}$ is a $16\times 16$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabc'cb'a'd'spqr'rq'p's')$,
- $m(h_0)= \begin{pmatrix} 0 & X_0^{-1} & 0 & 0\\ \frac{1}{c_1c_2c_3}\cdot X_0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{c_1c_2c_3}\cdot X_0\\ 0 & 0 & X_0^{-1} & 0 \end{pmatrix}$ is a $16\times 16$ matrix whose rows and columns are numbered according to the following sequence of indices of $V_{(p_1,p_2,p_3)}: \; (dabcs'p'q'r'd'a'b'c'spqr)$,
- $m(g^k\cdot h_0^{a_0} h_1^{a_1} h_2^{a_2} h_3^{a_3})=m(g)^k\cdot g^k \left( m(h_0)^{a_0}\cdot m(h_1)^{a_1}\cdot m(h_2)^{a_2}\cdot m(h_3)^{a_3} \right)$, where $0\leq a_i\leq 1$, $k\geq 0$.
Here we denoted $G=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}$, $1=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$ (in the definitions of $m(h_i)$) and $X_0=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & c_1 & 0 & 0\\ 0 & 0 & c_2 & 0\\ 0 & 0 & 0 & c_3 \end{pmatrix}$.\
Note that $m(h_i)\cdot m(h_j)=m(h_j)\cdot m(h_i)$, $m(h_i)^2=\frac{1}{c_i}$, $1\leq i\leq 3$, $m(h_0)^2=\frac{1}{c_1c_2c_3}$, $m(g)^3=1$ and $m(gh_ig^{-1})=m(g)\cdot g(m(h_i))\cdot m(g)^{-1}$.\
This implies that the class of $D$ in $H^2(S,F^{*})$ is represented by the $2$-cocycle $\lambda \colon S\times S\rightarrow F^{*}$ such that $${\lambda}(h_0^{a_0}h_1^{a_1}h_2^{a_2}h_3^{a_3},h_0^{b_0}h_1^{b_1}h_2^{b_2}h_3^{b_3})=(c_1c_2c_3)^{x_0}\cdot (c_2c_3)^{x_1}\cdot (c_1c_3)^{x_2}\cdot (c_1c_2)^{x_3}$$ and ${\lambda}(g^kh,g^lh')=g^{k+l}({\lambda}(g^{-l}hg^l,h'))$, where $0\leq a_i\leq 1$, $0\leq b_i\leq 1$, $x_i=1$ if $a_i=b_i=1$ and $0$ otherwise, and $h, h'$ are elements of the subgroup ${\mathbb Z}/2{\mathbb Z}\oplus({\mathbb Z}/2{\mathbb Z})^{\oplus 3}\subset S$ generated by $h_0,h_1, h_2, h_3$.\
Let us multiply $\lambda$ by the inverse of the coboundary of the 1-cochain given by the morphism $c\colon S\rightarrow F^{*}$ such that $$c(g^k\cdot h_0^{a_0} h_1^{a_1} h_2^{a_2} h_3^{a_3})=g^k\left( (\sqrt{c_1c_2c_3})^{a_0}\cdot (\sqrt{c_1})^{a_1}\cdot (\sqrt{c_2})^{a_2}\cdot (\sqrt{c_3})^{a_3} \right),$$ where $0\leq a_i\leq 1$, $k\geq 0$. Note that $c(gh_ig^{-1})=g(c(h_i))$.\
This changes $\lambda$ to a 2-cocycle ${\lambda}' \colon S\times S\rightarrow F^{*}$ such that $${\lambda}'(g^k\cdot h_0^{a_0}h_1^{a_1}h_2^{a_2}h_3^{a_3},g^l\cdot h_0^{b_0}h_1^{b_1}h_2^{b_2}h_3^{b_3})=(-1)^{a_0\cdot (b_0+b_1+b_2+b_3)},$$ where $0\leq a_i\leq 1$, $0\leq b_i\leq 1$.\
Let $H\subset S$ be the subgroup generated by $g, h_1h_2, h_1h_3, h_2h_3$ and $$F^H=k\left(\sqrt{-1},\sqrt{(a+\rho)(a+\frac{1}{1-\rho})(a+1-\frac{1}{\rho})} \right)=k(\sqrt{-1},\sqrt{-1-3a+a^3})$$ be the corresponding fixed subfield of $F$. Denote the generators of $Gal(F^H/k)\cong {\mathbb Z}/2{\mathbb Z}\oplus {\mathbb Z}/2{\mathbb Z}$ by $h_0$ and $h=h_1h_2h_3$.\
We see that the class of $D$ in $H^2(S,F^{*})$ is the image under the inflation homomorphism $H^2(Gal(F^H/k),(F^H)^{*})\rightarrow H^2(S,F^{*})$ of a class represented by the $2$-cocycle ${\lambda}''\colon Gal(F^H/k)\times Gal(F^H/k) \rightarrow k(\sqrt{-1},\sqrt{-1-3a+a^3})^{*}$ such that ${\lambda}''(h_0,h_0)={\lambda}''(h_0,h)=-{\lambda}''(h,h_0)=-{\lambda}''(h,h)=-1$. Multiplying it by the coboundary of the 1-cochain given by the morphism $c\colon Gal(F^H/k)\rightarrow (F^H)^{*}$ such that $c(h)=c(h_0)=\sqrt{-1}$, $c(hh_0)=1$, we obtain a 2-cocycle (also denoted by ${\lambda}''$) with the property ${\lambda}''(h_0h,-)={\lambda}''(-,h_0h)=1$ and ${\lambda}''(h_0,h_0)=1$. Note that $(F^H)^{<h_0h>}=k(\sqrt{1+3a-a^3})$ is a totally real quadratic field with Galois group ${\mathbb Z}/2{\mathbb Z}$ with generator $1$.\
This means that the cohomological class of $D$ can be obtained via the inflation homomorphism from the class in $H^2(Gal(k(\sqrt{1+3a-a^3})/k), k(\sqrt{1+3a-a^3})^{*})$ of the $2$-cocycle ${\lambda}_0\colon {\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z} \rightarrow k(\sqrt{1+3a-a^3})^{*}$ such that ${\lambda}_0(1,1)=-1$.\
Hence $D$ is a quaternion algebra over $\mathbb Q=k$ of degree $deg(D)=2$ split over ${\mathbb Q}(\sqrt{1+3a-a^3})$ with $4$ generators over $\mathbb Q$: $1,i,j,k$ such that $i^2=j^2=1+3a-a^3$, $k=ij=-ji$. In other words, $D=(1+3a-a^3,1+3a-a^3)_{\mathbb Q}$.\
So, in this example $End(KS(X))_{\mathbb Q}\cong Mat_{128\times 128}((1+3a-a^3,1+3a-a^3)_{\mathbb Q})$.\
\(4) If in the previous example we take $$\Phi=-(b\cdot\rho)\cdot X_1^2-(b\cdot\rho)\cdot X_2^2-X_3^2-X_4^2-X_5^2,$$ where $b> 0$ is a rational number which is not a square of another rational number, then the same computation as above gives: $$End(KS(X))_{\mathbb Q}\cong Mat_{128\times 128}((b,b)_{\mathbb Q})$$ for the corresponding $K3$ surface $X$.\
Acknowledgement.
================
We thank Yuri Zarhin for suggesting this problem and for pointing out an error in the Example section of the previous version. Many of our constructions were influenced by papers [@vanGeemen], [@vanGeemen1] and [@vanGeemen2], where Bert van Geemen studies endomorphism algebras of Kuga-Satake varieties in more special cases.\
|
[**An Appraisal of Muon Neutrino Disappearance\
at Short Baseline**]{} L. Stanco$^1$, S. Dusini$^1$, A. Longhin$^2$, A. Bertolin$^1$, M. Laveder$^3$
[*$^1$INFN-Padova, Via Marzolo, 8, I-35131 Padova, Italy,\
$^2$Laboratori Nazionali di Frascati, INFN, Via E. Fermi 40, I-00044 Frascati, Italy,\
$^3$Padova University and INFN-Padova, Via Marzolo, 8, I-35131 Padova, Italy*]{}
[**Abstract**]{}
Neutrino physics is nowadays receiving more and more attention as a possible source of information for the long–standing problem of new physics beyond the Standard Model. The recent measurement of the third mixing angle $\theta_{13}$ in the standard mixing oscillation scenario encourages us to pursue the still missing results on leptonic CP violation and absolute neutrino masses. However, several puzzling measurements exist which deserve an exhaustive evaluation. We will illustrate the present status of the muon disappearance measurements at small $L/E$ and the current CERN project to revitalize the neutrino field in Europe with emphasis on the search for sterile neutrinos. We will then illustrate the achievements that a double muon spectrometer can make with regards to discovery of new neutrino states, using a newly developed analysis.
[*To be published in “Advances in High Energy Physics”.*]{}
Introduction
============
The unfolding of the physics of the neutrino is a long and exciting story spanning the last 80 years. Over this time the interchange of theoretical hypotheses and experimental facts has been one of the most fruitful demonstrations of the progress of knowledge in physics. The work of the last decade and a half finally brought a coherent picture within the Standard Model (SM) (or some small extensions of it), namely the mixing of three neutrino flavour states with three $\nu_1$, $\nu_2$ and $\nu_3$ mass eigenstates. The last unknown mixing angle, $\theta_{13}$, was recently measured [@theta13-DB; @theta13-RE; @theta13-DC; @theta13-T2] but still many questions remain unanswered to completely settle the scenario: the absolute masses, the Majorana/Dirac nature and the existence and magnitude of leptonic CP violation. Answers to these questions will beautifully complete the (standard) three–neutrino model but they will hardly provide an insight into new physics Beyond the Standard Model (BSM). Many relevant questions will stay open: the reason for neutrinos, the relation between the leptonic and hadronic sectors of the SM, the origin of Dark Matter and, overall, where and how to look for BSM physics. Neutrinos may be an excellent source of BSM physics and their story is supporting that at length.
There are actually several experimental hints for deviations from the “coherent” picture described above. Many unexpected results, not statistically significant on a single basis, appeared also in the last decade and a half, bringing attention to the hypothesis of [*sterile neutrinos*]{} [@pontecorvo]. A recent White Paper [@whitepaper] contains a comprehensive review of these issues. In this paper we will focus on one of the most intriguing and long–standing unresolved result: the unexpected oscillation of neutrinos at relatively small values of the ratio $L/E$ (distance in km, energy in GeV), corresponding to a scale of $\mathcal{O}$(1) eV$^2$, incompatible with the much smaller values related to the atmospheric $|\Delta m_{32}^2|\simeq 2.4 \times 10^{-3}$ eV$^2$ and to the solar $\Delta m_{21}^2 \simeq 8 \times 10^{-5}$ eV$^2$ scales.
The first unexpected measurement came from an excess of $\overline\nu_e$ originating from an initial $\overline\nu_\mu$ beam from Decay At Rest (LNSD [@lsnd]). The LSND experiment saw a 3.8 $\sigma$ effect. The subsequent experiment with $\nu_{\mu}$ ($\overline\nu_\mu$) beam from accelerator, MiniBooNE [@larnessie_5], although confirming an independent 3.8 $\sigma$ effect after sustained experimental work, was unable to draw conclusive results on the origin of the LSND effect having observed an excess at higher $L/E$ in an energy region where background is high.
In recent years many phenomenological studies were performed by analyzing the LSND effect together with similar unexpected results coming from the measurement of lower than expected rates of $\overline{\nu}_e$ and $\nu_e$ interactions ([*disappearance*]{}), either from [**(a)**]{} near-by nuclear reactors ($\overline\nu_e$) [@reattori] or [**(b)**]{} from Mega-Curie K-capture calibration sources in the solar $\nu_e$ Gallium experiments [@larnessie_4]. These $\nu_e$ ($\overline\nu_e$) disappearance measurements, all at the statistical level of 3-4 $\sigma$, could also be interpreted [@giunti-laveder] as oscillations between neutrinos at large $\Delta m^2\simeq 1$ eV$^2$. Several attempts were then tried to reach a coherent picture in terms of mixing between active and sterile neutrinos, in $3+1$ and $3+2$ [@larnessie-6] or even $3+1+1$ [@treunouno] or $3+3$ [@tretre] models, as extensions of the standard three–neutrino model. We refer to [@models; @models1; @giuntinew] as the most recent and industrious works where a very crucial issue is raised: “[*a consistent interpretation of the global data in terms of neutrino oscillations is challenged by the non-observation of a positive signal in $\nu_{\mu}$ disappearance experiments*]{}” [@models1]. In fact, in any of the above models, essential information comes from the disappearance channels ($\nu_{\mu}$ or $\nu_e$), which is one of the cleanest channels to measure the oscillation parameters (see Sect. \[sect-3\] for details).
The presence of additional sterile states introduces quite naturally appearance and disappearance phenomena involving the flavour states in all channels. In particular, a $\nu_\mu$ disappearance effect has to be present and possibly measured. It turns out that only old experiments and measurements are available for Charged Current (CC) $\nu_\mu$ interactions at small $L/E$ [@CDHS]. The CDHS experiment reported in 1984 the non–observation of $\nu_\mu$ oscillations in the $\Delta m^2$ range $0.3$ eV$^2$ to $90$ eV$^2$. Their analyzed region of oscillation did not span however low values of the mixing parameter down to around $0.1$ in $\sin^2(2\theta)$. More recent results are available on $\nu_\mu$ disappearance from MiniBooNE [@mini-mu], a joint MiniBooNE/SciBooNE analysis [@mini-sci-mu1; @mini-sci-mu2] and the Long–Baseline MINOS experiment [@minos]. These results slightly extend the $\nu_\mu$ disappearance exclusion region, however still leaving out the small–mixing region. Similar additional constraints on $\nu_{\mu}$ disappearance could possibly come from the analysis of atmospheric neutrinos in IceCube [@icecube].
Despite this set of measurements being rather unsatisfactory when compared with the corresponding LSND allowed region that lies at somewhat lower values of the mixing angle, they are still sufficient to introduce tensions in all the phenomenological models developed so far (see e.g. [@whitepaper; @models; @models1; @giuntinew; @reviews] for comprehensive and recent reviews). Therefore it is mandatory to setup a new experiment able to improve the small–mixing angle region exclusion by at least one order of magnitude with respect to the current results. In such a way one could also rule out the idea that the mixing angle extracted from LSND is larger than the true value due to a data over–fluctuation. Once again, the $\nu_\mu$ disappearance channel should be the optimal one to perform a full disentangling of the mechanism given the strong tension between the $\nu_e$ appearance and $\nu_\mu$ disappearance around $\Delta m^2\simeq 1$ eV$^2$. In fact, whereas the LSND effect may be confirmed by a more accurate $\nu_e$ oscillation measurement, only the presence of a $\nu_\mu$ oscillation pattern could shed more light on the nature and the interpretation of the effect.
Further, in the paper we will briefly discuss the newly proposed CERN experiment, following a detailed analysis of possible ways to measure the $\nu_\mu$ disappearance channel. In particular the evaluation of the $\nu_\mu$ disappearance rates at two different sites leads us naturally to take their ratio, elucidating the possibility to observe depletions or/and excesses. Throughout the paper we will focus on the need for a new $\nu_\mu$ CC measurement corresponding to an increase of one order of magnitude in the sensitivity to the mixing parameter, exploiting the determination of the muon charge for a proper evaluation of the different models and the separation of $\nu_{\mu}$ and $\overline{\nu}_\mu$ in the same neutrino beam. Finally, we will pay particular attention to the consistency and robustness (in statistical terms) of the results.
The CERN experimental proposal
==============================
The need for a definitive clarification on the possible existence of a neutrino mass scale around 1 eV has brought up several proposals and experimental suggestions exploiting the sterile neutrino option by using different interaction channels and refurbished experiments. In the light of the considerations discussed in the previous section there are essentially two sets of experiments which must be redone: a) the measurement of $\overline\nu_{e}$ neutrino fluxes at reactors (primarily the ILL one) together with refined and detailed computations of the flux simulations (see e.g. [@huber]); b) the appearance/disappearance oscillation measurements at low $L/E$ with a standard muon neutrino beam with its intrinsic electron neutrino component.
There is actually another interesting option which comes from the Neutrino Factory studies and the very recently submitted EOI from $\nu$STORM [@nustorm]. We refer to [@winter] for a comprehensive review of the corresponding possible $\nu_e$ and $\nu_{\mu}$ disappearance effects. It is interesting to note that our figures of merit about $\nu_{\mu}$ disappearance are rather similar to or even slightly more competitive than those illustrated in [@winter] (e.g compare the exclusion regions in Fig. 6 of [@winter] and those in Fig. \[ster-5\] of this paper despite the use of different C.L.), not forgetting the rather long time needed to setup the $\nu$STORM project.
Coming to experiments with standard beams, investigations are underway at CERN where two Physics Proposals [@nessie; @icarus] were submitted in October 2011 and later merged into a single Technical Proposal (ICARUS-NESSiE, [@larnessie]). CERN has subsequently set up working groups for the proton beam extraction from the SPS, the secondary beam line and the needed infrastructure/buildings for the detectors. The work was reported in a recent LOI [@edms].
The experiment is based on two identical Liquid Argon (LAr)–Time Projection Chambers (TPC) [@icarus] complemented by magnetized spectrometers [@nessie] detecting electron and muon neutrino events at far and near positions, 1600 m and 460 m away from the proton target, respectively. The project will exploit the ICARUS T600 detector, the largest LAr-TPC ever built of about 600 ton mass, now presently in the LNGS underground laboratory where it was exposed to the CNGS beam. It is supposed to be moved at the CERN ÒfarÓ position. An additional 1/4 of the T600 detector (T150) would be constructed from scratch as a clone of the original one, except for the dimensions, and located in the near site. Two spectrometers would be placed downstream of the two LAr-TPC detectors to greatly enhance the physics reach. The spectrometers will exploit a bipolar magnet with instrumented iron slabs, and a newly designed air–core magnet, to perform charge identification and muon momentum measurements in an extended energy range (from 0.5 GeV or less to 10 GeV), over a transverse area larger than 50 m$^2$.
While the LAr-TPCs will mainly perform a direct measurement of electron neutrinos [@rubbia] the spectrometers will allow an extended exploitation of the muon neutrino component, with neutrino/antineutrino discrimination on an event-by-event basis.
The neutrino beam
-----------------
The proposed new neutrino beam will be constructed in the SPS North Area [@edms]. The setup is based on a 100 GeV proton beam with a fast extraction scheme providing about 3.5 $\cdot$ 10$^{13}$ protons/pulse in two pulses of 10.5 $\mu$s durations[^1] separated by 50 ms for a sample of about 4.5 $\cdot$ 10$^{19}$ protons on target (p.o.t.) per year. A target station will be located next to the so called TCC2 target zone, 11 m underground, followed by a cylindrical He-filled decay pipe with a length of about 110 m and a diameter of 3 m. The beam dump of 15 m in length, will be composed of iron blocks with a graphite inner core. Downstream of the beam dump a set of muon chambers stations will act as beam monitors. The beam will point upward, with a slope of about 5 mrad, resulting in a depth of 3 m for the detectors in the far site.
The current design of the focusing optics includes a pair of pulsed magnetic horns operated at relatively low currents. A graphite target of about 1 m in length is deeply inserted into the first horn allowing a large acceptance for the focusing of low momentum pions emitted at large angles. This design allows production of a spectrum peaking at about 2 GeV thus matching the most interesting domain of $\Delta
m^2$ with the detector locations at 460 and 1600 m from the target.
The charged current event rates for $\nu_\mu$ and $\bar{\nu}_\mu$ at the near and far detectors are shown in Fig. \[beam-AL\] for the positive and negative focusing configuration.
![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol1_EGeV_NEAR "fig:"){width="53.00000%"} ![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol-1_EGeV_NEAR "fig:"){width="53.00000%"} ![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol1_EGeV_FAR "fig:"){width="53.00000%"} ![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol-1_EGeV_FAR "fig:"){width="53.00000%"} ![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol1_NEAR "fig:"){width="53.00000%"} ![Expected neutrino CC interactions in the no–oscillation hypothesis for positive polarity (left) and negative polarity (right) for the new proposed CERN neutrino beam (elaborated from [@edms]) and for an integrated luminosity of 1 year. In the first (second) row the rates are shown at the near (far) position. In the third row the rates expected at the near site are displayed as function of $\log_{10}(1/E)$, used in the paper (see Sect. \[sect-3\]). Note that the distributions of the antineutrino rates in case of positive polarity are multiplied by a factor 5 to allow a better visual inspection.[]{data-label="beam-AL"}](Rates_pol-1_NEAR "fig:"){width="53.00000%"}
A relevant contamination of $\nu_\mu$ in the negative polarity configuration is visible especially at high energy. This component arises as a result of the decays of high energy poorly de–focused mesons produced at small angles. The charge discrimination of the magnetic system described below will allow an efficient discrimination of these two components with a charge confusion below or of the order of 1% from sub–GeV (0.3–0.5 GeV) up to momenta around 8–10 GeV [@nessie].
Spectrometer requirements
-------------------------
The main purpose of the spectrometers placed downstream of the LAr-TPC is to provide charge identification and momentum reconstruction for the muons produced in neutrino interactions occurring in the LAr volume or in the magnetized iron of the spectrometers. In order to perform this measurement with high precision and in a wide energy range, from sub-GeV to multi-GeV, a massive iron-core dipole magnet (ICM) is coupled to an air–core magnet (ACM) in front of it [@nessie]. Low momentum muons will be measured by the ACM while the ICM will be employed at higher momenta.
As considered in the previous sections the definition of two sites, near and far, constitutes a fundamental issue for each physics program which aims to perform any sterile neutrino search. The two layouts have to be as similar as possible in order to allow an almost complete cancellation of the systematic uncertainties when comparing the measurements made at the near and far sites. Hence the near spectrometer will be an exact clone of the far one, with identical thickness along the beam axis but a scaled transverse size. A sketch of the possible far site NESSiE detector is shown in Fig. \[nessie-far\].
![Sketch of the far site spectrometer that could be built by extensively reusing materials available from the OPERA spectrometers [@opera]. The basic elements of the new ACM concept are also depicted (courtesy of the NESSiE Collaboration). Neutrinos are traveling from right to left.[]{data-label="nessie-far"}](nessie-far-mod3-small){width="70.00000%"}
The key feature of the ACM is the large geometric and momentum acceptance. The need of a low momentum threshold for muon reconstruction can be met using a magnet in air. The only dead material along the muon path is given by the conductors needed to generate the magnetic field and the position detectors instrumenting the magnet itself. For the conductors the use of aluminum instead of copper is preferable due to the lower $Z$ and a density lower by a factor 3. The magnetic field needed in the low momentum range covered by the ACM is in the range 0.1 - 0.15 T. A spatial resolution in the range of 0.1 - 1 mm can be reached using drift tubes [@opera-hpt] as high precision trackers in combination with scintillator strip detectors [@opera-scint]. These could provide the external trigger needed by the drift tubes and a coarse spatial measurement in the non–bending direction i.e. the direction parallel to the drift tubes. Silicon photomultiplier (SiPM) devices may eventually be used to read out some scintillator planes embedded in the magnetic field.
The general layout of the two OPERA [@opera] iron spectrometers fulfills the requirements of the ICM detector, and they could eventually be used for the CERN project putting the two magnets one after the other in order to obtain a total of 4 m longitudinal thickness of iron.
The OPERA spectrometers are built assembling vertical iron plates (slabs) in a planar structure of 875 cm (width) $\times$ 800 cm (height). Each passive plane is made out of seven adjacent iron slabs. Resistive Plate Chambers (RPC) [@opera-rpc] are sandwiched between iron planes; 21 RPC detectors, arranged in seven rows and three columns, are used in each active plane. Each RPC detector has a rectangular shape and covers an area of about 3.2 m$^2$. The RPCs provide tracking measurements with about 1 cm resolution using the digital read-out of strips with a 2.6 cm pitch in the bending direction and 3.5 cm in the non–bending direction. The magnet is made of two arms with 22 RPC layers alternating with iron layers. Top and bottom iron yokes are connecting the arms. Copper coils surrounding the yokes are used to generate a magnetic field of about 1.5 T in the iron circuit. A total of 924 RPC chambers are needed to instrument each spectrometer. They could be recovered from the OPERA spectrometers and re-used. These RPC chambers are standard 2 mm gap chambers with bakelite electrodes with resistivity in the range from 10$^{11}$ to 5$\times$10$^{12}$ $\Omega$ $\cdot$ cm at $T=20 ^{\circ}$C. In OPERA they are operated in the streamer regime with a gas mixture made of $Ar/C_2H_2F_4/I$-$C_4H_{10}/SF_6$ in the volume ratios of 75.4/20/4/0.6 at five refills/day. The high-amplitude streamer signals produced by charged tracks crossing the gas volume allow to house the Front-End discriminators in racks placed on top of the spectrometer.
The $\nu_\mu$ disappearance analysis {#sect-3}
====================================
The use of at least two sites where neutrino interactions can occur is mandatory to observe the oscillation pattern, as established by almost all the most recent neutrino experiments. Indeed the disappearance probability due to an additional sterile neutrino is given by the usual two-flavour formula: $$P(\nu_\alpha\rightarrow\nu_\alpha)=1-\sin^2(2\theta)\cdot\sin^2(1.267\cdot\Delta m^2\cdot L/E), \, \text{\small ($L$ in km, $E$ in GeV)}$$ where the disappearance of flavour $\alpha$ is due to the oscillation of neutrino mass states at the $\Delta m^2$ scale and at an effective mixing angle $\theta$ that can be simply parametrized as a function of the elements of a $3+1$ extended mixing matrix. As $L$ is fixed by the experiment location, the oscillation is naturally driven by the neutrino energy, with an [*amplitude*]{} determined by the mixing parameter.
The disappearance of muon neutrinos due to the presence of an additional sterile state depends only on terms of the extended PMNS [@pmns] mixing matrix ($U_{\alpha i}$ with $\alpha= e,\mu,\tau$ and $i=1$,…,4) involving the $\nu_{\mu}$ flavor state and the additional mass eigenstate 4. In a 3+1 model at Short Baseline (SBL) we have: $$P(\nu_{\mu}\to\nu_{\mu})_{SBL}^{3+1} = 1 - 4 \vert U_{\mu 4} \vert^2 (1 - \vert U_{\mu 4} \vert^2) \sin^2 \frac{\Delta m^2_{41} L}{4E}$$ In contrast, appearance channels (i.e. $\nu_\mu \to \nu_e$) are driven by terms that mix up the couplings between the initial and final flavour states and the sterile state yielding a more complex picture: $$P(\nu_{\mu}\to\nu_e)_{SBL}^{3+1} = 4 \vert U_{\mu 4}\vert^2 \vert U_{e 4} \vert^2 \sin^2 \frac{\Delta m^2_{41} L}{4E}$$ This also holds in extended $3 + n$ models.
It is interesting to notice that the appearance channel is suppressed by two more powers in $\vert U_{\alpha 4}\vert$. Furthermore, since $\nu_e$ or $\nu_\mu$ appearance requires $\vert U_{e 4}\vert > 0$ and $\vert U_{\mu 4}\vert > 0$, it should be naturally accompanied by a corresponding $\nu_e$ and $\nu_\mu$ disappearance. In this sense the disappearance searches are essential for providing severe constraints on the models of the theory (a more extensive discussion on this issue can be found e.g. in Sect. 2 of [@winter]).
It must also be noted that the number of $\nu_e$ neutrinos depends on the $\nu_e\rightarrow\nu_s$ disappearance and $\nu_\mu\rightarrow\nu_e$ appearance, and, naturally, from the intrinsic $\nu_e$ contamination in the beam. On the other hand, the amount of $\nu_\mu$ neutrinos depends only on the $\nu_\mu\rightarrow\nu_s$ disappearance and $\nu_e\rightarrow\nu_\mu$ appearance but the latter is much smaller due to the fact that the $\nu_e$ contamination in $\nu_\mu$ beams is usually at the percent level. Therefore in the $\nu_{\mu}$ disappearance channel the oscillation probabilities in both near and far detectors can be measured without any interplay of different flavours, i.e. by the same probability amplitude.
By taking into account the lengths $L$ of the locations given in the CERN proposal, [*$L_{Near}=$*]{} 460 m and [*$L_{Far}=$*]{}1600 m, we may plot the disappearance probability as a function of the more [*convenient*]{} variable $\log_{10}(1/E)$, using $E$ in MeV to avoid singularities. As an example, in Fig. \[ster-1\] the disappearance probability is shown for the near and far sites, by assuming an amplitude[^2] that corresponds to the averaged reactor anomaly $\overline\nu_e$ disappearance, [@reattori], and $\Delta m^2=1$ eV$^2$. It appears that at $\Delta m^2\approx 1$ eV$^2$ oscillations are already visible at the near site. Such a behaviour may turn out to help the actual measurement by plotting the ratio between the far and near sites data as a function of $\log_{10}(1/E)$ (Fig. \[ster-2\]). The [*Far/Near*]{} ratio of the oscillation probabilities well illustrates the pattern that could be observed at the CERN Short–Baseline experiment: a decrease above 1 GeV and an increase below 1 GeV of the neutrino energy!
![Disappearance probabilities in the two–flavour limit at near (top) and far (bottom) sites, at 460 and 1600 m, respectively, by using the amplitude provided by the reactor anomaly, 0.146, and the mass scale $\Delta m^2=1$ eV$^2$. The x-axis corresponds to $\log_{10}(1/E_\nu)$, with $E$ in MeV.[]{data-label="ster-1"}](ster-1-Andrea-la){width="60.00000%"}
![The ratio of the disappearance probabilities at near and far sites, 460 and 1600 m, respectively, in the two–flavour limit, by using the amplitude provided by the reactor anomaly, 0.146, and the mass scale $\Delta m^2=1$ eV$^2$. The x-axis corresponds to $\log_{10}(1/E_\nu)$, with $E$ in MeV.[]{data-label="ster-2"}](ster-2-Andrea-rev){width="70.00000%"}
The disappearance patterns have to be evaluated after folding cross-sections and efficiencies with the neutrino fluxes and detector geometry. The [*Far/Near*]{} pattern does not depend at first order either on cross-sections and efficiencies or on the fluxes as far as they are similar at the two sites, and within the use of two [**]{} identical detector systems. However, due to the different $\nu$ and $\overline{\nu}$ cross-sections, errors are enhanced when neutrinos and anti-neutrinos are not identified and separated on an event-by-event basis. Therefore the use of magnetic detectors is mandatory to keep systematics under control. Then the largest source of error consists of the smearing introduced by the reconstruction procedure. Eventually this issue will be addressed by considering a conservative systematic effect in the muon momentum reconstruction.
Fig. \[ster-3\] (top) illustrates a realistic observation of the non–oscillation probability when convoluted with the estimated CERN $\nu_{\mu}$ beam flux and the neutrino cross–sections, by using the detector systems described above and one year of data taking. The flatness is lost as expected. However a specific case of oscillation is drawn, confirming that the workable behaviour is still present. The correlated estimator, the double ratio parameter, defined as [*(Far/Near)$_{oscillated}$/ (Far/Near)$_{unoscillated}$*]{}, is also shown in the bottom of the Figure. It can be used for the statistical analysis. In the following we will demonstrate how the oscillation behaviour of the [*Far/Near*]{} parameter will allow an increase of an order of magnitude in the sensitivity to the mixing angle, taking into account the systematic errors, too.
![The oscillated and unoscillated $\nu_{\mu}$ event distributions, parametrized in $\log_{10}(1/E_{\nu})$, for a luminosity of $4.5\times 10^{19}$ p.o.t. of the CERN $\nu_{\mu}$ beam flux over the two magnet system described in the text, are plotted with their statistical errors. On the top the [*Far/Near*]{} collected ratios are drawn. The no–oscillation shape is due to the convolution of the neutrino cross-section and the fluxes at the near and far site, while the oscillated shape is due to the sterile inclusion (with $\Delta m^2=1$ eV$^2$ and mixing amplitude equal to $0.146$). On the bottom the double ratio [*(Far/Near)$_{oscillated}$/(Far/Near)$_{unoscillated}$*]{} is shown for the same sterile assumption.[]{data-label="ster-3"}](dratio-ster-3){width="80.00000%"}
Another parameter can be used in case of measurement of Neutral Current (NC) events become available, either with the proposed LAr detectors or even with a less refined and massive detector. The NC/CC ratio becomes then a valuable estimator [@nessie] that may enter into the evaluation of the significance of the sterile neutrino observation. A study that includes the information from Neutral Current events is out of the scope of the present paper. However for completeness we would like to briefly outline here the issue as it was originally described in [@nessie].
The observation of a depletion in NC events would be a direct and manifest signature of the existence of sterile neutrinos. In fact the NC event rates are unaffected by standard neutrino mixing being flavour–blind such that their disappearance could only be explained by $\nu_{\mu}$ ($\nu_e$) $\rightarrow\nu_s$ transitions. Even if NC events, either from $\nu_e$ or $\nu_{\mu}$, are efficiently detected by a Liquid Argon detector or a less refined detector, the transition rate measured with NC events has to agree with the $\nu_{\mu}$ CC disappearance rate once the $\nu_{\mu}\rightarrow\nu_{\tau}$ and $\nu_{\mu}\rightarrow\nu_e$ contributions have been subtracted. Therefore the NC disappearance is measured at best by the double ratio: $({\frac{NC}{CC})_{Far}}/({\frac{NC}{CC})_{Near}}$. The double ratio is the most robust experimental quantity to detect NC disappearance, once $CC_{Near}$ and $CC_{Far}$ are precisely measured thanks to the spectrometers, at the near and far locations. For that it is mandatory to disentangle $\nu_{\mu}$ and $\overline\nu_{\mu}$ contributions.
Simulation and results {#sect-4}
======================
The magnetic detector system that has to be developed for the $\nu_{\mu}$ disappearance measurement should take into account all the considerations depicted in previous sections. The system developed by the NESSiE Collaboration [@nessie] is actually well suited since it couples a very powerful high-$Z$ magnet for the momentum measurement via [range]{} to a low-$Z$ magnet, to extend the useful muon momentum interval as low as possible, to allow charge discrimination on an event–by–event basis and to allow NC event measurement whether coupled to an adequate (but not necessarily highly performant and with large mass) detector to identify NC events.
In the following we concentrate on the results achievable by the [*Far/Near*]{} estimators described in the previous section, by exploiting a full neutrino beam simulation, an up–to–date neutrino interaction model (GENIE 2.6 [@genie]), with detailed interactions in low-$Z$ and high-$Z$ magnets and a generic geometry. The Iron Core Magnets have been simulated by a massive cube of iron with a mass equivalent to the OPERA magnets and by using a 90% inner fiducial volume (770 tons and 330 tons for the far and near locations, respectively). Single hits have been extracted at the position of the RPC detectors and a realistic resolution has been included (0.75 cm for the single hit). The ACM has been simulated as an empty 1 m deep region with 0.1 T magnetic field, limited by two vertical aluminum slabs. The overall simulation corresponds to the GLoBES software [@globes] (version 3.0.11), to which it was partially compared and cross checked.
The muon reconstruction requires a minimal penetration length that corresponds to a 0.5 GeV cut. A conservative resolution of 10% was used for the momentum evaluation, while the charge mis–identification was fixed at the 1% level. Systematic effects due to mis–calibration of the detectors have been conservatively taken at 0.5% and 1% levels, which include systematics due to the [*relative*]{} flux at the near and far positions.
In Fig. \[ster-4\] the double ratio [*Far/Near*]{} is shown as obtained by using the reconstructed muon momentum, for different values of $\Delta m^2$ scales. The relevant behaviour of the estimator is confirmed for $\Delta m^2$ values above $\approx 2$ eV$^2$ also in this very realistic and conservative approach. In fact, by using the muon reconstructed momentum in CC events instead of the true $\nu_{\mu}$ energy the oscillations properties will be affected by the kinematics and the interaction processes (it is worthwhile to note that there are mostly quasi–elastic interactions in the accounted–for energy range).
![The double ratio of the disappearance probabilities at the near and far sites, 460 and 1600 m, respectively, by using the amplitude provided by the reactor anomaly, 0.146, and mass scales $\Delta m^2$ ranging from 0.1 to 5 eV$^2$ as a function of $\log_{10}(1/p_\mu)$, where “[*p*]{}” is the reconstructed muon momentum in MeV/c. We apply cuts on the minimum muon path length (25 cm of range in iron) and the minimum momentum ($500$ MeV/c). A 90% fiducial acceptance of the magnet volume and an uncorrelated 1% systematic error (from OPERA experience [@opera-ele]) are also considered. Data collection corresponds to 1 year, i.e. $4.5\cdot 10^{19}$ p.o.t. The error bars correspond to the statistical errors. The blue band for $\Delta m^2=2$ eV$^2$ is obtained by including the systematic error summed quadratically to the statistical one, the same result being obtained for the other mass scales.[]{data-label="ster-4"}](fig6_stefano_rev_onesys_corretta){width="68.00000%"}
Finally, in Fig. \[ster-5\] the estimated limits at 95% C.L. on $\nu_{\mu}$ disappearance that can be achieved via the [*Far/Near*]{} estimator are shown for different data periods (3, 5 and 10 years, corresponding to $13.5\cdot 10^{19}$, $22.5\cdot 10^{19}$ and $45.0\cdot 10^{19}$ p.o.t., respectively). The different results for $\nu_{\mu}$ and $\overline\nu_{\mu}$ beams were evaluated using the two variables, $p$ and $\log_{10}(1/p)$. In negative polarity runs the muon charge identification allows an independent, simultaneous and similar–sensitivity measurements of the $\overline\nu_{\mu}$ and $\nu_{\mu}$ disappearance rates, due to the large $\nu_{\mu}$ contamination in the $\overline\nu_{\mu}$ beam.
![The estimated limits at 95% C.L. for $\nu_{\mu}$ disappearance at a Short–Baseline beam at CERN for several luminosity running periods and different beam polarities, with a two–site massive spectrometer (770 tons and 330 tons, respectively) with 90% inner fiducial volume.\
The top figure refers to the positive polarity beam. The continuous (dashed) lines correspond to the sensitivity limits obtained with the $\log_{10}(1/p)$ ($p$) variable. 3 years correspond to $13.5\cdot 10^{19}$ p.o.t., 5 years to $22.5\cdot 10^{19}$ p.o.t. and 10 years to $45.0\cdot 10^{19}$ p.o.t. The exclusion limit from combined MiniBooNE and SciBooNE $\nu_{\mu}$ disappearance result at 90% C.L. from Ref. [@mini-sci-mu1] is shown for comparison by the black curve in the right.\
The bottom figure refers to the negative polarity beam. Sensitivity limits are evaluated with the $\log_{10}(1/p)$ variable. Clearly the negative polarity run allows the contemporary analysis of the $\overline\nu_{\mu}$ and $\nu_{\mu}$ disappearance exclusion regions thanks to the disentangling of the muon charge on an event–by–event–basis. The black curve in the right shows for comparison the central value of the sensitivity at 90% C.L. from combined MiniBooNE and SciBooNE $\overline\nu_{\mu}$ disappearance result (Ref. [@mini-sci-mu2]).[]{data-label="ster-5"}](SD-2-rev "fig:"){width="70.00000%"} ![The estimated limits at 95% C.L. for $\nu_{\mu}$ disappearance at a Short–Baseline beam at CERN for several luminosity running periods and different beam polarities, with a two–site massive spectrometer (770 tons and 330 tons, respectively) with 90% inner fiducial volume.\
The top figure refers to the positive polarity beam. The continuous (dashed) lines correspond to the sensitivity limits obtained with the $\log_{10}(1/p)$ ($p$) variable. 3 years correspond to $13.5\cdot 10^{19}$ p.o.t., 5 years to $22.5\cdot 10^{19}$ p.o.t. and 10 years to $45.0\cdot 10^{19}$ p.o.t. The exclusion limit from combined MiniBooNE and SciBooNE $\nu_{\mu}$ disappearance result at 90% C.L. from Ref. [@mini-sci-mu1] is shown for comparison by the black curve in the right.\
The bottom figure refers to the negative polarity beam. Sensitivity limits are evaluated with the $\log_{10}(1/p)$ variable. Clearly the negative polarity run allows the contemporary analysis of the $\overline\nu_{\mu}$ and $\nu_{\mu}$ disappearance exclusion regions thanks to the disentangling of the muon charge on an event–by–event–basis. The black curve in the right shows for comparison the central value of the sensitivity at 90% C.L. from combined MiniBooNE and SciBooNE $\overline\nu_{\mu}$ disappearance result (Ref. [@mini-sci-mu2]).[]{data-label="ster-5"}](SD-4-rev "fig:"){width="70.00000%"}
The parametrization used for the neutrino energy, $\log_{10}(1/E)$, which we conservatively prefer to address as $\log_{10}(1/p_{\mu-rec})$, provides slightly better limits in almost all the ($\Delta m^2$, $\sin^2(2\theta)$) excluded regions. This is partially due to the gaussian shape sensitivity of $1/p$ when cuts are applied to the corresponding variable. Moreover, the elucidation of the behaviour of the [*Far/Near*]{} ratio in terms of depletions and excesses in different regions of the spectra allows a better comparison between unoscillated and oscillated hypotheses.
As a result more than an order of magnitude improvement can be obtained in the sensitivity to the mixing parameter space between standard neutrino and sterile ones with respect to today’s limits over the whole $\Delta m^2$ range investigated. Specifically the mixing angle sensitivity (at 95% C.L.) reaches $2.2\times 10^{-3}$ at $\Delta m^2 = 1$ eV$^2$, while at full mixing a $\Delta m^2=0.03$ eV$^2$ sensitivity at 95% C.L. is obtained already with only 3 years of data collection with the neutrino beam.
Slightly worse results are obtained by collecting data with a negative polarity beam. The limiting mixing angle sensitivity at 95% C.L. is around $10^{-2}$ at $\Delta m^2 = 1$ eV$^2$. At full mixing a $\Delta m^2=0.1$ eV$^2$ sensitivity at 95% C.L. is obtained with 3 years of data collection with the antineutrino beam, the limiting factor being the intensity of the $\overline\nu_{\mu}$ beam. However it is worthwhile to note that by operating in negative polarity similar sensitivities can be obtained at the same time also for the neutrino component due to the large contamination of $\nu$ in the anti-$\nu$ beams.
The relevance of statistics in presence of systematic effects is depicted in Figs. \[ster-6\]. It is evident that the overall error, dominated by the sensitivity on the measurement of the muon momentum, puts an intrinsic limitation to the data statistics which can be collected. As expected the useful luminosity that can be collected is asymptotically limited by systematics. The positive effect of the variable $\log_{10}(1/p_{\mu-rec})$ is more evident when larger errors are in place.
![Systematic effects for the estimated limits at 95% C.L. for $\nu_{\mu}$ disappearance at a Short–Baseline beam at CERN for 5 (positive polarity, top) and 10 (negative polarity, bottom) years of data taking, with a two–site massive spectrometer (770 tons and 330 tons, respectively) with 90% inner fiducial volume. The exclusion regions are evaluated by using the $\log_{10}(1/p)$ (continuous lines) and $p$ (dashed lines) variables in case of the positive polarity beam data taking.[]{data-label="ster-6"}](SD-1-rev "fig:"){width="70.00000%"} ![Systematic effects for the estimated limits at 95% C.L. for $\nu_{\mu}$ disappearance at a Short–Baseline beam at CERN for 5 (positive polarity, top) and 10 (negative polarity, bottom) years of data taking, with a two–site massive spectrometer (770 tons and 330 tons, respectively) with 90% inner fiducial volume. The exclusion regions are evaluated by using the $\log_{10}(1/p)$ (continuous lines) and $p$ (dashed lines) variables in case of the positive polarity beam data taking.[]{data-label="ster-6"}](SD-3-rev "fig:"){width="70.00000%"}
Conclusions
===========
Neutrino physics is receiving more and more attention as a venue for the long standing search for new physics beyond the Standard Model. The current anomalies which do not fit into the established standard scenario with 3 neutrinos deserve refined studies and experiments. The CERN proposal for a new Short–Baseline experimental project is a very valuable one. We illustrated the current critical tensions in the muon-neutrino disappearance field and the achievements that can be obtained within the CERN project. Specifically, by considering $\mathcal{O}$(1) kton massive spectrometers, an improvement by an order of magnitude can be obtained in the sensitivity to the mixing parameter space between standard neutrinos and sterile ones with respect to today’s limits. Conversely, a possible $\nu_{\mu}$ disappearance signal will be essential to measure the relevant physical parameters and to fully disentangle the different sterile models.
An effective analysis can be performed with a two-site experiment by using muon spectrometers with a low-$Z$ part that allows clean charge identification on an event-by-event basis, and with a massive part allowing clean momentum measurement through range. Such a kind of spectrometer is under study by the NESSiE Collaboration and might be available with a limited investment. The performances of these kinds of detectors, enlightened by the type of analysis developed in this paper, are suitable to put a definitive result on the sterile neutrino issue at the eV mass scale.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are heartily dependent of the contributions of the NESSiE, ICARUS and CERN-CENF groups in developing the CERN project and the experimental proposals. We wish also to warmly thank the encouragements of Marzio Nessi and Carlo Rubbia in supporting such studies. We are finally in debit to Maury Goodman for suggested criticisms and a careful proof–reading of the manuscript.
[9]{}
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[^1]: Pulses of 10.5 $\mu$s duration are normally put in coincidence with the fast response of the spectrometers’ detectors and efficiently used to reject the cosmic ray background. See e.g. [@opera-time] where a time resolution of less than 2 ns is reported for the detectors used in the OPERA experiment.
[^2]: It is worthwhile to note that the disappearance amplitude affects the sensitivity to the mixing angle that in turn depends mainly on the statistical extent of the data. Therefore the same evidence can be obtained for smaller amplitudes, i.e. smaller effective mixing angles, by increasing the data sample. Finally the ultimate limit of an experiment is set by the intrinsic systematic errors corresponding to the kinematics of the neutrino interaction and to the muon reconstruction sensitivity. Fig. \[ster-6\] below illustrates the issue.
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abstract: |
This paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.
Key words: K-theory, excision, singularities, cyclic homology, $p$-adic fields.
MSC: 19D55 (primary), 14H20 (secondary).
author:
- 'Matthew Morrow[^1]'
title: '$K$-theory of one-dimensional rings via pro-excision'
---
Introduction {#introduction .unnumbered}
============
In the first section of this paper we will show that if $A$ is a one-dimensional, Noetherian, reduced, semi-local ring for which the normalisation morphism $A\to{\widetilde}A$ is finite, then there is a long exact, Mayer–Vietoris, ‘pro-excision’ sequence of pro abelian groups $$\cdots\to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r)\to\cdots\tag{pro-MV},$$ where ${\mathfrak}m,{\mathfrak}M$ denote the Jacobson radicals of $A,{\widetilde}A$ respectively. There is also a similar sequence for the relative $K$-groups. Here $\operatorname*{``\varprojlim\!''}_{r}$ denotes a pro abelian group, i.e. a formal inverse system of groups, sometimes denoted, e.g., $\{K_n(A/{\mathfrak}m^r)\}_r$.
Before discussing the main results of this paper, we explain the source of (pro-MV). It has been known at least since work by R. Swan [@Swan1971 Thm. 3.1] that $K$-theory fails to satisfy excision; i.e., if $A\to B$ is a morphism of rings and $I$ is an ideal of $A$ mapped isomorphically to an ideal of $B$, then $K_n(A,I)\to K_n(B,I)$ need not be an isomorphism. Having fixed $I$ as a non-unital algebra, A. Suslin [@Suslin1995] showed, by building on earlier work of himself and M. Wodzicki [@Suslin1992], that $I$ satisfies excision for [*all*]{} such morphisms $A\to B$ if and only if $I$ is [*homologically unital*]{}, in Wodzicki’s sense that ${\operatorname}{Tor}^{{\mathbb}Z\ltimes I}_*({\mathbb}Z,{\mathbb}Z)=0$ for $*>0$. Unfortunately, this is not commonly satisfied for rings of algebraic geometry. A recent trend has therefore been to consider instead the problem of ‘pro-excision’: i.e., When is the map $\operatorname*{``\varprojlim\!''}_r K_n(A,I^r)\to\operatorname*{``\varprojlim\!''}_rK_n(B,I^r)$ an isomorphism? For example, if $A$ is a Noetherian ${\mathbb}Q$-algebra then these maps are isomorphisms by a recent theorem of the author [@Morrow_Birelative Thm. 0.1]. Moreover, T. Geisser and L. Hesselholt [@GeisserHesselholt2006; @GeisserHesselholt2011] have established a pro version of the Suslin–Wodzicki condition. In section \[section\_pro\_excision\] we use Geisser–Hesselholt’s results to show that if $I$ is the conductor ideal of a one-dimensional, Noetherian, reduced ring for which the normalisation map is finite, then it satisfies pro-excision, thereby resulting in long exact, Mayer–Vietoris, pro-excision sequences such as (pro-MV) above.
Such sequences have immediate global applications: If $X$ is a proper, reduced curve over a finite field, then pro-excision implies that $K_n(X)\to K_n({\widetilde}X)$ has finite kernel and cokernel for $n\ge 1$, whence $K_n(X)$ is finite by G. Harder [@Harder1977] and C. Soulé [@Soule1984]. In other words, Harder–Soulé’s finiteness result extends to singular curves. The arithmetic analogue, which also follows from pro-exicision, is that if $A\subseteq {\mathcal{O}}_F$ is an order in the ring of integers of a number field $F$, then $K_n(A)$ is finitely generated and of the same rank as $K_n({\mathcal{O}}_F)$; these ranks are of course known thanks to A. Borel [@Borel1974]. The proofs of these results are postponed until section \[subsection\_global\] with the other material on rings with finite residue fields.
However, the major theme of this paper is of a local nature, showing that such pro-excision sequences often break into short exact sequences and studying the many interesting consequences, especially to Geller’s conjecture in the finite residue characteristic case. In particular, sections \[subsection\_finite\_char\] – \[subsection\_main\_results\] are devoted to the proof of the following key theorem, which can be interpreted as an analogue for singular rings of the Gersten conjecture:
\[theorem\_intro\_1\] Let $A$ be a one-dimensional, Noetherian, reduced semi-local ring containing a field such that $A\to {\widetilde}A$ is a finite morphism; let ${\mathfrak}m$ and ${\mathfrak}M$ denote the Jacobson radicals of $A$ and ${\widetilde}A$. Then the relative version of (pro-MV) breaks into short exact sequences of pro abelian groups, yielding $$0 \to K_n(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n({\widetilde}A,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\to0$$ for all $n\ge 0$. If ${\widetilde}A\to{\widetilde}A/{\mathfrak}M$ splits (e.g., $A$ complete), then the non-relative sequence (pro-MV) also splits into short exact sequences: $$0\to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r)\to0.$$
The special case of the theorem when $A$ is essentially of finite type over a field of characteristic zero and $n=2$ was proved by A. Krishna [@Krishna2005 Thm. 3.6]; the $n=2$ assumption was removed in the author’s earlier work [@Morrow_Singular_Gersten] on this subject, but the characteristic and finite type assumptions remained. Therefore the theorem extends the results in [@Morrow_Singular_Gersten] on $K$-theoretic adèles to all one-dimensional, Noetherian, reduced schemes over a field for which the normalisation map is finite. I suspect that the theorem is true without the equal characteristic assumption; section \[subsection\_p\_adic\_orders\] gives partial results in mixed characteristic.
Informally, the theorem states that the contributions to the $K$-theory of $A$ coming from its singularities can be entirely captured using the $K$-theory of the quotients $A/{\mathfrak}m^r$, for $r\gg0$. For example, we use the theorem to prove the following in section \[subsection\_KH\], where $KH$ denotes C. Weibel’s homotopy invariant $K$-theory:
\[theorem\_intro\_1a\] Let $A$ be as in theorem \[theorem\_intro\_1\], and assume that $A\to A/{\mathfrak}m$ splits. Then, for each $n\ge0$, the kernel of $K_n(A)\to KH_n(A)$ embeds into $K_n(A/{\mathfrak}m^r)$ for $r\gg0$.
Moreover, precisely because (pro-MV) and the first theorem describe the singular contribution to the $K$-groups, they have important applications to Geller’s conjecture, which we interpret as the following, although this is not exactly what S. Geller asked in 1986 [@Geller1986]:
> “Let $A$ be a one-dimensional, Noetherian, reduced local ring, and suppose that $K_2(A)\to K_2(F)$ is injective, where $F=\operatorname{Frac}A$ is the total quotient ring of $A$. Then $A$ is necessarily regular.”
Apart from the seminormal, equal characteristic case, there has been no progress until now on the conjecture when $A$ has finite residue characteristic. We prove the following in section \[subsubsection\_Geller\]:
\[theorem\_intro\_2\] Geller’s conjecture is true if $A$ is an ${\mathbb}F_p$-algebra with perfect residue field for which $A\to {\widetilde}A$ is a finite morphism.
Meanwhile, in section \[subsection\_rel\_to\_HC\] we offer the following interesting alternative to Geller’s conjecture in characteristic zero:
\[theorem\_intro\_2a\] If $A$ is as in Geller’s conjecture, is essentially of finite type over a characteristic zero field, and is not regular, then $K_n(A)\to K_n(F)$ is not injective for some $n\ge 3$.
Next we turn our attention to mixed characteristic rings; section \[subsection\_p\_adic\_orders\] analyses consequences of the sequence (pro-MV) for reduced ${\mathbb}Z_p$-algebras which are finitely generated and torsion-free as ${\mathbb}Z_p$-modules, e.g. ${\mathbb}Z_p+p{\mathbb}Z_q$. If $A$ is such a ring then ${\widetilde}A$ is a finite product of rings of integers of local fields, whose $K$-groups are largely understood thanks to L. Hesselholt and I. Madsen’s proof of the Quillen–Lichtenbaum conjecture [@Hesselholt2003 Thm. A]. In particular, this implies that the following theorem holds for rings of integers of local fields; extending it to $A$ relies on the observation that pro-excision implies that $K_n(A)\to K_n({\widetilde}A)$ has finite kernel and cokernel for $n\ge1$.
\[theorem\_intro\_3\] Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module. Then $$K_n(A)=\begin{cases}
\mbox{divisible ${\mathbb}Z_{(p)}$-module }\oplus\mbox{ finite $p$-group} & n\ge 2\mbox{ even,}\\
\mbox{torsion-free ${\mathbb}Z_{(p)}$-module }\oplus\mbox{ finite group} & n\ge 1\mbox{ odd,}
\end{cases}$$
It follows from group theory that the groups appearing in these direct sum decompositions are determined, up to isomorphism, by $K_n(A)$. For example, if $n$ is even then the finite $p$-group appearing in the theorem is necessarily the quotient of $K_n(A)$ by its maximal divisible subgroup; in fact, if $n$ is even then the standard short exact sequence $$0\to{\operatorname}{Ext}^1_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,K_n(A))\to K_n(A;{\widehat}{{\mathbb}Z})\to\operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,K_{n-1}(A))\to 0$$ implies, in conjunction with the previous theorem, that the finite $p$-group is precisely the $K$-group with ${\widehat}{{\mathbb}Z}$ coefficients $K_n(A;{\widehat}{{\mathbb}Z})$. A similar structural description of $K_n(A;{\widehat}{{\mathbb}Z})$ when $n$ is odd is also given in corollary \[corollary\_profinite\_K\_groups\_of\_p\_adic\_order\], and these results are used to prove the analogue of theorem \[theorem\_intro\_1\] in odd degrees for $A$. In even degree we can reduce the problem to understanding the torsion in the even degree $K$-groups; this is well understood for $K_2$, but not in general, resulting in the following:
\[theorem\_intro\_4\] Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module. Then $K_2(A;{\widehat}{{\mathbb}Z})$ equals the finite $p$-group alluded to in theorem \[theorem\_intro\_3\] (and similarly for ${\widetilde}A$), and there is a short exact, Mayer–Vietoris sequence $$0\to K_2(A)\to K_2(A;{\widehat}{{\mathbb}Z})\oplus K_2({\widetilde}A)\to K_2({\widetilde}A;{\widehat}{{\mathbb}Z})\to 0$$
Finally, using similar methods as in our proof of Geller’s conjecture in equal finite characteristic, this theorem yields the first results towards Geller’s conjecture in mixed characteristic. Our methods do not see the element $p$, so rather than establishing regularity under the conditions of the conjecture, we instead bound the embedding dimension of $A$ by $2$:
\[theorem\_intro\_5\] Let $A$ be a one-dimensional, Noetherian, reduced local ring of mixed characteristic with finite residue field of characteristic $p>2$, and such that $A\to{\widetilde}A$ is finite. Suppose that at least one of the following is true:
1. $\operatorname{Frac}{\widehat}A$ contains no non-trivial $p$-power roots of unity; or
2. $A$ is seminormal, and ${\widehat}A$ is not a certain exceptional case (see theorem \[theorem\_geller\_in\_mixed\_char\] for details); or
3. ${\widetilde}A$ is local and all $p$-power roots of unity in $\operatorname{Frac}{\widehat}A$ belong to ${\widehat}A$.
If the map $K_2(A)\to K_2(\operatorname{Frac}A)$ is injective then ${\operatorname}{embdim}A\le 2$.
In the remainder of this introduction we describe our methods and the ingredients of the proofs of the above results, beginning with theorem \[theorem\_intro\_1\]. It is sufficient, using the relative version of (pro-MV), to show that $$K_n({\widetilde}A,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_rK_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$$ is surjective for all $n\ge 1$. Thus it is enough to show that the codomain is entirely symbolic and so, since ${\widetilde}A/{\mathfrak}M^r$ is a finite product of truncated polynomial rings, this reduces the theorem to proving that $$\operatorname*{``\varprojlim\!''}_rK_n(k[t]/{(t^r)},{(t)})$$ is entirely symbolic for any field $k$. When $k$ has characteristic zero this is proved directly in section \[subsection\_char\_zero\] by filtration arguments in cyclic homology after applying the following case of the Goodwillie isomorphism [@Goodwillie1986]: $K_n(k[t]/{(t^r)},{(t)})\cong HC_{n-1}^{{\mathbb}Q}(k[t]/{(t^r)},{(t)})$. The proof is philosophically similar in finite characteristic, in that we apply Hesselholt and Madsen’s [@Hesselholt2001; @Hesselholt2008] description of the $K$-theory of truncated polynomial rings in finite characteristic using topological cyclic homology via the McCarthy isomorphism [@McCarthy1997].
Sections \[subsection\_KH\] – \[subsection\_rel\_to\_HC\] cover the applications of (pro-MV) and the first main theorem; they may be read independently:
Firstly, theorem \[theorem\_intro\_1a\] follows in a straightforward way from theorem \[theorem\_intro\_1\] by relating the $KH$-theory of $A$ with the $K$-theory of ${\widetilde}A$.
Next, (pro-MV) reduces theorem \[theorem\_intro\_2\] to checking that $$\operatorname*{``\varprojlim\!''}_rK_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\to\operatorname*{``\varprojlim\!''}_rK_2({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$$ is injective if and only if $A$ is regular. It can be shown using perfectness of the residue field that the codomain of this map vanishes, so it becomes enough to construct a non-zero element of $K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)$ for some $r$ under the assumption that $A$ is singular; this is easily achieved using Dennis–Stein symbols. We emphasise that the key to the proof is that (pro-MV) reduces the problem to the $K$-theory of the quotients $A/{\mathfrak}m^r$ and ${\widetilde}A/{\mathfrak}M^r$. Krishna [@Krishna2005] studied Geller’s conjecture in a similar way in characteristic zero, introducing an ‘Artinian Geller’s Conjecture’ which mimicked G. Cortiñas, S. Geller, and C. Weibel’s [@Weibel1998] Artinian analogue of Berger’s conjecture on the torsion in differential forms of curve singularities.
Finally, theorem \[theorem\_intro\_2a\] is proved by establishing an intermediate result which is of interest in its own right: The maps $$K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M),\quad\quad HC_{n-1}(A,{\mathfrak}m)\to HC_{n-1}({\widetilde}A,{\mathfrak}M)$$ have isomorphic kernels (and cokernels). This is proved by establishing an analogue of theorem \[theorem\_intro\_1\] for cyclic homology and then appealing to the Goodwillie isomorphism. For example, the resulting isomorphism of the kernels is uniquely determined by the commutativity of the diagram $$\xymatrix@R=5mm{
{\operatorname{Ker}}(K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M))\ar@{^(->}[d]\ar[r]^\cong& {\operatorname{Ker}}(HC_{n-1}(A,{\mathfrak}m)\to HC_{n-1}({\widetilde}A,{\mathfrak}M))\ar@{^(->}[d]\\
K_n(A,{\mathfrak}m) \ar[d] & HC_{n-1}(A,{\mathfrak}m)\ar[d]\\
\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\ar[r]^\cong &\operatorname*{``\varprojlim\!''}_r HC_{n-1}(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)
}$$ where the bottom arrow is the Goodwillie isomorphism; the isomorphism of the cokernels is determined in a similar way. These isomorphisms reduce theorem \[theorem\_intro\_2a\] to the same claim for cyclic homology, which can be deduced from the ‘Hochschild homology criterion for smoothness’ [@Avramov1992].
Now we say something about our proofs for mixed characteristic rings with finite residue fields. If $A$ is a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, then pro-excision implies that the kernel and cokernel of $K_n(A)\to K_n({\widetilde}A)$ are finite; this reduces theorem \[theorem\_intro\_3\] to the normal case, where it is known. The weaker mixed characteristic analogue of theorem \[theorem\_intro\_1\] is deduced by explicitly examining the structural description offered by that theorem. We obtain the strongest result when $n=2$, namely theorem \[theorem\_intro\_4\], because the divisible summand of $K_2({\widetilde}A)$ is known to be torsion-free; the lack of this knowledge for the other even degree $K$-groups is what prevents us from fully proving theorem \[theorem\_intro\_1\] in the mixed characteristic setting.
Theorem \[theorem\_intro\_5\] can then be proved in a similar way as theorem \[theorem\_intro\_2\]: theorem \[theorem\_intro\_4\] reduces Geller’s conjecture to an Artinian analogue. This can often be directly tackled at the level of $K_2(A/{\mathfrak}m^2)$, which can be described in a classical style using differential forms and exterior powers of the cotangent space.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I am grateful to C. Weibel and T. Geisser for interesting and helpful conversations during the Algebraic $K$-theory and Arithmetic meeting in Bedlewo, Poland, during July 2012. I also thank the anonymous referee who, as well as making many comments which improved the paper, identified certain inaccuracies and omissions in the original manuscript.
I greatly appreciate the Simons Foundation’s support via a postdoctoral fellowship.
Notation, conventions, etc. {#notation-conventions-etc. .unnumbered}
---------------------------
Every ring in this paper is commutative, though we stress that the first two theorems of section \[section\_pro\_excision\] remain valid in the associative, non-commutative case. Every ring is moreover unital, with the strict exception of certain instances in section \[section\_pro\_excision\] where we write “non-unital ring”.
Given a reduced ring $A$, we denote its total quotient ring by $\operatorname{Frac}A$, and its integral closure in $\operatorname{Frac}A$ by ${\widetilde}A$. We often require ${\widetilde}A$ to be finitely generated as an $A$-module, i.e. that $A\to {\widetilde}A$ is finite (the term “Mori ring” is in the literature but we will not use it); this is true if $A$ is excellent [@EGA_IV_II 7.8.3]. A ring is said to be normal if and only if it is reduced and integrally closed in its total quotient ring, i.e. $A={\widetilde}A$.
Since most of this paper concerns local rings, $K_0$ is typically uninteresting and therefore we do not replace $K$-theory by its non-connective completion; in particular, our long exact Mayer–Vietoris sequences finish with a possibly non-surjective map between $K_0$ terms. However, the groups $K_n$, $n\le 0$, always satisfy excision, so there is no loss of generality in only working with the non-negative $K$-groups.
Cyclic homology with respect to the base ring ${\mathbb}Z$ is denoted $HC_*=HC_*^{{\mathbb}Z}$.
Pro-excision in $K$-theory in dimension one {#section_pro_excision}
===========================================
The aim of this foundational section is to describe pro-excision in algebraic $K$-theory, prove that it is satisfied when normalising one-dimensional rings, and state the consequences, namely the long exact sequences of proposition \[proposition\_standard\_consequences\] and corollary \[corollary\_main\_application\_of\_birelative\_vanishing\]. These consequences are essential for the remainder of the paper.
If $I$ is an ideal of a ring $A$, then $K(A,I)$ is defined to be the homotopy fibre of the map $K(A)\to K(A/I)$; its homotopy groups $K_n(A,I)$ are the [*relative $K$-groups*]{} associated to $A$ and $I$, and they fit into a long exact sequence $$\cdots\to K_n(A,I)\to K_n(A)\to K_n(A/I)\to\cdots$$ If $f:A\to B$ is a morphism of rings carrying $I$ isomorphically to an ideal of $B$, then there is a canonical map $K(A,I)\to K(B,I)$, whose homotopy fibre is denoted $K(A,B,I)$; its homotopy groups are the [*birelative $K$-groups*]{}, fitting into a long exact sequence $$\cdots\to K_n(A,B,I)\to K_n(A,I)\to K_n(B,I)\to\cdots$$
A non-unital ring $I$ is said to satisfy [*excision for $K$-theory*]{} if and only if whenever $I$ is embedded as an ideal into a unital ring $A$, and $f:A\to B$ satisfies the conditions of the previous paragraph, then $K_n(A,I)\to K_n(B,I)$ is an isomorphism for all $n\ge 0$; equivalently, $K_n(A,B,I)=0$ for $n\ge 0$. Informally, the groups $K_n(A,I)$ depends only on $I$, not $A$. Obvious modifications of this terminology will be used. There is a universal choice of such a ring $A$, namely $A={\mathbb}Z\ltimes I$, sometimes denoted ${\widetilde}I$ in the literature.
The following is A. Suslin’s celebrated solution of the excision problem in $K$-theory, building on earlier work of M. Wodzicki:
Let $I$ be a non-unital ring, and set $C={\mathbb}Z$, ${\mathbb}Q$, or ${\mathbb}Z/m{\mathbb}Z$ for any $m\in{\mathbb}Z$. Fix $p>0$. Then $I$ satisfies excision for $K_n(-,C)$, for $n\le p$, if and only if ${\operatorname}{Tor}^{{\mathbb}Z\ltimes I}_n({\mathbb}Z,C)=0$ for $n=1,\dots,p$.
Suslin–Wodzicki’s criterion is rarely satisfied for ideals $I$ occurring in algebraic geometry: it is more appropriate for non-unital $C^*$-algebras and other similar function algebras. Of greater interest to us will be when excision is satisfied as we pass to increasingly fat nilpotent thickenings of an ideal; the following theorem, which we only state in the integral case, is the necessary pro extension of Suslin’s result:
Let $f:A\to B$ be a morphism of rings, and $I\subset A$ an ideal mapped isomorphically by $f$ to an ideal to $B$. Suppose that the pro abelian group $$\operatorname*{``\varprojlim\!''}_r {\operatorname}{Tor}_n^{{\mathbb}Z\ltimes(I^r)}({\mathbb}Z,{\mathbb}Z)$$ is zero for all $n>0$. Then the pro abelian group $$\operatorname*{``\varprojlim\!''}_r K_n(A,B,I^r)$$ is zero for all $n\ge 0$, and so $\operatorname*{``\varprojlim\!''}_rK_n(A,I^r)\to\operatorname*{``\varprojlim\!''}_rK_n(B,I^r)$ is an isomorphism; i.e. $I$ satisfies ‘pro-excision’ in $K$-theory.
\[remark\_pro\_cats\] Before continuing we include a brief discussion about pro abelian groups.
Everything we need about categories of pro objects may be found in one of the standard references, such as the appendix to [@ArtinMazur1969], or [@Isaksen2002]. We will use ${\operatorname}{Pro}Ab$, the category of pro abelian groups. An object of this category is a contravariant functor $X:{\mathcal}I\to {\mathcal}{C}$, where ${\mathcal}I$ is a small cofiltered category (in this paper it is fine to assume ${\mathcal}I={\mathbb}N$); this object is denoted $\operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I} X(i)$. Morphisms are given by the rule $$\operatorname{Hom}_{{\operatorname}{Pro}Ab}(\operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I}X,\operatorname*{``\varprojlim\!''}_{j\in{\mathcal}J}Y):={\varprojlim}_{j\in {\mathcal}J}{\varinjlim}_{i\in {\mathcal}I}\operatorname{Hom}_{Ab}(X(i),Y(j)),$$ where the right side is a genuine pro-ind limit in the category of sets, and composition is defined in the obvious way. For example, a pro abelian group $\operatorname*{``\varprojlim\!''}_{r\ge 1}X(r)$ is isomorphic to zero if and only if for each $r\ge 1$ there exists $s\ge r$ such that the transition map $X(s)\to X(r)$ is zero.
There is a fully faithful embedding $Ab\to{\operatorname}{Pro}Ab$ with a right adjoint $${\operatorname}{Pro}Ab\to Ab,\quad \operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I} X(i)\mapsto {\varprojlim}_{i\in{\mathcal}I} X(i),\tag{\dag}$$ which is left exact but not right exact: its first derived functor is precisely ${\varprojlim}^1$. Moreover, ${\operatorname}{Pro}{\mathcal}A$ is an abelian category; given a inverse system of exact sequences $$\cdots{\longrightarrow}X_{n-1}(i){\longrightarrow}X_n(i){\longrightarrow}X_{n+1}(i){\longrightarrow}\cdots,$$ the limit $$\cdots {\longrightarrow}\operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I}X_{n-1}(i){\longrightarrow}\operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I}X_n(i){\longrightarrow}\operatorname*{``\varprojlim\!''}_{i\in{\mathcal}I}X_{n+1}(i){\longrightarrow}\cdots$$ is an exact sequence in ${\operatorname}{Pro}Ab$. This does [*not*]{} imply that $$\cdots {\longrightarrow}{\varprojlim}_{i\in{\mathcal}I}X_{n-1}(i){\longrightarrow}{\varprojlim}_{i\in{\mathcal}I}X_n(i){\longrightarrow}{\varprojlim}_{i\in{\mathcal}I}X_{n+1}(i){\longrightarrow}\cdots$$ is exact, since () is not an exact functor.
Before introducing methods to check the conditions of the Geisser–Hesselholt theorem, we collect together the standard consequences which we will use of the vanishing of the pro birelative $K$-groups $\operatorname*{``\varprojlim\!''}_r K_n(A,B,I^r)$:
\[proposition\_standard\_consequences\] Let $f:A\to B$ be a morphism of rings, and let $I\subset A$ be an ideal mapped isomorphically by $f$ to an ideal of $B$. Suppose that $\operatorname*{``\varprojlim\!''}_r K_n(A,B,I^r)=0$ for all $n\ge 0$. Then
1. There is a natural, long exact, Mayer–Vietoris sequence $$\cdots\to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/I^r)\oplus K_n(B)\to\operatorname*{``\varprojlim\!''}_r K_n(B/I^r)\to\cdots$$
2. Suppose that $J$ (resp. $J'$) is an ideal of $A$ (resp. of $B$) containing $I$ (resp. $f(I)$). Then there is a natural, long exact, Mayer–Vietoris sequence of relative $K$-groups $$\cdots\to K_n(A,J)\to\operatorname*{``\varprojlim\!''}_r K_n(A/I^r,J/I^r)\oplus K_n(B,J')\to\operatorname*{``\varprojlim\!''}_r K_n(B/I^r,J'/I^r)\to\cdots$$
3. Suppose $J\supseteq I$ is an ideal of $A$ mapped isomorphically to an ideal of $B$. For any $n\ge 0$, the canonical map $$K_n(A,B,J)\to\operatorname*{``\varprojlim\!''}_rK_n(A/I^r,B/I^r,J/I^r)$$ is an isomorphism.
4. Suppose $J\supseteq I$ is an ideal of $A$ mapped isomorphically to an ideal of $B$. For any $n\ge 0$, the map $K_n(A,B,J)\to K_n(A/I^r,B/I^r,J/I^r)$ is split injective for $r\gg 0$.
(i)–(iii) follow in a straightforward way by taking the limit over $r$ of exact sequences of homotopy groups, or using pro spectra. (iv) is a consequence of (iii).
The homotopy groups of $\operatorname*{\operatorname*{holim}}_rK(A,B,I^r)$ fit into short exact sequences $$0\to{{\varprojlim}_r}^1K_{n+1}(A,B,I^r)\to\pi_n(\operatorname*{\operatorname*{holim}}_rK(A,B,I^r))\to{\varprojlim}_rK_n(A,B,I^r)\to 0$$ If $I$ satisfies pro-excision then the two outer terms are zero for all $n\ge1$, and so we deduce that $\operatorname*{\operatorname*{holim}}_rK(A,B,I^r)$ is contractible. Thus [ $$\xymatrix{
K_n(A) \ar@{->}[r] \ar@{->}[d] & K_n(B) \ar@{->}[d]\\
\operatorname*{\operatorname*{holim}}_rK_n(A/I^r) \ar@{->}[r] & \operatorname*{\operatorname*{holim}}_rK_n(B/I^r)
}$$ ]{} is a homotopy cartesian square of spectra. This leads to variants of the long exact sequences of the previous proposition in which the pro abelian groups are replaced by homotopy groups of homotopy limits of spectra; however, it is much easier to work with pro abelian groups throughout.
The rest of this section is dedicated to the proof of the following consequence of the Geisser–Hesselholt theorem, implying the vanishing of the pro birelative $K$-groups in a wide variety of situations, the most important of which (for us) we give in the subsequent corollary. We will say that an ideal $I$ of a ring $D$ is [*locally invertible*]{} if and only if, for every prime ideal ${\mathfrak}p\subset D$, the ideal $I_{\mathfrak}p$ is generated by a single non-zero-divisor of $D_{\mathfrak}p$. If $D$ is Noetherian, then $I$ is invertible if and only if it is locally invertible.
\[proposition\_vanishing\_of\_birelatives\_for\_locally\_free\] Let $f:A\to B$ be a morphism of rings, and $I\subset A$ an ideal mapped isomorphically by $f$ to an ideal to $B$. Suppose moreover that there exists a ring $D$ with the following two properties: $I$ is isomorphic, as a non-unital ring, to a locally invertible ideal of $D$; and $D$ is flat over ${\operatorname}{Im}({\mathbb}Z\to D)$. Then $$\operatorname*{``\varprojlim\!''}_r K_n(A,B,I^r)=0$$ for all $n\ge 0$.
The following corollary of the proposition will be the case of interest to us. If $A$ is a one-dimensional, Noetherian, reduced ring such that $A\to{\widetilde}A$ is finite, then the [*conductor ideal*]{} ${\mathfrak}f:=\operatorname{Ann}_A({\widetilde}A/A)$ is non-zero; it is the largest ideal of ${\widetilde}A$ contained inside $A$, and the quotients $A/{\mathfrak}f$ and ${\widetilde}A/{\mathfrak}f$ are Artinian. The radical of ${\mathfrak}f$ inside ${\widetilde}A$, i.e. the ideal $\sqrt{{\mathfrak}f}=\{b\in{\widetilde}A:b^r\in{\mathfrak}f\mbox{ for }r\gg0\}$, is equal to the intersection of the finitely many maximal ideals ${\mathfrak}M$ of ${\widetilde}A$ for which $A_{A\cap{\mathfrak}M}$ is not normal.
\[corollary\_main\_application\_of\_birelative\_vanishing\] Let $A$ be a one-dimensional, Noetherian, reduced ring such that $A\to{\widetilde}A$ is finite; let $I$ be a non-zero ideal of $B:={\widetilde}A$ contained inside $A$, e.g. the conductor ideal ${\mathfrak}f=\operatorname{Ann}_A(B/A)$. Then $$\operatorname*{``\varprojlim\!''}_r K_n(A,B,I^r)=0$$ for all $n\ge 0$.
In particular, if $J'$ is an ideal of $B$ contained in the radical (taken inside $B$) of ${\mathfrak}f$, and $J:=A\cap J'$, then there are long exact Mayer–Vietoris sequences $$\cdots\to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/J^r)\oplus K_n(B)\to\operatorname*{``\varprojlim\!''}_r K_n(B/J'^r)\to\cdots$$ $$\cdots\to K_n(A,J)\to\operatorname*{``\varprojlim\!''}_r K_n(A/J^r,J/J^r)\oplus K_n(B,J')\to\operatorname*{``\varprojlim\!''}_r K_n(B/J',J'/J'^r)\to\cdots$$
To prove the birelative vanishing claim we first reduce to the case where $\operatorname{Spec}A$ is connected and of dimension one. Indeed, the spectrum of $A$ has finitely many components and some of these may be spectra of fields, since we have not insisted $A$ be equi-dimensional. Therefore we may write $A=A'\times\kappa$, and $I=I'\times J$, where $A'$ is an equi-dimensional, Noetherian, reduced ring, $\kappa$ is a finite product of fields, and $I'$ (resp. $J$) is an ideal of $A'$ (resp. $\kappa$). Then $B={\widetilde}{A'}\times\kappa$ and $I'$ is an ideal of ${\widetilde}{A'}$; since $K$-theory is additive for products of rings, we have $$K_n(A,B,I^r)\cong K_n(A',{\widetilde}{A'},I'^r)\oplus K_n(\kappa,\kappa,J^r)\cong K_n(A',{\widetilde}{A'},I'^r).$$ In the same way, if $\operatorname{Spec}A'$ has multiple components then we may treat them individually by writing $A'=A''\times A'''$ and proceeding inductively. This reduces us to the connected, one-dimensional case.
Under this assumption, we will show that the conditions of proposition \[proposition\_vanishing\_of\_birelatives\_for\_locally\_free\] are satisfied with $D=B$, from which the first part of the corollary follows. Firstly, $B$ is a finite product of Dedekind domains in each of which $I$ has non-zero image, and so $I$ is automatically a locally invertible ideal of $B$. Secondly, the image of ${\mathbb}Z$ inside $B$ is either equal to ${\mathbb}Z$ (in which case $B$ is a torsion-free, hence flat, ${\mathbb}Z$-module), or is equal to a field $k$ since $\operatorname{Spec}A$ is connected (in which case $B$ is automatically flat over $k$).
For the long exact sequences, take $I=J'{\mathfrak}f$ and just apply proposition \[proposition\_standard\_consequences\], noting that the chains of ideals $\{I^r\}$, $\{J^r\}$ and $\{J'^r\}$ are cofinal in one another.
\[example\_main\_application\_of\_birelative\_vanishing\] If $A$ is a one-dimensional, Noetherian, reduced semi-local ring such that $A\to B:={\widetilde}A$ is finite, then we may apply corollary \[corollary\_main\_application\_of\_birelative\_vanishing\] to the Jacobson radicals ${\mathfrak}m$, ${\mathfrak}M$ of $A$, $B$, yielding the long exact Mayer–Vietoris sequences alluded to in the introduction $$\cdots\to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r)\oplus K_n(B)\to\operatorname*{``\varprojlim\!''}_r K_n(B/{\mathfrak}M^r)\to\cdots$$ $$\cdots\to K_n(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n(B,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_r K_n(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\to\cdots$$
Here we explain some issues surrounding the assumed finiteness of the normalisation morphism $A\to{\widetilde}A$.
Let $A$ be one-dimensional, Noetherian, reduced ring, with normalisation $B={\widetilde}A$. According to the Krull–Akizuki theorem [@Huneke2006 Thm. 4.9.2], $B$ is again one-dimensional and Noetherian, hence is a product of finitely many Dedekind domains. Moreover, if ${\mathfrak}p$ is a prime ideal of $A$, then the prime ideals of $B$ containing ${\mathfrak}p$ are precisely those finitely many prime ideals occurring in the prime ideal factorisation of ${\mathfrak}pB$; so only finitely many prime ideals of $B$ contain ${\mathfrak}p$. In particular, if $A$ is semi-local then so is $B$.
If $A\to B$ is moreover assumed to be a finite morphism, in which case the Krull–Akizuki theorem is of course unnecessary to deduce that $B$ is Noetherian, then the quotient $B/A$ is a finitely generated $A$-module, and so the conductor ${\mathfrak}f=\operatorname{Ann}_A(B/A)$ is a non-zero ideal of $A$ such that $A/{\mathfrak}f$ has finite length. Conversely, if $I$ is an ideal of $B$ contained in $A$, and we assume that $I$ is not contained inside any minimal prime ideal of $B$ (to avoid the situation that $B=B'\times B''$ and $I=I'\times\{0\}$), then $I$ contains a non-zero-divisor $f$ of $B$; hence $fB\cong B$ is an ideal of $A$, whence it is finitely generated as an $A$-module.
In conclusion, the finiteness of $A\to B$ is the minimal condition required to formulate corollary \[corollary\_main\_application\_of\_birelative\_vanishing\].
Now we turn to the proof of proposition \[proposition\_vanishing\_of\_birelatives\_for\_locally\_free\]. We will check the conditions of the Geisser–Hesselholt theorem by explicitly calculating the Tor groups in question; this is achieved in a standard way using bar resolutions, which we now briefly review.
Let $k$ be a ring, $D$ a $k$-algebra possibly without unit, and let $M,N$ be $D$-modules (recall our convention, for simplicity, that all rings are commutative). The associated two-sided bar construction is the simplicial $D$-module $$\beta_\bullet^k(N,D,M)=N\otimes_k\underbrace{D\otimes_k\cdots\otimes_kD}_{{{\mbox{\scriptsize {{\mbox{\scriptsize $\bullet$ times}}}}}}}\otimes_kM,$$ with the obvious boundary and degeneracy maps. If $D$ is unital then the presence of the “extra degeneracy” (e.g. [@Loday1992 Prop. 1.1.2]) implies that $\beta_\bullet^k(D,D,M)$ is a resolution of $M$; if $D$ and $M$ are also flat over $k$, it follows that $\beta_\bullet^k(N,D,M)=N\otimes_D\beta_\bullet^k(D,D,M)$ calculates ${\operatorname}{Tor}_*^D(N,M)$.
Suppose now that $I$ is a non-unital, flat $k$-algebra. Then the normalised chain complex associated to $\beta_\bullet^k(k,k\ltimes I,k)$, which we have just shown calculates ${\operatorname}{Tor}_*^{k\times I}(k,k)$, can be identified with the subcomplex ${\overline}\beta{}^k_\bullet(I):=\beta_\bullet^{k\ltimes I}(k,I,k)$, where $I$ acts on $k$ as zero. Explicitly, ${\overline}\beta{}^k_\bullet(I)$ is the complex of $k$-modules $$\cdots{\longrightarrow}I\otimes_kI\otimes_kI{\longrightarrow}I\otimes_kI\stackrel0{{\longrightarrow}} k{\longrightarrow}0$$ where the $k$ sits in degree $0$ and the boundary maps are $b=\sum_{i=0}^{n-1}(-1)^id_i$, where $$d(a_0\otimes\cdots\otimes a_n)=a_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_n\qquad (i=0,\dots,n-1).$$
In light of this discussion, the key to proving proposition \[proposition\_vanishing\_of\_birelatives\_for\_locally\_free\] therefore rests with the following lemma:
Let $k\to D$ be a flat morphism of rings, and $I\subseteq D$ a locally invertible ideal of $D$. Then the canonical map $$H_*(\overline\beta{}^k_\bullet(I^2))\to H_*({\overline}\beta{}^k_\bullet(I))$$ is zero for $*>0$.
To illustrate the proof we first suppose that $I=tD$ for some non-zero-divisor $t\in D$. For each $n\ge 0$ define $$s:{\overline}\beta{}^k_n(I^2)\to {\overline}\beta{}^k_{n+1}(I),\quad t^2a_0\otimes\cdots\otimes t^2a_n\mapsto t\otimes ta_0\otimes t^2a_1\otimes\cdots\otimes t^2a_n,$$ where $a_0,\dots,a_n\in D$. Letting $f:{\overline}\beta{}^k_\bullet(I^2)\to {\overline}\beta{}^k_\bullet(I)$ denote the canonical map induced by the inclusion $I^2\subseteq I$, it is clear that $d_0s=f$ and $d_is=sd_{i-1}$ for $i=1,\dots,n$. So $b's+sb'=f$, whence $f$ induces the zero map on homology.
Now we consider the general case; we must reduce to the situation where have a contracting homotopy by localising enough. First observe that ${\overline}\beta{}^k_\bullet$ has the structure of a complex of left $D$-modules; moreover, for any maximal ideal ${\mathfrak}m$ of $D$, the flatness of $D\to D_{{\mathfrak}m}$ implies that $H_*(D_{{\mathfrak}m}\otimes_D {\overline}\beta{}^k_\bullet(I))=D_{{\mathfrak}m}\otimes_D H_*({\overline}\beta{}^k_\bullet(I))$. Moreover, $ID_{{\mathfrak}m}=tD_{{\mathfrak}m}$ for some non-zero-divisor $t\in I$, and we may further write $ID_{{\mathfrak}m}={\varinjlim}_{s\in D\setminus{\mathfrak}m} \frac{t}{s}D$. So, for $s\in D\setminus{\mathfrak}m$, let ${\mathcal}N_\bullet^s(I)$ denote the subcomplex of $D_{{\mathfrak}m}\otimes_D {\overline}\beta{}^k_\bullet(I)$ given by $${\mathcal}N_n^s(I)=\left(\frac{t}{s}D\right)\otimes_k I^{\otimes_k n}$$ (and similarly ${\mathcal}N_\bullet^s(I^2)$, replacing $t$ by $t^2$). Then $$D_{{\mathfrak}m}\otimes_D H_*({\overline}\beta{}^k_\bullet(I))={\varinjlim}_{s\in D\setminus{\mathfrak}m} H_*({\mathcal}N_\bullet^s(I))$$ (and similarly for $I^2$), so we have reduced the problem to proving that the canonical map $$f:{\mathcal}N_\bullet^s(I^2)\to{\mathcal}N_\bullet^s(I)$$ induces zero on homology. But this follows similarly to the invertible case, using the contracting homotopy $$s:{\mathcal}N_n^s(I^2)\to{\mathcal}N_{n+1}^s(I),\quad \frac{t^2}{s}a\otimes b_1\otimes\cdots\otimes b_n\mapsto \frac{t}{s}\otimes ta\otimes b_1\otimes\cdots\otimes b_n,$$ where $a\in D$ and $b_1,\dots,b_n\in I^2$.
Let $k,D,I$ be as in the previous lemma, and let $n>0$. Then $${\operatorname}{Tor}_n^{k\ltimes(I^2)}(k,k)\to {\operatorname}{Tor}_n^{k\ltimes I}(k,k)$$ is zero, and so $$\operatorname*{``\varprojlim\!''}_r{\operatorname}{Tor}_n^{k\ltimes(I^r)}(k,k)=0.$$
The first claim is just a statement of the previous lemma, since we had previously shown $H_*({\overline}\beta{}^k_\bullet(I))\cong{\operatorname}{Tor}_*^{k\ltimes I}(k,k)$ (and similarly for $I^2$). The second claim follows by replacing $I$ by successively higher powers of itself and repeatedly applying the first claim, which remains applicable since $I^r$ is still a locally invertible ideal of $D$.
We need one more lemma, for which we refer to Geisser and Hesselholt’s paper for a proof. In fact, their result is much more general and the proof is based on manipulations of Eilenberg–Maclane spectra; for our special case, a direct analysis of the Tor groups using spectral sequences is possible and is due to appear in forthcoming work.
Let $k'\to k$ be a surjective map of rings, and $I$ a non-unital $k$-algebra. Suppose that $\operatorname*{``\varprojlim\!''}_r{\operatorname}{Tor}_n^{k\ltimes(I^r)}(k,k)=0$ for all $n>0$. Then $\operatorname*{``\varprojlim\!''}_r{\operatorname}{Tor}_n^{k'\ltimes(I^r)}(k',k')=0$ for all $n>0$.
[@GeisserHesselholt2011 Prop. 3.6].
Now we may prove that the pro birelative $K$-groups vanish in situations of interest to us:
By assumption there is a ring $D$ which contains $I$ as a locally invertible ideal and which is flat over $k:={\operatorname}{Im}({\mathbb}Z\to D)$. The previous corollary implies that $$\operatorname*{``\varprojlim\!''}_r{\operatorname}{Tor}_n^{k\ltimes(I^r)}(k,k)=0$$ for all $n>0$, and then the previous lemma, with $k'={\mathbb}Z$, implies $\operatorname*{``\varprojlim\!''}_r{\operatorname}{Tor}_n^{{\mathbb}Z\ltimes(I^r)}({\mathbb}Z,{\mathbb}Z)=0$ for all $n>0$. Finally we apply the Geisser–Hesselholt theorem from the start of the section to complete the proof.
Applications in equal characteristic
====================================
In this section we apply corollary \[corollary\_main\_application\_of\_birelative\_vanishing\] to the study of one-dimensional local rings of equal characteristic. Sections \[subsection\_finite\_char\] and \[subsection\_char\_zero\] contain preliminary lemmas, and we then establish in section \[subsection\_main\_results\] when the pro-excision, Mayer–Vietoris sequences breaks into short exact sequences. In particular, we show that the $K$-theory of a complete, equal charactersitic, one-dimensional local ring is captured by its generic $K$-theory and by all thickenings of its closed point (theorem \[theorem\_main\_theorem\_II\]), an idea also encapsulated by our application in section \[subsection\_KH\] to homotopy invariant $K$-theory.
Using pro-excision we prove Geller’s conjecture in the finite characteristic, perfect residue field case in section \[subsubsection\_Geller\], and then turn to characteristic zero and a relation with cyclic homology in section \[subsection\_rel\_to\_HC\].
Sections \[subsection\_KH\]–\[subsection\_rel\_to\_HC\] are independent of each other.
Finite charactersitic {#subsection_finite_char}
---------------------
We begin with a result in finite characteristic. The following does not seem to be available in exactly this form anywhere in the literature, but readily follows from deep results on the $K$-theory of truncated polynomial algebras by L. Hesselholt and I. Madsen, building on older work by S. Bloch.
If $A$ is any ring then we write $K^{{\mbox{\scriptsize sym}}}_n(A)\subseteq K_n(A)$ to denote the symbolic part of the $K$-group, i.e. the image of the canonical map from Milnor $K$-theory to Quillen $K$-theory; moreover, we write $K_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)},{(t)}):={\operatorname{Ker}}(K_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)})\to K_n^{{\mbox{\scriptsize sym}}}(A))$.
\[proposition\_symbolic\_in\_char\_p\] Let $A$ be a regular, Noetherian, local ${\mathbb}F_p$-algebra, and let $n\ge1$. Then the canonical injection $$\operatorname*{``\varprojlim\!''}_rK_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)},{(t)})\to \operatorname*{``\varprojlim\!''}_rK_n(A[t]/{(t^r)},{(t)})$$ is an isomorphism.
For a moment let $A$ be any ring, and let $r\ge 1$. We denote by ${\mathbb}W_r(A)$ the length $r$ big Witt vectors; this is a commutative, unital ring whose underlying abelian group structure is $$1+tA[t]/1+t^{r+1}A[t].$$ Next let ${\mathbb}W_r\Omega_A^*$ be the big de Rham Witt complex (e.g., [@Hesselholt2001]); it is a differential graded algebra over ${\mathbb}W_r(A)$, equipped with a canonical surjection $$\Omega_{{\mathbb}W_r(A)}^*{\twoheadrightarrow}{\mathbb}W_r\Omega_A^*$$ of dg algebras, which is an isomorphism when $*=0$.
Now let $A$ be a regular, Noetherian ${\mathbb}F_p$-algebra. In [@Hesselholt2001 Thm. A], Hesselholt and Madsen determined the $K$-theory of truncated polynomial algebras over $A$ using R. McCarthy’s [@McCarthy1997] comparison theorem between relative $K$-groups and topological cyclic homology: their primary result takes the form of a long exact sequence $$\cdots{\longrightarrow}\bigoplus_{i\ge 1}{\mathbb}W_i\Omega_A^{n+1-2i} \stackrel{\oplus_iV_r}{{\longrightarrow}}\bigoplus_{i\ge 1}{\mathbb}W_{ir}\Omega_A^{n+1-2i}\stackrel{{\varepsilon}}{{\longrightarrow}} K_n(A[t]/{(t^r)},{(t)}){\longrightarrow}\cdots,$$ where $V_r:{\mathbb}W_i\Omega_A^*\to{\mathbb}W_{ir}\Omega_A^*$ denotes the Verschiebung map and we will describe a special case of ${\varepsilon}$ below. To be precise, the result in [@Hesselholt2001] is stated only when $A$ is smooth over a perfect field of characteristic $p$, but it was observed in later work that it extended to the more general regular case using Neron–Popescu desingularisation [@Popescu1985; @Popescu1986].
The behaviour of this complex with $r$ was later given in [@Hesselholt2008 Thm. A]: given $s\ge r$, there is a morphism of complexes $$\xymatrix{
\cdots \ar[r] & \bigoplus_{i\ge 1}{\mathbb}W_i\Omega_A^{n+1-2i} \ar[r]\ar[d]_0 & \bigoplus_{i\ge 1}{\mathbb}W_{is}\Omega_A^{n+1-2i}\ar[r]\ar[d] & K_n(A[t]/{(t^s)},{(t)})\ar[r]\ar[d] & \cdots \\
\cdots \ar[r] & \bigoplus_{i\ge 1}{\mathbb}W_i\Omega_A^{n+1-2i} \ar[r] & \bigoplus_{i\ge 1}{\mathbb}W_{ir}\Omega_A^{n+1-2i}\ar[r] & K_n(A[t]/{(t^r)},{(t)})\ar[r] & \cdots
}$$ where the left vertical arrow is zero, the right vertical arrow is the canonical map, and the central vertical arrow is $\bigoplus_{i\ge 1}$ of the maps $$\xymatrix{
{\mathbb}W_{is}\Omega_A^{n+1-2i}\ar[r] &
{\mathbb}W_{ir}\Omega_A^{n+1-2i}\ar[r]^{\times {\alpha}(s,r)} &
{\mathbb}W_{ir}\Omega_A^{n+1-2i},
}$$ where the first map is the canonical reduction map and the second map is multiplication by a certain element ${\alpha}(s,r)\in{\mathbb}W({\mathbb}F_p)={\varprojlim}_r{\mathbb}W_r({\mathbb}F_p)$. Moreover, \[Thm. 6.3, op. cit.\] states that for any $r\ge 1$ there exists $s_0\ge r$ such that if $s\ge s_0$ and $i>1$ then $\times {\alpha}(s,r)$ is the zero map on ${\mathbb}W_{ir}\Omega_A^{n+1-2i}$.
Applying $\operatorname*{``\varprojlim\!''}_r$ to the system of complexes, one now quickly sees that the limit of the $i=1$ parts of the ${\varepsilon}$ maps $$\operatorname*{``\varprojlim\!''}_r{\mathbb}W_r\Omega_A^{n-1} {\longrightarrow}\operatorname*{``\varprojlim\!''}_rK_n(A[t]/{(t^r)},{(t)})\tag{\dag}$$ is an isomorphism, which we will denote $u$. Now, assuming in addition that $A$ is local, we must more carefully describe $u$; we refer the reader to [@Illusie1979 I.5] or [@Stienstra1985 §8] for more details on what we are about to say.
In [@Bloch1977 II], Bloch called the relative $K$-group $C_rK_n(A):=K_n(A[t]/{(t^{r+1})},{(t)})$ the [*curves of length $r$ on $K_n$*]{}; let $SC_rK_n(A)=K_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^{r+1})},{(t)})$ be its symbolic part. Bloch (when $p\neq 2$, an assumption that was removed by [@Kato1980 §2.2]; see also [@Stienstra1982 §11]) showed that $S{\widehat}C_rK_{*+1}(A):=\operatorname*{``\varprojlim\!''}_r\bigoplus_{n\ge 0}SC_rK_{n+1}(A)$ can be equipped with the structure of pro differential graded associative algebra in such a way that, looking at its degree $0$ component, the inverse of the determinant map $$\{\,\}:\operatorname*{``\varprojlim\!''}_r{\mathbb}W_r(A)=\operatorname*{``\varprojlim\!''}_r1+tA[t]/1+t^{r+1}A[t]\to S{\widehat}C_rK_1(A)$$ is an isomorphism of pro rings. The universal property of $\operatorname*{``\varprojlim\!''}_r\Omega_{{\mathbb}W_r(A)}^*$ as a pro differential graded algebra therefore implies that there is an induced homomorphism of $\operatorname*{``\varprojlim\!''}_r{\mathbb}W_r(A)$-algebras $$\phi:\operatorname*{``\varprojlim\!''}_r\Omega_{{\mathbb}W_r(A)}^*\to S{\widehat}C_rK_{*+1}(A)$$ (see [@Bloch1977 Thm. II.§6.2.1] for a detailed proof). This descends to the quotient $\operatorname*{``\varprojlim\!''}_r{\mathbb}W_r\Omega_A^*$ by [@Illusie1979 Thm. I.5.2], inducing $$u:\operatorname*{``\varprojlim\!''}_r{\mathbb}W_r\Omega_A^* \to S{\widehat}C_rK_{*+1}(A)= \operatorname*{``\varprojlim\!''}_rK_{*+1}^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)},{(t)}),$$ which is the desired map. Thus the isomorphism () has image inside $\operatorname*{``\varprojlim\!''}_rK_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)},{(t)})$, which completes the proof.
\[remark\_vanishing\_of\_even\_K\_groups\_for\_truncated\_polys\] Suppose that $k$ is a perfect field of characteristic $p\neq0$. Then $\Omega_{{\mathbb}W_r(k)}^*=0$ for $*>0$, so the surjection $\Omega_{{\mathbb}W_r(k)}^*{\twoheadrightarrow}{\mathbb}W_r\Omega_k^*$ shows that ${\mathbb}W_r\Omega_k^*=0$ for $*>0$. Applying the isomorphism () of the previous proof, with $A=k$, we deduce that $$\operatorname*{``\varprojlim\!''}_rK_n(k[t]/{(t^r)},{(t)})=0$$ for all $n\ge 2$. In fact, if $n\ge 2$ is even then already $K_n(k[t]/{(t^r)},{(t)})=0$ for all $r\ge 1$ by [@Hesselholt1997a Thm. A]. In our proof of Geller’s conjecture we will need the vanishing of this relative group when $n=2$, which can proved in a straightforward classical way by manipulating Steinberg symbols: see e.g., [@Dennis1975 Lemma 3.4].
Suppose that $A$ is a one-dimensional, Noetherian, reduced semi-local ${\mathbb}F_p$-algebra such that $A\to{\widetilde}A$ is finite, and that the residue fields of $A$ are perfect. Then ${\widetilde}A/{\mathfrak}M^r$ may be identified with a finite product of truncated polynomial rings (see the proof of corollary \[corollary\_surjectivity\_in\_limit\]) and so the previous paragraph implies that $\operatorname*{``\varprojlim\!''}_rK_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)=0$ for all $n\ge 2$. From corollary \[corollary\_main\_application\_of\_birelative\_vanishing\] we deduce that $$K_n(A,{\mathfrak}m)\cong K_n(A,{\mathfrak}M)\oplus\operatorname*{``\varprojlim\!''}_rK_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)$$ for $n\ge 2$. It follows that the map $K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M)$ is surjective and that its kernel is isomorphic to a direct summand of $K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)$ for $r\gg0$.
Characteristic zero {#subsection_char_zero}
-------------------
Now we establish the analogue of the previous proposition in characteristic zero. The proof is inspired by ideas from two papers by A. Krishna [@Krishna2005; @Krishna2010], especially section 3 of [@Krishna2005]; parts of the proof are also presented in special cases in the author’s work [@Morrow_Singular_Gersten].
The following lemma and proposition make use of the Hodge decomposition of Hochschild and cyclic homology, which we now briefly review; more details may be found in [@Loday1992 §4.5–4.6]. Let $k$ be a ring containing ${\mathbb}Q$ and let $R$ be a $k$-algebra; all Hochschild and cyclic homologies will be taken with respect to $k$.
The action of the Eulerian idempotents $e_n^{(i)}\in{\mathbb}Q[{\operatorname}{Sym}_n]$, for $1\le i\le n$, on the Hochschild complex $C_\bullet(R)$ are compatible with the boundary maps, thereby resulting in a direct sum decomposition of the Hochschild complex $$C_\bullet(R)=\bigoplus_{i\ge 1}C_\bullet^{(i)}(R),\quad\quad\mbox{where } C_n^{(i)}(R):=e_n^{(i)}C_n(R).$$ Thus the Hochschild homology groups decompose as $HH_n(R)=\bigoplus_{i=1}^n HH_n^{(i)}(R)$, where $HH_n^{(i)}(R):=H_n(C_\bullet^{(i)}(R))$. The cyclic homology groups decompose in a similar way: $HC_n(R)=\bigoplus_{i=1}^n HC_n^{(i)}(R)$.
The canonical surjection $HH_n(R)\to\Omega_{R/k}^n$ and anti-symmetrisation map ${\varepsilon}_n:\Omega_{R/k}^n\to HH_n(R)$ induce isomorphisms and .
The following lemma describes the cyclic homology of graded algebras in the limit; I am grateful to C. Weibel for explaining this style of argument to me.
\[lemma\_HC\_of\_graded\_ring\] Let $k$ be a ring (with respect to which all Hochschild/cyclic homologies in the lemma will be taken), and $A=\bigoplus_{w\ge 0}A_w$ a positively graded $k$-algebra. Write $A_{\ge r}=\bigoplus_{w\ge r}A_w\subseteq A$ for each $r\ge 0$. Then for any $n\ge 0$, the canonical map $$HH_n(A)\to\operatorname*{``\varprojlim\!''}_r HH_n(A/A_{\ge r})$$ is surjective.
Suppose further that $k$ contains ${\mathbb}Q$ and that $A$ is a filtered inductive limit of smooth, finite-type $k$-algebras. Then $$\operatorname*{``\varprojlim\!''}_r {\widetilde}{HC}_n^{(i)}(A/A_{\ge r})=0$$ for all $0\le i<n$, where the notation means the $i^{{\mbox{\scriptsize th}}}$ piece of the Hodge decomposition of the reduced cyclic homology ${\widetilde}{HC}_*=HC_*/HC_*(A_0)$. Hence $$\operatorname*{``\varprojlim\!''}_r{\widetilde}{HC}_n(A/A_{\ge r})\cong\operatorname*{``\varprojlim\!''}_r{\widetilde}\Omega^n_{(A/A_{\ge r})/k}/d{\widetilde}\Omega^{n-1}_{(A/A_{\ge r})/k},$$ where ${\widetilde}{\Omega}^*$ denotes the quotient by $\Omega_{A_0/k}^*$.
For any $k$-algebra $R$ we let $C_\bullet(R)=C^k_\bullet(R)=R^{\otimes_k\bullet+1}$ denote its Hochschild complex, whose homology is $HH_*(R)$. If $R$ is positively graded, then $C_\bullet(R)$ breaks into a direct sum of subcomplexes, $C_\bullet(R)=\bigoplus_{w\ge 0}C_\bullet(R)_w$, where the weight $w$ piece is $$C_n(R)_w:=\bigoplus_{i_0+\cdots+i_n=w}R_{i_0}\otimes_k\cdots\otimes_kR_{i_n}\subseteq C_n(R)$$ This in turn induces a decomposition of the Hochschild homology, $HH_*(R)=\bigoplus_{w\ge 0}HH_*(R)_w$. Also, write $F^pC_\bullet(R)=\bigoplus_{w\ge p}C_\bullet(R)_w$ for the associated filtration on the complex.
Let $r\ge 0$ be given. By considering two cases we will show that if $s>(n+1)r$ then the map $$\frac{HH_n(A/A_{\ge s})_w}{{\operatorname}{Im}HH_n(A)_w}{\longrightarrow}\frac{HH_n(A/A_{\ge r})_w}{{\operatorname}{Im}HH_n(A)_w}\tag{\dag}$$ is zero for all $w\ge 0$:
case $\mathbf{w>(n+1)r}$:
: Then it is obvious that $C_n(A/A_{\ge r})_w=0$, and so $HH_n(A/A_{\ge r})_w=0$, which clearly suffices to prove our claim.
case $\mathbf{w<s}$:
: Notice that $C_n(A,A_{\ge s}):={\operatorname{Ker}}(C_n(A)\to C_n(A/A_{\ge s}))$ is additively generated by symbols $a_0\otimes\cdots\otimes a_n$ where $|a_i|\ge s$ for at least one $i$, and therefore $C_\bullet(A,A_{\ge s})\subseteq F^sC_\bullet(A)$. Hence $HH_n(A)\to HH_n(A/A_{\ge s})$ is an isomorphism on the weight $w$ pieces whenever $0\le w<s$; so in this case the left side of () vanishes, which is again sufficient to prove our claim.
Thus for each $n,r\ge 0$ we have found $s\ge r$ such that the map $$\frac{HH_n(A/A_{\ge s})}{{\operatorname}{Im}HH_n(A)}{\longrightarrow}\frac{HH_n(A/A_{\ge r})}{{\operatorname}{Im}HH_n(A)}$$ is zero; this is precisely the statement that $HH_n(A)\to\operatorname*{``\varprojlim\!''}_rHH_n(A/A_{\ge r})$ is surjective, thereby proving the first claim of the lemma.
Next, assuming that $k$ contains ${\mathbb}Q$ (so that the Hodge decomposition exists) and that $A$ is a filtered inductive limit of smooth, finite-type $k$-algebras (so that the Hochschild–Konstant–Rosenberg theorem [@Loday1992 Thm. 3.4.4] implies $HH_n^{(i)}(A)=0$ for $i\neq n$), we see that $$\operatorname*{``\varprojlim\!''}_r HH_n^{(i)}(A/A_{\ge r})=0$$ whenever $0\le i<n$. So certainly $\operatorname*{``\varprojlim\!''}_r{\widetilde}{HH}_n^{(i)}(A/A_{\ge r})=0$ whenever $0\le i<n$.
Extending this to cyclic homology is straightforward. Indeed, the SBI sequence for the reduced Hochschild and cyclic homology of a graded ring $R$ breaks into short exact sequences [@Loday1992 Thm. 4.1.13], and the $S$, $B$, and $I$ maps respect the Hodge grading in a suitable way \[Prop. 4.6.9, op. cit.\]: $$0\to{\widetilde}{HC}_{n-1}^{(i-1)}(R){\xrightarrow}{B} {\widetilde}{HH}_n^{(i)}(R){\xrightarrow}{I}{\widetilde}{HC}_n^{(i)}(R)\to 0.$$ Thus we obtain short exact sequences of pro $A$-modules, $$0\to\operatorname*{``\varprojlim\!''}_r{\widetilde}{HC}_{n-1}^{(i-1)}(A/A_{\ge r})\to \operatorname*{``\varprojlim\!''}_r{\widetilde}{HH}_n^{(i)}(A/A_{\ge r})\to\operatorname*{``\varprojlim\!''}_r{\widetilde}{HC}_n^{(i)}(A/A_{\ge r})\to 0,$$ and we have just proved that the central term vanishes when $0\le i<n$; hence the right term vanishes, proving the desired vanishing claim for the limit of cyclic homology.
The final claim is immediate from the standard identification of $HC_n^{(n)}$ with $\Omega^n/d\Omega^{n-1}$.
\[proposition\_limit\_is\_symbols\_char\_0\] Let $A$ be a regular, Noetherian, local ${\mathbb}Q$-algebra, and let $n\ge1$. Then the canonical injection $$\operatorname*{``\varprojlim\!''}_rK_n^{{\mbox{\scriptsize sym}}}(A[t]/{(t^r)},{(t)})\to \operatorname*{``\varprojlim\!''}_rK_n(A[t]/{(t^r)},{(t)})$$ is an isomorphism.
According to the previous lemma with $k={\mathbb}Q$ and $A[t]$ in place of $A$ (which is a filtered inductive limit of smooth, finite-type ${\mathbb}Q$-algebras by Neron–Popescu desingularisation), $$\operatorname*{``\varprojlim\!''}_r HC_n^{(i)}(A[t]/{(t^r)},{(t)})=0\tag{\dag}$$ whenever $0\le i<n$.
Notice that $K_n(A[t]/{(t^r)},{(t)})$ is a relative $K$-group for a nilpotent ideal in a ${\mathbb}Q$-algebra, hence is a ${\mathbb}Q$-vector space by Weibel [@Weibel1982 1.5]. So T. Goodwillie’s celebrated isomorphism [@Goodwillie1986] takes the form $$K_n(A[t]/{(t^r)},{(t)}){\stackrel{\simeq}{\to}}HC_{n-1}(A[t]/{(t^r)},{(t)}).$$ This isomorphism respects the Adams/Hodge decompositions by [@Cathelineau1990; @Cortinas2009][^2], thus inducing $$K_n^{(i)}(A[t]/{(t^r)},{(t)}){\stackrel{\simeq}{\to}}HC_{n-1}^{(i-1)}(A[t]/{(t^r)},{(t)}).$$ The vanishing of () therefore implies that the canonical inclusion $$\operatorname*{``\varprojlim\!''}_r K_n^{(n)}(A[t]/{(t^r)},{(t)})\to \operatorname*{``\varprojlim\!''}_r K_n(A[t]/{(t^r)},{(t)})$$ is an isomorphism.
It remains to show that $K_n^{(n)}(A[t]/{(t^r)},{(t)})$ is entirely symbolic. We start with the following classical Nesterenko–Suslin result [@Suslin1989 Thm. 4.1]: if $R$ is a local ring with an infinite residue field, then $K_n^{(n)}(R)_{{\mathbb}Q}\cong K_n^M(R)_{{\mathbb}Q}$. Therefore $$K_n^{(n)}(A[t]/{(t^r)},{(t)})={\operatorname{Ker}}(K_n^M(A[t]/{(t^r)})\to K_n^M(A))\otimes_{{\mathbb}Z}{\mathbb}Q.$$ Next we use the following standard result concerning Milnor $K$-theory: If $R$ is a ring and $J\subseteq R$ is an ideal contained inside its Jacobson radical, then $\kappa:={\operatorname{Ker}}(K_n^M(R)\to K_n^M(R/J))$ is generated by Steinberg symbols of the form $\{a_1,a_2\dots,a_n\}$, where $a_1\in1+J$ and $a_2,\dots,a_n\in{A^{\!\times}}$. (Indeed, if we let $\Lambda$ denote the subgroup of $K_n^M(R)$ generated by such elements, then it is enough to check that $$K_n^M(R/J)\to K_n^M(R)/\Lambda,\quad\{a_1,\dots,a_n\}\mapsto \{{\widetilde}a_1,\dots,{\widetilde}a_n\}$$ is well-defined, where ${\widetilde}a\in{R^{\!\times}}$ denotes an arbitrary lift of $a\in{(R/J)^{\!\times}}$.) If moreover $J$ is nilpotent and $R\supseteq{\mathbb}Q$, then $1+J$ is a divisible group, whence $\kappa$ is also a divisible group and so $${\operatorname{Ker}}(K_n^M(R)\to K_n^M(R/J))\to{\operatorname{Ker}}(K_n^M(R)\to K_n^M(R/J))\otimes_{{\mathbb}Z}{\mathbb}Q$$ is surjective. Taking $R=A[t]/{(t^r)}$ and $J={(t)}$ completes the proof.
(An alternative way to show that $K_n^{(n)}(A[t]/{(t^r)},{(t)})$ is entirely symbolic is to show that $HC_{n-1}^{(n-1)}(A[t]/{(t^r)},{(t)})$ consists entirely of logarithmic forms: these correspond to symbols in $K$-theory via the Goodwillie isomorphism.)
The short exact Mayer–Vietoris sequences {#subsection_main_results}
----------------------------------------
The propositions of sections \[subsection\_finite\_char\] and \[subsection\_char\_zero\] yield the following essential corollary:
\[corollary\_surjectivity\_in\_limit\] Let $B$ be a one-dimensional, normal, semi-local ring containing a field; let ${\mathfrak}M$ denote its Jacobson radical. Then the canonical map $$K_n(B,{\mathfrak}M)\to \operatorname*{``\varprojlim\!''}_rK_n(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$$ is surjective for all $n\ge0$.
Each of the groups on the right is zero if $n=0$, so assume $n>0$. Let ${\widehat}B$ be the ${\mathfrak}M$-adic completion of $B$. There is an isomorphism ${\widehat}B\cong \prod_{{\mathfrak}n}{\widehat}{B_{{\mathfrak}n}}$, where ${\mathfrak}n$ varies over the finitely many maximal ideals of $B$, and each ${\widehat}{B_{{\mathfrak}n}}$ is a complete discrete valuation ring containing a field, hence isomorphic to $k_{{\mathfrak}n}[[t]]$ for some field $k_{{\mathfrak}n}$. Then $$K_n(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\cong\bigoplus_{{\mathfrak}n}K_n(k_{{\mathfrak}n}[t]/{(t^r)},{(t)})$$ and so the propositions of sections \[subsection\_finite\_char\] and \[subsection\_char\_zero\] imply that $$\operatorname*{``\varprojlim\!''}_rK_n^{{\mbox{\scriptsize sym}}}(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r){\stackrel{\simeq}{\to}}\operatorname*{``\varprojlim\!''}_rK_n(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r).$$
This reduces the claim to showing that $K_n^{{\mbox{\scriptsize sym}}}(B,{\mathfrak}M)\to K_n^{{\mbox{\scriptsize sym}}}(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$ is surjective for each $n,r\ge 1$. But this is clear: ${\mathfrak}M^r$ is contained in the Jacobson radical of $B$, so ${B^{\!\times}}\to{(B/{\mathfrak}M^r)^{\!\times}}$ is surjective.
We are now prepared to prove our first main result, stating that the long exact, pro-excision, Mayer–Vietoris sequence breaks into short exact sequences in dimension one; special cases were established in [@Morrow_Singular_Gersten Prop. 3.8] and [@Krishna2005 Thm. 3.6]. This serves as a singular analogue of the Gersten conjecture:
\[theorem\_main\_theorem\] Let $A$ be a one-dimensional, Noetherian, reduced semi-local ring containing a field, and such that $A\to{\widetilde}A$ is finite; let ${\mathfrak}m$ and ${\mathfrak}M$ denote the Jacobson radicals of $A$ and ${\widetilde}A$. For any $n\ge 0$, there is a natural short exact sequence $$0 \to K_n(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n({\widetilde}A,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\to0$$ of pro abelian groups. In other words, the square [ $$\xymatrix{
K_n(A,{\mathfrak}m) \ar@{->}[r] \ar@{->}[d] & K_n({\widetilde}A,{\mathfrak}M) \ar@{->}[d]\\
\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r) \ar@{->}[r] & \operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)
}$$ ]{} is bicartesian in Pro$Ab$.
According to example \[example\_main\_application\_of\_birelative\_vanishing\], there is a long exact Mayer–Vietoris sequence of relative $K$-groups $$\cdots\to K_n(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n({\widetilde}A,{\mathfrak}M)\stackrel{(\ast)}{\to}\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\to\cdots$$ But $B$ satisfies the conditions of the previous corollary, so arrow ($\ast$) is surjective for all $n\ge 0$, which completes the proof.
\[corollary\_to\_main\_theorem\] Let $A,{\mathfrak}m,{\mathfrak}M$ be as in the previous theorem, and fix $n\ge0$.
1. The kernel (resp. cokernel) of the map $K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M)$ is isomorphic to a direct summand of the kernel (resp. cokenel) of the map $K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\to K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$ for $r\gg0$. In particular, the canonical map $$K_n(A,{\mathfrak}m)\to K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n({\widetilde}A,{\mathfrak}M)$$ is injective for $r\gg 0$.
2. Let $A'$ also satisfy the conditions of the previous theorem, and suppose that $A\to A'$ is an $S$-analytic isomorphism [@Weibel1986], where $S$ is the set of non-zero-divisors of $A$ (e.g., $A'$ could be an étale extension with the same residue field, or the Henselization or completion of $A$). Then the $S$-analytic, relative Mayer–Vietoris sequence breaks into short exact sequences $$0\to K_n(A,{\mathfrak}m)\to K_n(A',{\mathfrak}m')\oplus K_n({\widetilde}A,{\mathfrak}M)\to K_n({\widetilde}{A'},{\mathfrak}M')\to 0$$
In other words, the kernel and cokernel of the map $K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M)$ are unchanged after replacing $A$ by $A'$.
According to the theorem, the square of groups [ $$\xymatrix{
K_n(A,{\mathfrak}m) \ar@{->}[r] \ar@{->}[d] & K_n({\widetilde}A,{\mathfrak}M) \ar@{->}[d]\\
\operatorname*{``\varprojlim\!''}_rK_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r) \ar@{->}[r] & \operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)
}$$ ]{} is bicartesian, and so the kernels (resp. cokernels) of the two horizontal arrows are the same. This easily implies (i). For (ii), notice that an $S$-analytic isomorphism $A\to A'$ does not change the bottom row of the diagram, and thus [ $$\xymatrix{
K_n(A,{\mathfrak}m) \ar@{->}[r] \ar@{->}[d] & K_n({\widetilde}A,{\mathfrak}M) \ar@{->}[d]\\
K_n(A',{\mathfrak}m') \ar@{->}[r] & K_n({\widetilde}{A'},{\mathfrak}M')
}$$ ]{} is also bicartesian.
The following is the non-relative version of the theorem, in which we suppose that $K_n({\widetilde}A)\to K_n({\widetilde}A/{\mathfrak}M)$ is surjective for all $n\ge 0$; e.g., perhaps ${\widetilde}A\to{\widetilde}A/{\mathfrak}M$ splits, which is automatic if $A$ is complete but also holds for unibranch, rational singularities.
\[theorem\_main\_theorem\_II\] Let $A,{\mathfrak}m,{\mathfrak}M$ be as in the previous theorem, and suppose further that $K_n({\widetilde}A)\to K_n({\widetilde}A/{\mathfrak}M)$ is surjective for all $n\ge0$. Then, for any $n\ge 0$, there is a natural short exact sequence $$0 \to K_n(A)\to\operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r)\to0.$$
Using the same argument as the previous theorem, it is enough to show that $K_n({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_rK_n({\widetilde}A/{\mathfrak}M^r)$ is surjective. The long exact $K$-theory sequences for ${\widetilde}A\to{\widetilde}A/{\mathfrak}M$ and ${\widetilde}A/{\mathfrak}M^r\to{\widetilde}A/{\mathfrak}M$ split, yielding short exact sequences $$\xymatrix{
0 \ar[r] & K_n({\widetilde}A,{\mathfrak}M) \ar[r]\ar[d] & K_n({\widetilde}A) \ar[r]\ar[d] & K_n({\widetilde}A/{\mathfrak}M)\ar[r]\ar@{=}[d] & 0\\
0 \ar[r] & K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r) \ar[r] & K_n({\widetilde}A/{\mathfrak}M^r) \ar[r] & K_n({\widetilde}A/{\mathfrak}M)\ar[r] & 0
}$$ After taking $\operatorname*{``\varprojlim\!''}_r$ of the bottom row the left vertical arrow becomes surjective by corollary \[corollary\_surjectivity\_in\_limit\], whence the central vertical arrow also becomes surjective.
The following is the non-relative version of corollary \[corollary\_to\_main\_theorem\]:
Let $A,{\mathfrak}m,{\mathfrak}M$ satisfy the conditions of the previous theorem, and fix $n\ge0$.
1. The kernel (resp. cokernel) of the map $K_n(A)\to K_n({\widetilde}A)$ is isomorphic to a direct summand of the kernel (resp. cokenel) of the map $K_n(A/{\mathfrak}m^r)\to K_n({\widetilde}A/{\mathfrak}M^r)$ for $r\gg0$. In particular, the canonical map $$K_n(A)\to K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)$$ is injective for $r\gg 0$.
2. Let $A'$ also satisfy the conditions of the previous theorem, and suppose that $A\to A'$ is an $S$-analytic isomorphism [@Weibel1986], where $S$ is the set of non-zero-divisors of $A$ (e.g., $A'$ could be an étale extension with the same residue field, or the Henselization or completion of $A$). Then the $S$-analytic, relative Mayer–Vietoris sequence breaks into short exact sequences $$0\to K_n(A)\to K_n(A')\oplus K_n({\widetilde}A)\to K_n({\widetilde}{A'})\to 0$$
In other words, the kernel and cokernel of the map $K_n(A)\to K_n({\widetilde}A)$ are unchanged after replacing $A$ by $A'$.
One repeats the proof of corollary \[corollary\_to\_main\_theorem\], using absolute rather than relative $K$-groups.
Suppose that $A$ satisfies the conditions of the second of the previous theorems. Then Quillen’s proof of the Gersten conjecture in the geometric case, and its extension to the general equal characteristic case by Neron–Popescu desingularisation [@Panin2003] (in fact, the case of an arbitrary equal charactersitic discrete valuation ring had already been treated by C. Sherman [@Sherman1978]), tells us that $K_n({\widetilde}A)\to K_n(F)$ is injective, where $F$ is the total quotient ring of $A$. The corollary therefore implies that ${\operatorname{Ker}}(K_n(A)\to K_n(F))$ is ‘small enough’ to embed into $K_n(A/{\mathfrak}m^r)$ for some sufficiently large $r$. Thus the philosophy of the second theorem is the following:
> The $K$-theory of $A$ is determined by its generic information together with all infinitesimal thickenings of its closed point.
This philosophy will be made even more precise by our result on $KH$-theory in the next section.
Let $A,{\mathfrak}m,{\mathfrak}M$ be as in the theorems; we will finish this section by showing that the conclusions of the second theorem and its corollary are not always valid without some sort of surjectivity/splitting assumption on ${\widetilde}A$. Indeed, supposing that $K_n(A)\to K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)$ is injective for $r\gg 0$, we see from the second long exact sequence of corollary \[corollary\_main\_application\_of\_birelative\_vanishing\] that $$\operatorname*{``\varprojlim\!''}_r K_{n+1}(A/{\mathfrak}m^r)\oplus K_{n+1}({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_r K_{n+1}({\widetilde}A/{\mathfrak}M^r)$$ is surjective. Since ${\widetilde}A\to{\widetilde}A/{\mathfrak}M$ splits after completion, the map $\operatorname*{``\varprojlim\!''}_r K_{n+1}({\widetilde}A/{\mathfrak}M^r)\to K_{n+1}({\widetilde}A/{\mathfrak}M)$ is also surjective, and so we conclude that $$K_{n+1}(A/{\mathfrak}m)\oplus K_{n+1}({\widetilde}A)\to K_{n+1}({\widetilde}A/{\mathfrak}M)$$ is surjective. Supposing that $A\to A/{\mathfrak}m$ splits (i.e., $A$ has a coefficient field), the map $K_{n+1}(A/{\mathfrak}m)\to K_{n+1}({\widetilde}A/{\mathfrak}M)$ factors through $K_{n+1}({\widetilde}A)$, and so we see that in fact $$K_{n+1}({\widetilde}A)\to K_{n+1}({\widetilde}A/{\mathfrak}M)$$ is surjective, which of course need not be true. The following provides a specific example:
Let $A$ be the local ring of the singular point on the nodal curve $Y^2=X^2(X+1)$ over a field $k$. We will show that the map $$K_2(A)\to K_2(A/{\mathfrak}m^r)\oplus K_2(F)$$ is not injective for any $r\ge 0$; to prove this using the above argument we must show that $K_3({\widetilde}A)\to K_3({\widetilde}A/{\mathfrak}M)$ is not surjective:
Well, $B:={\widetilde}A$ is the semi-local ring obtained by localising $C:=k[t]$ away from two distinct points $x_1,x_2\in {\mathbb}A_k^1$. Quillen’s localisation theorem implies that there is a short exact sequence $$0\to K_*(k)\to K_*(B)\to \bigoplus_{x\in{\mathbb}A_k^1\setminus\{x_1,x_2\}} K_{*-1}(k)\to 0.$$
However, in the case $*=3$, the boundary map $\bigoplus_{x\neq x_1,x_2}{\partial}_x:K_3(B)\to\bigoplus_{x\neq x_1,x_2}K_2(k)$ is already surjective when restricted to the symbolic part $K_3^{{\mbox{\scriptsize sym}}}(B)$ of $K_3(B)$; this is because $K_2(k)$ is generated by symbols and the tame symbols satisfy $${\partial}_x\{\theta_1,\theta_2,t_y\}=\begin{cases}\{\theta_1,\theta_2\}&x=y\\0&x\neq y,\end{cases}$$ if $x,y\in{\mathbb}A_k^1$, $\theta_i\in {k^{\!\times}}$, and $t_y\in k[t]$ is a local parameter at $y$.
Writing $K_3^{{\mbox{\scriptsize ind}}}=K_3/K_3^{{\mbox{\scriptsize sym}}}$ as usual, this implies that $K_3^{{\mbox{\scriptsize ind}}}(k)\to K_3^{{\mbox{\scriptsize ind}}}(B)$ is surjective. So, if $$K_3^{{\mbox{\scriptsize ind}}}(B)\to K_3^{{\mbox{\scriptsize ind}}}(B/{\mathfrak}M)=K_3^{{\mbox{\scriptsize ind}}}(k)\oplus K_3^{{\mbox{\scriptsize ind}}}(k)$$ were surjective (which would certainly follow from the surjectivity we are aiming to disprove), we would deduce that the diagonal map $K_3^{{\mbox{\scriptsize ind}}}(k)\to K_3^{{\mbox{\scriptsize ind}}}(k)\oplus K_3^{{\mbox{\scriptsize ind}}}(k)$ were surjective. However, $K_3^{{\mbox{\scriptsize ind}}}(k)$ is non-zero: for example, its $n$-torsion is $H^0(k,\mu_n^{\otimes 2})$ for any $n$ not divisible by ${\operatorname}{char}k$ by [@Levine1987], and this is non-zero by picking any such $n$ such that $\mu_n\subseteq{k^{\!\times}}$. This completes the proof.
In fact, with $A$ still being the local ring of the singular point at a node, one can even show that $K_2(A)\to K_2({\widehat}A)\oplus K_2({\widetilde}A)$ is not injective. The argument is given in [@Morrow_Singular_Gersten Prop. 2.14]: it is very similar, using the excision sequence for Weibel’s homotopy invariant $K$-theory to reduce to the same non-surjectivity assertion.
Application to $KH$-theory {#subsection_KH}
--------------------------
Now we turn to applications of the main theorems of the previous section.
For a ring $R$, we denote by $KH(R)$ Weibel’s homotopy invariant $K$-theory [@Weibel1989a], and we write $K^{{\mbox{\scriptsize sing}}}(X)$ for the homotopy fibre of $K(X)\to KH(X)$ (there does not appear to be a standardised name for this fibre, but it certainly captures information about the singularities of $R$; if $R$ is $K_0$-regular then $KH(R)\simeq KV(R)$ and so $K^{{\mbox{\scriptsize sing}}}(R)\simeq K^{{\mbox{\scriptsize nil}}}(R)$). Recall that $KH$-theory satisfies excision, is invariant under nilpotent extensions, and agrees with $K$-theory on regular rings.
If $A$ is a one-dimensional, Noetherian, reduced, semi-local ring such that $A\to{\widetilde}A$ is finite, then the $S$-analytic isomorphism $A\to{\widehat}A$ yields homotopy cartesian squares in both $K$-theory and $KH$-theory: $$\xymatrix{
K(A) \ar[r]\ar[d] & K(\operatorname{Frac}A)\ar[d]\\
K({\widehat}A)\ar[r] & K(\operatorname{Frac}{\widehat}A)
}\quad
\xymatrix{
KH(A) \ar[r]\ar[d] & KH(\operatorname{Frac}A)\ar[d]\\
KH({\widehat}A)\ar[r] & KH(\operatorname{Frac}{\widehat}A)
}$$ Taking homotopy fibres shows that $K^{{\mbox{\scriptsize sing}}}(A)\simeq K^{{\mbox{\scriptsize sing}}}({\widehat}A)$; i.e., $K^{{\mbox{\scriptsize sing}}}$ is an analytic invariant of $A$. The main purpose of this section is to establish corollary \[corollary\_KH\]; first we need a lemma, which we prove in greater generality than required:
\[lemma\_K\_sing\] Let $A$ be a one-dimensional, Noetherian, reduced ring such that $A\to{\widetilde}A$ is finite, and let ${\mathfrak}f=\operatorname{Ann}_A({\widetilde}A/A)\subseteq A$ be the conductor ideal; let $J'$ be a radical ideal of ${\widetilde}A$ contained in $\sqrt{{\mathfrak}f}$ (the radical of ${\mathfrak}f$ inside ${\widetilde}A$), and set $J=A\cap J'$.
1. Then $KH(A,J)\simeq K({\widetilde}A,J')$.
2. Moreover, the groups $K_n^{{\mbox{\scriptsize sing}}}(A)$ fit into a long exact sequence $$\cdots\to K_n^{{\mbox{\scriptsize sing}}}(A)\to \operatorname*{``\varprojlim\!''}_rK_n(A/J^r,J/J^r)\to \operatorname*{``\varprojlim\!''}_rK_n({\widetilde}A/J'^r,J'/J'^r) \to\cdots$$
${\mathfrak}f':=J'\cap {\mathfrak}f$ is an ideal of both $A$ and ${\widetilde}A$, whose radical in $A$ is $J$ and whose radical in ${\widetilde}A$ is $J'$. We claim that all the following arrows are weak equivalences: $$KH(A,J)\to KH(A,{\mathfrak}f')\to KH({\widetilde}A,{\mathfrak}f')\to KH({\widetilde}A,J')\leftarrow K({\widetilde}A,J')$$ Indeed, the first and third are weak equivalences because $KH$ is nil-invariant; the second is because $KH$ satisfies excision; the fourth is because ${\widetilde}A$ and ${\widetilde}A/J'$ are regular (the latter is a finite products of fields). This proves (i).
For the second claim, we offer a quick proof using pro spectra, which we have not properly discussed; the cautious reader may replace our homotopy cartesian diagrams of pro spectra by statements about the pro relative groups.
The square of spectra [ $$\xymatrix{
KH(A) \ar@{->}[r] \ar@{->}[d] & KH({\widetilde}A) \ar@{->}[d]\\
KH(A/J) \ar@{->}[r] & KH({\widetilde}A/J')
}$$ ]{} is homotopy cartesian by the proof of the first part. Meanwhile, according to corollary \[corollary\_main\_application\_of\_birelative\_vanishing\], the square [ $$\xymatrix{
K(A) \ar@{->}[r] \ar@{->}[d] & K({\widetilde}A) \ar@{->}[d]\\
\operatorname*{``\varprojlim\!''}_rK(A/J^r) \ar@{->}[r] & \operatorname*{``\varprojlim\!''}_rK({\widetilde}A/J'^r)
}$$ ]{} is homotopy cartesian; taking homotopy fibres from the second square to the first, we see that [ $$\xymatrix{
K^{{\mbox{\scriptsize sing}}}(A) \ar@{->}[r] \ar@{->}[d] & \ast \ar@{->}[d]\\
\operatorname*{``\varprojlim\!''}_r{\operatorname}{hofib}(K(A/J^r)\to KH(A/J)) \ar@{->}[r] & \operatorname*{``\varprojlim\!''}_r{\operatorname}{hofib}(K({\widetilde}A/J'^r)\to KH({\widetilde}A/J'))
}$$ ]{} is homotopy cartesian. Since the rings $A/J$ and ${\widetilde}A/J'$ are products of fields, we may replace each $KH$ in the bottom row by $K$, completing the proof.
Let $A$ be as in the previous lemma, and assume further that it is semi-local, contains a field, and that $K_n(A)\to K_n(A/{\mathfrak}m)$ is surjective for all $n\ge 1$ (e.g., $A$ local and containing a coefficient field). Then the kernels (resp. cokernels) of the maps $$K_n(A)\to KH_n(A),\quad\quad \operatorname*{``\varprojlim\!''}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\to\operatorname*{``\varprojlim\!''}_r K_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)$$ are canonically isomorphic for all $n\ge 1$. Here we use our standard notation that ${\mathfrak}m,{\mathfrak}M$ are the Jacobson radicals of $A,{\widetilde}A$.
The surjectivity assumption implies that the long exact sequences for the $K$ and $KH$-theories of $A\to A/{\mathfrak}m$ break into short exact sequences: $$\xymatrix{
0 \ar[r] & K_n(A,{\mathfrak}m) \ar[r]\ar[d] & K_n(A) \ar[r]\ar[d] & K_n(A/{\mathfrak}m)\ar[r]\ar@{=}[d] & 0\\
0 \ar[r] & KH_n(A, {\mathfrak}m) \ar[r] & KH_n(A) \ar[r] & KH_n(A/{\mathfrak}m)\ar[r] & 0
}$$ Thus the left square in the diagram is bicartesian, and so the vertical arrows have the same kernels and cokernels. By the previous lemma we may replace $KH_n(A,{\mathfrak}m)$ by $K_n({\widetilde}A,{\mathfrak}M)$, and then theorem \[theorem\_main\_theorem\] completes the proof.
\[corollary\_KH\] Let $A$ satisfy the conditions of the previous theorem. Then, for any $n\ge1$, the map $$K_n(A)\to K_n(A/{\mathfrak}m^r)\oplus KH_n(A)$$ is injective for $r\gg 0$.
This is an immediate consequence of the previous theorem since the assumption that $K_{n+1}(A)\to K_{n+1}(A/{\mathfrak}m)$ is surjective implies that $K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\subseteq K_n(A/{\mathfrak}m^r)$.
Geller’s conjecture in finite characteristic {#subsubsection_Geller}
--------------------------------------------
Now we turn to applications to Geller’s conjecture. S. Geller’s conjecture, raised in 1986 [@Geller1986], is the following:
> “Let $A$ be a one-dimensional, Noetherian, reduced local ring, with total quotient ring $F$. Then the map $K_2(A)\to K_2(F)$ is injective if and only if $A$ is regular.”
The ‘if’ direction is a classical theorem of K. Dennis and M. Stein [@Dennis1975 Thm. 2.2]. Geller herself established the conjecture provided that $A$ has equal characteristic, perfect residue field, and seminormal singularities. Except for this seminormal case there has been no progress on Geller’s conjecture in finite characteristic. As discussed in the introduction, Krishna [@Krishna2005] reduced Geller’s conjecture in characteristic zero to an Artinian analogue, just as Cortiñas, Geller, and Weibel [@Weibel1998] had done earlier for Berger’s conjecture on differential forms.
Using similar ideas we offer the following complete solution to Geller’s conjecture for equal characteristic rings with perfect residue field of finite characteristic:
\[theorem\_Geller\] Let $A$ be a one-dimensional, Noetherian, reduced, local ${\mathbb}F_p$-algebra whose residue field is perfect and such that $A\to{\widetilde}A$ is finite. Suppose that the map $$K_2(A){\longrightarrow}K_2(F)$$ is injective, where $F$ is the total quotient ring of $A$. Then $A$ is regular.
As usual, let ${\mathfrak}m$ denote the maximal ideal of $A$, and ${\mathfrak}M$ the Jacobson radical of ${\widetilde}A$. Let ${\widehat}A$ be the ${\mathfrak}m$-adic completion of $A$; we will show it is sufficient to prove the theorem for ${\widehat}A$ in place of $A$. We first claim that ${\widehat}A$ is reduced and that ${\widehat}A\to{\widetilde}{{\widehat}A}$ is finite. Indeed, by flatness of completion, ${\widehat}A$ embeds into ${\widehat}{{\widetilde}A}={\widetilde}A\otimes_A{\widehat}A$, which is a finite product of complete discrete valuation rings since ${\widetilde}A$ is a finite product of discrete valuation rings; therefore ${\widehat}A$ is reduced. This shows moreover that ${\widehat}{{\widetilde}A}$ is normal and finite over ${\widehat}A$, whence it is equal to ${\widetilde}{{\widehat}A}$, completing the proof of our claim.
The map $A\to {\widehat}A$ is an $S$-analytic isomorphism [@Weibel1986], where $S$ is the set of non-zero-divisors of $A$, and so there is a resulting long-exact, Mayer–Vietoris sequence $$\cdots \to K_n(A)\to K_n(F)\oplus K_n({\widehat}A)\to K_n({\widehat}F)\to \cdots,$$ where ${\widehat}F$ is the total quotient ring of ${\widehat}A$. From the assumption that $K_2(A)\to K_2(F)$ is injective, and the fact that $K_1(A)\cong{A^{\!\times}}\to K_1(F)\cong{F^{\!\times}}$ is injective, this long-exact sequence yields a short exact sequence $$0\to K_2(A)\to K_2(F)\oplus K_2({\widehat}A)\to K_2({\widehat}F)\to0,$$ whence $K_2({\widehat}A)\to K_2({\widehat}F)$ is also injective; moreover, $A$ is regular if and only if ${\widehat}A$ is regular, since the two rings have the same dimension and isomorphic cotangent spaces. Therefore we may replace $A$ by ${\widehat}A$ in the rest of the proof. Since any complete, Noetherian, local ring of equal characteristic contains a coefficient field [@Cohen1946 Thm. 9], and since ${\widetilde}{{\widehat}A}$ is a finite product of such rings, we may therefore henceforth assume that the morphisms $A\to A/{\mathfrak}m$ and ${\widetilde}A\to{\widetilde}A/{\mathfrak}M$ split.
Using these splitting assumptions we will now deduce that the map $K_2(A,{\mathfrak}m)\to K_2({\widetilde}A,{\mathfrak}M)$ is injective. Indeed, the splitting assumptions imply that $K_2(A,{\mathfrak}m)\subseteq K_2(A)$ and $K_2({\widetilde}A,{\mathfrak}M)\subseteq K_2({\widetilde}A)$, so it is enough to prove that $K_2(A)\to K_2({\widetilde}A)$ is injective; but we are assuming the stronger result that $K_2(A)\to K_2(F)$ is injective.
Moreover, by [@Dennis1975 Lemma 3.4] (see also remark \[remark\_vanishing\_of\_even\_K\_groups\_for\_truncated\_polys\]), the pro abelian group $\operatorname*{``\varprojlim\!''}_rK_2(k[t]/{(t^r)},{(t)})$ vanishes if $k$ is any perfect field of finite characteristic. Since ${\widetilde}A/{\mathfrak}M^r$ is a finite product of such truncated polynomial rings, we deduce that $\operatorname*{``\varprojlim\!''}_rK_2({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)=0$. The short exact sequence of theorem \[theorem\_main\_theorem\] therefore yields a surjection $$K_2(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_2({\widetilde}A,{\mathfrak}M)$$ (even an isomorphism, but we don’t need this). So, considering the maps $$K_2(A,{\mathfrak}m)\to\operatorname*{``\varprojlim\!''}_r K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_2({\widetilde}A,{\mathfrak}M){\xrightarrow}{{{\mbox{\scriptsize proj}}}} K_2({\widetilde}A,{\mathfrak}M),$$ we have proved that the composition is injective and the first arrow is surjective. It follows that $$\operatorname*{``\varprojlim\!''}_r K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)=0,\tag{\dag}$$ which is the key to completing the proof.
Each group $K_2(A/{\mathfrak}m^r)$ is generated by Steinberg symbols, whence the transition maps $K_2(A/{\mathfrak}m^{r+1})\to K_2(A/{\mathfrak}m^r)$ are surjective; since $K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)$ is contained in $K_2(A/{\mathfrak}m^r)$ for all $r$, thanks to the splitting of $A/{\mathfrak}m^r\to A/{\mathfrak}m$, it follows that the transition maps $$K_2(A/{\mathfrak}m^{r+1},{\mathfrak}m/{\mathfrak}m^{r+1})\to K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)$$ are also surjective. Therefore the vanishing of the pro abelian group () implies the vanishing of the individual groups: $K_2(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)=0$ for all $r\ge 1$. To complete the proof, it therefore suffices to prove the following claim:
If $A$ is not regular, then $K_2(A/{\mathfrak}m^2,{\mathfrak}m/{\mathfrak}m^2)$ is non-zero.
Well, if $A$ is not regular, then $\dim_\kappa{\mathfrak}m/{\mathfrak}m^2\ge 2$, where $\kappa=A/{\mathfrak}m$; let $x,y\in{\mathfrak}m/{\mathfrak}m^2$ be linearly independent elements. There are various ways to show that the Dennis–Stein symbol $$\langle x,y\rangle\in K_2(A/{\mathfrak}m^2,{\mathfrak}m/{\mathfrak}m^2)$$ is non-zero, which will complete the proof, the easiest of which is probably the following. The Maazen–Steinstra presentation of $K_2(A/{\mathfrak}m^2,{\mathfrak}m/{\mathfrak}m^2)$ using Dennis–Stein symbols [@MaazenSteinstra1977 Thm. 3.1] readily implies that there is a well-defined homomorphism (having fixed a splitting of $A/{\mathfrak}m^2\to\kappa$) $$K_2(A/{\mathfrak}m^2,{\mathfrak}m/{\mathfrak}m^2){\longrightarrow}\left(\bigwedge\nolimits_\kappa^2A/{\mathfrak}m^2\right)\bigg/\langle a\wedge b\,:a\mbox{ or } b \mbox{ is in }\kappa\rangle\,\cong\bigwedge\nolimits_\kappa^2{\mathfrak}m/{\mathfrak}m^2,\quad \langle a,b\rangle\mapsto a\wedge b,$$ where at least one of $a, b\in A/{\mathfrak}m^2$ belongs to ${\mathfrak}m/{\mathfrak}m^2$; this takes $\langle x,y\rangle$ to $x\wedge y\in \bigwedge_\kappa^2{\mathfrak}m/{\mathfrak}m^2$, where it is non-zero by choice of $x$ and $y$.
Relation to cyclic homology in characteristic zero {#subsection_rel_to_HC}
--------------------------------------------------
The aim of this section is theorem \[theorem\_higher\_Geller\] below, offering an alternative to Geller’s conjecture in characteristic zero. The key tool however, which likely has other applications and depends crucially on our main theorem \[theorem\_main\_theorem\], is corollary \[corollary\_K\_vs\_HC\], stating that the kernel of a map between $K$-groups is isomorphic to the analogous kernel for cyclic homology.
We start with the cyclic homology version of our main theorem \[theorem\_main\_theorem\]:
Let $A$ be a one-dimensional, Noetherian, reduced, semi-local ${\mathbb}Q$-algebra such that $A\to{\widetilde}A$ is finite, and let $n\ge 0$. Then the natural square [ $$\xymatrix{
HC_n(A,{\mathfrak}m) \ar@{->}[r] \ar@{->}[d] & HC_n({\widetilde}A,{\mathfrak}M) \ar@{->}[d]\\
\operatorname*{``\varprojlim\!''}_r HC_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r) \ar@{->}[r] & \operatorname*{``\varprojlim\!''}_r HC_n({\widetilde}A/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)
}$$ ]{} is bicartesian in Pro$Ab$, where all cyclic homologies are taken with respect to ${\mathbb}Q$.
The proof is essentially the same as for $K$-theory, so we will be brief. Firstly, G. Cortiñas’ [@Cortinas2006] proof of the KABI conjecture implies that we may replace $K_n$ by $HC_{n-1}$ in the vanishing result of corollary \[corollary\_main\_application\_of\_birelative\_vanishing\]; hence the subsequent long-exact, Mayer–Vietoris sequences remain valid for cyclic homology in place of $K$-theory. In particular, there is a long exact, Mayer–Vietoris sequence $$\cdots\to HC_n(A,{\mathfrak}m)\to \operatorname*{``\varprojlim\!''}_rHC_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus HC_n({\widetilde}A,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_rHC_n({\widetilde}A/{\mathfrak}M,{\mathfrak}M/{\mathfrak}M^r)\to\cdots,$$ so to complete the proof it is sufficient to show that $$HC_n({\widetilde}A,{\mathfrak}M)\to\operatorname*{``\varprojlim\!''}_rHC_n({\widetilde}A/{\mathfrak}M,{\mathfrak}M/{\mathfrak}M^r)$$ is surjective. Fortunately, lemma \[lemma\_HC\_of\_graded\_ring\] implies that the right side of this morphism is $$\operatorname*{``\varprojlim\!''}_r{\operatorname{Ker}}\big(\Omega^n_{{\widetilde}A/{\mathfrak}M^r}/d\Omega^{n-1}_{{\widetilde}A/{\mathfrak}M^r}\to \Omega^n_{{\widetilde}A/{\mathfrak}M}/d\Omega^{n-1}_{{\widetilde}A/{\mathfrak}M}\big)$$ Since ${\operatorname{Ker}}(HC_n({\widetilde}A)\to HC_n({\widetilde}A/{\mathfrak}M))$ contains a direct summand isomorphic to ${\operatorname{Ker}}(\Omega^n_{{\widetilde}A}/d\Omega^{n-1}_{{\widetilde}A}\to\Omega^n_{{\widetilde}A/{\mathfrak}M}/d\Omega^{n-1}_{{\widetilde}A/{\mathfrak}M})$, one sees that the desired morphism is surjective.
\[corollary\_K\_vs\_HC\] Let $A$ be as in the previous proposition and let $n\ge 1$. Then the kernel (resp. cokernel) of the maps $$K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M),\qquad HC_{n-1}(A,{\mathfrak}m)\to HC_{n-1}({\widetilde}A,{\mathfrak}M)$$ are canonically isomorphic.
This is a consequence of the previous proposition, the analogous theorem for $K$-theory (namely theorem \[theorem\_main\_theorem\]), and the Goodwillie isomorphism.
Before we can use the corollary to prove the main theorem of the section, we need a ‘cyclic homology criterion for smoothness’. In the following result we denote by $\Omega^\bullet_{R/k}$ the de Rham mixed complex of a $k$-algebra $R$, whose Hochschild and cyclic homologies are respectively $HH_n(\Omega^\bullet_{R/k})=\Omega_{R/k}^n$ and $HC_n(\Omega^\bullet_{R/k})=\Omega^n_{R/k}/d\Omega^{n-1}_{R/k}\oplus\bigoplus_{p\ge1} H_{{\mbox{\scriptsize dR}}}^{n-2p}(R/k)$.
Let $k\subseteq K$ be an extension of characteristic zero fields, and let $R$ be an essentially finite type $K$-algebra which is not smooth over $K$. Then the canonical map $$HC_n^k(R)\to HC_n(\Omega^\bullet_{R/k})$$ is not injective for some $n\ge 2$.
The Hochschild homology criterion for smoothness [@Avramov1992], which offers an converse to the Hoschschild–Konstant–Rosenberg theorem, says that if $R$ is a finitely generated algebra over a field $K$, then $\Omega_{R/K}^n\to HH_n^K(R)$ is an isomorphism for all $n\ge 0$ if and only if $R$ is smooth over $K$. Obviously this remains valid if $R$ is merely essentially of finite type over $K$, but we will also need to be able to replace $K$ by $k$, which we do as follows.
According to the Künneth decomposition and base change for Hochschild homology, one has isomorphisms of graded algebras $$\begin{aligned}
HH_*^k(R)\otimes_kK\cong HH_*^K(R\otimes_kK)
&\cong HH_*^K(R)\otimes_KHH_*^K(K\otimes_kK)\\
&\cong HH_*^K(R)\otimes_k\Omega_{K/k}^*\end{aligned}$$ Using the same decomposition for de Rham complexes, we see from faithfully flat descent that if $\Omega^*_{R/k}\to HH_*^k(R)$ were to be an isomorphism, then so would be $\Omega^*_{R/K}\to HH_*^K(R)$. In conclusion, since we are assuming $R$ is not smooth over $K$, we deduce that $\Omega^n_{R/k}\to HH_n^k(R)$ is not an isomorphism for some $n\ge 0$.
Next, the following result may be proved by a straightforward induction using the SBI sequences: if $C_\bullet\to D_\bullet$ is a morphism of mixed complexes such that $HH_n(C_\bullet)\to HH_n(D_\bullet)$ is surjective for all $n\ge 0$ and such that $HC_n(C_\bullet)\to HC_n(D_\bullet)$ is injective for all $n\ge 0$, then in fact $HC_n(C_\bullet)\to HC_n(D_\bullet)$ is an isomorphism for all $n\ge 0$ (and so $HH_n(C_\bullet)\to HH_n(D_\bullet)$ is also an isomorphism for all $n\ge 0$).
So, letting $C_\bullet^k(R)$ denotes the Hochschild complex of $R$ as a $k$-algebra, the usual morphism of mixed complexes [@Loday1992 §2.3]$$\pi:C_\bullet^k(R)\to\Omega_{R/k}^\bullet,\quad r_0\otimes\cdots\otimes r_n\mapsto r_0\,dr_1\wedge\cdots\wedge r_n$$ induces a surjection on the associated Hochschild homologies, but not an isomorphism by what we saw above; therefore the induced map on the cyclic homologies $$HC_n^k(R)\to HC_n(\Omega_{R/k}^\bullet)$$ is not injective for some $n\ge 2$ (it is an isomorphism for $n=0,1$), as desired.
Now we are equipped to prove our higher degree alternative to Geller’s conjecture:
\[theorem\_higher\_Geller\] Let $A$ be a one-dimensional, reduced, semi-local ring which is essentially of finite type over some characteristic zero field, and assume that $A$ is not regular. Then the map $$K_n(A)\to K_n(\operatorname{Frac}A)$$ is not injective for some $n\ge3$.
Just as we replaced $A$ by its completion in the proof of theorem \[theorem\_Geller\], here we may replace $A$ by a large enough finite extension to ensure that $A\to A/{\mathfrak}m$ and ${\widetilde}A\to {\widetilde}A/{\mathfrak}M$ split. Then $K_n(A,{\mathfrak}m)\subseteq K_n(A)$ and $K_n({\widetilde}A,{\mathfrak}M)\subseteq K_n({\widetilde}A)$, so it is enough to prove that $K_n(A,{\mathfrak}m)\to K_n({\widetilde}A,{\mathfrak}M)$ is not injective for some $n\ge 3$. According to the previous corollary, it is therefore sufficient to prove that $HC_n(A,{\mathfrak}m)\to HC_n({\widetilde}A,{\mathfrak}M)$ is not injective for some $n\ge 2$.
Well, by the previous lemma we may find $n\ge 2$ and non-zero $x\in HC_n(A)$ such that $x$ vanishes in $HC_n(\Omega_{A/{\mathbb}Q}^\bullet)$. Since $HC_n(A/{\mathfrak}m)=HC_n(\Omega_{(A/{\mathfrak}m)/{\mathbb}Q}^\bullet)$ and $HC_n({\widetilde}A)=HC_n(\Omega_{{\widetilde}A/{\mathbb}Q}^\bullet)$, we deduce that $x$ vanishes in both $HC_n({\widetilde}A)$ and $HC_n(A/{\mathfrak}m)$, so belongs to $HC_n(A,{\mathfrak}m)\subseteq HC_n(A)$; this completes the proof.
Using the same techniques, it appears it may be possible to strengthen the conclusion of the previous theorem to ‘not injective for infinitely many $n\ge 3$’.
The case of finite residue fields {#section_finite_res_field}
=================================
In this section we use the pro-excision results from section \[section\_pro\_excision\] to study one-dimensional, Noetherian, reduced rings with finite normalisation map, all of whose residue fields are finite: e.g., orders in number fields, affine reduced curves over finite fields, and local versions of these. Notice that corollary \[corollary\_main\_application\_of\_birelative\_vanishing\] applies to such rings. Before specialising to the arithmetic setting, we begin by establishing some general results.
We need the following:
\[lemma\_thanks\_to\_Vigleik\] Let $R$ be a finite ring. Then $K_n(R)$ is finite for all $n\ge1$.
In particular, if $A$ is a one-dimensional, Noetherian ring all of whose residue fields are finite, and $I\subseteq A$ is an ideal such that $A/I$ has finite length, then $K_n(A/I^r)$ is finite for all $r,n\ge1$. Hence ${\varprojlim}_rK_n(A/I^r)$ is a profinite group.
I am grateful to V. Angeltveit for explaining the argument to me. Firstly, Bass stability implies that, for any fixed $n$, $H_n(BGL(R)^+,{\mathbb}Z)=H_n(GL(R),{\mathbb}Z)=H_n(GL_m(R),{\mathbb}Z)$ for $m$ sufficiently large, and $H_n(GL_m(R),{\mathbb}Z)$ is finite for $n\ge 1$ since $GL_m(R)$ is a finite group. Thus all the integral homology groups of degree $\ge 1$ of the $K$-theory space $BGL(R)^+$ are finite.
Since $BGL(R)^+$ is an infinite loop space, its $\pi_1$ acts trivially on its $\pi_n$ for all $n\ge 1$, so the theory of Serre classes tells us that $$\pi_n(BGL(R)^+)\mbox{ is finite for all }n\ge 1\Longleftrightarrow H_n(BGL(R)^+,{\mathbb}Z)\mbox{ is finite for all }n\ge 1,$$ completing the first part of the proof.
The ‘in particular’ claim follows from the fact that $A/I^r$ is finite for all $r\ge 1$.
\[proposition\_finite\_kernel\_and\_cokernel\] Let $A$ be a one-dimensional, Noetherian, reduced ring such that $A\to{\widetilde}A$ is finite, and all of whose residue fields are finite. Then $$K_n(A)\to K_n({\widetilde}A)$$ has finite kernel and cokernel for all $n\ge 1$.
Moreover, if $\ell$ is a prime number invertible in $A/{\mathfrak}f$, where ${\mathfrak}f$ denotes the conductor of $A\to {\widetilde}A$, then the kernel and cokernel have no $\ell$-torsion.
Consider the following diagram of spectra $$\xymatrix{
K(A,B,{\mathfrak}f) \ar[r] & K(A,{\mathfrak}f)\ar[r]\ar[d] & K(B,{\mathfrak}f)\ar[d]\\
&K(A)\ar[r]\ar[d] & K(B)\ar[d]\\
&K(A/{\mathfrak}f)\ar[r] & K(B/{\mathfrak}f)
}$$ in which the two columns and the top row are homotopy fibre sequences. According to proposition \[proposition\_standard\_consequences\](iv), if $n\ge 1$ then $K_n(A,B,{\mathfrak}f)$ embeds into $K_n(A/{\mathfrak}f^r,B/{\mathfrak}f^r,{\mathfrak}f/{\mathfrak}f^r)$ for $r\gg 0$; but this latter group is finite by the previous lemma, and so $K_n(A,B,{\mathfrak}f)$ is finite. Hence, in the Serre quotient category $Ab/FinAb$ we have $K_n(A,{\mathfrak}f)\cong K_n(B,{\mathfrak}f)$ and $K_n(A/{\mathfrak}f)\cong K_n(B/{\mathfrak}f)\cong 0$ for all $n\ge1$; the claim follows.
If $\ell$ is a prime number invertible in $A/{\mathfrak}f$, then the relative groups $K_n(A,{\mathfrak}f^r)$, $K_n(B,{\mathfrak}f^r)$ have no $\ell$-torsion by [@Weibel1982 Consequence 1.4], so we may repeat the previous argument, replacing the category $FinAb$ by the category of finite abelian groups without $\ell$-torsion.
Our next aim is to show that the long exact, Mayer–Vietoris sequences of pro abelian groups from section \[section\_pro\_excision\] can actually be realised to the level to profinite groups; this is a formal consequence of the following lemma:
\[lemma\_mittag\_leffler\] Let $$\cdots \to A_n\to\operatorname*{``\varprojlim\!''}_r A_n(r)\to \operatorname*{``\varprojlim\!''}_r B_n(r)\to A_{n-1}\to\cdots$$ be a long exact sequence in ${\operatorname}{Pro}Ab$, where $A_n, A_n(r), B_n(r)\in Ab$. Suppose that the group $B_n(r)$ is finite for all $n,r$. Then the resulting complex of groups $$\cdots\to A_n\to{\varprojlim}_r A_n(r)\to {\varprojlim}_r B_n(r)\to A_{n-1}\to\cdots$$ is exact.
Firstly, there is no loss of generality in assuming that the long exact sequence in ${\operatorname}{Pro}Ab$ arises from an inverse system of complexes of abelian groups $$C_\bullet(r)=\qquad \cdots\to A_n\to A_n(r)\to B_n(r)\to A_{n-1}\to \cdots$$ For each $n$, the pro abelian group $\operatorname*{``\varprojlim\!''}_r{\operatorname}{Im}(A_n\to A_n(r))$ has surjective transition maps, while $\operatorname*{``\varprojlim\!''}_r{\operatorname}{Im}(A_n(r)\to B_n(r))$ is a limit of finite groups and hence satisfies the Mittag-Leffler condition; it easily follows that $\operatorname*{``\varprojlim\!''}_r A_n(r)$ also satisfies the Mittag-Leffler condition.
Thus all terms in the inverse system of complexes $\cdots\to C_\bullet(2)\to C_\bullet(1)$ satisfy the Mittag-Leffler condition, so a standard result on hyper derived functors (e.g., [@Weibel1994 Thm. 3.5.8]) states that there are short exact sequences in homology $$0\to {{\varprojlim}_r}^1H_{n+1}(C_\bullet(r))\to H_n({\varprojlim}_rC_\bullet(r))\to{\varprojlim}_rH_n(C_\bullet(r))\to 0.$$ The two outer groups vanish since the pro abelian groups $\operatorname*{``\varprojlim\!''}_r H_n(C_\bullet(r))=H_n(\operatorname*{``\varprojlim\!''}_rC_\bullet(r))$ are zero; hence $H_n({\varprojlim}_rC_\bullet(r))$ vanishes for all $n$, as desired.
\[corollary\_group\_versions\_of\_les\] Let $A$ be a one-dimensional, Noetherian, reduced semi-local ring such that $A\to{\widetilde}A$ is finite, and all of whose residue fields are finite; let ${\mathfrak}m$, ${\mathfrak}M$ denote the Jacobson radicals of $A$, ${\widetilde}A$. Then there are long exact, Mayer–Vietoris sequences of abelian groups $$\cdots\to K_n(A)\to{\varprojlim}_r K_n(A/{\mathfrak}m^r)\oplus K_n(B)\to{\varprojlim}_r K_n(B/{\mathfrak}M^r)\to\cdots$$ $$\cdots\to K_n(A,{\mathfrak}m)\to{\varprojlim}_r K_n(A/{\mathfrak}m^r,{\mathfrak}m/{\mathfrak}m^r)\oplus K_n(B,{\mathfrak}M)\to{\varprojlim}_r K_n(B/{\mathfrak}M^r,{\mathfrak}M/{\mathfrak}M^r)\to\cdots$$
This follows by applying the previous two lemmas to the exact sequences of example \[example\_main\_application\_of\_birelative\_vanishing\].
In order to prove the analogue in the finite residue field case of our main theorem \[theorem\_main\_theorem\_II\], we would like to check condition (i) of the following corollary; the corollary shows that this would follow from showing, informally, that ${\varprojlim}_r K_{n+1}(A/{\mathfrak}m^r)$ is ‘open’ in ${\varprojlim}_r K_{n+1}({\widetilde}A/{\mathfrak}M^r)$ and that $K_{n+1}({\widetilde}A)$ is ‘dense’. Despite being a conceivable ‘continuity’ property of $K$-theory, we will only be able to prove this in special cases.
\[corollary\_application\_of\_group\_version\_of\_les\] Let notation be as in the previous corollary. Then the following are equivalent:
1. $K_n(A)\to K_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)$ is injective for $r\gg0$.
2. ${\varprojlim}_r K_{n+1}(A/{\mathfrak}m^r)\oplus K_{n+1}({\widetilde}A)\to{\varprojlim}_r K_{n+1}({\widetilde}A/{\mathfrak}M^r)$ is surjective.
Since the kernel of $K_n(A)\to K_n({\widetilde}A)$ is finite by proposition \[proposition\_finite\_kernel\_and\_cokernel\], one sees that (i) is equivalent to the injectivity of $K_n(A)\to {\varprojlim}_rK_n(A/{\mathfrak}m^r)\oplus K_n({\widetilde}A)$. The result therefore follows from the first long exact sequence of the previous corollary.
The global case {#subsection_global}
---------------
Here we apply pro-excision and the preliminary observations of the previous section to deduce finiteness properties of $K$-groups of orders in number fields and non-smooth curves over finite fields.
We begin with the geometric case. A celebrated theorem due to G. Harder [@Harder1977] and C. Soulé [@Soule1984] states that the higher $K$-groups of a smooth projective curve over a finite field are finite groups; we can remove the smoothness hypothesis:
Let $k$ be a finite field and $X$ a one-dimensional, reduced scheme, separated and of finite type over $k$. Then $K_n(X)\to K_n({\widetilde}X)$ has finite kernel and cokernel for all $n\ge 1$. In particular, if $X/k$ is proper then $K_n(X)$ is finite for all $n\ge 1$.
The first claim is a straightforward induction, using proposition \[proposition\_finite\_kernel\_and\_cokernel\], on the number of affine patches required to cover $X$. Indeed, let $\pi:{\widetilde}X\to X$ be the normalisation map, and $U,V\subseteq X$ an open cover of $X$ such that the claim has been proved for $U$, $V$, and $U\cap V$. Then Thomason–Trobaugh Zariski descent [@Thomason1990 Thm. 8.1] provides us with long exact Mayer–Vietoris sequences $$\xymatrix{
\cdots\ar[r] & K_n(X) \ar[r]\ar[d] & K_n(U)\oplus K_n(V) \ar[r]\ar[d] & K_n(U\cap V) \ar[r]\ar[d] & \cdots\\
\cdots\ar[r] & K_n({\widetilde}X) \ar[r] & K_n({\widetilde}U)\oplus K_n({\widetilde}V) \ar[r] & K_n({\widetilde}U\cap {\widetilde}V) \ar[r] & \cdots
}$$ By hypothesis the right and central vertical arrows have finite kernel and cokernel for $n\ge 1$, whence the same is true of the left vertical arrow.
The second claim follows by applying the Harder–Soulé theorem to ${\widetilde}X$.
The analogue in the arithmetic case of Harder–Soulé’s result is that of A. Borel [@Borel1974] and D. Quillen [@Quillen1973], stating that the $K$-groups of the ring of integers of a number field are finitely generated and precisely identifying their ranks.
Let $F$ be a number field with ring of integers ${\mathcal{O}}$, and let $A\subseteq {\mathcal{O}}$ be an order (i.e. any subring with fractions $F$). Then $K_n(A)$ is a finitely generated abelian group and $K_n(A)\otimes{\mathbb}Q\to K_n({\mathcal{O}})\otimes{\mathbb}Q$ is an isomorphism for all $n\ge 1$.
$A$ automatically satisfies the conditions of proposition \[proposition\_finite\_kernel\_and\_cokernel\] and ${\widetilde}A={\mathcal{O}}$, whence the claims follows from Borel–Quillen.
The local case: finite ${\mathbb}Z_p$-algebras {#subsection_p_adic_orders}
----------------------------------------------
In this section we will apply pro-excision to the study of certain finite ${\mathbb}Z_p$-algebras; to be precise, we will be interested in rings $A$ satisfying the following equivalent conditions:
1. $A$ is a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module.
2. $A$ is a one-dimensional, Noetherian, reduced, complete semi-local ring, of mixed characteristic $(0,p)$, and having finite residue fields.
If $A$ satisfies these conditions then its total quotient ring $\operatorname{Frac}A$ is a finite product of finite extensions of ${\mathbb}Q_p$, and its normalisation ${\widetilde}A$ is the product of their rings of integers; the Jacobson radicals of $A$ and ${\widetilde}A$ will always be denoted ${\mathfrak}m$ and ${\mathfrak}M$ respectively. Note that $A\to{\widetilde}A$ is a finite morphism because $A$, being complete, is excellent and hence has finite normalisation [@EGA_IV_II 7.8.3].
It would be more straightforward (and intuitive) to assume in addition that $A$ is a local domain, but this restriction would later cause problems.
\[examples\_p\_adic\_orders\] Here we offer some examples and basic properties of such rings:
1. Let ${\mathcal{O}}$ be the ring of integers of a finite extension of ${\mathbb}Q_p$, and let ${\mathfrak}p$ be the maximal ideal of ${\mathcal{O}}$. Then $A:={\mathbb}Z_p+{\mathfrak}p^s$ satisfies the above conditions for any $s\ge 1$; moreover, ${\widetilde}A={\mathcal{O}}$ and $A/{\mathfrak}m={\mathbb}F_p$.
2. Let ${\mathcal{O}}_1,\dots,{\mathcal{O}}_n$ be rings of integers of finite extensions of ${\mathbb}Q_p$, and let ${\mathfrak}p_1,\dots,{\mathfrak}p_n$ denote their maximal ideals. Let $k$ be a finite field contained in all ${\mathcal{O}}_1/{\mathfrak}p_1,\dots,{\mathcal{O}}_n/{\mathfrak}p_n$; e.g., $k={\mathbb}F_p$ suffices. Then $$A:=\big\{(f_i)\in\prod\nolimits_i{\mathcal{O}}_i:f_i\mbox{ mod } {{\mathfrak}p_i}\mbox{ belongs to $k$ and does not depend on }i\big\}$$ is a seminormal local ring satisfying the above conditions, with normalisation $\prod_i{\mathcal{O}}_i$ and residue field $k$.
3. Suppose $A$ is local and satisfies the above conditions. Then Hensel’s lemma implies that $A$ contains the Teichmüller lifts of its residue field $A/{\mathfrak}m={\mathbb}F_q$. Since $A$ also contains ${\mathbb}Z_p$, we deduce it contains ${\mathbb}Z_q:=W({\mathbb}F_q)$.
4. If ${\mathcal{O}}$ is the ring of integers of a finite extension of ${\mathbb}Q_p$ and $G$ is a finite group, then the group algebra ${\mathcal{O}}G$ satisfies the above conditions. The only condition which is not immediate is that ${\mathcal{O}}G$ is reduced, but it is enough to check this when $G$ is cyclic, in which case ${\mathcal{O}}G={\mathcal{O}}[X]/{(X^n-1)}$ for some $n\ge 1$, and this is reduced since $X^n-1$ has no repeated factors in the UFD ${\mathcal{O}}[X]$. For example, if $G$ is the cyclic group of order $p$, then ${\mathbb}Z_pG$ is the seminormal ring constructed in (ii) from the fields ${\mathbb}Q_p$ and ${\mathbb}Q_p(\zeta_p)$; i.e., $${\mathbb}Z_pG\cong\{(f,g)\in{\mathbb}Z_p\times{\mathbb}Z_p[\zeta_p]:f\mbox{ mod }{(p)}=g\mbox{ mod } {(1-\zeta)}\}.$$
The [*topological $K$-groups*]{} of a semi-local ring $A$ with Jacobson radical ${\mathfrak}m$ are defined by $$K_n^{{\mbox{\scriptsize top}}}(A):=\pi_n(K^{{\mbox{\scriptsize top}}}(A)),\quad K^{{\mbox{\scriptsize top}}}(A):=\operatorname*{\operatorname*{holim}}_rK(A/{\mathfrak}m^r)$$ The following isomorphisms, the second of which is a deep theorem of A. Suslin and A. Yufryakov, relate completed $K$-groups, topological $K$-groups, and $K$-groups with ${\widehat}{{\mathbb}Z}$-coefficients (which we comment on after the lemma); these isomorphisms will be essential tools for our study of the $K$-groups of finite ${\mathbb}Z_p$-algebras.
\[lemma\_homotopy\_description\_of\_K\_groups\] Let $A$ be a ${\mathbb}Z_p$-algebra which is finitely generated as a ${\mathbb}Z_p$-module; let ${\mathfrak}m$ be the Jacobson radical of $A$. For all $n\ge1$, there are canonical isomorphisms $${\varprojlim}_r K_n(A/{\mathfrak}m^r)\cong K_n^{{\mbox{\scriptsize top}}}(A)\cong K_n(A;{\widehat}{{\mathbb}Z}).$$
The topological $K$-groups fit into short exact sequences $$0\to {{\varprojlim}_r}^1 K_{n+1}(A/{\mathfrak}m^r)\to K_n^{{\mbox{\scriptsize top}}}(A)\to{\varprojlim}_rK_n(A/{\mathfrak}m^r)\to 0.$$ But $K_{n+1}(A/{\mathfrak}m^r)$ is finite for all $r$ (lemma \[lemma\_thanks\_to\_Vigleik\]), so the ${\varprojlim}^1$ term vanishes and we get isomorphisms $K_n^{{\mbox{\scriptsize top}}}(A)\cong{\varprojlim}_rK_n(A/{\mathfrak}m^r)$ for $n\ge 0$.
Next, since $A$ is a finite ${\mathbb}Z_p$-algebra, A. Suslin and A. Yufryakov [@Suslin1984a; @Suslin1986] proved that the canonical map $K(A)\to K^{{\mbox{\scriptsize top}}}(A)$ induces a weak equivalence after profinite completion: $K(A)^{{{\widehat}{\phantom{o}}}}\stackrel{\sim}{\to} K^{{\mbox{\scriptsize top}}}(A)^{{{\widehat}{\phantom{o}}}}$ (the full argument can be found in the appendix of [@Hesselholt1997]). But profinite completion commutes with homotopy limits, and so $$K^{{\mbox{\scriptsize top}}}(A)^{{{\widehat}{\phantom{o}}}}=\operatorname*{\operatorname*{holim}}_r\left( K(A/{\mathfrak}m^r)^{{{\widehat}{\phantom{o}}}}\right)\stackrel{(\ast)}{=}\operatorname*{\operatorname*{holim}}_r K(A/{\mathfrak}m^r),$$ where the final equality follows again from the fact that $K(A/{\mathfrak}m^r)$ has finite homotopy groups, at least if we ignore $\pi_0$: thus ($\ast$) is actually only an equality if we restrict to a connected component of each side. Hence $\pi_n(K(A)^{{{\widehat}{\phantom{o}}}})=K^{{\mbox{\scriptsize top}}}_n(A))$ for $n>0$, establishing the second isomorphism.
\[remark\_Z\_hat\_coefficients\] $K$-theory $K_*(-;{\widehat}{{\mathbb}Z})=\pi_*(K(-)^{{{\widehat}{\phantom{o}}}})$ with ${\widehat}{{\mathbb}Z}$-coefficients is defined to be the homotopy groups of the profinite completion of the $K$-theory spectrum. It is described by short exact sequences $$0\to{\operatorname}{Ext}^1_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,K_n(-))\to K_n(-;{\widehat}{{\mathbb}Z})\to\operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,K_{n-1}(-))\to 0$$ $$0\to{{\varprojlim}_\lambda}^1 K_n(-)[\lambda]\to {\operatorname}{Ext}^1_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,K_n(-))\to K_n(-)^{{{\widehat}{\phantom{o}}}}\to 0$$ where $\lambda$ varies over positive integers ordered by divisibility, $C[\lambda]$ denotes the $\lambda$-torsion of an abelian group $C$, and $C^{{{\widehat}{\phantom{o}}}}={\varprojlim}_\lambda C/\lambda C$ denotes the ${\widehat}{{\mathbb}Z}$-completion of $C$.
Let $F$ be a finite extension of ${\mathbb}Q_p$ with ring of integers ${\mathcal{O}}$. We must review the structure of the $K$-groups of ${\mathcal{O}}$. For more details we refer the reader to the survey [@Weibel2005 §5].
Let $i\ge 1$, and set $$w_i(F)=\#H^0(F,\mu(i)),\quad\quad w_i^{(p)}(F)=\#H^0(F,\mu_{p^\infty}(i)),$$ where $\mu$ (resp. $\mu_{p^\infty}$) denote the group of all (resp. $p$-power) roots of unity in $F^{{\mbox{\scriptsize alg}}}$; then $${\mathbb}Z/w_i(F){\mathbb}Z\cong H^0(F,\mu(i)),\quad\quad {\mathbb}Z/w_i^{(p)}(F){\mathbb}Z\cong H^0(F,\mu_{p^\infty}(i)).$$ Then $K_{2i}({\mathcal{O}})$ decomposes into a direct sum $$K_{2i}({\mathcal{O}})\cong D_i({\mathcal{O}})\oplus{\mathbb}Z/w_i^{(p)}(F){\mathbb}Z,$$ where $D_i({\mathcal{O}})$ is a divisible ${\mathbb}Z_{(p)}$-module. On the other hand, the $e$-invariant $e:K_{2i-1}({\mathcal{O}})\to{\mathbb}Z/w_i(F){\mathbb}Z$ induces a direct sum decomposition $$K_{2i-1}({\mathcal{O}})\cong T_i({\mathcal{O}})\oplus {\mathbb}Z/w_i(F){\mathbb}Z,$$ where $T_i({\mathcal{O}})$ is a torsion-free ${\mathbb}Z_{(p)}$-module.
\[example\_K2\_groups\_of\_local\_field\] Consider the case $n=2$ as an example. Let $F$ be a finite extension of ${\mathbb}Q_p$ and let $\mu_F\subset F$ be the group of roots of unity inside it; put $m=|\mu_F|$. Then the Hilbert symbol induces a surjective homomorphism $H:K_2(F)\to\mu_F$. A theorem of C. Moore [@Moore1968] states that ${\operatorname{Ker}}H=mK_2(F)$ and that this kernel is an uncountable, divisible group (even uniquely-divisible, by [@Merkurjev1983]) contained inside $K_2({\mathcal{O}})$; moreover, $K_2(F)\to\mu_F$ splits.
Restricting to $K_2({\mathcal{O}})$ one obtains a split short exact sequence $$0{\longrightarrow}{\operatorname{Ker}}H=D_1({\mathcal{O}}){\longrightarrow}K_2({\mathcal{O}}){\longrightarrow}\mu^{(p)}_F\cong{\mathbb}Z/w_1^{(p)}(F){\mathbb}Z{\longrightarrow}0$$ where $\mu^{(p)}_F$ denotes the $p$-power roots of unity inside $F$.
Using pro-excision we may generalise these structural descriptions to all finite ${\mathbb}Z_p$-algebras satisfying the equivalent conditions (i)–(ii) above:
\[theorem\_description\_of\_K\_thy\_of\_p-adic\_orders\] Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, and let $i\ge 1$. Then $K_{2i}(A)$ decomposes as a direct sum $$K_{2i}(A)\cong D_i(A)\oplus W_i^{(p)}(A)$$ where $D_i(A)$ is a divisible ${\mathbb}Z_{(p)}$-module and $W_i^{(p)}(A)$ is a finite $p$-group. On the other hand, $K_{2i-1}(A)$ decomposes as a direct sum $$K_{2i-1}(A)\cong T_i(A)\oplus W_i(A)$$ where $T_i(A)$ is a torsion-free ${\mathbb}Z_{(p)}$-module and $W_i(A)$ is a finite group.
Let $F=\operatorname{Frac}A$ be the total quotient ring of $A$ and let ${\mathcal{O}}={\widetilde}A$ be its normalisation. The claims are clearly true for ${\mathcal{O}}$ since it is a finite product of rings of integers of finite extensions of ${\mathbb}Q_p$; therefore we may write $D_i({\mathcal{O}})$, $W_i^{(p)}({\mathcal{O}})$, etc. having the claimed properties.
Proposition \[proposition\_finite\_kernel\_and\_cokernel\] implies that the kernel and cokernel of $$K_{2i}(A)\to K_{2i}({\mathcal{O}})=D_i({\mathcal{O}})\oplus W_i^{(p)}({\mathcal{O}})$$ are finite $p$-groups. Therefore the kernel and cokernel of the composition $K_{2i}(A)\to D_i({\mathcal{O}})$ are finite $p$-groups; but divisible groups have no non-trivial finite images, so this map is actually surjective. The first claim now follows from the algebraic lemma \[lemma\_on\_groups\](ii) which we have postponed until the end of the section to avoid disrupting the exposition.
For the odd case, the same argument implies that the kernel (resp. cokernel) of the composition $K_{2i-1}(A)\to K_{2i-1}({\mathcal{O}}){\twoheadrightarrow}T_i({\mathcal{O}})$ is a finite group (resp. finite $p$-group). Its image is therefore a torsion-free ${\mathbb}Z_{(p)}$-submodule of $T_i({\mathcal{O}})$, and so we deduce that $K_{2i-1}(A)_{{\mbox{\scriptsize tors}}}$ is a finite subgroup of $K_{2i-1}(A)$. Finally we use algebraic lemma \[lemma\_on\_groups\](i) to see that $$0\to K_{2i-1}(A)_{{\mbox{\scriptsize tors}}}\to K_{2i-1}(A)\to K_{2i-1}(A)/K_{2i-1}(A)_{{\mbox{\scriptsize tors}}}\to 0$$ splits.
We stress that although the direct sum decompositions appearing in the proposition are not unique, the summands themselves are. Firstly, $D_i(A)=\bigcap_{n\ge 1}nK_{2i}(A)$ is the maximal divisible subgroup of $K_{2i}(A)$, and $W_i^{(p)}(A)$ is the quotient. Secondly, $W_i(A)$ is the torsion subgroup of $K_{2i-1}(A)$, and $T_i(A)$ is the quotient.
Gabber rigidity [@Gabber1992] (and Quillen’s calculation of the $K$-theory of finite fields) implies that $W_i(A)\otimes_{{\mathbb}Z}{\mathbb}Z[\tfrac{1}{p}]\cong K_{2i-1}(A/{\mathfrak}m)$.
From the proposition we obtain structural descriptions of the $K$-theory of $A$ with ${\widehat}{{\mathbb}Z}$ coefficients, again analogous to what is know for rings of integers of local fields; also, recall from lemma \[lemma\_homotopy\_description\_of\_K\_groups\] that $K_n(A;{\widehat}{{\mathbb}Z})\cong{\varprojlim}_r K_n(A/{\mathfrak}m^r)$ for all $n\ge 1$:
\[corollary\_profinite\_K\_groups\_of\_p\_adic\_order\] Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, let $i\ge 1$, and continue to use the notation introduced in the previous theorem. Then there is a natural isomorphism $$K_{2i}(A;{\widehat}{{\mathbb}Z})\cong W_i^{(p)}(A)$$ and a short exact sequence $$0\to K_{2i+1}(A)^{{{\widehat}{\phantom{o}}}}\to K_{2i+1}(A;{\widehat}{{\mathbb}Z})\to\operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_i(A))\to 0.$$
These readily follow from the standard short exact sequences for $K_*(-;{\widehat}{{\mathbb}Z})$ given in remark \[remark\_Z\_hat\_coefficients\].
Now we may prove an arithmetic analogue of the main results in section \[subsection\_main\_results\]; unfortunately we can only prove it in odd degrees:
Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, and let $i\ge 1$. Then there is a short exact sequence $$0\to K_{2i-1}(A)\to\operatorname*{``\varprojlim\!''}_rK_{2i-1}(A/{\mathfrak}m^r)\oplus K_{2i-1}({\widetilde}A)\to\operatorname*{``\varprojlim\!''}_rK_{2i-1}({\widetilde}A/{\mathfrak}M^r).$$ In particular, $$K_{2i-1}(A)\to K_{2i-1}(A/{\mathfrak}m^r)\oplus K_{2i-1}({\widetilde}A)$$ is injective for all $r\gg 0$.
Applying lemma \[lemma\_homotopy\_description\_of\_K\_groups\] and the previous corollary to ${\widetilde}A$, we see that $$K_{2i}({\widetilde}A)\to {\varprojlim}_rK_{2i}({\widetilde}A/{\mathfrak}M^r)=K_{2i}({\widetilde}A;{\widehat}{{\mathbb}Z})=W_i^{(p)}(A)$$ is surjective. The claimed injectivity now follows from corollary \[corollary\_application\_of\_group\_version\_of\_les\], and then the short exact sequence follows from example \[example\_main\_application\_of\_birelative\_vanishing\].
We would like to prove the injectivity claim of the previous theorem in even degrees; such a result would imply that the first long exact Mayer–Vietoris sequence of example \[example\_main\_application\_of\_birelative\_vanishing\] breaks into short exact sequences, thereby fully extending the main results of section \[subsection\_main\_results\] to such finite ${\mathbb}Z_p$-algebras. Unfortunately, the best that we can offer in even degree is a list of equivalent conditions which reduces the problem to understanding the torsion in $K_{2i}(A)$, which is unfortunately a difficult problem whose solution is only know when $i=1$:
\[proposition\_equivalent\_conditions\_in\_even\_case\] Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, and let $i\ge 1$. Then the following are equivalent:
1. $K_{2i}(A)\to K_{2i}(A/{\mathfrak}m^r)\oplus K_{2i}({\widetilde}A)$ is injective for all $r\gg 0$.
2. The canonical map $\operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_i(A))\to \operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_i({\widetilde}A))$ is surjective.
3. The canonical map $D_i(A)\to D_i({\widetilde}A)$ is injective.
4. The canonical map $D_i(A)\to D_i({\widetilde}A)$ is an isomorphism.
Corollary \[corollary\_profinite\_K\_groups\_of\_p\_adic\_order\] implies that $W_i^{(p)}(A)=K_{2i}(A)/D_i(A)$ embeds into $K_{2i}(A/{\mathfrak}m^r)$ for $r\gg 0$, from which (i)$\Leftrightarrow$(iii) easily follows. Next notice that the map $D_i(A)\to D_i({\widetilde}A)$ has finite kernel and cokernel by proposition \[proposition\_finite\_kernel\_and\_cokernel\]; since the cokernel is divisible it is actually zero, and so this map is surjective. This proves (iii)$\Leftrightarrow$(iv) and gives a short exact sequence $$0\to G\to D_i(A)\to D_i({\widetilde}A)\to 0$$ where $G$ is a finite group. The long exact sequence for ${\operatorname}{Ext}_{{\mathbb}Z}^*({\mathbb}Q/{\mathbb}Z,-)$ of this sequence degenerates to $$0\to \operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_i(A))\to \operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_i(A))\to G\to 0,$$ which proves (ii)$\Leftrightarrow$(iii).
We finish this section by collecting together the various results from abstract algebra which were required in the proof of theorem \[theorem\_description\_of\_K\_thy\_of\_p-adic\_orders\]:
\[lemma\_on\_groups\]
1. Let $G$ be an abelian group and suppose that the subgroup of torsion elements $G_{{\mbox{\scriptsize tors}}}$ has finite exponent. Then $G_{{\mbox{\scriptsize tors}}}$ is a direct summand of $G$.
2. Let $$0\to B\to C\to D\to 0$$ be a short exact sequence of abelian groups, where $B$ is finite and $D$ is divisible. Then $C$ is isomorphic to the direct sum of a divisible group and a finite group which is a quotient of $B$.
\(i) is a (perhaps unfamiliar) result from the theory of pure subgroups; e.g., see [@Robinson1996 4.3.9].
(ii): Since $B$ is finite, its lattice of subgroups $nC\cap B$, $n\ge 1$, is eventually constant, equal to $B_\infty\subseteq B$, say. In other words, there is a fixed integer $m\ge 1$ such that $nmC\cap B=B_\infty$ for all $n\ge 1$. Since $D$ is divisible, the map $mC\to D$ is surjective, and so we obtain an isomorphism $mC/B_\infty\cong D$; moreover, $B_\infty\subseteq\bigcap_{n\ge1}nmC$ by construction, and so it easily follows that $mC$ is a divisible group.
Since $C/B$ is $m$-divisible, the map $B\to C/mC$ is surjective, and thus induces an isomorphism $B/B_\infty{\stackrel{\simeq}{\to}}C/mC$. In conclusion we obtain an exact sequence $$0\to mC\to C\to B/B_\infty\to 0,$$ where $mC$ is a divisible group and $B/B_\infty$ is a finite group. But Baer’s well-known criterion states that divisible groups are injective in the category $Ab$, and so this exact sequence splits.
Finite ${\mathbb}Z_p$-algebras continued: $K_2$ and Geller’s conjecture {#subsection_mixed_char_Geller}
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We continue to study pro-excision for finite ${\mathbb}Z_p$-algebras, now focussing on $K_2$ and applications to Geller’s conjecture in mixed characteristic. For a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, lemma \[lemma\_homotopy\_description\_of\_K\_groups\] and corollary \[corollary\_profinite\_K\_groups\_of\_p\_adic\_order\] tell us that $$K_2(A;{\widehat}{{\mathbb}Z})=W_1^{(p)}(A)={\varprojlim}_r K_2(A/{\mathfrak}m^r)=K_2(A/{\mathfrak}m^r)\quad(r\gg0),$$ which is a finite $p$-group. Moreover, example \[example\_K2\_groups\_of\_local\_field\] implies that if $A$ is normal then this group is simply the group of $p$-power roots of unity inside $A$.
For $K_2$ of such ${\mathbb}Z_p$-algebras we can prove the full analogue of the main theorems of section \[subsection\_main\_results\]:
Let $A$ be a reduced ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module. Then $D_1(A)$ is torsion-free and there is a short exact, Mayer–Vietoris sequence $$0\to K_2(A)\to K_2(A;{\widehat}{{\mathbb}Z})\oplus K_2({\widetilde}A)\to K_2({\widetilde}A;{\widehat}{{\mathbb}Z})\to 0.$$
Example \[example\_K2\_groups\_of\_local\_field\] implies that $D_1({\widetilde}A)$ is not merely divisible, but is also torsion-free. Hence $\operatorname{Hom}_{{\mathbb}Z}({\mathbb}Q/{\mathbb}Z,D_1({\widetilde}A))=0$, so condition (ii) of proposition \[proposition\_equivalent\_conditions\_in\_even\_case\] implies that $K_2(A)\to K_2(A/{\mathfrak}m^r)\oplus K_2({\widetilde}A)$ is injective for $r\gg 0$ and that $D_1(A)\cong D_1({\widetilde}A)$. Also, $K_1(A)={A^{\!\times}}\to K_1({\widetilde}A)={{\widetilde}A^{\!\times}}$ is injective. Combining these two results with the long exact sequence of corollary \[corollary\_group\_versions\_of\_les\] implies that there is a short exact sequence $$0\to K_2(A)\to {\varprojlim}_rK_2(A/{\mathfrak}m^r)\oplus K_2({\widetilde}A)\to {\varprojlim}_rK_2({\widetilde}A/{\mathfrak}M^r)\to 0,$$ which is the desired result.
A useful diagrammatic way to restate the theorem is the following: $$\xymatrix{
0 \ar[r] & D_1(A) \ar[r]\ar[d]^{\cong} & K_2(A) \ar[r]\ar[d] & K_2(A;{\widehat}{{\mathbb}Z})\ar[r]\ar[d] & 0\\
0 \ar[r] & D_1({\widetilde}A) \ar[r] & K_2({\widetilde}A) \ar[r] & K_2({\widetilde}A;{\widehat}{{\mathbb}Z})\ar[r] & 0
}$$ Thus the right square is bicartesian and all reasonable questions concerning the central vertical arrow can be reduced to an analogous question for the right vertical arrow. In particular we obtain the following, which reduces Geller’s conjecture in mixed charactersitic to an Artinian version (c.f. the opening paragraph of section \[subsubsection\_Geller\]):
\[corollary\_Artinian\_Geller\_mixed\_char\] Let $A$ be a one-dimensional, Noetherian, reduced local ring of mixed characteristic $(0,p)$, with finite residue field, and such that $A\to{\widetilde}A$ is finite. Consider the following statements:
1. $K_2(A){\longrightarrow}K_2(\operatorname{Frac}A)$ is injective.
2. $K_2({\widehat}A){\longrightarrow}K_2(\operatorname{Frac}{\widehat}A)$ is injective.
3. $K_2({\widehat}A;{\widehat}{{\mathbb}Z})\to K_2({\widehat}{{\widetilde}A};{\widehat}{{\mathbb}Z})$ is injective.
4. $K_2(A/{\mathfrak}m^r){\longrightarrow}K_2({\widetilde}A/{\mathfrak}M^r)$ is injective for $r\gg 0$.
Then (i)$\Rightarrow$(ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv).
(i)$\Rightarrow$(ii) is proved exactly as in the proof of theorem \[theorem\_Geller\]. The remaining equivalences are clear in light of the above bicartesian square since ${\widehat}A$ satisfies the conditions of the previous theorem.
Using the corollary and a handful of lemmas which we postpone until afterwards, we can now present the first ever results on Geller’s conjecture in mixed characteristic. We can rarely show that $A$ is regular, i.e., ${\operatorname}{embdim}A=1$, but only that ${\operatorname}{embdim}A\le 2$. This is a consequence of the inability of our methods to detect the crucial element $p\in A$.
Additionally, in case (ii) of the theorem we must exclude one possibility, which we now explain. If $q$ is a power of $p$ and ${\mathcal{O}}$ is the ring of integers of a finite extension of $\operatorname{Frac}{\mathbb}Z_q$, then $$\{(f,g)\in{\mathbb}Z_q\times{\mathcal{O}}:f\mbox{ mod }p{\mathbb}Z_q=g\mbox{ mod }{\mathfrak}p\}\tag{\dag}$$ (${\mathfrak}p$ is the maximal ideal of ${\mathcal{O}}$) is a seminormal finite ${\mathbb}Z_p$-algebra as in example \[examples\_p\_adic\_orders\](ii); if moreover ${\mathcal{O}}/{\mathfrak}p$ is a strict extension of ${\mathbb}F_q$ and ${\mathcal{O}}$ also contains non-trivial $p$-power roots of unity, then we say that ($\dag$) is [*bad*]{}.
\[theorem\_geller\_in\_mixed\_char\] Let $A$ be a one-dimensional, Noetherian, reduced local ring of mixed characteristic $(0,p)$ such that $A\to{\widetilde}A$ is finite, and with finite residue field. Assume $p\neq2$ and suppose that at least one of the following is true:
1. $\operatorname{Frac}{\widehat}A$ contains no non-trivial $p$-power roots of unity; or
2. $A$ is seminormal, but ${\widehat}A$ is not isomorphic to a bad ring in the above sense; or
3. ${\widetilde}A$ is local and all $p$-power roots of unity in $\operatorname{Frac}{\widehat}A$ belong to ${\widehat}A$.
If the map $K_2(A)\to K_2(\operatorname{Frac}A)$ is injective then ${\operatorname}{embdim}A\le 2$.
In fact, in case (i), if $p\in{\mathfrak}m^2$ then we actually prove that ${\operatorname}{embdim}A=1$, i.e. that $A$ is regular.
Using corollary \[corollary\_Artinian\_Geller\_mixed\_char\] we see that we may replace $A$ by its completion, which is a reduced, local ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module. Letting ${\mathbb}F_q=A/{\mathfrak}m$, example \[examples\_p\_adic\_orders\](iii) says that ${\mathbb}Z_q\subseteq A$.
(i): Assume first that $\operatorname{Frac}A$, hence ${\widetilde}A$, contains no non-trivial $p$-power roots of unity. Then $K_2({\widetilde}A;{\widehat}{{\mathbb}Z})=0$, so again using the corollary we deduce that if $K_2(A)\to K_2(\operatorname{Frac}A)$ is injective then $K_2(A/{\mathfrak}m^r)=0$ for $r\gg 0$; but since $K_2(A/{\mathfrak}m^r)\to K_2(A/{\mathfrak}m^2)$ is surjective, this would imply that $K_2(A/{\mathfrak}m^2)= 0$.
Therefore, according to proposition \[proposition\_K\_2\_via\_diff\_forms\] below, $\bigwedge^2_{{\mathbb}F_q}{\mathfrak}m/({\mathfrak}m^2+p{\mathbb}Z_q)=0$; i.e., $$\dim_{{\mathbb}F_q}{\mathfrak}m/({\mathfrak}m^2+p{\mathbb}Z_q)\le1,$$ from which (i) and the final claim about $p\in{\mathfrak}m^2$ follow.
(ii): Now assume instead that $A$ is seminormal. By standard theory of seminormal rings (we refer the reader to any of [@Davis1978; @Weibel1989; @Roberts1976; @Weibel1980] for such standard theory), $A$ has the following description: if ${\mathfrak}q_1,\dots,{\mathfrak}q_m$ are the minimal prime ideals of $A$, and $I_i:=\bigcap_{j\neq i}{\mathfrak}q_j$, then the maximal ideal of $A$ is ${\mathfrak}m=I_1+\cdots+I_m$ and this sum is direct.
We first treat the case $m>2$. In this case there clearly exist indices ${\alpha}\neq \beta$ and elements $x\in I_{\alpha}$, $y\in I_\beta$ such that $x,y,p$ are linearly independent in ${\mathfrak}m/{\mathfrak}m^2=I_1/I_1^2\oplus\cdots\oplus I_m/I_m^2$. Then the Dennis–Stein symbol ${\langle x,y\rangle}\in K_2(A)$ vanishes in $K_2(A/{\mathfrak}q_i)$ for all $i$, since each ${\mathfrak}q_i$ contains $x$ or $y$; hence ${\langle x,y\rangle}$ vanishes in $K_2({\widetilde}A)$. But ${\langle x,y\rangle}$ has non-zero image in $K_2(A/{\mathfrak}m^2)$ by proposition \[proposition\_K\_2\_via\_diff\_forms\] and by choice of $x,y$. This shows that $K_2(A)\to K_2({\widetilde}A)$ cannot be injective when $m>2$.
Next suppose that $m=2$, so that $$A=\{(f_1,f_2)\in{\widetilde}{A/I_1}\times{\widetilde}{A/I_2}:f_1\mbox{ mod } {\mathfrak}p_1=f_2\mbox{ mod } {\mathfrak}p_2\in {\mathbb}F_q\},$$ where ${\mathfrak}p_i$ denotes the maximal ideal ${\widetilde}{A/I_i}$, which is the ring of integers of a finite extension of ${\mathbb}Q_p$. There are two subcases to consider. Firstly, if the images of $p$ in $I_1/I_1^2$ and $I_2/I_2^2$ span neither of these ${\mathbb}F_q$-spaces, then there exist $x\in I_1$ and $y\in I_2$ such that $x,y,p$ are linearly independent in ${\mathfrak}m/{\mathfrak}m^2$, and the same proof as in the case $m>2$ works. Secondly suppose that the image of $p$ in $I_2/I_2^2$ spans this space; this case is trickier. Since $I_2/I_2^2$ is the tangent space of the local ring $A/I_1$, we deduce that $A/I_1$ is both regular and unramified, i.e. $A/I_1\cong {\mathbb}Z_q$. Letting ${\mathcal{O}}=A/I_2$, it follows that $A$ is exactly of type () above; so, by our assumption that $A$ is not bad, either ${\mathcal{O}}$ has residue field ${\mathbb}F_q$ or ${\mathcal{O}}$ contains no non-trivial $p$-power roots of unity. The second case is covered by (i). In the first case it is straightforward to check that ${\mathfrak}m$ is generated by the elements $(p,0),(0,\pi)$, where $\pi$ is a uniformiser of ${\mathcal{O}}$ (c.f. example \[example\_geller\] below), whence ${\operatorname}{embdim}A\le 2$. This completes the proof of part (ii) of the theorem.
(iii): Finally, suppose that ${\widetilde}A$ is local and all $p$-power roots of unity in $\operatorname{Frac}A$ belong to $A$. Let $\zeta$ be a generator of the (possibly trivial) group of $p$-power roots of unity in ${\widetilde}{A}$ and let $A/{\mathfrak}m={\mathbb}F_q$. Then $A$ contains ${\mathcal{O}}:={\mathbb}Z_q[\zeta]$, which is the ring of integers in ${\mathbb}Q_q(\zeta)$. Thus the composition $${\langle \zeta\rangle}=K_2({\mathcal{O}};{\widehat}{{\mathbb}Z})\to K_2(A;{\widehat}{{\mathbb}Z})\to K_2({\widetilde}{A};{\widehat}{{\mathbb}Z})={\langle \zeta\rangle}$$ is an isomorphism. Assuming henceforth that $K_2(A)\to K_2({\widetilde}A)$ is injective implies, using corollary \[corollary\_Artinian\_Geller\_mixed\_char\], that the second arrow in this composition is injective; therefore the second arrow is actually an isomorphism and the first arrow is a split surjection.
Hence $K_2({\mathcal{O}}/{\mathfrak}p^2)\to K_2(A/{\mathfrak}m^2)$ is surjective, where ${\mathfrak}p$ denotes the maximal ideal of ${\mathcal{O}}$. Rewriting these $K_2$ groups in terms of differential forms using lemma \[lemma\_relative\_to\_absolute\_HC\] and applying the standard exact sequence for differential forms, we see that $\Omega_{(A/{\mathfrak}m^2)/({\mathcal{O}}/{\mathfrak}p^2)}^1/d(A/{\mathfrak}m^2)=0$. Finally, from a slight modification of lemma \[lemma\_square\_zero\], this can be rewritten as $$\bigwedge\nolimits_{{\mathbb}F_q}^2{\mathfrak}m/({\mathfrak}m^2+\pi{\mathcal{O}})=0,$$ where $\pi$ is a uniformiser of ${\mathcal{O}}$. Just as we finished the proof of part (i), this implies ${\operatorname}{embdim}A\le 2$.
\[example\_geller\] Let $p>2$ be prime and let $q=p^l$ be a power of $p$. Then $A:={\mathbb}Z_p+p^s{\mathbb}Z_q$ is a reduced, local ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module; $A$ has normalisation ${\mathbb}Z_q$, maximal ideal ${\mathfrak}m=p{\mathbb}Z_p+p^s{\mathbb}Z_q$, residue field ${\mathbb}F_p$, and embedding dimension $\dim_{{\mathbb}F_p}{\mathfrak}m/{\mathfrak}m^2=l+1$. Indeed, an ${\mathbb}F_p$ basis for ${\mathfrak}m/{\mathfrak}m^2$ is given by $p$, $p^s\theta_i$, $i=1,\dots,l$, where $\{\theta_i\}$ are Teichmüller lifts of a basis of ${\mathbb}F_q$ as a ${\mathbb}F_p$-space.
So, assuming that $l\neq 1$, part (i) of the theorem implies that $K_2(A)\to K_2({\mathbb}Z_q)$ is not injective. However, if $l=1$ then $\dim_{{\mathbb}F_p}{\mathfrak}m/(p{\mathbb}Z_p+{\mathfrak}m^2)=1$ and so $K_2(A/{\mathfrak}m^2)=0$, telling us nothing about the putative injectivity of $K_2(A)\to K_2({\mathbb}Z_q)$. It seems likely that the Dennis–Stein symbol ${\langle p,p^s\theta_1\rangle}\in K_2(A/{\mathfrak}m^r)$ will be non-zero for $r\gg0$, which would prove non-injectivity of $K_2(A)\to K_2({\mathbb}Z_q)$, but I cannot prove it.
The theorem required various explicit descriptions of $K_2(A/{\mathfrak}m^r)$, especially when $r=2$, which we establish in the remainder of this section by modifying classical results such as those in [@MaazenSteinstra1977] and [@Weibel1980a]:
\[lemma\_generalisation\_of\_MS\] Let $R$ be a ring containing a nilpotent ideal $I$; let $N$ be the smallest integer for which $I^N=0$, and assume that $N!\in{R^{\!\times}}$. Then there is a natural isomorphism $$K_2(R,I)\cong HC_1(R,I).$$
If $N=1$ then both sides vanish; assume henceforth that $N>1$. So, in particular, $2$ is invertible in $R$, which implies that $HC_0, HC_1$ and $HC_2$ may be defined using Connes’ complex rather than the cyclic bicomplex: see the remark in [@Loday1992 §2.1]. So the relative group $HC_1(R,I)$ admits the following description: First let $C_1(R,I)$ be the submodule of $R\otimes_{\mathbb}ZR$ generated by symbols $a\otimes b$ where at least one of $a,b$ lies in $I$; then $HC_1(R,I)$ is the abelian group obtained by quotienting $C_1(R,I)$ by the relations $$\begin{aligned}
&ab\otimes c-a\otimes bc + ca\otimes b=0\quad\quad (a,b,c\in R,\mbox{ at least one in }I)\\
&a\otimes b+b\otimes a=0\quad\quad (a,b\in R,\mbox{ at least one in }I)\end{aligned}$$
On the other hand, F. Keune [@Keune1978 Thm. 15] proved that $K_2(R,I)$ admits the following description by Dennis–Stein symbols: It is the abelian group generated by symbols ${\langle a,b\rangle}$, where $a,b\in R$ and at least one of $a,b$ lies in $I$, modulo the relations $$\begin{aligned}
&{\langle a,b\rangle}=-{\langle -b,-a\rangle}\\
&{\langle a,b\rangle}+{\langle a,c\rangle}={\langle a,b+c+abc\rangle}\\
&{\langle a,bc\rangle}={\langle ab,c\rangle}+{\langle ac,b\rangle}\end{aligned}$$
In the case when $R\to R/I$ is split, and with the same hypothesis that $N!\in{R^{\!\times}}$, H. Maazen and J. Stienstra [@MaazenSteinstra1977 E.g. 3.12] explicitly constructed an isomorphism $$K_2(R,I)\cong {\operatorname{Ker}}(\Omega_R^1\to\Omega_{R/I}^1)/dI=HC_1(R,I),\quad {\langle a,b\rangle}\mapsto l(a,b)\,da,$$ where $l(X,Y)$ is a formal logarithm function. Their proof works verbatim in the general situation when $R\to R/I$ is not necessarily split, replacing $l(a,b)\,da$ by $l(a,b)\otimes a\in HC_1(R,I)$.
Next we pass from the relative groups to the absolute ones:
\[lemma\_relative\_to\_absolute\_HC\] Let $R$ be a finite ring with Jacobson radical ${\mathfrak}M$, and suppose that $R/{\mathfrak}M$ is a finite product of finite fields of characteristic $p$. Assume that ${\mathfrak}M^{p-1}=0$. Then there is a natural isomorphism $$K_2(R)\cong \Omega_R^1/dR$$
The relative group $K_2(R,{\mathfrak}M)$ is a ${\mathbb}Z_{(p)}$-module, while $K_3(R/{\mathfrak}M)$ is a finite group of order prime to $p$ (thanks to Quillen’s calculation of the $K$-theory of finite fields); therefore the map $K_3(R/{\mathfrak}M)\to K_2(R,{\mathfrak}M)$ is zero. Moreover, $K_2(R/{\mathfrak}M)=0$, again because $R/{\mathfrak}M$ is a finite product of finite fields, and so we have proved that $K_2(R,{\mathfrak}M)\cong K_2(R)$.
Now we will prove the analogous result for cyclic homology, which will finish the proof (using the previous lemma). Notice that it does not matter whether we compute $HH$ and $HC$ with respect to ${\mathbb}Z$ or with respect to the image of ${\mathbb}Z$ inside the ring under question; we will freely pass between the two without indictation. Since $R/{\mathfrak}M$ is a finite product of finite fields of characteristic $p$, it is smooth over ${\mathbb}F_p$ and $\Omega_{R/{\mathfrak}M}^*=0$ for $*\ge1$. The Hochschild–Kostant–Rosenberg theorem [@Loday1992 Thm. 3.4.4] and SBI sequence therefore implies that $HC_2(R/{\mathfrak}M)=HC_0(R/{\mathfrak}M)=R/{\mathfrak}M$. The existence of Teichmüller lifts easily implies that $HC_0(R)\to HC_0(R/{\mathfrak}M)$ is surjective, and therefore $HC_2(R)\to HC_2(R/{\mathfrak}M)$ is surjective. Moreover, $HC_1(R/{\mathfrak}M)=0$, completing the proof that $HC_1(R,{\mathfrak}M)=HC_1(R)=\Omega_R^1/dR$.
Next we specialise to the case of a square-zero ideal:
\[lemma\_square\_zero\] Let $R$ be a ring containing an ideal $I$ such that $I^2=0$ and such that $2\in{R^{\!\times}}$. Assume that the composition $H_{{\mbox{\scriptsize dR}}}^0(R)\to R\to R/I$ is surjective, and let $k$ be any subring of $H_{{\mbox{\scriptsize dR}}}^0(R)$ which surjects onto $R/I$. Then there is a natural isomorphism of $k$-modules $$\bigwedge\nolimits_{R/I}^2{\overline}I\cong\Omega_R^1/dR,\quad a\wedge b\mapsto a\,db,$$ where ${\overline}I:=I/I\cap k$.
First notice that $R=k+I$, though not necessarily as a direct sum, and that $\Omega_R^1=\Omega_{R/k}^1$, so we may work with $\Omega_{R/k}^1$ throughout; we will identify $\Omega_{R/k}^1/dR$ with $HC_1^k(R)$ via $a\,db\leftrightarrow a\otimes b$, which has the following presentation: it is the quotient of $R\otimes_kR$ by the $k$-submodule generated by $$\begin{aligned}
&ab\otimes c-a\otimes bc + ca\otimes b=0\quad\quad (a,b,c\in R),\\
&a\otimes b+b\otimes a=0\quad\quad (a,b\in R).\end{aligned}$$
Let $\Lambda$ be the $k$-submodule of $\bigwedge_k^2R$ generated by terms $a\wedge b$ where at least one of $a,b$ belongs to $k$. We claim that there is an isomorphism $$HC_1^k(R)\cong \big(\bigwedge\nolimits_k^2R\big)/\Lambda,\quad a\otimes b\leftrightarrow a\wedge b.$$ It is clear that $(\bigwedge_k^2R)/\Lambda\to\Omega_{R/k}^1/dR$, $a\wedge b\mapsto a\,db$ is well-defined, thereby defining the isomorphism in one direction. In the other direction, it is evident that $a\otimes b\mapsto a\wedge b\mod\Lambda$ sends $a\otimes b+b\otimes a$ to zero, so it remains only to check that $$ab\wedge c-a\wedge bc+ca\wedge b=0\mod\Lambda$$ for all $a,b,c\in R$. Since the identity is linear and symmetric in $a,b,c$ it is sufficient to prove it in the following two cases:
1. $a\in k$: Then the identity becomes $$ab\wedge c-a\wedge bc-ab\wedge c=-a\wedge bc\equiv0\mod\Lambda.$$
2. $a,b,c\in I$: Then the identity vanishes since $I^2=0$.
This proves that $HC_1^k(R)\to (\bigwedge_k^2R)/\Lambda$ is well-defined, completing the proof of our claimed isomorphism.
Finally, it is straightforward to see that the surjection $\bigwedge_k^2I\to(\bigwedge_k^2R)/\Lambda$ descends to an isomorphism $\bigwedge_{k/k\cap I}^2{\overline}I\cong(\bigwedge_k^2R)/\Lambda$.
Now we reach the main application of the lemmas; recall from example \[examples\_p\_adic\_orders\](iii) that if a finite local ${\mathbb}Z_p$-algebra has residue field ${\mathbb}F_q$ then it contains ${\mathbb}Z_q$.
\[proposition\_K\_2\_via\_diff\_forms\] Let $A$ be a reduced local ${\mathbb}Z_p$-algebra which is finitely generated and torsion-free as a ${\mathbb}Z_p$-module, with residue field ${\mathbb}F_q$, and assume $p>2$. Then there is a natural isomorphism $$K_2(A/{\mathfrak}m^2)\cong\bigwedge\nolimits^2_{{\mathbb}F_q}{\mathfrak}m/{\mathfrak}m',\quad{\langle x,y\rangle}\leftrightarrow y\wedge x$$ where ${\mathfrak}m'={\mathfrak}m^2+p{\mathbb}Z_q$.
Combine the previous three lemmas, with $R=A/{\mathfrak}m^2$ and $k={\mathbb}Z_q/{\mathbb}Z_q\cap{\mathfrak}m^2$.
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-theory of [$1$]{}-dimensional schemes. In [*Applications of algebraic [$K$]{}-theory to algebraic geometry and number theory, [P]{}art [I]{}, [II]{} ([B]{}oulder, [C]{}olo., 1983)*]{}, vol. 55 of [*Contemp. Math.*]{} Amer. Math. Soc., Providence, RI, 1986, pp. 811–818.
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Matthew Morrow,\
University of Chicago,\
5734 S. University Ave.,\
Chicago,\
IL, 60637,\
USA\
[mmorrow@math.uchicago.edu]{}\
<http://math.uchicago.edu/~mmorrow/>\
[^1]: University of Chicago, supported by a Simons Postdoctoral Fellowship.
[^2]: I am grateful to the referee for pointing out that this statement is often not clearly credited. For any ring $R$ there are natural maps , where $ch_n$ is the Chern character and $B$ is the boundary map from cyclic homology to negative cyclic homology.
Now assume that $I$ is a nilpotent ideal of a ${\mathbb}Q$-algebra $R$. Then a result of Goodwillie [@Goodwillie1985 Thm. II.5.1] states that the induced map on relative groups $B:HC_{n-1}(R,I)\to HN_n(R,I)$ is an isomorphism, thereby inducing the relative Chern character $ch_n:K_n(R,I)\to HC_{n-1}(R,I)$. Moreover, Goodwillie [@Goodwillie1986] also proved that this relative Chern character is an isomorphism; his proof relied on a “modified relative Chern character” $ch_n':K_n(R,I)\to HC_{n-1}(R,I)$, which was only recently proved, by Cortiñas and Weibel [@Cortinas2009a], to in fact be equal to $ch_n$. However, J.-L. Cathelineau [@Cathelineau1990] had already proved that the modified character $ch'_n$ respects the Adams/Hodge decompositions. In conclusion, $ch_n:K_n(R,I)\to HC_{n-1}(R,I)$ respects the decompositions.
|
---
abstract: 'To accommodate the seemingly “anti-hierarchical” properties of galaxies near the upper end of the mass function within our hierarchical paradigm, current models of galaxy evolution postulate a phase of vigorous AGN feedback at high redshift, which effectively terminates star formation by quenching the supply of cold gas. Using the SINFONI IFU on the VLT, we identified kpc-sized outflows of ionized gas in z$\sim$2$-$3 radio galaxies, which have the expected signatures of being powerful AGN-driven winds with the potential of terminating star formation in the massive host galaxies. The bipolar outflows contain up to few$\times 10^{10}$ M$_{\odot}$ in ionized gas with velocities near the escape velocity of a massive galaxy. Kinetic energies are equivalent to $\sim 0.2$% of the rest mass of the supermassive black hole. We discuss the results of this on-going study and the global impact of the observed outflows.'
address:
- 'GEPI, Observatoire de Paris, CNRS, Universite Denis Diderot; 5, Place Jules Janssen, 92190 Meudon, France'
- 'Marie-Curie Fellow'
author:
- 'Nesvadba, N., P., H.$^{1,}$'
- 'Lehnert, M. D.$^1$'
title: 'Outflows, Bubbles, and the Role of the Radio Jet: Direct Evidence for AGN Feedback at z$\sim$2'
---
The role of AGN feedback for galaxy evolution in the early universe
===================================================================
AGN feedback is now a critical element of state-of-the-art models of galaxy evolution tailored to solve some of the outstanding issues at the upper end of the galaxy mass function. Observationally, a picture emerges where AGN feedback is most likely related to the mechanical energy output of the synchrotron emitting, relativistic plasma ejected during the radio-loud phases of AGN activity: Giant cavities in the hot, X-ray emitting halos of massive galaxy clusters filled with radio plasma are robust evidence for AGN feedback heating the gas on scales of massive galaxy clusters (e.g., McNamara & Nulsen, 2007). Best et al. (2006) analyzed a large sample of early-type galaxies from the SDSS catalog with FIRST and NVSS radio data and found that heating by the radio source may well balance gas cooling over 2 orders of magnitude in radio power and in stellar mass.
However, since most of the growth of massive galaxies was completed during the first few Gyrs after the Big Bang, observations at low redshift can only provide evidence that AGN feedback is able to [*maintain*]{} the hot, hydrostatic halos of massive early-type galaxies ([*“maintenance mode”*]{}). If we want to observe directly whether AGN feedback indeed quenched star formation and terminated galaxy growth in the early universe ([*“quenching mode”*]{}), we have to search at high redshift. With this goal, we started a detailed analysis of the rest-frame optical line emission in powerful, z$\sim$2$-$3 radio galaxies with integral field spectroscopy, where we may plausibly expect the strongest signatures of AGN-driven winds.
Powerful radio galaxies at z$\sim$ 2$-$3: Dying starbursts in the most massive galaxies?
========================================================================================
The observed properties of HzRGs suggest they may be ideal candidates to search for strong, AGN-driven winds: They have large stellar (Seymour et al. 2007) and dynamical (Nesvadba et al. 2007a) masses of $\sim$$10^{11-12}$ M$_{\odot}$ and reside in significant overdensities of galaxies suggesting particularly massive underlying dark-matter halos (e.g., Venemans et al. 2007). Large molecular gas masses in some sources (e.g., Papadopoulos et al. 2000) and submillimeter observations suggest that some HzRGs at redshifts z$\ge 3-4$ are dust-enshrouded, strongly star-forming galaxies with FIR luminosities in the ULIRG regime. Interestingly, the fraction of submillimeter-bright HzRGs shows a rapid decline from $>$50% at z$>$2.5 to $\le$15% at z$<$2.5 (Reuland et al. 2004). This suggests that HzRGs may be particularly massive galaxies near the end of their phase of active star formation. They also host particularly powerful AGN. Thus, they are good candidates to search for the kinematic signatures of AGN-driven winds.
Observational evidence for AGN-driven winds in z$\sim$2$-$3 radio galaxies
==========================================================================
To directly investigate whether HzRGs may be the sites of powerful, AGN driven winds, we collected a sample of HzRGs at redshifts z$\sim$2$-$3 with rest-frame optical near-infrared spectral imaging obtained with SINFONI on the VLT. Including scheduled observations, our total sample will consist of 29 galaxies spanning wide ranges in radio power and radio size. We also include galaxies with compact, and probably young, radio sources. We will in the following concentrate on the analysis of a first subsample of 6 galaxies, 4 with extended jets with radii between 10 and 50 kpc, 2 with more compact radio sources $<$10 kpc in radius. For details see Nesvadba et al. (2006, 2007a, 2008).
![[*(left to right:)*]{} \[OIII\]$\lambda$5007 emission line morphologies of MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4 and TXS0828+193 at z$=$2.6. Contours indicate the line-free continuum morphology for MRC0406-244 and TXS0828+193, and the 4.8 GHz radio core for MRC0316-257, where we did not detect the continuum. []{data-label="nesvadba2_fig1"}](nesvadba2_fig1a.ps "fig:"){width="37.00000%"} ![[*(left to right:)*]{} \[OIII\]$\lambda$5007 emission line morphologies of MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4 and TXS0828+193 at z$=$2.6. Contours indicate the line-free continuum morphology for MRC0406-244 and TXS0828+193, and the 4.8 GHz radio core for MRC0316-257, where we did not detect the continuum. []{data-label="nesvadba2_fig1"}](nesvadba2_fig1b.ps "fig:"){width="32.00000%"} ![[*(left to right:)*]{} \[OIII\]$\lambda$5007 emission line morphologies of MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4 and TXS0828+193 at z$=$2.6. Contours indicate the line-free continuum morphology for MRC0406-244 and TXS0828+193, and the 4.8 GHz radio core for MRC0316-257, where we did not detect the continuum. []{data-label="nesvadba2_fig1"}](nesvadba2_fig1c.ps "fig:"){width="29.00000%"}
Continuum and emission line morphologies
----------------------------------------
Using an integral-field spectrograph, we were able to extract continuum-free line images as well as line-free continuum images from our three-dimensional data cubes (Fig. 1). We find that in all cases, the continuum emission is relatively compact, but spatially resolved in some cases, with half-light radii $\le 5$ kpc. Radio-loud AGN activity is often related to an on-going merger. However, we only identify one continuum knot per galaxy. For the merger scenario, this may suggest an advanced stage where the galaxies are seperated by less than the $\sim$ 4 kpc spatial resolution of our data. Alternatively, since SINFONI is relatively inefficient in detecting low surface-brightness continuum emission, nuclear activity may have been triggered by other processes like minor mergers or perhaps cooling flows in cluster environments.
The extended, distorted morphologies of HzRGs with extended jets seen in broad-band imaging are mostly due line contamination, originating from emission line regions that extend over several 10s of kpc, and are significantly larger than the continuum (Nesvadba et al. 2008), but extend to smaller radial distances than the radio lobes. The same is found from Ly$\alpha$ longslit spectroscopy (e.g., Villar-Martin 2003). Overall, different emission lines in the same galaxy show similar morphologies. In the galaxies with [*compact*]{} radio sources, the line emission appears also compact. This may suggest a causal relationship between the advance of the jet and the extent of the high surface brightness emission line gas.
Kinematics, outflow energies, and physical properties of the ionized gas {#ssec:kinematics}
------------------------------------------------------------------------
We fitted spectra extracted from individual spatial resolution elements to construct two-dimensional maps of the relative velocities and line widths (Fig. 2). Typically, the velocity maps show two bubbles with relatively homogeneous internal velocity, and projected velocities relative to each other of 700$-$1000 km s$^{-1}$, reminiscent of back-to-back outflows extending from near the radio core. MRC1138-262 has a more complex structure with at least 3 bubbles. Line widths are generally large, indicating strong turbulence, with typical FWHMs $\sim$500$-$1200 km s$^{-1}$. Areas with wider lines may be due to partial overlap between bubbles.
Filamentary morphologies and low gas filling factors suggest that the UV-optical line emission may originate from clouds of cold gas that are being swept up by an expanding hot medium, most likely related to the overpressurized ’cocoon’ of gas heated by the radio jet. In such a scenario the velocity of the clouds may yield an estimate of the kinetic energy injection rate necessary to accelerate the gas to the observed velocities of up to $\sim 10^{45}$ erg s$^{-1}$ (Nesvadba et al. 2006). The size and velocities of the outflow suggest dynamical timescales of few $\times 10^7$ yrs. Maintaining the observed outflows over such timescales requires total energy injections of $\sim 10^{60}$ erg. This is in the range of what is observed for AGN driven bubbles in massive clusters at low redshift (e.g., McNamara & Nulsen, 2006, and references therein). The observed velocities and kinetic energies are also in the range of escape velocities and binding energies expected for galaxies with masses of few $\times 10^{11}$ M$_{\odot}$ (Nesvadba et al. 2006). This may suggest that much of the gas participating in the outflows may ultimately be unbound from the underlying gravitational potential.
![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2a.ps "fig:"){height="27.00000%"} ![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2b.ps "fig:"){height="25.00000%"} ![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2c.ps "fig:"){height="25.00000%"}\
![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2d.ps "fig:"){height="27.00000%"} ![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2e.ps "fig:"){height="25.00000%"} ![[*top, left to right:*]{} Maps of relative velocities (in km s$^{-1}$) for MRC0316-257 at z$=$3.1, MRC0406-244 at z$=$2.4, and TXS0828+193 at z$=$2.6. [*bottom, left to right*]{}: Maps of FWHMs (in km s$^{-1}$) for the same galaxies. Contours show the jet morphologies. For TXS0828+193, the lobes are outside of the area shown.[]{data-label="nesvadba2_fig2"}](nesvadba2_fig2f.ps "fig:"){height="25.00000%"}
Molecular and ionized gas budgets
---------------------------------
Having measured H$\alpha$ line fluxes, we are able to roughly estimate ionized gas masses assuming case B recombination (see Nesvadba et al. 2008 for details). For galaxies where we also measured H$\beta$, we correct for extinction of A$_V\sim$1$-$4 mag and find ionized gas masses of up to few $\times$ $10^{10}$ M$_{\odot}$. (Without the correction, estimates are few $\times 10^9$ M$_{\odot}$, Nesvadba et al. 2008.) This exceeds the amount of ionized gas found in any other high-redshift galaxy population by several orders of magnitudes, including galaxies with starburst-driven winds. Nesvadba et al. (2007b) investigated a spatially-resolved, starburst driven wind in a strongly star-forming submillimeter-selected galaxy at z$\sim$2.6 with of order few$\times$$10^6$ M$_{\odot}$ in ionized gas in the wind. Compact radio galaxies have lower entrained gas masses, but in the range of what would be expected for less evolved outflows with similar entrainment rates as the galaxies with large radio lobes (Nesvadba et al. 2007a).
Molecular gas masses in strongly star-forming galaxies at high redshift are also typically in the range of few $\times 10^{10}$ M$_{\odot}$ (e.g., Neri et al. 2003), and are a necessary prerequisite to fuel the observed starbursts with star formation rates of few 100 M$_{\odot}$ yr$^{-1}$. However, not all HzRGs have been detected in CO. TXS0828+193 specifically, which is part of our sample, appears to have less than $\sim 10^{10}$ M$_{\odot}$ in molecular gas (Nesvadba et al., in prep.). This illustrates that the AGN winds may affect a significant fraction of the overall interstellar medium of strongly star-forming, massive galaxies in the early universe. Since the velocities are near the expected escape velocity of a massive galaxy and underlying dark-matter halo (§\[ssec:kinematics\]), much of this gas may actually escape.
Global impact of AGN driven winds
=================================
Four out of four HzRGs with extended radio jets show evidence for outflows with with kinetic energies of up to 10$^{60}$ erg over dynamical timescales of $10^7$ yrs, and the preliminary analysis of our full sample suggests that this is far from being unusual. Nesvadba et al. (2006, 2008) estimate that the outflow energies correspond to $\sim 10$% of the jet kinetic luminosity. If this coupling efficiency between jet and interstellar medium is typical for HzRGs with similarly powerful radio sources, then the redshift-dependent luminosity function of Willott et al. (2001) suggests that at redshifts z$\sim$ 1$-$3, AGN-winds release an overall energy density of about $10^{57}$ erg s$^{-1}$ Mpc$^{-3}$. Some of this energy release may contribute to heating and increasing the entropy in extra-galactic gas surrounding the HzRG, and to enhance gas stripping in satellite galaxies, so that subsequent merging with satellites will be relatively dissipationless. This may later contribute to preserving the low content in cold gas and old, luminosity weighted ages of the highly metal-enriched stellar population in massive galaxies to the present day, in spite of possible continuous accretion of satellite galaxies over cosmologically significant periods (Nesvadba et al. 2008).
If the outflows are related to the nuclear activity, then the ultimate energy source powering the outflow is accretion onto the supermassive black hole in the center of the galaxy. Thus, models of galaxy evolution often parameterize the efficency of AGN feedback by the energy equivalent of the rest mass of the black hole. Since we have reason to believe that HzRGs approximately fall onto the low-redshift M-$\sigma$ relationship between the mass of the supermassive black hole and velocity dispersion of the host, we can use the stellar mass estimates of Seymour et al. (2007) to roughly estimate the black hole mass of our targets. We find that of order 0.1% of the energy equivalent of the black hole mass in HzRGs is being released in kinetic energy of the outflows. A similar estimate based on the global energy density released by powerful radio galaxies estimated above, and the local black hole mass density yields a very similar result, $\sim 0.2$%. This is very close to what is assumed in galaxy evolution models (e.g., Di Matteo et al. 2005), and highlights the likely importance of the observed outflows on galaxy evolution.
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|
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abstract: |
We present measurements of $CP$-violation parameters in $b \to s\gamma$ transitions based on a sample of $\NBB05$ $B\bbar$ pairs collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB energy-asymmetric $e^+e^-$ collider. One neutral $B$ meson is fully reconstructed in the $\bz\to\ks\piz\gamma$ decay channel irrespective to the $\ks\piz$ intermediate state. The flavor of the accompanying $B$ meson is identified from its decay products. $CP$-violation parameters are obtained from the asymmetries in the distributions of the proper-time intervals between the two $B$ decays.
We obtain the following results for the $\ks\piz$ invariant mass covering the full range up to $1.8\GeVcc$: $$\begin{aligned}
{{\mathcal S}}_{\ks\piz\gamma}&=&\SkspizgmResultSS,\nonumber\\
{{\mathcal A}}_{\ks\piz\gamma}&=&\AkspizgmResultSS.\nonumber\end{aligned}$$
title: |
\
Time-Dependent [$CP$]{} Asymmetries in $\bz\to\ks\piz\gamma$ transition\
---
04[27510\^6]{} 05[38610\^6]{}
Introduction
============
In the Standard Model (SM), $CP$ violation arises from an irreducible phase, the Kobayashi-Maskawa (KM) phase [@Kobayashi:1973fv], in the weak-interaction quark-mixing matrix. The phenomena of time-dependent $CP$ violation in decays through radiative penguin processes such as $b\to s\gamma$ are sensitive to physics beyond the SM. Within the SM, the photon emitted from a $\bz$ ($\bzb$) meson is dominantly right-handed (left-handed). Therefore the polarization of the photon carries information on the original $b$-flavor and the decay is, thus, almost flavor-specific. As a result, the SM predicts a small asymmetry [@Atwood:1997zr; @Grinstein:2004uu] and any significant deviation from this expectation would be a manifestation of new physics. It was pointed out that in decays of the type $\bz\to P^0Q^0\gamma$, where $P^0$ and $Q^0$ represent any $CP$ eigenstate spin-0 neutral particles (e.g. $P^0 = \ks$ and $Q^0 = \piz$), many new physics effects on the mixing-induced $CP$ violation do not depend on the resonant structure of the $P^0Q^0$ system [@Atwood:2004jj].
At the KEKB energy-asymmetric $e^+e^-$ (3.5 on 8.0$\GeV$) collider [@bib:KEKB], the $\Upsilon(4S)$ is produced with a Lorentz boost of $\beta\gamma=0.425$ along the $z$ axis, which is defined as the direction antiparallel to the $e^+$ beam direction. In the decay chain $\Upsilon(4S)\to \bz\bzb \to {f_{\rm sig}}{f_{\rm tag}}$, where one of the $B$ mesons decays at time ${t_{\rm sig}}$ to a final state ${f_{\rm sig}}$, which is our signal mode, and the other decays at time ${t_{\rm tag}}$ to a final state ${f_{\rm tag}}$ that distinguishes between $B^0$ and $\bzb$, the decay rate has a time dependence given by $$\begin{aligned}
\label{eq:psig}
{\cal P}(\Dt) =
\frac{e^{-|\Dt|/{\taubz}}}{4{\taubz}}
\biggl\{1 + {\ensuremath{q}}\Bigl[ {{\mathcal S}}\sin({\Delta m_d}\Dt)
+ {{\mathcal A}}\cos({\Delta m_d}\Dt)
\Bigr]
\biggr\}.\end{aligned}$$ Here ${{\mathcal S}}$ and ${{\mathcal A}}$ are $CP$-violation parameters, $\taubz$ is the $B^0$ lifetime, ${\Delta m_d}$ is the mass difference between the two $B^0$ mass eigenstates, $\Dt$ is the time difference ${t_{\rm sig}}- {t_{\rm tag}}$, and the $b$-flavor charge ${\ensuremath{q}}$ = +1 ($-1$) when the tagging $B$ meson is a $B^0$ ($\bzb$). Since the $B^0$ and $\bzb$ mesons are approximately at rest in the $\Upsilon(4S)$ center-of-mass system (c.m.s.), $\Dt$ can be determined from the displacement in $z$ between the ${f_{\rm sig}}$ and $f_{\rm tag}$ decay vertices: $\Delta t \simeq ({z_{\rm sig}}- {z_{\rm tag}})/(\beta\gamma c)
\equiv \Delta z/(\beta\gamma c)$.
For $\bz\to\ks\piz\gamma$, the $\ks$ vertex is displaced from the $B$ vertex and often lies outside of the silicon vertex detector (SVD). When the $\ks$ vertex can be reconstructed inside the SVD, the time-dependent $CP$ asymmetry can be measured. Measurements of such $CP$ asymmetries were previously reported by [[BaBar]{}]{} [@Aubert:2004pe] and Belle [@bib:ushiroda05] in the $\bz\to\kstarz(\to\ks\piz)\gamma$ decay: $$\begin{aligned}
{{\mathcal S}}_{\kstarz\gamma} &=& 0.25 \pm 0.63 \pm 0.14 \phantom{{\mbox {\rm Belle}}}
({\mbox {{\sc B\hspace*{-0.2ex}a\hspace*{-0.2ex}B\hspace*{-0.2ex}a\hspace*{-0.2ex}r}}})\\
{{\mathcal S}}_{\kstarz\gamma} &=& \SkstarzgmResultlast \phantom{{\mbox {{\sc B\hspace*{-0.2ex}a\hspace*{-0.2ex}B\hspace*{-0.2ex}a\hspace*{-0.2ex}r}}}}
({\mbox {\rm Belle}}).\end{aligned}$$ Belle also measured these asymmetries with an extended $M_{\ks\piz}$ mass region[@bib:ushiroda05] ($M_{\ks\piz}<1.8\GeVcc$): $${{\mathcal S}}_{\ks\piz\gamma} = \SkspizgmResultlast \phantom{{\mbox {{\sc B\hspace*{-0.2ex}a\hspace*{-0.2ex}B\hspace*{-0.2ex}a\hspace*{-0.2ex}r}}}}
({\mbox {\rm Belle}}).$$ In this analysis, we update the $CP$ measurements for $\bz\to\ks\piz\gamma$ in the mass region $M_{\ks\piz}<1.8\GeVcc$ with an additional dataset of $\NBBadd$ $B\bbar$ pairs.
The Belle detector is a large-solid-angle magnetic spectrometer that consists of an SVD, a 50-layer central drift chamber (CDC), an array of aerogel threshold Čerenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect $K_L^0$ mesons and to identify muons (KLM). The detector is described in detail elsewhere [@Belle]. Two inner detector configurations were used. A 2.0 cm beampipe and a 3-layer silicon vertex detector (SVD1) was used for the first sample of $\NBBsvdI$ $B\bbar$ pairs, while a 1.5 cm beampipe, a 4-layer silicon detector (SVD2) and a small-cell inner drift chamber were used to record the remaining $\NBBsvdII$ $B\bbar$ pairs [@Ushiroda].
Event Selection, Flavor Tagging and Vertex Reconstruction
=========================================================
Event Selection for $\ks\piz\gamma$
-----------------------------------
For high energy prompt photons, we select an isolated cluster in the ECL that has no corresponding charged track, and has the largest energy in the c.m.s. We require the shower shape to be consistent with that of a photon. In order to reduce the background from $\piz$ and $\eta$ mesons, we exclude photons compatible with $\piz\to\gamma\gamma$ or $\eta\to\gamma\gamma$ decays; we reject photon pairs that satisfy $\mathcal{L}_{\piz}\ge 0.18$ or $\mathcal{L}_{\eta}\ge 0.18$, where $\mathcal{L}_{\piz(\eta)}$ is a $\piz$ ($\eta$) likelihood described in detail elsewhere [@Koppenburg:2004fz]. The polar angle of the photon direction in the laboratory frame is restricted to the barrel region of the ECL ($33^\circ < \theta_\gamma < 128^\circ$), but is extended to the end-cap regions ($17^\circ < \theta_\gamma < 150^\circ$) for the second data sample due to the reduced material in front of the ECL.
Neutral kaons ($\ks$) are reconstructed from two oppositely charged pions that have an invariant mass within $\pm 6\MeVcc$ ($2\sigma$) of the $\ks$ nominal mass. The $\pip\pim$ vertex is required to be displaced from the interaction point (IP) in the direction of the pion pair momentum [@Abe:2004xp]. Neutral pions ($\piz$) are formed from two photons with the invariant mass within $\pm 16\MeVcc$ ($3\sigma$) of the $\piz$ mass. The photon momenta are then recalculated with a $\piz$ mass constraint and we require the momentum of $\piz$ candidates in the c.m.s. to be greater than $0.3\GeVc$. The $\ks\piz$ invariant mass, $M_{\ks\piz}$, is required to be less than $1.8\GeVcc$.
$\bz$ mesons are reconstructed by combining $\ks$, $\piz$ and $\gamma$ candidates. We form two kinematic variables: the energy difference $\dE\equiv E_B^{\rm c.m.s.}-E_{\rm beam}^{\rm c.m.s.}$ and the beam-energy constrained mass $\mb\equiv\sqrt{(E_{\rm beam}^{\rm c.m.s.})^2-(p_B^{\rm
c.m.s.})^2}$, where $E_{\rm beam}^{\rm c.m.s.}$ is the beam energy, $E_B^{\rm
c.m.s.}$ and $p_B^{\rm c.m.s.}$ are the energy and the momentum of the candidate in the c.m.s. Candidates are accepted if they have $\mb >
5.2\GeVcc$ and $-0.5\GeV < \dE < 0.5\GeV$.
We reconstruct $\bp\to\ks\pip\gamma$ candidates in a similar way as the $\bz\to\ks\piz\gamma$ decay in order to reduce the cross-feed background from $\bp\to\ks\pip\gamma$ in $\bz\to\ks\piz\gamma$. The $\bp\to\ks\pip\gamma$ events are also used for various crosschecks. For a $\pip$ candidate, we require that the track originates from the IP and that the transverse momentum is greater than $0.1\GeVc$. We also require that the $\pip$ candidate cannot be identified as any other particle species ($K^+, p^+, e^+,$ and $\mu^+$).
Candidate $\bp\to\ks\pip\gamma$ and $\bz\to\ks\piz\gamma$ decays are selected simultaneously; we allow only one candidate for each event. The best candidate selection is based on the event likelihood ratio $\rsigbkg$ that is obtained from a Fisher discriminant $\calf$ [@Fisher], which uses the extended modified Fox-Wolfram moments [@Abe:2003yy] as discriminating variables. We select the candidate that has the largest $\rsigbkg$. The signal region is defined as $-0.2\GeV < \dE < 0.1\GeV$ and $5.27\GeVcc < \mb < 5.29\GeVcc$.
We use events outside the signal region as well as large Monte Carlo (MC) samples to study the background components. The dominant background is from continuum light quark pair production ($e^+e^-\to
q\,\bar{q}$ with $q = u,d,s,c$), which we refer to as $\qq$ hereafter. In order to reduce the $\qq$ background contribution, we form another event likelihood ratio $\rsigbkgBH$ by combining $\rsigbkg$ with $\cos\theta_H$ and $\cos\theta_B$, where $\theta_B$ is the polar angle of the $B$ meson candidate momentum in the laboratory frame, and $\theta_H$ is the angle between the $B$ candidate momentum and the daughter $\ks$ momentum in the rest frame of the $\ks\pi$ system. Since the relative background contribution will be smaller in the region of the $K^*$, we introduce two $\ks\piz$ invariant mass regions: MR1, defined as $0.8\GeVcc < M_{\ks\piz} < 1.0\GeVcc$, and MR2 which is defined as $M_{\ks\piz} < 1.8\GeVcc$ after excluding MR1. The specific $\rsigbkgBH$ selection criteria applied depend on both the mass region and flavor tagging information. After applying all other selection criteria described so far, 77% of the $\qq$ background is rejected while 87% of the $\kstarz\gamma$ signal is retained in MR1; in MR2, 87% of the $\qq$ is rejected while 68% of the $\ktwostarz\gamma$ signal is retained. Background contributions from $B$ decays, which are considerably smaller than $\qq$, are dominated by cross-feed from other radiative $B$ decays including $\bp\to\ks\pip\gamma$.
Flavor Tagging
--------------
The $b$-flavor of the accompanying $B$ meson is identified from inclusive properties of particles that are not associated with the reconstructed signal decay. The algorithm for flavor tagging is described in detail elsewhere [@bib:fbtg_nim]. We use two parameters, ${\ensuremath{q}}$ defined in Eq. (\[eq:psig\]) and $r$, to represent the tagging information. The parameter $r$ is an event-by-event flavor-tagging dilution factor that ranges from 0 to 1; $r=0$ when there is no flavor discrimination and $r=1$ implies unambiguous flavor assignment. It is determined by using MC data and is only used to sort data into six $r$ intervals. The wrong tag fraction $w$ and the difference $\Delta w$ in $w$ between the $\bz$ and $\bzb$ decays are determined for each of the six $r$ intervals from data [@Abe:2004xp].
Vertex Reconstruction
---------------------
The vertex position of the signal-side decay is reconstructed from the $\ks$ trajectory with a constraint on the IP; the IP profile ($\sigma_x\simeq 100\rm\,\mu m$, $\sigma_y\simeq 5\rm\,\mu m$, $\sigma_z\simeq 3\rm\,mm$) is convolved with the finite $B$ flight length in the plane perpendicular to the $z$ axis. Both pions from the $\ks$ decay are required to have enough hits in the SVD in order to reconstruct the $\ks$ trajectory with high resolution: at least one layer with hits on both sides and at least one additional hit in the $z$ side of the other layers for SVD1, and at least two layers with hits on both sides for SVD2. The reconstruction efficiency depends not only on the $\ks$ momentum but also on the SVD geometry. The efficiency with SVD2 (51%) is significantly higher than with SVD1 (40%) because of the larger detection volume. The other (tag-side) $B$ vertex determination is the same as that for the $\bz\to\phi\ks$ analysis [@Abe:2004xp].
Signal Yield Extraction
=======================
Figure \[fig:mb\] shows the $\mb$ ($\dE$) distribution for the reconstructed $\ks\piz\gamma$ candidates within the $\dE$ ($\mb$) signal region after flavor tagging and vertex reconstruction. The signal yield is determined from an unbinned two-dimensional maximum-likelihood fit to the $\dE$-$\mb$ distribution. The fit region is chosen as $-0.4\GeV <
\dE < 0.5\GeV$ and $5.2\GeVcc < \mb$ to avoid other $B\bbar$ background events that populate the low-$\dE$ high-$M_{\ks\piz}$ region. The signal distribution is represented by a PDF obtained from an MC simulation of $\bz\to\kstarz\gamma$ and $\bz\to\ktwostarz\gamma$ that accounts for a small correlation between $\mb$ and $\dE$. The background from $B$ decays are also modeled with an MC simulation. For the $\qq$ background, we use the ARGUS parameterization [@bib:ARGUS] for $\mb$ and a second-order polynomial for $\dE$. The normalizations of the signal and background distributions and the $\qq$ background shape are the five free parameters in the fit. We observe a total of $\NevtMRI$ candidates in the signal box in MR1, which decreases to $\NevtVMRI$ after flavor tagging and $B$ vertex reconstruction, and obtain $\NsigMRI$ signal events from the fit; the average signal purity over the six $r$ intervals is $\PurityMRI$%. In MR2, corresponding numbers are $\NevtMRII$, $\NevtVMRII$, $\NsigMRII$, and $\PurityMRII$%.
[$CP$]{} Asymmetry Measurements
===============================
We determine ${{\mathcal S}}$ and ${{\mathcal A}}$ from an unbinned maximum-likelihood fit to the observed $\Dt$ distribution. The probability density function (PDF) expected for the signal distribution, ${\cal P}_{\rm
sig}(\Dt;{{\mathcal S}},{{\mathcal A}},{\ensuremath{q}},w,\Delta w)$, is given by the time dependent decay rate \[Eq. (\[eq:psig\])\] modified to incorporate the effect of incorrect flavor assignment. The distribution is convolved with the proper-time interval resolution function $\Rsig$, which takes into account the finite vertex resolution. The parametrization of $\Rsig$ is the same as that used for the $\bz\to\ks\piz$ decay [@Abe:2004xp]. $\Rsig$ is first derived from flavor-specific $B$ decays [@bib:BELLE-CONF-0436] and modified by additional parameters that rescale vertex errors to account for the fact that there is no track directly originating from the $B$ meson decay point.
For each event, the following likelihood function is evaluated: $$\begin{split}
P_i
=& (1-\fol)\int_{-\infty}^{+\infty} \biggl[
\fsig{\cal P}_{\rm sig}(\Dt')\Rsig (\Dt_i-\Dt') \\
&+(1-\fsig){\cal P}_{\rm bkg}(\Dt')\Rbkg (\Dt_i-\Dt')\biggr]
d(\Dt') \\
&+\fol P_{\rm ol}(\Dt_i),
\label{eq:likelihood}
\end{split}$$ where $P_{\rm ol}$ is a Gaussian function that represents a small outlier component with fraction $\fol$ [@bib:resol]. The signal probability $\fsig$ is calculated on an event-by-event basis from the function which we obtained as the result of the two-dimensional $\dE$-$\mb$ fit for the signal yield extraction. A PDF for background events, ${\cal P}_{\rm bkg}$, is modeled as a sum of exponential and prompt components, and is convolved with a Gaussian which represents the resolution function $\Rbkg$ for the background. All parameters in ${\cal P}_{\rm bkg}$ and $\Rbkg$ are determined by a fit to the $\Dt$ distribution of a background-enhanced control sample, i.e. events outside of the $\dE$-$\mb$ signal region. We fix $\tau_\bz$ and ${\Delta m_d}$ at their world-average values [@bib:HFAG].
The only free parameters in the final fit are ${{\mathcal S}}_{\ks\piz\gamma}$ and ${{\mathcal A}}_{\ks\piz\gamma}$, which are determined by maximizing the likelihood function $L =
\prod_iP_i(\Dt_i;{{\mathcal S}},{{\mathcal A}})$ where the product is over all events. We obtain $$\begin{aligned}
{{\mathcal S}}_{\ks\piz\gamma} &=& \SkspizgmResultSS,\nonumber \\
{{\mathcal A}}_{\ks\piz\gamma} &=& \AkspizgmResultSS.\nonumber\end{aligned}$$
We define the raw asymmetry in each $\Dt$ bin by $(N_{q=+1}-N_{q=-1})/(N_{q=+1}+N_{q=-1})$, where $N_{q=+1(-1)}$ is the number of observed candidates with $q=+1(-1)$. Figure \[fig:asym\] shows the raw asymmetries for the $\ks\piz\gamma$ events. Note that these are simple projections onto the $\Delta t$ axis, and do not reflect other event-by-event information (such as the signal fraction, the wrong tag fraction and the vertex resolution), which is in fact used in the unbinned maximum-likelihood fit for ${{\mathcal S}}$ and ${{\mathcal A}}$.
Systematic Error
----------------
Primary sources of the systematic error are (1) uncertainties in the resolution function ($\pm 0.06$ for ${{\mathcal S}}_{\ks\piz\gamma}$ and $\pm 0.03$ for ${{\mathcal A}}_{\ks\piz\gamma}$), (2) uncertainties in the vertex reconstruction ($\pm 0.03$ for ${{\mathcal S}}_{\ks\piz\gamma}$ and $\pm 0.04$ for ${{\mathcal A}}_{\ks\piz\gamma}$) and (3) uncertainties in the background fraction ($\pm 0.07$ for ${{\mathcal S}}_{\ks\piz\gamma}$ and $\pm 0.03$ for ${{\mathcal A}}_{\ks\piz\gamma}$). Effects of tag-side interference [@Long:2003wq] contribute $\pm 0.07$ for ${{\mathcal A}}_{\ks\piz\gamma}$. Also included are effects from uncertainties in the wrong tag fraction and physics parameters (${\Delta m_d}$, $\taubz$ and ${{\mathcal A}}_{\kstarz\gamma}$). Fitting a large sample of MC events revealed no bias in the fit procedure. The statistical errors from the MC fit are assigned as systematic errors. The total systematic error is obtained by adding these contributions in quadrature.
Crosschecks
-----------
Various crosschecks of the measurement are performed. We apply the same fit procedure to the $\bz\to\jpsi\ks$ sample without using $\jpsi$ daughter tracks for the vertex reconstruction [@bib:sqq05]. We obtain ${{\mathcal S}}_{\jpsi\ks} = +0.73\pm 0.08$(stat) and ${{\mathcal A}}_{\jpsi\ks} =
+0.01\pm 0.04$(stat), which are in good agreement with the world-average values [@bib:PDG2004]. We perform a fit to $\bp\to\ks\pip\gamma$ , which is a good control sample of the $\bz\to\ks\piz\gamma$ decay, without using the primary $\pip$ for the vertex reconstruction. The result is consistent with no $CP$ asymmetry, as expected. Lifetime measurements are also performed for these modes, and values consistent with the world-average values are obtained. Ensemble tests are carried out with MC pseudo-experiments using ${{\mathcal S}}$ and ${{\mathcal A}}$ obtained by the fit as the input parameters. We find that the statistical errors obtained in our measurements are all within the expectations from the ensemble tests. Fits to the two $M_{\ks\piz}$ regions yield ${{\mathcal S}}= {{{\SkspizgmMRIVal}\pm {\SkspizgmMRIStat\mbox{(stat)}}\pm {\SkspizgmMRISyst\mbox{(syst)}}}}$ and ${{\mathcal A}}= {{{\AkspizgmMRIVal}\pm {\AkspizgmMRIStat\mbox{(stat)}}\pm {\AkspizgmMRISyst\mbox{(syst)}}}}$ for MR1, and ${{\mathcal S}}= {{{\SkspizgmMRIIVal}\pm {\SkspizgmMRIIStat}}}$(stat) and ${{\mathcal A}}= {{{\AkspizgmMRIIVal}\pm {\AkspizgmMRIIStat}}}$(stat) for MR2. The results are consistent with those from the full $M_{\ks\piz}$ sample.
Summary
=======
We have performed a measurement of the time-dependent $CP$ asymmetry in the decay $\bz\to\ks\piz\gamma$ with $\ks\piz$ invariant mass up to $1.8\GeVcc$, based on a sample of $\NBB05$ $B\bbar$ pairs. We obtain $CP$-violation parameters ${{\mathcal S}}_{\ks\piz\gamma}=\SkspizgmResultSS$ and ${{\mathcal A}}_{\ks\piz\gamma}=\AkspizgmResultSS$. We do not find any significant $CP$ asymmetry, and therefore no indication of new physics from right handed currents, with the present statistics.
Acknowledgment
==============
We thank the KEKB group for the excellent operation of the accelerator, the KEK cryogenics group for the efficient operation of the solenoid, and the KEK computer group and the National Institute of Informatics for valuable computing and Super-SINET network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Education, Science and Training; the National Science Foundation of China under contract No. 10175071; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation; the Polish State Committee for Scientific Research under contract No. 2P03B 01324; the Ministry of Science and Technology of the Russian Federation; the Ministry of Higher Education, Science and Technology of the Republic of Slovenia; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S.Department of Energy.
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|
---
abstract: |
We exploit the decoherence of electrons due to magnetic impurities, studied via weak localization, to resolve a longstanding question concerning the classic Kondo systems of Fe impurities in the noble metals gold and silver: which Kondo-type model yields a realistic description of the relevant multiple bands, spin and orbital degrees of freedom? Previous studies suggest a fully screened spin $S$ Kondo model, but the value of $S$ remained ambiguous. We perform density functional theory calculations that suggest $S =
3/2$. We also compare previous and new measurements of both the resistivity and decoherence rate in quasi 1-dimensional wires to numerical renormalization group predictions for $S=1/2,1$ and $3/2$, finding excellent agreement for $S=3/2$.
author:
- |
T. A. Costi$^{1,2}$, L. Bergqvist$^{1}$, A. Weichselbaum$^{3}$, J. von Delft$^{3}$, T. Micklitz$^{4,7}$, A. Rosch$^{4}$,\
P. Mavropoulos$^{1,2}$, P. H. Dederichs$^{1}$, F. Mallet$^{5}$, L. Saminadayar$^{5,6}$, and C. Bäuerle$^{5}$
title: 'Kondo decoherence: finding the right spin model for iron impurities in gold and silver '
---
[*Introduction.—*]{} The Kondo effect of magnetic impurities in non-magnetic metals, e.g. Mn, Fe or Co in Cu, Ag or Au, first manifested itself in the early 1930’s as an anomalous rise in resistivity with decreasing temperature, leading to a resistivity minimum [@deHaasvandenBergPhysica34]. In 1964 Kondo explained this effect [@KondoProgTheorPhys64] as resulting from an antiferromagnetic exchange coupling between the spins of localized magnetic impurities and delocalized conduction electrons. However, for many dilute magnetic alloys a fundamental question has remained unresolved to this day: which effective low-energy Kondo-type model yields a realistic description of the relevant multiple bands, spin and orbital degrees of freedom [@NozieresBlandinJPhys80]? Cases in point are Fe impurities in Au and Ag, the former being the very first magnetic alloy known to exhibit an anomalous resistivity minimum [@deHaasvandenBergPhysica34]. Previous attempts to fit experimental data on, for example, Fe impurities in Ag (abbreviated as AgFe) with exact theoretical results for thermodynamics, by assuming a fully screened low-energy effective Kondo model [@DesgrangesJPhysC85; @HewsonBook97], have been inconclusive: specific heat data is absent and the local susceptibility of Fe in Ag obtained from Mössbauer spectroscopy [@SteinerHuefnerPRB75] indicated a spin of $S=3/2$ while a fully screened $S=2$-model has been used to fit the temperature dependence of the local susceptibility [@SchlottmannSacramentoAdvPhys93]. A promising alternative route to identify the model for Fe in Au or Ag is offered by studying transport properties of high purity quasi-one dimensional mesoscopic wires of Au and Ag, doped with a carefully controlled number of Fe impurities by means of ion-implantation [@PierreBirgePRB03; @SchopferSaminadayarPRL03; @BauerleSaminadayarPRL05; @MalletBauerlePRL06; @AlzoubiBirgePRL06; @CapronBauerlePRB08]. Magnetic impurities affect these in two different ways. Besides causing the afore-mentioned resistivity anomaly, they also make an anomalous contribution ${\gamma_{\rm m}}(T)$ to the electronic phase decoherence rate ${\gamma_\phi}(T)$ measured in weak (anti)localization: an itinerant electron which spin-flip-scatters off a magnetic impurity leaves a mark in the environment and thereby suffers decoherence. By checking model predictions for both effects against experimental observations over several decades in temperature, decoherence can thus be harnessed as a highly sensitive probe of the actual form of the effective exchange coupling. Experiments along these lines [@MalletBauerlePRL06; @AlzoubiBirgePRL06] were consistent with a Kondo model in which the impurity spin is fully screened and inconsistent with underscreened or overscreened Kondo models [@MalletBauerlePRL06]. A consistent description of [*both*]{} resistivity and decoherence measurements using the simplest fully screened Kondo model, the $S=1/2$ single-channel Kondo model, was, however, not possible: different Kondo scales were required for fitting the resistivity and decoherence rates [@MalletBauerlePRL06; @AlzoubiBirgePRL06].
In this Letter we address the above problem via the following strategy: (i) We carry out density functional theory calculations within the local density approximation (LDA) for Fe in Au and Ag to obtain information that allows us to prescribe a low-energy effective model featuring 3 bands coupling to impurities with spin $S=3/2$. (ii) We calculate the resistivity ${\rho_{\rm m}}(T)$ and decoherence rate $\gamma_{\rm m} (T)$ due to magnetic impurities for three fully screened Kondo models, with $n=2S = 1$, 2 and 3, using Wilson’s numerical renormalization group (NRG) approach. (iii) We compare these predictions to experimental data: extracting the characteristic Kondo temperature ${T_{\rm K}^S}$ for each choice of $n$ from fits to ${\rho_{\rm m}}(T)$ and using these ${T_{\rm K}^S}$ to obtain parameter-free predictions for ${\gamma_{\rm m}}(T)$, we find that the latter agree best with experiment for $n=3$.
[*LDA calculations.—*]{} Fully relaxed density functional theory calculations employing the VASP code [@KresseFurthmuellerPRB96] showed that low-symmetry Fe configurations (split-interstitials [@VoglDederichsPRL76]) are energetically unfavorable: Fe impurities prefer an environment with cubic symmetry. As the calculated defect formation energy of an Fe interstitial was found to be about 2 eV higher than the energy of a substitutional defect, we discuss the latter case in the following. This is in line with experiments on Fe-implantation in AgAu alloys, where only substitutional Fe-atoms are found [@KirschFrotaPessoaEPL02].
Fig. \[fig1\] shows the d-level local density of states (LDOS) of substitutional Fe in Ag and Au, obtained by spin-polarized calculations using a 108 atom supercell, with similar results being found for a 256 atom supercell. The cubic local symmetry leads to e$_{\rm g}$ (doublet) and t$_{\rm 2g}$ (triplet) components with a e$_{\rm g}$-t$_{\rm 2g}$ splitting, $\Delta\gtrsim 0.15$eV in LDA (Fig. \[fig1\](a-b)). The widths $\Gamma_{\rm e_g}$ and $\Gamma_{\rm t_{2g}}$ of the e$_{\rm g}$ and t$_{\rm 2g}$ states close to the Fermi level ($E_{\rm F}$) are of the order of 1eV, resulting from a substantial coupling to the conduction electrons. The large t$_{\rm 2g}$ component at $E_{\rm F}$ persists within LDA+U (Fig. \[fig1\](c-d) using $U=3$eV and a Hund’s coupling $J_H=0.8$eV).
![ (color online). The d-level local density of states (LDOS) of substitutional Fe in Ag and Au within spin-polarized LDA (a,b) and LDA+U (c,d), with inclusion of spin-orbit interactions, and showing the e$_g$ (red) and t$_{\rm 2g}$ (black) components of the d-level LDOS of FeAg (left panels) and FeAu (right panels). Majority/minority contributions are shown positive/negative. Legends give the spin ($\mu_{S}$) and orbital ($\mu_{L}$) magnetic moments in units of the Bohr magneton ${\mu_{\rm B}}$, and the splitting ($\Delta$) between the e$_{\rm g}$ and t$_{\rm 2g}$ components of the d-level LDOS. []{data-label="fig1"}](fig1.eps){width="0.9\linewidth"}
The spin and orbital moments are given in the legends of Fig. \[fig1\] (spin-polarized Korringa-Kohn-Rostoker calculations yielded similar values [@KirschFrotaPessoaEPL02]): Within spin-polarized LDA a large spin moment $\mu_S$ of approximately 3-3.1${\mu_{\rm B}}$ forms spontaneously, consistent with Mösbauer measurements that give 3.1-3.2${\mu_{\rm B}}$ for the spin moment for Fe in Ag [@SteinerHuefnerPRB75]. In contrast, there is no tendency for a sizable orbital moment (or a Jahn-Teller distortion). The small orbital moments $\mu_L$ of $< 0.1 \, {\mu_{\rm B}}$ (consistent with experimental results [@brewer.04]) arise only due to weak spin-orbit coupling. We therefore conclude that the orbital degree of freedom is quenched on an energy scale set by the width $\Gamma_{t_{\rm 2g}}$ of the t$_{\rm
2g}$ orbitals. Moreover, since the spin-orbit splitting of the localized spin in the cubic environment is proportional to $\mu_L^4$, it is tiny, well below our numerical precision of $0.01$meV, and, therefore, smaller than the relevant Kondo temperatures.
[*Low-energy effective models.—*]{} The above results justify formulating an effective low-energy model in terms of the spin-degree of freedom only. The large spin moment $\mu_S$ of 3-3.1${\mu_{\rm B}}$ suggests an effective spin $S=3/2$. Our LDA results thus imply as effective model a spin-3/2 3-channel Kondo model, involving local and band electrons of t$_{\rm 2g}$ symmetry. An alternative possibility, partially supported by the large (almost itinerant) t$_{\rm 2g}$ component at $E_{\rm F}$, would be to model the system as a spin-1 localized in the e$_{\rm g}$ orbitals, that is perfectly screened by two conduction electron channels of e$_{\rm g}$ symmetry. This spin is then also coupled to (almost itinerant) t$_{\rm
2g}$ degrees of freedom via the ferromagnetic $J_H$. At high temperature, the latter binds an itinerant $t_{\rm 2g}$ spin 1/2 to the local spin 1 to yield an effective spin-3/2, consistent with the spin-moment of 3-3.1 ${\mu_{\rm B}}$ obtained within LDA, whereas in the low temperature limit, the irrelevance of $J_{H}$ under renormalization [@HewsonBook97] leads to the stated effective spin-1, 2-band model. Though such a model is well justified only for $J_{H}\ll
\Gamma_{\rm t_{2g}}$, which is not the case here where $J_{H}\sim
\Gamma_{\rm t_{2g}}$, our LDA results do not completely exclude such a model. To identify which of the models is most appropriate, we shall confront their predictions with experimental data below.
We thus describe Fe in Ag and Au using the following fully screened Kondo model: $$\label{eq:NChannelKondo}
H = \sum_{k\alpha \alpha} \varepsilon_k c^\dagger_{k \alpha \sigma}
c_{k \alpha \sigma} + J \sum_\alpha {\bf S} \cdot {\bf s}_\alpha \ .$$ It describes $n$ channels of conduction electrons with wave vector $k$, spin $\sigma$ and channel index $\alpha$, whose spin density $\Sigma_\alpha {\bf s}_\alpha$ at the impurity site is coupled antiferromagnetically to an Fe impurity with spin $S=n/2$. Whereas our LDA results suggest $n=3$, we shall also consider the cases $n=1$ and 2.
[*NRG calculations.—*]{} The resistivity ${\rho_{\rm m}}(T)$ and decoherence rate $ {\gamma_{\rm m}}(T)$ induced by magnetic impurities can be obtained from the temperature and frequency dependence of the impurity spectral density [@MicklitzRoschPRL06; @ZarandAndreiPRL04]. We have calculated these quantities using the NRG [@NRGspectra; @NRGspectraDevelopments; @WeichselbaumVonDelftPRL07]. While such calculations are routine for $n=1$ and 2 [@NRGspectra], they are challenging for $n=3$. Exploiting recent advances in the NRG [@WeichselbaumVonDelftPRL07] we were able to obtain accurate results also for $n=3$ (using a discretization parameter of $\Lambda=2$ and retaining 4500 states per NRG iteration).
Fig. \[fig2\] shows ${\rho_{\rm m}}(T)$ and ${\gamma_{\rm m}}(T)$ for $n=2S=1,2$ and $3$. For $T \gtrsim {T_{\rm K}^S}$, enhanced spin-flip scattering causes both ${\rho_{\rm m}}(T)$ and ${\gamma_{\rm m}}(T)$ to increase with decreasing temperature. For $T \lesssim {T_{\rm K}^S}$ the effective exchange coupling becomes so strong that the impurity spins are fully screened by conduction electrons, forming spin singlets, causing ${\rho_{\rm m}}(T)$ to saturate to a constant and ${\gamma_{\rm m}}(T)$ to drop to zero. While these effects are well-known [@KondoProgTheorPhys64; @PierreBirgePRB03; @SchopferSaminadayarPRL03; @BauerleSaminadayarPRL05; @MalletBauerlePRL06; @AlzoubiBirgePRL06], it is of central importance for the present study that they depend quite significantly on $S = n/2$, in such a way that *conduction electrons are scattered and decohered more strongly the larger the local spin $S$*: With increasing $S$, (i) both resistivities and decoherence rates decay more slowly with $T$ at large temperatures $(\gg {T_{\rm K}^S})$, and (ii) the “plateau” near the maximum of ${\gamma_{\rm m}}(T)$ increases slightly in maximum height ${\gamma_{\rm m}}^{\rm max}$ and significantly in width. These changes turn out to be sufficient to identify the proper value of $S$ when comparing to experiments below.
![ (color online). (a) Resistivity ${\rho_{\rm m}}(T)$ (solid lines), and, (b) decoherence rate ${\gamma_{\rm m}}(T)$ for $2S = n =1,2,3$; ${\rho_{\rm m}}(0)=2\tau\bar{\rho}/\pi\hbar\nu_{0}$, ${\gamma_{\rm m}}^0=2/\pi\hbar\nu_{0}$, where $\bar{\rho}$ is the residual resistivity, $\nu_{0}$ the density of states per spin and channel, $\tau$ the elastic scattering time, and ${\gamma_{\rm m}}^{\rm max}$ is the maximum value of ${\gamma_{\rm m}}(T)$. We defined the Kondo scale ${T_{\rm K}^S}$ for each $S$ via ${\rho_{\rm m}}({T_{\rm K}^S})={\rho_{\rm m}}(0)/2$. Dashed lines in (a) show that the empirical form ${\rho_{\rm m}}(T)/{\rho_{\rm m}}(0)\approx f_{S}(T/{T_{\rm K}^S})$ with $ f_S(x)=(1+(2^{1/\alpha_S}-1)x^{2})^{-\alpha_S}$, used to fit experimental to NRG results for $S=1/2$ [@GoldhaberGordonMeiravPRL98], also adequately fits the NRG results for $S=1$ and $S=3/2$. []{data-label="fig2"}](fig2.eps){width="0.9\linewidth"}
[*Comparison with experiment.—*]{} We compared our theoretical results for $\rho_{\rm m}(T)$ and $\gamma_{\rm m}(T)$ to measurements on quasi 1-dimensional, disordered wires, for two AgFe samples [@MalletBauerlePRL06], (AgFe2 and AgFe3 having $27 \pm 3$ and $67.5 \pm 7$ ppm Fe impurities in Ag, respectively), with a Kondo scale ${T_{\rm K}}\approx 5{\rm K}$ (for $S=3/2$, see below). These measurements extend up to $T \lesssim {T_{\rm K}}$ allowing the region $T/{T_{\rm K}}\lesssim 1$ of the scaling curves in Fig. \[fig2\] to be compared to experiment. At $T\gtrsim {T_{\rm K}}\approx 5$K (i.e. $T/{T_{\rm K}}\ge 1$) the large phonon contribution to the decoherence rate prohibits reliable extraction of $\gamma_{\rm m}(T)$ for our Ag samples (see below) . In order to compare theory and experiment for temperatures $T/{T_{\rm K}}\ge 1$, above the maximum in the decoherence rate, we therefore carried out new measurements on a sample (AuFe3) with $7 \pm 0.7$ ppm Fe impurities in Au with a lower Kondo scale ${T_{\rm K}}\approx 1.3$[K]{} but, as discussed above, described by the same Kondo model. Combining both sets of measurement thereby allows a large part of the scaling curves in Fig. \[fig2\] to be compared with experiment.
Following [@MalletBauerlePRL06], we subtract the electron-electron contribution [@AltshulerKhmelnitskyJPhysC82] from the total resistivity $\rho$, yielding $\Delta \rho$ due to magnetic impurities (m) and phonons (ph): $$\label{eq:delta-rho}
\Delta \rho(T) = {\rho_{\rm m}}(T) + \rho_{\rm ph} (T) + \delta \; .$$ Here $\delta$ is an (unknown) offset [@note-unknown-offset] and $\Delta \rho(T)$ is expressed per magnetic impurity. For temperatures low enough that $\rho_{\rm ph}(T)$ can be neglected, $\Delta\rho(T)-\delta$ corresponds to the theoretical curve $\rho_{\rm m} (T) = \rho_{\rm m} (0) f_S (T/{T_{\rm K}^S})$ \[cf. caption of Fig. \[fig2\]\], where $\rho_{\rm m} (0) =\Delta\rho(0)-\delta $ is the unitary Kondo resistivity. Fig. \[fig3\] illustrates how we extract the Kondo scale ${T_{\rm K}^S}$ and $\rho_{\rm m} (0)$ from the experimental data, by fitting the Kondo-dominated part of $\Delta \rho (T)$ in a fixed temperature range (specified in the caption of Fig. \[fig3\]) to the NRG results of Fig. \[fig2\](a), using the Ansatz $$\label{eq:fitting-ansatz-1}
\Delta \rho(T) \approx \delta +
(\Delta\rho(0) - \delta)f_{S}(T/{T_{\rm K}^S}) \; .$$ Such fits are made for each of the fully screened Kondo models, using ${T_{\rm K}^S}$ and $\delta$ as fit parameters. Importantly, the values for ${T_{\rm K}^S}$ and $\rho_{{\rm m}}(0)$ obtained from the fits, given in the inset and caption of Fig. \[fig3\], respectively, show a significant $S$-dependence: both ${T_{\rm K}^S}$ and $\rho_{{\rm m}}(0)$ increase with $S$, since the slope of the logarithmic Kondo increase of the theory curves for $\rho_{\rm m}$ \[cf. Fig. \[fig2\]\] decreases significantly in magnitude with $S$. Nevertheless, all three models fit the Kondo contribution very well, as shown in Fig. \[fig3\], so a determination of the appropriate model from resistivity data alone is not possible.
To break this impasse, we exploit the remarkably sensistive $S$-dependence of the spin-flip-induced decoherence rate $\gamma_{m}(T)$. Fig. \[fig4\] shows the measured dimensionless decoherence rate ${\gamma_{\rm m}}(T) / {\gamma_{\rm m}}^{\rm max}$ for Ag and Au samples (symbols) as function of $T/{T_{\rm K}^S}$ for $S=1/2, 1 $ and $3/2$, using the ${T_{\rm K}^S}$ values extracted from the resistivities, together with the corresponding *parameter-free* theoretical predictions (lines), taken from Fig. \[fig2\](b). The agreement between theory and experiment is poor for $S=1/2$, better for $S=1$ but excellent for $S=3/2$, confirming the conclusion drawn above from [*ab initio*]{} calculations. The dependence on $S$ is most strikingly revealed through the width of the plateau region (in units of $T/{T_{\rm K}^S}$), which grows with $S$ for the theory curves but shrinks with $S$ for the experimental data (for which ${T_{\rm K}^S}$ grows with $S$), with $S=3/2$ giving the best agreement.
[*Conclusions.—*]{} In this Letter we addressed one of the fundamental unresolved questions of Kondo physics: that of deriving and solving the effective low-energy Kondo model appropriate for a realistic description of Fe impurities in Au and Ag. Remarkably, for both Ag and Au samples, the use of a fully screened $S=3/2$ three channel Kondo model allows a *quantitatively consistent* description of both the resistivity and decoherence rate [*with a single ${T_{\rm K}}$*]{} (for each material). Our results set a benchmark for the level of quantitative understanding attainable for the Kondo effect in real materials.
L. B. acknowledges support from the EU within the Marie Curie Actions for Human Resources and Mobility; P.M. from the ESF programme SONS, contract N. ERAS-CT-2003-980409; T. M. from the U.S. Dept. of Energy, Office of Science, Contract No. DE- AC02-06CH11357; L.S. and C.B. acknowledge technical support from the Quantronics group, Saclay and A.D. Wieck and financial support from ANR PNANO “QuSPIN”. Support from the John von Neumann Institute for Computing (Jülich), the DFG (SFB 608, SFB-TR12 and De730/3-2) and from the Cluster of Excellence *Nanosystems Initiative Munich* is gratefully acknowledged.
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abstract: 'We examine in some detail the influence of the systematics in different data sets including type Ia supernova sample, baryon acoustic oscillation data and the cosmic microwave background information on the fitting results of the Chevallier-Polarski-Linder parametrization. We find that the systematics in the data sets does influence the fitting results and leads to different evolutional behavior of dark energy. To check the versatility of Chevallier-Polarski-Linder parametrization, we also perform the analysis on the Wetterich parametrization of dark energy. The results show that both the parametrization of dark energy and the systematics in data sets influence the evolutional behavior of dark energy.'
author:
- Yungui Gong
- Bin Wang
- 'Rong-Gen Cai'
title: Probing the cosmic acceleration from combinations of different data sets
---
Introduction
============
The convincing fact that our universe is experiencing accelerated expansion [@acc1; @acc2] has become one of the most important and mysterious issues of modern cosmology. In the framework of general relativity, the cosmic acceleration is attributed to an exotic energy called dark energy (DE). The simplest candidate of the DE is the cosmological constant with equation of state (EOS) $w=-1$. Despite some severe problems such as the cosmological constant problem and the coincidence problem, the cosmological constant is favored by current astronomical observations. Besides, there are some other DE models with EOS varying with cosmic time either above $-1$ or below $-1$. Interestingly, some current observations even give the hint that EOS of DE has crossed $-1$ at least once [@gong07; @gong]. To narrow down the DE candidates, one has to examine the EOS carefully by confronting different observational data sets including the type Ia supernova (SNIa) luminosity distance, the cosmic microwave background (CMB) temperature anisotropy and the baryon acoustic oscillations (BAO) in the galaxy power spectrum, and this could be the best what we can do to approach the truth of DE.
In addition to the EOS, a new diagnostic of DE was introduced in [@sahni] which is called $Om$ diagnostic. It is a combination of the Hubble parameter and the cosmological redshift, which depends upon the first derivative of the luminosity distance and is less sensitive to observational errors than the EOS. If the value of $Om$ is the same at different redshifts, then the DE is the cosmological constant. The slope of $Om$ can differentiate between different DE models with $w>-1$ or $w<-1$ even if the value of the matter density is not accurately known.
Analyzing the Constitution SNIa sample [@consta] together with the BAO data [@sdss6; @sdss7] by using the popular Chevallier-Polarski-Linder (CPL) parametrization [@cpl1; @linder03], it was revealed that there appears the increase of $Om$ and $w$ at redshifts $z<0.3$ [@star]. This suggests that the cosmic acceleration may have already been over the peak and now the acceleration is slowing down. However, including the CMB shift parameter which is another independent observable at high redshift, it was found that the result changes dramatically and the value of $Om$ becomes un-evolving which is consistent with the $\Lambda$CDM model. Further check shows that the CPL ansatz is unable to fit the data simultaneously at low and high redshifts. It was argued that this could either due to the systematics in some of the data sets which is not sufficiently understood or because the CPL parametrization is not versatile to accommodate the cosmological evolution of DE suggested by the data [@star]. The data sets used in the analysis of [@star] are limited to the Constitution set of 397 SNIa in combination with BAO distance ratio of the distance measurements obtained at $z=0.2$ and $z=0.35$ and the CMB shift parameter. It is expected that if more combinations of new data sets are included, these arguments can be clarified. In [@gong], it was showed that the result of the analysis on the CPL model [@star] heavily depends on the choice of BAO data. The result obtained by using the BAO distance ratio data was found not consistent with that by using other observational data, and this inconsistency can be overcome if the BAO $A$ parameter [@baoa] is employed instead [@gong]. In this work we are going to investigate this problem further by comparing different data set combinations among SNIa, BAO and CMB.
Observational data
==================
For the SNIa data, we use the Constitution sample [@consta] and the first year Sloan digital sky survey-II (SDSS-II) SNIa (hereafter Sdss2) [@sdss2]. The Constitution sample consists of the Union sample [@union] together with 185 CfA3 SNIa data, which totally contains 397 SNIa. The CfA3 addition makes the cosmologically useful sample of nearby SNIa much larger than before, which reduces the statistical uncertainty to the point where systematics plays the largest role. To test the systematic differences and consistencies, in [@consta] four light curve fitters, $SALT$, $SALT2$, $MLCS2k2$ with $R_V = 3.1$ (MLCS31), $MLCS2k2$ with $R_V = 1.7$ (MLCS17), have been used. For the Constitution SNIa data using the template SALT (hereafter Csta), the intrinsic uncertainty of 0.138 mag for each CfA3 SNIa, the peculiar velocity uncertainty of $400$km/s, and the redshift uncertainty have been considered [@consta]. These data were suggested by observers to be the best data for model independent analysis of the expansion history. The Constitution SNIa data using the template SALT2 (hereafter Cstb), excludes the SNIa with $z<0.01$ or $t_{1st}>10d$, so it has 351 SNIa data. Using the template MLCS17 on the Constitution data (hereafter Cstc), the SNIa with $A_v\geq 1.5$ and $t_{1st}>10d$ has been cut out and it excludes those SNIa whose MLCS17 fit has a reduced $\chi^2_\nu$ being 1.6 or higher, thus Cstc has only 372 SNIa data. The Cstd sample is formed by using the template MLCS31 on the Constitution sample and cutting out the SNIa with $A_v\geq
1.5$ and $t_{1st}>10d$. It excludes any SNIa whose MLCS31 fit has a reduced $\chi^2_\nu$ being 1.5 or higher, so that it contains 366 SNIa data. We will examine the systematic differences brought by the Constitution sample with different light curve fitters. In addition we will also consider Sdss2 sample and investigate the systematic difference it brought, to compare with the Constitution sample. The Sdss2 consists of 103 new SNIa with redshifts $0.04<z<0.42$, discovered during the first season of the SDSS-II supernova survey, 33 nearby SNIa with redshfits $0.02<z<0.1$, 56 SNIa with redshifts $0.16<z<0.69$ from ESSENCE [@riess; @essence], 62 with redshifts $0.25<z<1.01$ SNIa from SNLS [@snls], and 34 SNIa with redshifts $0.21<z<1.55$ from HST. For simplicity, we only use the Sdss2 data with the template MLCS2K2. For the 288 Sdss2 SNIa data, we also consider the redshift uncertainties from spectroscopic measurements and from peculiar motions of the host galaxy. For the redshift uncertainty, $\sigma_{z,pec}$, arises from peculiar velocities of and within host galaxies, we take $\sigma_{z,pec} = 0.0012$. In addition, the intrinsic error of 0.16 mag is added to the uncertainty of the distance modulus [@sdss2].
To fit the SNIa data, we define $$\label{chi1} \chi^2(\bm{p},H_0)=\sum_{i=1}\frac{[\mu_{obs}(z_i)-\mu(z_i,\bm{p},H_0)]^2}{\sigma^2_i},$$ where the extinction-corrected distance modulus $\mu(z,\bm{p},H_0)=5\log_{10}[d_L(z)/{\rm Mpc}]+25$, $\sigma_i$ is the total uncertainty in the SNIa data, and the luminosity distance for a flat universe is $$\label{lum} d_L(z,\bm{p},H_0)=\frac{1+z}{H_0} \int_0^z \frac{dx}{E(x)},$$ where the dimensionless Hubble parameter $E(z)=H(z)/H_0$. Because the normalization of the luminosity distance is unknown, the nuisance parameter $H_0$ in the SNIa data is not the observed Hubble constant. We marginalize over the nuisance parameter $H_0$ with a flat prior, which leads to [@gong08] $$\label{chi} \chi^2_{sn}(\bm{p})=\sum_{i=1}\frac{\alpha_i^2}{\sigma^2_i}-\frac{(\sum_i\alpha_i/\sigma_i^2
-\ln 10/5)^2}{\sum_i 1/\sigma_i^2}
-2\ln\left(\frac{\ln 10}{5}\sqrt{\frac{2\pi}{\sum_i 1/\sigma_i^2}}\right),$$ where $\alpha_i=\mu_{obs}(z_i)-25-5\log_{10}[H_0 d_L(z_i)]$.
Besides considering these SNIa data sets individually, our analysis will also investigate the combination of SNIa data with the BAO data. For the BAO data, we first use the BAO distance measurements obtained at $z=0.2$ and $z=0.35$ from joint analysis of the 2dFGRS and SDSS data [@sdss7]. Defining $d_z(\bm{p},H_0)=r_s(z_d)/D_V(z)$, where the comoving sound horizon and effective distance are $$\label{rshordef} r_s(z,\bm{p})=\int_z^\infty \frac{dx}{c_s(x)E(x)},$$ $$\label{dvdef} D_V(z,\bm{p},H_0)=\left[\frac{d_L^2(z)}{(1+z)^2}\frac{z}{H(z)}\right]^{1/3}
=H_0^{-1}\left[\frac{z}{E(z)}\left(\int_0^z\frac{dx}{E(x)}\right)^2\right]^{1/3}.$$ The redshift $z_d$ at the baryon drag epoch is fitted with the formula [@hu98] $$\label{zdfiteq} z_d=\frac{1291(\Omega_{m0} h^2)^{0.251}}{1+0.659(\Omega_{m0} h^2)^{0.828}}[1+b_1(\Omega_b h^2)^{b_2}],$$ $$\label{b1eq} b_1=0.313(\Omega_{m0} h^2)^{-0.419}[1+0.607(\Omega_{m0} h^2)^{0.674}],\quad b_2=0.238
(\Omega_{m0} h^2)^{0.223},$$ where $\Omega_{m0}$ is the current value of the dimensionless matter energy density, $\Omega_b$ is the dimensionless baryon matter energy density, and $h=H_0/100$. The sound speed $c_s(z)=1/\sqrt{3[1+\bar{R_b}/(1+z)}]$, where $\bar{R_b}=315000\Omega_b h^2(T_{cmb}/2.7{\rm K})^{-4}$ and $T_{cmb}=2.726$K. In [@sdss7], two points $d_{0.2}$ and $d_{0.35}$ and their covariance matrix are given. We will use $d_{0.2}=0.1905\pm 0.0061$, $d_{0.35}=0.1097\pm 0.0036$ and their covariance matrix (hereafter Bao2) as the second sample of BAO data in the data fitting. When we use these data, we add two more parameters $\Omega_b h^2$ and $h$. Based on these two BAO distance measurements, we can derive the BAO distance ratio $D_V(0.35)/D_V(0.2)=1.736\pm 0.065$ (hereafter BaoR), which is relatively model independent quantity. This BAO distance ratio was employed in the analysis in [@star]. In our case we will further use another two radial BAO data at redshifts $z=0.24$ and $z=0.43$ [@baoz] by using $\Delta
z(z)=H(z)r_s(z_d)/c$ (hereafter BaoZ). Combining BAO data sets Bao2 and BaoZ, we have the data set called Bao4. Therefore, for different BAO data sets, we can perform $\chi^2$ statistics for the model parameter $\bm{p}$ as follows $$\label{baoR} \chi^2_{Baor}(\bm{p})=\frac{[D_V(0.35)/D_V(0.2)-1.736]^2}{0.065^2},$$ $$\label{baoz} \chi^2_{BaoZ}(\bm{p},h,\Omega_b h^2)=\frac{[\Delta z(0.24)-0.0407]^2}{0.0011^2}+\frac{[\Delta z(0.43)-0.0442]^2}{0.0015^2},$$ and $$\label{bao2} \chi^2_{Bao2}(\bm{p},h,\Omega_b h^2)=\sum_{ij}\Delta d_i{\rm Cov}^{-1}(d_i,d_j)\Delta d_j,$$ where $i,j=0.2,0.35$, $\Delta d_{0.2}=d_{0.2}-0.1905$, $\Delta d_{0.35}=d_{0.35}-0.1097$.
Since the SNIa and BAO data contain information about the universe at relatively low redshifts, we will include the CMB information by implementing the Wilkinson microwave anisotropy probe 5 year (WMAP5) data to probe the entire expansion history up to the last scattering surface. In addition to employing the CMB shift parameter $R$ by defining the reduced distance in the flat universe to the last scattering surface $z_*$ as done in [@star] $$\label{shift} R(\bm{p})=\sqrt{\Omega_{m0}}\int_0^{z_*}\frac{dz}{E(z)}=1.71\pm 0.019,$$ we will add the acoustic scale $l_a$ in the data analysis. The acoustic scale $l_a$ is defined by $$\label{ladefeq} l_a=\frac{\pi d_L(z_*)}{(1+z_*)r_s(z_*)}=302.1\pm
0.86,$$ where the redshift $z_*$ is given by [@hu96] $$\label{zstareq} z_*=1048[1+0.00124(\Omega_b h^2)^{-0.738}][1+g_1(\Omega_{m0} h^2)^{g_2}]=1090.04\pm 0.93,$$ $$g_1=\frac{0.0783(\Omega_b h^2)^{-0.238}}{1+39.5(\Omega_b h^2)^{0.763}},\quad g_2=\frac{0.560}{1+21.1
(\Omega_b h^2)^{1.81}}.$$ We have three fitting parameters $x_i=(R,\ l_a,\ z_*)$ to include the CMB information now. Thus we have $\chi^2_{cmb}(\bm{p},h,\Omega_b
h^2)=\sum_{ij}\Delta x_i {\rm Cov}^{-1}(x_i,x_j)\Delta x_j$, $\Delta x_i=x_i-x_i^{obs}$ and Cov$(x_i,x_j)$ is the covariance matrix for the three parameters [@wmap5]. The CMB information provides a systematic check of the DE model by combining with the low redshift SNIa and BAO data sets.
Fitting Results
===============
We employ the Monte-Carlo Markov Chain (MCMC) method to explore the parameter space $\bm{p}$ and the nuisance parameters $h$ and $\Omega_b h^2$ in the data analysis. Our MCMC code [@gong08] is based on the publicly available package COSMOMC [@cosmomc].
We first explore the widely used CPL parametrization [@cpl1; @linder03] $$\label{cplwz}
w(z)=w_0+\frac{w_a \, z}{1+z}.$$ In this model, the dimensionless Hubble parameter for a flat universe is $$\label{cplez} E^2(z)=\frac{H^2(z)}{H^2_0}=\Omega_{m0}(1+z)^3+(1-\Omega_{m0})(1+z)^{3(1+w_0+w_a)}\exp(-3w_a z/(1+z)).$$ In this model, we have three parameters $\bm{p}=(\Omega_{m0},\ w_0,\ w_a)$. Knowing the expansion history of the universe, we can construct the $Om$ diagnostic by defining $$\label{omzeq} Om(z)=\frac{E^2(z)-1}{(1+z)^3-1}.$$ $Om$ diagnostic is useful in establishing the properties of DE at low redshifts. The constant $Om$ indicates that the DE is the cosmological constant and the bigger value of $Om$ shows that $w$ is bigger [@sahni]. $Om(z)$ is less sensitive to observational errors than EOS $w(z)$. Due to the degeneracy between $\Omega_{m0}$ and $w$, the property of $w(z)$ depends on the current value of matter energy density $\Omega_{m0}$. However, $Om$ diagnostic provides a null test of the cosmological constant without knowing the exact value of $\Omega_{m0}$. To reconstruct $Om(z)$, we need to apply the specific model and model parameters. Following [@star], we consider the uncertainties of $\Omega_{m0}$, $w_0$ and $w_a$ when we we reconstruct $Om(z)$.
$\begin{array}{c}
\subfigure[]{\includegraphics[width=3in]{cstaw0wacont.eps}
\label{cstadcpl:a}}
\subfigure[]{ \includegraphics[width=3in]{cstbw0wacont.eps}
\label{cstadcpl:b}}\\
\subfigure[]{\includegraphics[width=3in]{cstcw0wacont.eps}
\label{cstadcpl:c}}
\subfigure[]{ \includegraphics[width=3in]{cstdw0wacont.eps}
\label{cstadcpl:d}}
\end{array}$
$\begin{array}{c}
\subfigure[]{\includegraphics[width=3in]{sd2w0wacont.eps}
\label{sd2fits:a}}
\subfigure[]{ \includegraphics[width=3in]{cstawetw0wa.eps}
\label{sd2fits:b}}\\
\subfigure[]{\includegraphics[width=3in]{cstdwetw0wa.eps}
\label{sd2fits:c}}
\subfigure[]{ \includegraphics[width=3in]{sd2wetw0wa.eps}
\label{sd2fits:d}}
\end{array}$
Fig. 1 shows contours of the fitting results for the CPL parametrization by combining different SNIa data together with different BAO data and the combination of the WMAP5 data. The SNIa data used in Figs. 1a-1d and 2a are Csta, Cstb, Cstc, Cstd, and Sdss2, respectively. The green lines are for using just SNIa data alone with different templates. The dashed black lines are for SN+BaoR, the cyan lines are for SN+Bao2. Since the BaoR data is derived from Bao2 data, so the result using BaoR data is compatible with that of Bao2. Also we can see that the constraint is a little better with Bao2 than that with BaoR. The magenta lines are for SN+BaoZ, we see that the constraint from BaoZ is in general consistent with that from Bao2 although the former gives a little tighter constraint. These behaviors keep the same for SNIa data with different templates. The blue lines are for SN+Bao4, where Bao4 is the combination of Bao2+BaoZ. The combination of the SN+Bao4+WMAP5 is shown in the red solid lines. The observation that the CPL ansatz is strained to describe the DE behavior suggested by data at low and high redshifts by comparing Csta+BaoR and Csta+BaoR+CMB shift parameter [@star] has been reduced by using the same SN data (Csta) with BaoZ or Bao4, or other SN data sets (Cstb-Cstd, Sdss2) with arbitrary combinations of BAO data. This confirms that the systematics in some of the data sets really matters the fitting results.
![The marginalized $1\sigma$ and $2\sigma$ errors of $Om(z)$ reconstructed in the CPL model by using the Csta SNIa data. (a) uses the combination of SNIa+BaoR, (b) uses the combination of SNIa+BaoZ, (c) uses the combination of SNIa+Bao4 and (d) uses the combination of SNIa+Bao4+WMAP5.[]{data-label="cstaomzcpl"}](cstaomzcpl.eps){width="6in"}
![The marginalized $1\sigma$ and $2\sigma$ errors of $Om(z)$ reconstructed in the CPL model with SNIa+BaoR. The SNIa data used in (a)-(d) are Cstb, Cstc, Cstd, and Sdss2, respectively.[]{data-label="snbrcpl"}](snbrcplomz.eps){width="6in"}
For the Csta data, the reconstructed $Om(z)$ for SNIa data in combination with BAO and CMB data is shown in Fig. \[cstaomzcpl\]. Since $Om(z)$ is not a constant at $1\sigma$ level if we combine Cata SNIa with BaoR, BaoZ or Bao4, so the $\Lambda$CDM model is excluded at $1\sigma$ level if use the combination of Csta SNIa and BAO data. However, when the WMAP5 data is added, the $\Lambda$CDM model is consistent at $1\sigma$ level. The growth in the value of $Om(z)$ at low redshifts becomes smaller when we change the BAO data from BaoR to BaoZ or Bao4. In Fig. \[snbrcpl\], we show the marginalized $1\sigma$ and $2\sigma$ errors of $Om(z)$ reconstructed using the BaoR and different SNIa data sets. The growth in the value of $Om(z)$ at low redshifts in Fig. 3a by using Csta+BaoR as observed in [@star] gets flattened if we change SNIa data sets from Csta to Cstb, Cstc, Cstd and Sdss2. This shows that the finding in [@star] is not a general behavior even at low $z$. The evolutional behavior of $Om(z)$ at low redshifts is changed for different selections of the SNIa data. Thus the striking observation that our universe is slowing down [@star] based on the fitting at low redshifts for CPL ansatz is caused by the systematics of the specially chosen SNIa and BAO data sets. By choosing some other data sets, the CPL parametrization is in compatible with combinations of data sets at low and high redshifts. The un-evolving $Om(z)$ is also allowed even at low redshifts. From Figs. 3a and 4, we see that the $\Lambda$CDM model is not allowed at $1\sigma$ level for the combination of BaoR and Csta, Cstb or Cstc, although the cosmological constant is inside the $1\sigma$ contours of $w_0$ and $w_a$ in Figs. 1b and 1c. This shows the advantage of $Om$ diagnostic because the uncertainty of $\Omega_{m0}$ is taken accounted for.
In the above discussions we have focused on the commonly used functional form for DE, the CPL parametrization, and argued that the systematics in SNIa and BAO data sets really affects the evolution of the DE and the acceleration of the universe. Besides the systematics of data sets, whether the influences on the evolution of the DE and the acceleration are caused by the specific choice of the parametrization of the DE is another interesting question to ask. In [@star], the incompatibility of the CPL ansatz fitting to the data simultaneously at low and high redshifts was attributed to the versatility of the CPL parametrization. Choosing another ansatz of DE parametrization, it was found significantly better than that of CPL, that ansatz can provide a good fit to the combination of Csta+BaoR and the CMB shift data. In the rest of this work we are going to further examine the versatility of the CPL parametrization by considering a different ansatz of DE parametrization and confronting it with different combinations of various data sets as used above.
What we are going to study is the Wetterich parametrization of the form [@par1] $$\label{wetwz}
w(z)=\frac{w_0}{1+w_a \ln(1+z)}.$$ When $z=0$, $w(z)=w_0$, and at large redshift $z\gg 1$, $w(z)\approx
0$. The Wetterich parametrization becomes the constant parametrization $w(z)=w_0$ when $w_a=0$.
In this model, the dimensionless Hubble parameter for a flat universe is $$\label{wetez}
E^2(z)=\Omega_{m0}(1+z)^3+(1-\Omega_{m0})(1+z)^3[1+w_a\ln(1+z)]^{3w_0/w_a}.$$ To ensure the positivity of DE, we require that $w_a\ge 0$. For the convenience of numerical calculation, we take the independent parameters as $w_0^a=w_0/w_a$ and $w_a$. The parameters in this model are $\bm{p}=(\Omega_{m0},\ w_0^a,\ w_a)$, the number is the same as those in the case of CPL ansatz.
![The marginalized $1\sigma$ and $2\sigma$ errors of $Om(z)$ reconstructed in the Wetterich model. (a) uses the combination of Csta SNIa+BaoR, (b) uses the combination of Csta SNIa+BaoZ, (c) uses the combination of Sdss2 SNIa+BaoR, and (d) uses the combination of Sdss2 SNIa+BaoZ.[]{data-label="wetomz"}](wetomz.eps){width="6in"}
The compatibility checks in confronting with different SNIa, BAO data sets together with CMB data are shown in Figs. 2b-2d. In Figs. 2b-2d, we have used the Csta, Cstd and Sdss2 SNIa data sets, respectively. As indicated in Fig. 1 and Fig. 2, the green lines are just for SNIa data alone, the dashed black lines are for SN+BaoR, the cyan lines are results for SNIa+Bao2, the magenta lines indicate SNIa+BaoZ, the blue lines are for SNIa+Bao4, and the solid red lines are for the combination of SNIa+Bao4+WMAP5. It is interesting to see that different from the CPL ansatz, this parametrization allows the compatibility of different SNIa, Bao and CMB data sets, even for comparing the combinations of Csta+BaoR, Csta+BaoZ and Csta+Bao4+WMAP5. The Wetterich ansatz is found able to fit different data sets both at low and high redshifts. In Fig. \[wetomz\], we present the behaviors of the $Om(z)$ reconstructed by comparing the influence of different SNIa and BAO data sets. Interestingly, the evolution of $Om(z)$ is not affected much by different SNIa and BAO data sets. The $Om(z)$ is perfectly consistent with $\Lambda$CDM for different data sets.
For the comparison of different data sets and different parametrizations, we summarize the minimum value of $\chi^2$ in Table \[table1\]. From Table \[table1\], we observe the reliability of parametrizations when using different data sets. Comparing two different parametrizations, we see that both of them can fit well of the data, while the CPL model is a little better than the Wetterich parametrization for the Csta SNIa and the combination of Csta SNIa with BAO data.
Data CPL ($\chi^2$/DOF) Wetterich
------------------ -------------------- ------------
Csta 462.06/394 466.33/394
Csta+BaoR 462.43/395 467.62/395
Csta+Bao2 462.45/394 467.73/394
Csta+BaoZ 462.10/394 466.60/394
Csta+Bao4 464.11/396 468.57/396
Csta+Bao4+WMAP5 468.73/399 468.69/399
Sdss2 227.55/285 227.09/285
Sdss2+BaoR 229.77/286 229.45/286
Sdss2+Bao2 229.93/285 229.65/285
Sdss2+BaoZ 228.77/285 228.57/285
Sdss2+Bao4 231.15/287 231.02/287
Sdss2+Bao4+WMAP5 231.71/290 231.94/290
: The minimum value of $\chi^2$ for different combinations of data sets and models.[]{data-label="table1"}
Conclusion
==========
To summarize, we have examined the influence of the systematics in different data sets in SNIa and BAO on the fitting results of the CPL parametrization. We found that the tension observed in [@star] between low $z$ (Csta+BaoR) and the high $z$ (CMB) data is not a general behavior. By using SNIa with other templates and other BAO data sets, the incompatibility of the CPL parametrization will disappear. This result supports the speculation that the systematics in the data sets can affect the fitting results and leads to different evolution of the DE model [@star]. However this answer is still not definite. The different evolutions in $Om(z)$ as we observed by using different combinations of various SNIa and BAO data sets disappear when we use the Wetterich parametrization to replace the CPL parametrization. This again brings the problem as raised in [@star] that the CPL parametrization might be blamed to be not so versatile. In order to disclose the exact answer, we have to examine more models of DE. This work offers the attempt of using different combinations of various data sets at low and high redshifts to examine the effects of data systematics on the evolution of the DE and the acceleration of the universe. Further investigation along this line by including more data sets and examining more DE models are called for and we will report our progress in this direction in the future work.
RGC and BW thank Chongqing University of Posts and Telecommunications for the warm hospitality during their visits. This work was partially supported by NNSF of China (Nos. 10821504, 10878001, 10975168 and 10935013) and the National Basic Research Program of China under grant No. 2010CB833004. YG was partially supported by the Natural Science Foundation Project of CQ CSTC under grant No. 2009BA4050.
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|
---
author:
- |
Ramesh NARAYAN\
[*Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA*]{}\
Woong-Tae KIM\
[*Astronomy Program, SEES, Seoul National University, Seoul 151-742, Korea*]{}\
[*Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA*]{}
---
Conduction and Turbulent Mixing in Galaxy Clusters
==================================================
Abstract {#abstract .unnumbered}
--------
We discuss hydrostatic models of galaxy clusters in which heat diffusion balances radiative cooling. We consider two different sources of diffusion, thermal conduction and turbulent mixing, parameterized by dimensionless coefficients, $f$ and $\alpha_{\rm
mix}$, respectively. Models with thermal conduction give reasonably good fits to the density and temperature profiles of several cooling flow clusters, but some clusters require unphysically large values of $f>1$. Models with turbulent mixing give good fits to all clusters, with reasonable values of $\alpha_{\rm mix} \sim 0.01-0.03$. Both types of models are found to be essentially stable to thermal perturbations. The mixing model reproduces the observed scalings of various cluster properties with temperature, and also explains the entropy floor seen in galaxy groups.
Introduction
------------
For many years, it was thought that the strong X-ray emission observed in the cores of rich galaxy clusters results in a cooling flow in which gas settles in the gravitational potential and drops out as cold condensations [@NK_fab94]. Mass inflow rates were estimated to be $\sim10^2-10^3M_\odot$ yr$^{-1}$ in some clusters. However, recent X-ray observations with [*Chandra*]{} and [*XMM-Newton*]{} have found very little emission from gas cooler than about one-third of the virial temperature [@NK_pet01; @NK_pet03], suggesting that some heating source must prevent gas from cooling below this temperature. Candidate heating mechanisms include (1) energy injection from a central active galactic nucleus (AGN) [@NK_cio01; @NK_chu02; @NK_bru02; @NK_kai03], and (2) diffusive transport of heat from the outer regions of the cluster to the center via conduction [@NK_tuc83; @NK_bre88; @NK_nar01; @NK_voi02; @NK_zak03] or turbulent mixing [@NK_cho03; @NK_kim03b; @NK_voi04].
Heating by a central AGN is an attractive idea since many cooling flow clusters show radio jets and lobes that are apparently interacting with the cluster gas [@NK_beg01]. The power associated with the jets is often comparable to the total X-ray luminosity of the cluster. However, there are some difficulties with this model. Observations reveal that radio lobes are surrounded by X-ray-bright shells of relatively cool gas [@NK_sch02], which is a little surprising if this gas is being heated by the bubble. In addition, if the heating rate (per unit volume) of the gas by the AGN varies as $\rho^\alpha$, thermal stability requires $\alpha>1.5$ [@NK_zak03]; such a heating law does not seem natural. (Stability is not an issue if AGN heating is episodic [@NK_kai03]). Finally, no good correlation is seen between the AGN radio luminosity and the X-ray cooling rate [@NK_voi04].
Since the cooling cores of clusters have a lower temperature than the rest of the cluster, diffusive processes can bring heat to the center from the outside, provided the diffusion coefficient is large enough. An ordered magnetic field would strongly suppress cross-field diffusion of thermal electrons, and this argument has been traditionally invoked for ignoring thermal conduction. However, if the field lines are chaotically tangled over a wide range of length scales, the isotropic conduction coefficient $\kappa_{\rm cond}$ can be as much as a few tens of per cent of the Spitzer value $\kappa_{\rm
Sp}$ [@NK_nar01; @NK_cha03], which may be sufficient to supply the necessary heat to the cluster core. Turbulent mixing is another diffusive process that can transport energy efficiently to the center [@NK_cho03]. The turbulence might be sustained by the infall of small groups or subclusters, the motions of galaxies \[K. Makishima, this conference\], or energy input from AGNs [@NK_dei96; @NK_ric01]. The diffusion coefficient required to balance radiative cooling is typically $\kappa_{\rm mix}\sim 1-6\;\rm kpc^2\;Myr^{-1}$, which is similar to values inferred from observations of turbulence in clusters [@NK_kim03b; @NK_voi04].
In a series of papers [@NK_zak03; @NK_kim03b; @NK_kim03a], we have studied equilibrium models of galaxy clusters with thermal conduction and turbulent mixing. We summarize here the main results of this work.
Model {#NK_model}
-----
We assume that the hot gas in a galaxy cluster is in hydrostatic equilibrium and that it maintains energy balance between radiative cooling and diffusive heating, $$\frac{1}{\rho}\nabla P = -\nabla \Phi,
\qquad %%{\rm and}\qquad
\nabla\cdot\mathbf{F} = -j,$$ where $P$ is the thermal pressure, $\rho$ is the density, $\Phi$ is the gravitational potential, $\mathbf{F}$ is the local diffusive heat flux, and $j$ is the radiative energy loss rate per unit volume. For $kT{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}2$keV, $j$ is dominated by free-free emission, while for lower temperatures it is mostly due to line cooling.
We consider two diffusive processes: thermal conduction and turbulent mixing. In the case of the former, the heat flux is proportional to the temperature gradient. In the case of the latter, turbulent motions cause gas elements with different specific entropies to move around and mix with one another, causing a heat flux proportional to the entropy gradient. Thus, we write the net heat flux as $$\label{NK_Flux}
\mathbf{F}=-\kappa_{\rm cond}\nabla T -\kappa_{\rm mix}\rho T\nabla s,
\;\;\; \kappa_{\rm cond}=f\kappa_{\rm Sp},
\;\;\; \kappa_{\rm mix}=\alpha_{\rm mix}c_sH_p,$$ where $T$ is the temperature, and $s$ is the specific entropy. We assume that the conductivity $\kappa_{\rm cond}$ is a fraction $f$ of the Spitzer value $\kappa_{\rm Sp}$ in an unmagnetized plasma, and the mixing coefficient $\kappa_{\rm mix}$ is a fraction $\alpha_{\rm mix}$ of the product of the sound speed $c_s$ and pressure scale height $H_p$. We take $H_p\approx(r_c^2+r^2)^{1/2}$, where $r$ is the local radius and $r_c$ is the core radius [@NK_zak03], and set $s=c_v\ln(P\rho^{-\gamma})$, where $c_v$ is the specific heat at constant volume and $\gamma=5/3$ is the adiabatic index. For simplicity, we have considered models with either pure conduction or pure mixing.
Results {#NK_res}
-------
![Observed and modeled profiles of electron number density and temperature for ([*a*]{}) A1795, ([*b*]{}) A2390, ([*c*]{}) A2597, and ([*d*]{}) Hydra A. The data are from [*Chandra*]{}. The solid and dotted lines represent best-fit models based on pure thermal conduction and pure turbulent mixing, respectively. $H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ have been adopted. While the conduction model requires unphysically large values of $f>1$ for A2597 and Hydra A, the mixing model gives good fits to all four clusters with reasonable values of $\alpha_{\rm mix}\sim0.01-0.03$. ](a1795bw.ps "fig:"){height="18pc" width="16pc"} ![Observed and modeled profiles of electron number density and temperature for ([*a*]{}) A1795, ([*b*]{}) A2390, ([*c*]{}) A2597, and ([*d*]{}) Hydra A. The data are from [*Chandra*]{}. The solid and dotted lines represent best-fit models based on pure thermal conduction and pure turbulent mixing, respectively. $H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ have been adopted. While the conduction model requires unphysically large values of $f>1$ for A2597 and Hydra A, the mixing model gives good fits to all four clusters with reasonable values of $\alpha_{\rm mix}\sim0.01-0.03$. ](a2390bw.ps "fig:"){height="18pc" width="16pc"}
![Observed and modeled profiles of electron number density and temperature for ([*a*]{}) A1795, ([*b*]{}) A2390, ([*c*]{}) A2597, and ([*d*]{}) Hydra A. The data are from [*Chandra*]{}. The solid and dotted lines represent best-fit models based on pure thermal conduction and pure turbulent mixing, respectively. $H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ have been adopted. While the conduction model requires unphysically large values of $f>1$ for A2597 and Hydra A, the mixing model gives good fits to all four clusters with reasonable values of $\alpha_{\rm mix}\sim0.01-0.03$. ](a2597bw.ps "fig:"){height="18pc" width="16pc"} ![Observed and modeled profiles of electron number density and temperature for ([*a*]{}) A1795, ([*b*]{}) A2390, ([*c*]{}) A2597, and ([*d*]{}) Hydra A. The data are from [*Chandra*]{}. The solid and dotted lines represent best-fit models based on pure thermal conduction and pure turbulent mixing, respectively. $H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ have been adopted. While the conduction model requires unphysically large values of $f>1$ for A2597 and Hydra A, the mixing model gives good fits to all four clusters with reasonable values of $\alpha_{\rm mix}\sim0.01-0.03$. ](hydra_bw.ps "fig:"){height="18pc" width="16pc"}
We integrate the basic equations described above to calculate the radial profiles of the electron number density $n_e(r)$ and temperature $T(r)$. For each cluster, we assume that the observed gas temperature $T_{\rm obs}$ in the region outside the cooling core is the virial temperature and use this to determine the gravitational potential, assuming an NFW distribution for the dark matter [@NK_mao97; @NK_afs02]. We also use $T_{\rm obs}$ as a boundary condition for the gas at large radius. We vary the central density $n_e(0)$ and temperature $T(0)$, along with either $f$ (for the conduction model) or $\alpha_{\rm mix}$ (for the mixing model), to find the solution that best fits the observed density and temperature distributions of the cluster.
We have analyzed ten clusters (A1795, A1835, A2052, A2199, A2390, A2597, Hydra A, RX J1347.5$-$1145, Sersic 159-03, and 3C 295) for which high resolution data are available. Figure 1 shows the results of the model fitting for four of these clusters. Solid lines indicate the best-fit conduction models, while dotted lines show the best-fit mixing models. Overall, both models explain the observed data reasonably well.
Of the ten clusters, five (A1795, A1835, A2199, A2390, RX J1347.5$-$1145) are well described by a pure conduction model with $f\sim0.2-0.4$, while the other five (A2052, A2597, Hydra A, Sersic 159-03, and 3C 295, e.g., see Fig. 1$c,d$) require unphysically large values of $f>1$. The latter five clusters exhibit strong AGN activity in their centers and extended radio emission, which might indicate that the gas receives extra heat energy from the AGN [@NK_zak03].
The turbulent mixing model fits all ten clusters quite well, with a surprisingly narrow range of $\alpha_{\rm mix}\sim0.01-0.03$ [@NK_kim03b]. The five clusters that were incompatible with the conduction model tend to need a larger value of $\alpha_{\rm mix}$ by a factor of 2 than the other clusters (perhaps because the nuclear activity and the associated jets in these clusters cause enhanced turbulent transport). The values of $\alpha_{\rm mix}$ found from the model fitting correspond to a turbulent diffusion coefficient of $\kappa_{\rm mix}\sim1-6$ kpc$^2$ Myr$^{-1}$ at $r\sim50-300$ kpc, which is similar to the value one infers from typical parameters for intracluster turbulence: turbulent velocities $v_{\rm turb}\sim
100-300$ km s$^{-1}$ and eddy sizes $l_B\sim5-20$ kpc [@NK_ric01; @NK_car02].
Thermal Stability {#NK_stab}
-----------------
Since optically-thin gas at X-ray temperatures is known to be thermally unstable, it is necessary to check the stability of the equilibrium models discussed in §3. The absence of cold material in the centers of clusters indicates that the thermal instability is either absent or at least very weak. Since diffusive processes in general tend to stabilize thermal instability on small scales [@NK_fie65], it is interesting to ask whether thermal conduction with $f\sim 0.2-0.4$ or turbulent mixing with $\alpha_{\rm
mix}\sim0.01-0.03$ can suppress the growth of large-scale unstable modes in clusters.
We begin with a discussion of local linear modes, where we assume that the perturbations have rapid spatial variations. It is straightforward to derive a dispersion relation for such modes. Using equation (\[NK\_Flux\]) for the total heat flux, we find $$\label{NK_sig}
\sigma = \sigma_\infty - \kappa_{\rm mix} ( 1 + q) k_r^2,$$ where $\sigma$ is the growth rate of the model, $\sigma_\infty\equiv
3(\gamma-1)j/(\gamma P)$ is the growth rate of isobaric perturbations in the absence of diffusion [@NK_kim03a; @NK_fie65], $k_r$ is the radial wavenumber of the mode, and the dimensionless parameter $q\equiv(\gamma-1)\kappa_{\rm cond}T/ (\gamma\kappa_{\rm mix}P)$ measures the stabilizing effect of conduction relative to mixing. Putting in numerical values, clusters with pure conduction should be marginally stable to local perturbations [@NK_zak03]. Since $q
\sim 0.1 (f/0.2)(0.02/\alpha_{\rm mix}) (r/20\,{\rm kpc})^{-1}
(n_e/0.05\,{\rm cm^{-3}})^{-1}$ is normally less than unity in the region $r<20$ kpc where most of the cooling occurs, we expect turbulent mixing to have a stronger stabilizing effect relative to conduction.
We have confirmed these predictions by explicitly analyzing the global stability of the equilibrium models. By applying Lagrangian perturbations and solving the perturbed equations as a boundary value problem, we searched for all unstable/overstable modes and calculated their growth times $t_{\rm grow}$. In the presence of conduction, we find that all global modes become stable except for the fundamental, nodeless mode. The lone unstable mode has a very long growth time, e.g., A1795 with $f=0.2$ has $t_{\rm grow}\sim4.1$ Gyr, while Hydra A with $f=3.5$ has $t_{\rm grow}\sim9.3$ Gyr [@NK_kim03a]. Turbulent mixing suppresses the instability even more significantly; A1795 with $\alpha_{\rm mix}=0.011$ has $t_{\rm grow}$ much longer than the Hubble time, and Hydra A with $\alpha_{\rm mix}=0.021$ is completely stable [@NK_kim03b]. These results suggest that thermal instability is not a serious issue for clusters that achieve thermal balance through diffusive heat transport.
Scaling Laws
------------
The theory of cosmic structure formation indicates that the mass $M$ of a halo should scale with the virial temperature $T$ as $M\propto
T^{3/2}$, and that the X-ray luminosity and the entropy should scale as $L_X\propto T^2$ and $S\equiv Tn_e^{-2/3}\propto T$. However, cluster observations show different scaling laws: $M\propto
T^{1.7\sim1.9}$, $L_X\propto T^{2.5\sim3}$, $S\propto T^{0.6\sim0.7}$, for rich clusters with $kT{\lower.5ex\hbox{$\; \buildrel > \over \sim
\;$}} 2$ keV [@NK_all98; @NK_san03; @NK_pon03]; and $L_X\propto
T^{4\sim5}$, $S\propto T^{-0.7\sim 0.2}$, for small clusters or galaxy groups with $kT{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}} 1$ keV [@NK_pon03; @NK_hel00]. That is, not only are the observed power-law indices different from the self-similar predictions, there is also a clear break in cluster properties at a characteristic temperature $kT\sim1-2$ keV. The fact that smaller clusters or groups have relatively constant entropy has been recognized as an “entropy floor.” The prevailing explanations for the rather high entropy at low temperatures include pre-heating of intracluster gas [@NK_kai91; @NK_evr91], removal of cold low-entropy gas via galaxy formation in clusters [@NK_bry00], and supernova feedback [@NK_voi02]. Although some of these suggestions are fairly successful in reproducing the entropy floor and the observed scalings, none of them includes thermal conduction or turbulent mixing. If these processes are at all important in clusters, they should have a large effect on the scaling laws.
It is straightforward to derive scaling relationships that the equilibrium cluster models of §3 should obey. For rich clusters with $kT{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}} 2$ keV, where thermal bremsstrahlung ($j\propto n_e^2T^{1/2}$) dominates, heating by conduction leads to $L_X\propto T^4$ and $S\propto T^{0.3}$, while heating by turbulent mixing predicts $L_X\propto T^3$ and $S\propto
T^{0.6}$. On the other hand, for small clusters or groups ($kT{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}} 1$ keV), where cooling is dominated by line transitions ($j\propto
n_e^2T^{-0.7\sim-1}$), $L_X\propto T^4$ and $S\propto T^{-0.2\sim0}$ for the thermal conduction model, and $L_X\propto T^{4.2\sim4.5}$ and $S\propto T^{-0.3\sim-0.1}$ for the turbulent mixing model [@NK_kim03b]. We see that the scaling relations predicted by the mixing model are in remarkably good agreement with the observations. The dramatic change of cluster properties at $kT\sim(1-2)$ keV arises because of the change in the cooling mechanism above and below this temperature. Also, the entropy floor observed in groups is reproduced naturally.
Conclusion
----------
The thermal conduction and turbulent mixing models have certain attractive properties which ultimately are due to the fact that both models involve diffusive transport. Diffusion not only allows heat to move into the cluster center from the outside, it also irons out perturbations and thereby helps to control thermal instability. What is interesting is that the amount of diffusion required to fit the observations is comparable to that predicted by theoretical arguments.
Two caveats need to be mentioned. First, the presence of cold fronts in many clusters [@NK_mar00; @NK_vik01] indicates that large temperature and entropy jumps are able to survive in some regions of the hot gas. Diffusion is clearly suppressed across these surfaces. It is possible that cold fronts are special regions where the magnetic field is combed out parallel to the front, thereby suppressing cross-field conduction temporarily [@NK_vik01; @NK_zak03].
Second, all we have shown is that a cluster with the observed density and temperature profile would be in hydrostatic and thermal equilibrium and would be fairly stable. However, we have not explained how the cluster reaches the observed state starting from generic initial conditions. Time-dependent simulations show that a cluster with thermal conduction would either slowly evolve to an isothermal state if its initial density is less than a critical density, or develop a catastophic cooling flow otherwise [@NK_bre88]. Does the current observed state result from an initial rapid mass dropout (which decreases the density) and subsequent slow evolution with diffusive heating of an once overdense cluster [@NK_kim03a]? Are other heating mechanisms, e.g., AGNs, necessary to explain the present state of clusters? Answers to these questions are of fundamental importance to understanding clusters and more generally galaxy formation.
[**Acknowledgments.**]{} The work reported here was supported in part by NASA grant NAG5-10780 and NSF grant AST 0307433.
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|
---
abstract: 'In emergency situations, actions that save lives and limit the impact of hazards are crucial. In order to act, situational awareness is needed to decide what to do. Geolocalized photos and video of the situations as they evolve can be crucial in better understanding them and making decisions faster. Cameras are almost everywhere these days, either in terms of smartphones, installed CCTV cameras, UAVs or others. However, this poses challenges in big data and information overflow. Moreover, most of the time there are no disasters at any given location, so humans aiming to detect sudden situations may not be as alert as needed at any point in time. Consequently, computer vision tools can be an excellent decision support. The number of emergencies where computer vision tools has been considered or used is very wide, and there is a great overlap across related emergency research. Researchers tend to focus on state-of-the-art systems that cover the same emergency as they are studying, obviating important research in other fields. In order to unveil this overlap, the survey is divided along four main axes: the types of emergencies that have been studied in computer vision, the objective that the algorithms can address, the type of hardware needed and the algorithms used. Therefore, this review provides a broad overview of the progress of computer vision covering all sorts of emergencies.'
author:
- 'Laura Lopez-Fuentes'
- Joost van de Weijer
- 'Manuel González-Hidalgo'
- Harald Skinnemoen
- 'Andrew D. Bagdanov'
bibliography:
- 'refs.bib'
date: 'Received: date / Accepted: date'
title: 'Review on Computer Vision Techniques in Emergency Situations [^1] '
---
Introduction {#intro}
============
Emergencies are a major cause of both human and economic loss. They vary in scale and magnitude, from small traffic accidents involving few people, to full-scale natural disasters which can devastate countries and harm thousands of people. *Emergency management* has become very important to reduce the impact of emergencies. As a consequence, using modern technology to implement innovative solutions to prevent, mitigate, and study emergencies is an active field of research.
-- -- -- --
-- -- -- --
The presence of digital cameras has grown explosively over the last two decades. Camera systems perform real-time recording of visual information in hotspots where an accident or emergency is likely to occur. Examples of such systems are cameras on the road, national parks, inside banks, shops, airports, metros, streets and swimming pools. In Figure \[fig:examples\_surveillance\_cameras\] we give some examples of frames from videos that could potentially be recorded by some of these monitoring systems. It is estimated that as of 2014 there were over 245 million active, operational and professionally installed video cameras around the world [@IHS2015surveillance]. Mounting these video cameras has become very cheap but there are insufficient human resources to observe their output. Therefore, most of the data is being processed *after* an emergency has already occurred, thus losing the benefit of having a real-time monitoring tool in the first place.
The problem of real-time processing of the visual output from monitoring systems in potential emergency scenarios has become a hot topic in the field of computer vision. As we will show in this review, computer vision techniques have been used to assist in various stages of emergency management: from prevention, detection, assistance of the response, to the understanding of emergencies. A myriad of algorithms has been proposed to prevent events that could evolve into disaster and emergency situations. Others have focused on fast response after an emergency has already occurred. Yet other algorithms focus on assisting during the response to an emergency situation. Finally, several methods focus on improving our understanding of emergencies with the aim of predicting the likelihood of an emergency or dangerous situation occurring.
This review provides a broad overview of the progress of computer vision covering all sorts of emergencies. The review is organized in four main axes along which researchers working on the topic of computer vision for emergency situations have organized their investigations. The first axis relates to the *type of emergency* being addressed, and the most important division along this axis is the distinction between *natural* and *man-made* emergencies. The second axis determines the objective of a system: if it is going to detect risk factors to prevent the emergency, help detecting the emergency once it has occurred, provide information to help emergency responders to react to the emergency, or model the emergency to extract valuable information for the future. The third axis determines where the visual sensors should be placed, and also what type of sensors are most useful for the task. The last axis determines which algorithms will be used to assist in analyzing the data. This typically involves a choice of computer vision algorithms combined with machine learning techniques.
When organizing emergency situations along these axes, our review shows that there is significant overlap across emergency research areas, especially in the sensory data and algorithms used. For this reason this review is useful for researchers in emergency topics, allowing them to more quickly find relevant works. Furthermore, due to its general nature, this article aims to increase the flow of ideas among the various emergency research fields.
Since it is such an active field of research, there are several reviews on computer vision algorithms in emergency situations. However these reviews focus on a single, specific topic. In the field of fire detection, a recent survey was published summarizing the different computer vision algorithms proposed in the literature to detect fire in video [@ccetin2013video]. Also Gade et al. [@gade2014thermal] published a survey on applications of thermal cameras in which fire detection is also considered. Another related research direction is human activity recognition and detection of abnormal human actions. This field has many applications to video surveillance, as shown in a recent review on this topic [@ke2013review]. Work in the area of human activity recognition has focused on the specific topic of falling person detection. Mubashir et al. [@gade2014thermal] describe the three main approaches to falling person detection, one of them being computer vision. Another important application area of emergency management is traffic. The number of vehicles on roads has significantly increased in recent years, and simultaneously the number of traffic accidents has also increased. Therefore a large number of road safety infrastructure and control systems have been developed. Three main reviews have been recently published on the topic of road vehicles monitoring and accident detection using computer vision [@buch2011review; @kanistras2015survey; @kovacic2013computer]. As discussed here, there are several review papers about different specific emergency situations, however there is no *one* work reviewing computer vision in emergency situations considering many types of emergency situations. By doing so, we aspire to draw a wider picture of the field, and to promote cross-emergency dissemination of research ideas.
Computer vision is just one of several scientific fields which aims to improve emergency management. Before going into depth on computer vision techniques for emergency management, we briefly provide some pointers to other research fields for interested readers. Emergency management is a hot topic in the field of Internet of Things (IoT) [@maalel2013reliability; @yang2013internet]. The goal of the IoT is to create an environment in which information from any object connected to the network be efficiently shared with others in real-time. The authors of [@yang2013internet] provide an excellent study on how IoT technology can enhance emergence rescue operations. Robotics research is another field which already is changing emergency management. Robots are being studied for handling hazardous situations with radiation [@nagatani2013emergency], for assisting crowd evacuation [@sakour2017robot], and for search and rescue operations [@shah2004survey]. Finally, we mention reviews on human centered sensing for emergency evacuation [@radianti2013crowd] and research on the impact of social media on emergency management. The main advantage of computer vision compared to these other research fields is the omni-presence of cameras in modern society, both in the public space and mobile phones of users. However, for some problems computer vision might currently be still less accurate than other techniques.
Organization of this Review {#sec:organization}
===========================
Before we enter into the organization of the review it is important to define what we consider as an *emergency* and, therefore, what situations and events will be encompassed in this definition. We consider as an emergency any sudden or unexpected situation that meets at least one of the following conditions:
- it is a situation that represents an imminent risk of health or life to a person or group of persons or damage to properties or environment;
- it is a situation that has already caused loss of health or life to a person or group of persons or damage to properties or environment; or
- it is a situation with high probability of scaling, putting in risk the health or life of a person or group of persons or damage to properties or environment.
An emergency situation, therefore, requires situational awareness, rapid attention, and remedial action to prevent degradation of the situation. In some cases emergency situations can be prevented by avoiding and minimizing risks. Also, in some situations, the occurrence of an emergency situation can help to understand risk factors and improve the prevention mechanisms. Taking this definition in consideration, any type of uncontrolled fire would be considered as an emergency situation as it is an unexpected situation that could represent a risk for persons, the environment, and property. A controlled demolition of a building would not be considered an emergency situation, as by definition if it is controlled it is not an unexpected situation any more. However, if someone is injured or something unforeseen was damaged during the demolition, that would then be considered an emergency situation.
[ccc]{} & [**Emergency sub-group**]{} & [**Emergency Type**]{}\
Natural emergencies & &
---------------------
Fire
Flood
Drought
Earthquake
Hurricane/Tornado
Landslide/Avalanche
---------------------
\
& & Falling person\
&& Drowning\
&& Injured civilians\
(r)[2-3]{} & & Road accident\
&& Crowd related\
&& Weapon threaten\
After having defined what we consider an emergency situation we briefly explain the organization of this review. We begin in Section \[sec:Definition\] with an overview and classification into subcategories of emergency situations. Here, we briefly introduce the three axes along which we will organize the existing literature on computer vision for emergency situations, which corresponds to Sections 4 to 6. The organization is also summarized in Figure \[fig:graph\].
[ [**The Monitoring Objective Axis (Section \[sec:objective\]):**]{} ]{} the management of an emergency has a life cycle which can be divided into four main phases [@petak1985emergency]:
- **Preparedness**: during this phase there are risk factors that could trigger an emergency at some point. In order to reduce the chance of an emergency happening, it is important to estimate these factors and avoid or minimize them as much as possible. Not all emergencies can be prevented, but in some cases preventive measures can help to reduce the impact of a disaster (like teaching people how to behave during a hurricane).
- **Response**: response activities are performed after the emergency has occurred. At this point, it is very important to detect and localize the emergency and to maintain situational awareness to have a rapid and efficient reaction. Often the minutes immediately after the occurrence of an emergency can be crucial to mitigating its effects.
- **Recovery**: recovery activities take place until normality is recovered. In this phase actions must be taken to save lives, prevent further damaging or escalation of the situation. We consider that normality has been recovered when the immediate threat to health, life, properties or environment has subsided.
- **Mitigation**: this phase starts when normality has been re-established. During this phase it is important to study and understand the emergency, how it could have been avoided or what responses could have helped to mitigate it.
[ [**The Sensors and Acquisition System Axis (Section \[sec:acquisition\]):**]{} ]{} An emergency has four main management phases: prevention, detection, response and understanding of an emergency. For a computer vision algorithm to help in one or more of these phases, it is important to have visual sensors to record the needed information. There are many different types of devices for this, as also many types of devices in which they may be installed depending on the type of data being captured. Some of the most common are: fixed monitoring cameras, satellites, robots and Unmanned Aerial Vehicles (UAVs). These devices can capture different types of visual data, such as stereo or monocular images, infrared or thermal. Depending on the rate of data acquisition this data can be in the form of video or still images, and the video can be either from a static or moving source.
[ [**The Algorithmic Axis (Section \[sec:algorithms\]):**]{} ]{} The visual data collected by the sensor is then subject to a feature extraction process. Finally the features extracted are used as an input to a modeling or classification algorithm. In Figure \[fig:graph\] the main feature extraction and classification algorithms are listed.
Types of Emergencies {#sec:Definition}
====================
-- ------ -------------------
Fire
Flood
Drought
Falling person
Drowning
Injured civilians
Road accident
Crowd related
Weapon threaten
-- ------ -------------------
There are many types of emergencies and a great deal of effort has been invested in classifying and defining them [@abdallah2000public; @below2009disaster]. However there are many emergencies that due to their nature have not been studied in computer vision, such as famine or pollution. These types of emergencies do not have clear visual consequences that can be easily detected by visual sensors. In this paper we focus only on emergencies relevant in the computer vision field. In Table \[tab:types of emergencies\] we classify the types of emergencies that we consider in this survey. We distinguish two generic emergency groups: natural emergencies and man-made emergencies. Natural emergencies are divided into three emergency types: fire, flood and drought. Emergencies caused by humans are divided into two sub-types depending on the scope: emergencies that cause dangers to multiple persons and emergencies that cause dangers to a single person.
Natural emergencies
-------------------
Natural emergencies cover any adverse event resulting from natural processes of the earth. Natural emergencies can result in a loss of lives and can have huge economic costs. The severity of the emergency can be measured by the losses and the ability of the population to recuperate. Although natural emergencies are external to humans and therefore can not be prevented, society has been increasingly able to reduce the impact of these natural events by building systems for early detection or assistance during the event. Among these systems built to detect or assist natural emergencies, computer vision has played an important role in emergencies related to fire, flood and drought.
[ [**Fire:**]{} ]{} electric fire detectors have been integrated into buildings since the late 1980s. These devices detect smoke, which is a typical fire indicator. Currently, two major types of electric smoke detector devices are used [@reisinger1980smoke]: ionization smoke detectors and photoelectric smoke detectors. Although the devices achieve very good performance in closed environments, their reliability decreases in open spaces because they rely on smoke particles arriving to the sensor. For this reason, fire detection in monitoring cameras has become an important area of research in computer vision. The use of video as input in fire detection makes it possible to use in large and open spaces. Hence, most computer vision algorithms that study fire detection focus mainly on open spaces [@martinez2008computer; @gunay2010fire]. However, some articles [@ko2010early; @truong2012fire] also study fire detection in closed spaces, claiming to have several advantages over conventional smoke detectors: visual fire detectors can trigger the alarm before conventional electric devices, as they do not rely on the position of fire particles, and they can give information about the position and size of the fire.
[ [**Flood:**]{} ]{} flood results from water escaping its usual boundaries in rivers, lakes, oceans or man-made canals due to an accumulation of rainwater on saturated ground, a rise in the sea level or a tsunami. As a result, the surrounding lands are inundated. Floods can result in loss of life and can also have huge economic impact by damaging agricultural land and residential areas. Flood detection or modeling algorithms are still at a very early stage. In computer vision, there are two main trends: algorithms based on still cameras monitoring flood-prone areas [@lai2007real] and systems that process satellite images [@martinis2010automatic; @mason2012near; @mason2010flood]. On the one hand, algorithms based on earth still cameras cover a limited area but do not have hardware and software restrictions. On the other hand, algorithms based on satellite imagery cover a much wider area, but if the system is to be installed on board the satellite, there are many factors that have to be considered when deciding the type of hardware and software to be used on board a satellite [@kogan1995droughts; @mason2012near; @song2013drought].
[ [**Drought:**]{} ]{} droughts are significant weather-related disasters that can result in devastating agricultural, health, economic and social consequences. There are two main characteristics that determine the presence of drought in an area: a low surface soil moisture and a decrease in rainfall. Soil moisture data is rarely computed with visual data, however in [@hassan2015assessment] they use high-resolution remotely sensed data to determine the water balance of the soil. Another technique used to determine the beginning of drought is to compute the difference between the average precipitation, or some other climatic variable, over some time period and comparing the current situation to a historical record. Observation of rainfall levels can help prevent or mitigate drought. Although rainfall does not have a clear visual consequence that can be monitored through cameras, there have been some image processing algorithms studied to map annual rainfall records [@fu2012drought; @song2013drought]. These algorithms help to determine the extension and severity of a drought and to determine regions likely to suffer from this emergency.
[ [**Earthquake:**]{} ]{} earthquakes are sudden shakes or vibrations of the ground caused by tectonic movement. The magnitude of an earthquake can vary greatly from almost imperceptible earthquakes (of which hundreds are registered every day) up to very violent earthquakes that can be devastating. Earthquake prediction is a whole field of research called seismology. However, due to the lack of distinctive visual information about an earthquake, this emergency has received relatively little attention from the computer vision community. Several works address the robustness of motion tracking techniques under seismic-induced motions [@doerr2005methodology; @hutchinson2004monitoring], with the aim of assisting rescue and reconnaissance crews. The main difficulty for these systems is the fact that the camera itself is also undergoing motion. In [@doerr2005methodology] this problem is addressed by introducing a dynamic camera reference system which is shared between cameras and allows tracking of objects with respect to this reference coordinate system. Also, computer vision plays a role during the rescue of people affected by these natural disaster by detecting people with robots or UAVs in the damaged areas, this is tackled in the “injured civilians” section. [ [**Hurricane/tornado:**]{} ]{} hurricanes and tornadoes are violent storms characterized by heavy wind and rain. Similarly to earthquakes, depending on their magnitude they can be almost unnoticeable or devastating and at the same time the visual information derived from these natural disasters has not yet been studied in the field of computer vision. However, similarly to earthquakes, robots and UAVs can be used to find people in the areas affected by the disaster using human detection algorithms.
The more general work on “injured civilians” detection in affected areas which is shared among any other emergency that involves people rescue like earthquakes and hurricanes, is again applicable to landslide and avalanche emergencies. We found one work which evaluates this in particular for avalanches, and which takes into account the particular environment where avalanches occur [@bejiga2017convolutional].
Emergencies caused by humans
----------------------------
Emergencies caused by humans cover adverse events and disasters that result from human activity or man-made hazards. Although natural emergencies are considered to generate a greater number of deaths and have larger economic impact, emergencies caused by humans are increasing in importance and magnitude. These emergencies can be unintentionally caused, such as road accidents or drowning persons, but they can also be intentionally provoked, such as armed conflicts. In this survey, emergencies caused by humans are divided in two types depending on their scope: emergencies affecting multiple persons and emergencies affecting a single person.
### Emergencies affecting multiple persons
These emergencies are human-caused and affects several persons.
[ [**Road accident emergencies:**]{} ]{} Although depending on the extent of the emergency they can affect one or multiple persons, we consider road accidents to be multiple-person emergencies. Transport emergencies result from human-made vehicles. They occur when a vehicle collides with another vehicle, a pedestrian, animal or another road object. In recent years, the number of vehicles on the road has increased dramatically and accidents have become one of the highest worldwide death factors, reaching the number one cause of death among those aged between 15 and 29 [@world2015global]. In order to prevent these incidents, safety infrastructure has been developed, many monitoring cameras have been installed on roads, and different sensors have been installed in vehicles to prevent dangerous situations. In computer vision there are three main fields of study in which have focused on the detection or prevention of transport emergencies.
- Algorithms which focus on detecting anomalies, as in most cases they correspond to dangerous situations [@jiang2011anomalous; @saligrama2012video; @sultani2010abnormal].
- Algorithms specialized on detecting a concrete type of road emergency, such as a collision [@kamijo2000traffic; @veeraraghavan2003computer; @yun2014motion].
- Algorithms based on on-board cameras which are designed to prevent dangerous situations, such as algorithms that detect pedestrians and warn the driver [@gavrila2007multi; @guo2012robust; @hachisuka2011facial].
[ [**Crowd-related emergencies:**]{} ]{} These type of emergencies are normally initiated by another emergency when man persons are gathered in a limited area. They normally occur in multitudinous events such as sporting events, concerts or strikes. These type of events pose a great risk because when any emergency strikes it can lead to collective panic, thus aggravating the initial emergency. This is a fairly new area of research in computer vision. In computer vision for crowd-related emergencies we find different sub-topics like crowd density estimation, analysis of abnormal events [@andrade2006detection], crowd motion tracking [@ihaddadene2008real] and crowd behaviour analysis [@garate2009crowd], among others.
[ [**Weapon threat emergencies:**]{} ]{} Although detecting weapons through their visual features and characteristics has not yet been studied in computer vision, some studies have been done in the direction of detecting the heat produced by them. A dangerous weapon that threatens innocent people every year are mines. Millions of old mines are hidden under the ground of many countries which kill and injure people. These mines are not visible from the ground but they emit heat that can be detected using thermal cameras [@muscio2004land; @siegel2002land; @wasaki2001smart]. Another weapon that can be detected through thermal sensors are guns, due to the heat that they emit when they are fired [@price2004system].
### Emergencies affecting a single person
These are emergencies are caused by or affect one person. The classic example of this is detecting human falling events for monitoring the well-being of the elderly.
[ [**Drowning:**]{} ]{} Drowning happens when a person lacks air due to an obstruction of the respiratory track. These situations normally occur in swimming pools or beach areas. Although most public swimming pools and beaches are staffed with professional lifeguards, drowning or near-drowning incidents still occur due to staff reaction time. Therefore there is a need to automate the detection of these events. One way of automating detection of these events is the installation of monitoring cameras equipped with computer vision algorithms to detect drowning persons in swimming pools and beach areas. However, drowning events in the sea are difficult to monitor due to the extent of the hazardous area, so most computer vision research on this topic has been focused in drowning people in swimming pools [@eng2003automatic; @zecha2012swimmer]. [ [**Injured person:**]{} ]{} An injured person is a person that has come to some physical harm. After an accident, reaction time is crucial to minimize human losses. Emergency responders focus their work on the rescue of trapped or injured persons in the shortest time possible. A method that would help speed up the process is the use of aerial and ground robots with visual sensors that record the affected area at the same time as detect human victims. Also, in the event of an emergency situation, knowing if there are human victims in the area affected by the event may help determine the risk the responders are willing to take. For this purpose autonomous robots have been built to automatically explore the area and detect human victims without putting at risk the rescue staff [@andriluka2010vision; @castillo2005method; @kleiner2007genetic; @leira2015automatic; @rudol2008human; @soni2013victim]. Detecting injured or trapped humans in a disaster area is an extremely challenging problem from the computer vision perspective due to the articulated nature of the human body and the uncontrolled environment.
[ [**Falling person:**]{} ]{} With the growth of the elderly population there is a growing need to create timely and reliable smart systems to help and assist seniors living alone. Falls are one of the major risks the elderly living alone, and early detection of falls is very important. The most common solution used today are call buttons, which when the senior can press to directly call for help. However these devices are only useful if the senior is conscious after the fall. This problem has been tackled from many different perspectives. On the one hand wearable devices based on accelerometers have been studied and developed [@chen2005wearable; @zhang2006fall], some of these devices are in process of being commercialized [@fallDet2015]. On the other hand, falling detection algorithms have been studied from a computer vision perspective [@liao2012slip; @rougier2011fall].
Monitoring Objective {#sec:objective}
====================
As stated in Section \[sec:organization\], an emergency has four main phases which at the same time lead to four emergency management phases: prevention, detection, response/assistance and understanding of emergencies. It is important to understand, study and identify the different phases of an emergency life cycle since the way to react to it changes consequently. In this section, we will classify computer vision algorithms for emergency situations depending on the emergency management phase in which they are used.
-- -- --
-- -- --
Emergency Prevention {#subsec:prevention}
--------------------
This phase focuses on preventing hazards and risk factors to minimize the likelihood that an emergency will occur. The study of emergency prevention on computer vision can not be applied to any emergency since not all of them are induced by risk factors that have relevant visual features that can be captured by optical sensors. For example, the only visual feature which is normally present before a fire starts is smoke. Therefore research on fire prevention in computer vision focuses on smoke detection. As an example, in Figure \[fig:example prevention\] (a) we see a region of forest with a dense smoke that looks like the starting of a fire, however a fire detection algorithm would not detect anything until the fire is present in the video. Some early work on this topic started at the beginning of the century [@chen2004early; @gomez2003smoke]. Since then, smoke detection for fire prevention or early fire detection has been an open topic in computer vision. Most of the work done on smoke detection is treated as a separate problem from fire detection and aims to detect smoke at the smoldering phase to prevent fire [@gubbi2009smoke; @xiong2007video; @ye2015dynamic; @yuan2011video]. However there is also some work on smoke and flame detection when the fire has already started since smoke in many cases can be visible from a further distance [@kolesov2010natural; @yu2013real].
Traffic accidents are considered as one of the main sources of death and injuries globally. Traffic accidents can be caused by three main risk factors: the environment, the vehicle and the driver. Among these risk factors, it is estimated that the driver accounts for around 90% of road accidents [@Traffic2015]. To avoid these type of accidents, a lot of security devices have been incorporated on vehicles in the past years. There are two types of security devices: “active” whose function is to decrease the risk of an accident occurring, and “passive” which try to minimize the injuries and deaths in an accident. Among the active security devices, there is work on the usage of visual sensors to detect situations that could lead to an accident and force actions that could be taken to prevent the accident [@trivedi2007looking].
There are many different kind of traffic accidents but one of the most common and at the same time more deadly ones are pedestrians being run over. Due to the variant and deformable nature of humans, the environment variability, the reliability, speed and robustness that a security system is required, the detection of unavoidable impact with pedestrians has become a challenge in computer vision. This field of study is well defined and with established benchmarks and it has served as a common topic in numerous review articles [@benenson2014ten; @dollar2012pedestrian; @geronimo2009survey; @wojek2009multi]. Now, with the increase of training data and computing power this investigation has had a huge peak in the past years and even the first devices using this technology have already been commercialized [@2014mobileeye; @2016flir]. Figure \[fig:example prevention\] (b) shows an example of the output from a pedestrian detection algorithm on board a vehicle.
Another contributing factor to many accidents is the drivers fatigue, distraction or drowsiness. A good solution to detect the state of the driver in a non intrusive way is using an on board camera able to detect the drivers level of attention to the road. There are different factors that can be taken into account to determine the level of attention to the road of a driver. In [@choi2014head] they determine if the driver is looking at the road through the head pose and gaze direction, in Figure \[fig:example prevention\] (c) we show some examples of the images they used to train their algorithm. While in [@hachisuka2011facial] they determine 17 feature points from the face that they consider relevant to detect the level of drowsiness of a person. In [@hu2009driver] they study the movements of the eyelid to detect sleepy drivers.
Although not so widely studied, there are other type of road accidents that could also be prevented with the help of computer vision. For example, in [@hayashi2009predicting] they propose a framework to predict unusual driving behavior which they apply to right turns at intersections. Algorithms that use on board cameras to detect the road have also been studied [@guo2012robust], this could help to detect if a vehicle is about to slip on the road. In [@barth2010tracking] they propose a system with on board stereo cameras that assist drivers to prevent collisions by detecting sudden accelerations and self-occlusions. Multiple algorithms for traffic sign detection with on board cameras have also widely proposed in the literature [@houben2013detection; @mogelmose2014traffic], these algorithms mostly go in the direction of creating autonomous vehicles, however they can also be used to detect if the driver is not following some traffic rules and is therefore increasing the risk of provoking an accident.
In order to prevent falls, in [@liao2012slip] they propose a method to detect slips arguing that a slip is likely to lead to a fall or that a person that just had a slip is likely to have balance issues that increase the risk of falling down in the near future. They consider a slip as a sequence of unbalanced postures.
-- --
-- --
Emergency Detection
-------------------
The field of emergency detection through computer vision in contrast with the field of prevention is much wider and has been studied in more depth. In this section we will present the visual information that is being studied to detect different emergencies.
Fire detection has been widely studied in computer vision since it is very useful on open spaces, where other fire detection algorithms fail. In some cases it is useful to not only detect the fire but also some properties of the fire such as the flame height, angle or width [@martinez2008computer]. In order to detect fire on visual content, the algorithms studied exploit visual and temporal features of fire. In early research, flame was the main visual feature that was studied in fire detection [@celik2009fire; @gunay2010fire; @ko2010early; @rinsurongkawong2012fire; @truong2012fire; @van2010fire] however, as stated in Section \[subsec:prevention\], recently smoke has also been studied in the field of fire detection mainly because it appears before fire and it spreads faster so in most cases smoke will appear faster in the camera field of view [@kolesov2010natural; @yu2013real]. As fire has a non-rigid shape with no well defined edges, the visual content used to detect fire is related to the characteristic color of fire. This makes the algorithms in general vulnerable to lighting conditions and camera quality.
Water, the main cue for flood detection, has similar visual difficulties as fire in order to be detected since it has a non-rigid shape and the edges are also not well defined. Moreover, water does not have a specific characteristic color because it might be dirty and it also reflects the surrounding colors. Due to the mentioned difficulties flood detection in computer vision has not been so widely studied. In [@lai2007real] they use color information and background detail (because as said before the background will have a strong effect on the color of the water) and the ripple pattern of water. In [@borges2008probabilistic] they analyse the rippling effect of water and they use the relationship between the averages for each color channel as a detection metric.
Detecting person falls has become a major interest in computer vision, specially motivated to develop systems for elderly care monitoring at home. In order to detect a falling person it is important to first detect the persons which are visible in the data and then analyse the movements. Most algorithms use background subtraction to detect the person since in indoor situations, most of the movement come from persons. Then, to determine if the movement of the person corresponds to a fall some algorithms use silhouette or shape variation [@liao2012slip; @mirmahboub2013automatic; @rougier2007fall], direction of the main movement [@hazelhoff2008video], or relative height with respect to the ground [@mogelmose2014traffic]. However, sometimes it is difficult to distinguish between a person falling down and a person sitting, lying or sleeping, to address this issue, most systems consider the duration of the event since a fall is usually a faster movement [@lin2007automatic; @mogelmose2014traffic], for example, in [@chua2015simple] they consider 10 frames as a fast movement and use this threshold to differentiate two visually similar movements such as the ones given in Figure \[fig:example detection\] (a) which corresponds to sitting and falling.
The problem of drowning person detection is fairly similar to the falling person detection, since it is also in a closed environment and related to human motion. Similarly to fall detection algorithms, the first step to detect a drowning person is to detect the persons visible in the camera range, which is also done by background subtraction [@fei2009drowning; @kam2002video; @lu2004vision; @zecha2012swimmer]. In Figure \[fig:example detection\] we give an example of a detection of the persons in a swimming pool using a background modelling algorithm. However, in this case aquatic environments have some added difficulties because it is not static [@chen2011framework].
0.5 pt
[l\*[4]{}[c]{}l]{} &\
(lr)[2-5]{} [**Location**]{}& & & &\
Fixed location & & & &\
Robot or UAV & & & &\
Satellite & & & &\
Emergency Response/Assistance
-----------------------------
After an emergency has already occurred, one of the most important tasks of emergency responders is to help injured people. However, depending on the magnitude of the disaster, it may be difficult to effectively localize all these persons so emergency responders may find it useful to use robots or UAVs to search for these persons. In order to automatically detect injured people, some of these robots use cameras together with computer vision algorithms. When considering disasters taking place on the ground one of the main characteristic of an injured person is that they are normally lying on the floor so these algorithms are trained to detect lying people. The main difference between them is the perspective from which these algorithms expect to get the input, depending if the camera is mounted on a terrestrial robot [@castillo2005method; @kleiner2007genetic; @soni2012classifier; @soni2013victim] or on a UAV [@andriluka2010vision; @leira2015automatic; @rudol2008human].
A fast and efficient response the in the moments after an emergency has occur ed is critical to minimize damage. For that it is essential that emergency responders have situational awareness, which can be obtained through sensors in the damaged areas, by crowd sourcing applications, and from social media. However, extracting useful information from huge amounts of data can be a slow and tedious work. Therefore, organization of video data that facilitates the browsing on the data is very important [@schoeffmann2015video]. Since during an emergency it is common to have data on the same event from different angles and perspectives, in [@tompkin2012videoscapes] the authors combine interactive spatio-temporal exploration of videos and 3D reconstruction of scenes. In [@huang2014videoweb] the authors propose a way of representing the relationship among clips for easy video browsing. To extract relevant information from videos, in [@diem2016video] the authors find recurrent regions in clips which is normally a sign of relevance. In [@lopez2017bandwidth] the authors introduce a framework of consecutive object detection from general to fine that automatically detects important features during an emergency. This also minimizes bandwidth usage, which is essential in many emergency situations where critical communications infrastructure collapses.
Emergency Understanding
-----------------------
Keeping record of emergencies is very important to analyse them and extract information in order to understand the factors that have contributed to provoke the emergency, how these factors could have been prevented, how the emergency could have been better assisted to reduce the damages or the effects of the emergency. The systems developed to with this objective should feed back into the rest of the systems in order to improve them.
Acquisition {#sec:acquisition}
===========
---------------- -- -- -- --
Fixed location
Robot or UAV
Satellite
---------------- -- -- -- --
Choosing an appropriate acquisition setup is one of the important design choices. To take this decision one must take into account the type of emergency addressed (see Section \[sec:Definition\]) and objectives to fulfill (see Section \[sec:objective\]). One of the things to consider is the location where the sensor will be placed (e.g. in a fixed location, mounted on board a satellite, a robot, or a UAV, etc). Then, depending on the type of sensor we acquire different types of visual information: monocular, stereo, infrared, or thermal. Depending on the frequency in which the visual information is acquired we can obtain videos or still images. Finally, this section is summarized in Table \[tab:sensors\] where we include the advantages and disadvantages of the types of sensors studied and the locations at which they can be placed The type of sensors chosen for the system will have an important impact on the types of algorithms that can be applied, the type of information that can be extracted and the cost of the system, which is also many times a limitation.
Sensor location
---------------
The physical location where the sensor is installed determines the physical extent in the real world that will be captured by the sensor. The location of the sensor depends primarily on the expected physical extent and location of the emergency that is to be prevented, detected, assisted or understood.
[ [**In a fixed location:**]{} ]{} \[minisec:static\] When the type of emergency considered has a limited extent and the location where it can occur (the area of risk) is predictable, a fixed sensor is normally installed. In many cases this corresponds to a monitoring camera mounted on a wall or some other environmental structure. These are the cheapest and most commonly used sensors. They are normally mounted in a high location, where no objects obstruct their view and pointing to the area of risk.
Static cameras have been used to detect drowning persons in swimming pools, as in this case the area of risk is limited to the swimming pool [@chen2010hidden; @chen2011framework; @eng2003automatic; @fei2009drowning; @kam2002video; @lu2004vision; @zecha2012swimmer]. They have also been used to monitor dangerous areas on roads, inside vehicles either pointing to the driver to estimate drowsiness [@choi2014head; @hachisuka2011facial; @kumar2009application; @lee2011real], or pointing to the road to process the area in which it is being driven [@geronimo2009survey; @guo2012robust; @hayashi2009predicting; @houben2013detection; @ma2015real; @mogelmose2014traffic]. Fixed cameras are also widely used in closed areas to detect falling persons [@chua2015simple; @jiang2013real; @mirmahboub2013automatic; @rougier2007fall]. Finally, they can occasionally also be used to detect floods or fire in hotspots [@borges2008probabilistic; @celik2009fire; @lai2007real; @rinsurongkawong2012fire].
[ [**On board robots or UAVs:**]{} ]{} For emergencies that do not take place in a specific area, or that need a more immediate attention, sensors can be mounted on mobile devices such as robots or UAVs. The main difference between these two types of devices is the perspective from which the data can be acquired: terrestrial in the case of a robot, and aerial for UAVs. On the one hand, the main advantage of these data sources is that they are flexible and support real time interaction with emergency responders. On the other hand, their main drawback is their limited autonomy due to battery capacity. Robots and UAVs have been used extensively for human rescue. In such cases, a visual sensor is mounted on board the autonomous machine which can automatically detect human bodies [@andriluka2010vision; @castillo2005method; @leira2015automatic; @soni2012classifier; @soni2013victim]. The authors of [@hassan2015assessment] use a UAV with several optical sensors to capture aerial images of the ground, estimate the soil moisture, and determine if an area suffers or is close to suffering drought.
[ [**On board satellites:**]{} ]{} For emergencies of a greater extent, such as drought, floods or fire, sometimes the use of sensors on board satellites is more convenient [@hassan2015assessment; @mason2012near; @mason2010flood; @song2013drought]. By using satellites, it is possible to visualize the whole globe. However, the biggest drawback of this type of sensor is the frequency at which the data can be acquired since the satellites are in continuous movement and it may be necessary to wait up to a few days to obtain information about a specific area, as an example the Sentinel 2 which is a satellite in the Copernicus program for disaster management revisits every location within the same view angle every 5 days [@sentinel2esa]. Due to this, the information acquired by this type of sensor is mainly used for assistance or understanding of an emergency rather than to prevent or detect it.
[ [**Multi-view cameras:**]{} ]{} In some scenarios, like in the street, it is useful to have views from different perspectives in order to increase the range or performance of the algorithm. However, using data from different cameras also entails some difficulties in combining the different views. Since the most common scenario where multiple cameras are present is in the street, this problem has been tackled in pedestrian detection [@utasi2012multi] and traffic accidents [@forczmanski2016multi].
Type of sensor
--------------
Another important decision to take when working with computer vision for emergency management is the type of optical sensor that should be used. When taking this decision it is important to take into account the technical characteristics of the available sensors, the type of data that they will capture, and the cost.
[ [**Monocular RGB or greyscale cameras:**]{} ]{} Monocular cameras are devices that transform light from the visible spectrum into an image. This type of visual sensor is the cheapest and therefore by far the most widely used in computer vision and monitoring systems. Although these sensors may not be the optimal in some scenarios (such as fire detection), due to their low cost and the experience of researchers on the type of images generated by such sensors, they are used in almost all emergencies [@anderson2009falling; @eng2003automatic; @hu2006system; @lai2007real; @mehran2009abnormal; @toreyin2006computer].
[ [**Stereo RGB or greyscale cameras:**]{} ]{} Stereo cameras are optical instruments that, similarly to monocular cameras, transform light from the visible spectrum into an image. The difference between stereo and monocular cameras is that stereo sensors have at least two lenses with separate image sensors. The combination of the output from the different lenses can be used to generate stereo images from which depth information can be extracted. This type of sensor are especially useful on board vehicles where the distance between the objects and the vehicle is essential to estimate if a collision is likely to occur and when [@barth2010tracking]. They are also used on board of vehicles to detect lanes [@danescu2009probabilistic; @guo2012robust].
[ [**Infrared/thermal cameras:**]{} ]{} Infrared or thermal cameras are devices that detect the infrared radiation emitted by all objects with a temperature above absolute zero [@gade2014thermal]. Similarly to common cameras that transform visible light into images, thermal cameras convert infrared energy into an electronic signal which is then processed to generate a thermal image or video. This type of camera has the advantage that they eliminate the illumination problems that normal RGB or greyscale cameras have. They are of particular interest for emergency management scenarios such as fire detection since fire normally has a temperature higher than its environment.
It is also well known that humans have a body temperature that ranges between 36$^{\circ}$C and 37$^{\circ}$C in normal conditions. This fact can be exploited to detect humans, and has been used for pedestrian detection to prevent vehicle collisions with pedestrians [@suard2006pedestrian; @wang2012pedestrian], to detect faces and pupils [@kumar2009application], and for fall detection for elderly monitoring [@sixsmith2004smart]. Although not so extensively used, thermal and infrared cameras have also been used for military purposes, such as to detect firing guns [@price2004system] and to detect mines under ground [@muscio2004land; @siegel2002land; @wasaki2001smart].
One of the disadvantages of thermal cameras compared with other type of optical sensors is that their resolution is lower than many other visual sensors. For this reason some systems use thermal cameras in combination with other sensors. The authors of [@hassan2015assessment] use a thermal camera among other visual sensors to study the surface soil moisture and the water balance of the ground, which is in turn used to estimate, predict and monitor drought.
In order to obtain depth information in addition to thermal information, it is also possible to use infrared stereo cameras such as in [@bertozzi2005infrared] where the authors use infrared cameras to detect pedestrians by detecting their corporal heat, and then estimate pedestrian position using stereo information.
Acquisition rate
----------------
Depending on the rate at which optical data is acquired we can either obtain video or still images. The main difference between video and a still image is that still images are taken individually while video is an ordered sequence of images taken at a specific and known rate. To be considered a video, the rate at which images are taken must be enough for the human eye to perceive movement as continuous.
[ [**Video:**]{} ]{} The main advantage of videos over images is that, thanks to the known acquisition rate, they provide temporal information which is invaluable for some tasks. Also, most of the systems studied in this survey are monitoring systems meant to automatically detect risky situations or emergencies. As these situations are rare but could happen at any moment and it is important to detect them as soon as possible, so it is necessary to have a high acquisition rate.
For the study of many of the emergencies considered in this survey it is necessary to have temporal information. For example, merely detecting a person lying on the ground in a single image is not sufficient to infer that that person has *fallen*. Many systems distinguish between a person lying on the floor and a person that has fallen by using information about the speed at which this person reached the final posture [@lin2007automatic; @rougier2011fall]. Also, systems that estimate likelihood of a car crash must have temporal information to know at which speed the external objects are approaching the vehicle. To estimate driver drowsiness or fatigue, many systems take information into account such as the frequency at which the driver closes his eyes and for how long they remain closed [@kumar2009application].
Moreover, many algorithms developed for the detection of specific situations also require temporal information. In order to detect fire, most algorithms exploit the movement of the fire to differentiate it from the background. Similarly, for drowning person detection, many algorithms use temporal information to easily segment the person from the water [@chen2011framework; @eng2003automatic; @fei2009drowning; @lu2004vision]. Also, many falling person detection systems use tracking algorithms to track the persons in the scene [@hazelhoff2008video] or background subtraction algorithms to segment persons [@mirmahboub2013automatic].
[ [**Still images:**]{} ]{} Still images, in contrast with video, do not give information about the movement of the objects in the scene and does not provide any temporal information. Although most of the algorithms that study the problem of fire detection use temporal information and therefore require video, David Van et al. [@van2010fire] study a system that detects fire on still images to broaden the applicability of their system. In some cases due to the sensor used by the system it is not possible or convenient to obtain videos for example in systems that use satellite imagery [@hassan2015assessment; @mason2012near; @mason2010flood; @song2013drought], also in systems that use UAVs where in many cases is more convenient to take still images instead of videos in order to effectively send the data in real time or get higher resolution imagery [@castillo2005method; @soni2012classifier].
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
\
Fixed location& [ ]{}\$50 to \$2000 & [ ]{}\$200 to \$6000 & [ ]{}\$200 to \$2000 &\
& [ ]{}Up to 30 MP & [ ]{}Up to 1.2 MP & [ ]{}Up to 30 MP &\
& [ ]{}Real time data acquisition & [ ]{}Up to 0.02ºC sensitivity & [ ]{}Real time data acquisition &\
& [ ]{}Limited area of visualization & [ ]{}Real time data acquisition & [ ]{}Limited area of visualization &\
& & [ ]{}Limited area of visualization &&\
Robot or UAV & [ ]{}\$100 to \$2000 & [ ]{}\$250 to \$2000 & [ ]{}\$250 to \$2000 &\
& [ ]{}Up to 30 MP & [ ]{}Up to 1.2 MP & [ ]{}Up to 30 MP &\
&[ ]{}Portable & [ ]{}Up to 0.02ºC sensitivity & [ ]{}Portable &\
& [ ]{}Bad quality data acquitision in real time & [ ]{}Portable & [ ]{}Bad quality data acquitision in real time &\
& [ ]{}Limited autonomy & [ ]{}Bad quality data acquitision in real time & [ ]{}Limited autonomy &\
&& [ ]{}Limited autonomy &&\
Satellite & & & & [ ]{}Open data access through many research programs\
&&&& Spatial resolution of around 10m\
&&&& Can cover the entire globe\
&&&& May have to wait a few days until the satellite covers the entire globe\
Sensor cost
-----------
When implementing the final system, another important factor that should be taken into account before designing it is the cost from setting up the system. In general, the cost of the final system is going to be the cost of the visual sensor, the cost of the installation of the system and the cost of the data transmission. In this section we will do a brief study of the three sources of costs.
[**Description**]{}
-- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Example of a commonly used UAV. This UAV has an integrated camera of 12 Mpx, a flying autonomy of 28 minutes, a maximum transmission distance of 5km a live view working frequency of 2.4GHz and a live view quality of 720P and 30fps. This device can be bought on the market for around \$1200.
Example of a thermal camera that can be attached to a smart phone. It has an VGA resolution a sensibility of 0.1ºC and works with temperatures between -20ºC to 120ºC. A device if these characteristics can be bought for around \$300.
Example of a built in stereo camera of 10 MP resolution and a cost of \$600.
The most common visual sensors are monocular RGB and grayscale cameras. Since these sensors are so common and have a big demand, their technical features have improved greatly over time and their price has dramatically decreased. Therefore there exists a wide range of monocular cameras in the market with different technical features and prices. Depending on the quality of the device, its price can vary between approximately \$50 to \$2000. Although there also exists a wide variety of infrared cameras, these devices are less common and in general more expensive. Depending on the quality of the camera, its price fluctuates between \$200 and \$6000. To illustrate that, in the second row from Table \[tab:table of types of sensors\] we give an example of a thermal camera of \$300 and its specifications. Finally, stereo cameras are the less common visual sensors and there is fewer variety of these devices. Their price oscillates between \$200 and \$2000. In the third row from Table \[tab:table of types of sensors\] we present an example of an average stereo camera. However, it is fairly common to build a stereo device by attaching two monocular cameras which makes its price double the price of a monocular camera.
The cost of the installation of the system and the transmission of data varies greatly depending on the localization in which the sensor is installed. In this case, the cheapest installation would be at a fixed location, for which it would only be necessary to buy a holder to fix the sensor and acquire a broadband service in order to transmit the data in real time. On the other hand, it is also possible to install the sensor on a mobile device such as a robot or UAV. In this case it is necessary to first buy the device, whose price also have big fluctuations depending on their quality and technical characteristics such as its autonomy, dimensions and stability. A low cost robot or UAV can be bought for around \$50 while high quality ones can cost around \$20000. Moreover, these devices require a complex transmission system since the sensor is going to be in continuous movement. In fact, since these systems are meant to acquire data over a limited period of time, it is common not to install any transmission data system but saving all the data in the device and processing it after the acquisition is over. It is also possible to install a antenna on the device that transmits these data in real time over 2.4GHz or 5.8GHz frequency. In the first row from Table \[tab:table of types of sensors\] we show an example of a commonly used drone with an integrated camera and with a 2.4GHz antenna for live data transmission.
Algorithms {#sec:algorithms}
==========
In this section we review the most popular algorithms that have been applied in computer vision applications for emergency situations. We have considered the nine different types of emergencies that we introduced in Section \[sec:Definition\]. When selecting the articles to review in this survey we have considered a total of 114 papers that propose a computer vision system for some emergency management. However, not all the types of emergencies have bee equally studied in the literature nor in the survey, in Figure \[graph:types of emergencies\] we show a bar graph of the distribution of the articles among the different types of emergencies.
This section is divided in two main parts: feature extraction and machine learning algorithms. In the feature extraction part various approaches to extracting relevant information from the data are elaborated. Once that information is extracted, machine learning algorithms are applied to incorporate spatial consistency (e.g. Markov Random Fields), to model uncertainty (e.g. Fuzzy logic) and to classify the data to high-level semantic information, such as the presence of fire or an earthquake. We have focused on methods which are (or could be) applied to a wider group of applications in emergency management.
Feature extraction
------------------
Feature extraction in computer vision is the process of reducing the information in the data into a more compact representation which contains the relevant information for the task at hand. This is a common practice in computer vision algorithms related to emergency situations. We divide the features in three main groups: color, shape and temporal.
### Color features
The understanding of emergency situations often involves the detection of materials and events which have no specific spatial extent or shape. Examples include water, fire, and smoke. However, these textures often have a characteristic color, and therefore color features can provide distinctive features.
Color analysis is a well established computer vision research field and has been broadly applied in applications in emergency management. It is frequently used to detect fire and smoke since they have characteristic colors and – although not so frequently – it has also been used to detect flooded regions. Fire color may vary depending on the combustion material and the temperature. However most flames are yellow, orange and especially red. Smoke tends to be dark or white and can be more or less opaque. Water in flooded regions, according to [@borges2008probabilistic], presents a smooth dark surface or strong reflective spots much brighter than the surroundings.
One of the principle choices in color feature extraction is the color space or color model from which to compute the features. This choice influences characteristics of the color feature, including photometric invariance properties [@gevers2012color], device dependence, and whether it is perceptually uniform or not [@ibraheem2012understanding]. Here we will shortly review the most used color models and briefly describe some of their advantages and disadvantages.
The data produced by camera devices is typically in **RGB** color model. Due to the effortless availability of this color model and the separation of the red values, this color model has been often used for fire, smoke and flood detection [@borges2010probabilistic; @chen2010multi; @rinsurongkawong2012fire; @toreyin2006computer]. The main disadvantage of this color model is its lack of illumination invariance and dependence on the sensor. The **HSI** and **HSV** color models were proposed to resemble the color sensing properties of human vision and separate the luminance component from the chromatic ones. They have been applied to smoke detection [@barmpoutis2014smoke], to detect light gray water [@borges2008probabilistic] and to detect deep water [@lai2007real]. Several systems combine RGB and HSI color models [@chen2004early; @rinsurongkawong2012fire; @yu2013real]. The **YCbCr** color model has three components, one of which encodes the luminance information and two encode the chromaticity. Having the luminance component separated from chromatic components makes it robust to changes in illumination [@celik2009fire; @lai2007real]. Similarly to HSI and YCbCr, the **CIELab** color model makes it possible to separate the luminance information from the chromaticity. Moreover, it is device independent and perceptually uniform. This color model has been used to detect fire [@celik2010fast; @truong2012fire] and to improve the background models for swimming pools [@eng2003automatic].
### Shape and texture features
There are elements which are very characteristic of some of the emergencies presented in this survey, such as smoke or flames to detect fire, pedestrians to prevent traffic collisions or persons to detect victims in disaster areas. Detecting these characteristic elements is common to many of the algorithms proposed for emergency management with computer vision. The primary goal of many of these algorithms is to prevent or detect human harm and therefore humans are a common characteristic element of these emergency scenarios. The histogram of oriented gradients (HOG) descriptor was introduced to detect pedestrians and is one of the most used shape descriptors to detect humans. Among the algorithms studied in this survey, HOG was used to detect persons in disaster areas [@andriluka2010vision; @soni2012classifier; @soni2013victim], to detect pedestrians for on board vehicle cameras [@suard2006pedestrian; @wojek2009multi] but also to characterize other objects such as flames, smoke and traffic signs [@barmpoutis2014smoke; @wang2013spatial; @zaklouta2011warning] and as a descriptor for a tracking algorithm [@garate2009crowd] in crowd event recognition. Local binary patterns is also a well known shape descriptor which is used to describe textures and was successfully applied to the task of smoke detection [@tian2011smoke; @ye2015dynamic].
### Temporal features
Most of the input data used in emergency management comes in form of videos, which provide visual information as well as temporal information. Here we briefly summarize the main features which exploit the available temporal information in the data. [ [**Wavelets:**]{} ]{} Flicker is an important characteristic of smoke and flames and has been used in several systems to detect fire and smoke in video data. Fourier analysis is the standard tool for analysis of spatial frequency patterns. The extension of frequency analysis to spatial-temporal signals was first studied by Haar [@haar1910theorie] and later led to the wavelet transform [@mallat1989theory]. In [@toreyin2006computer] the authors study the periodic behaviour of the boundaries of the moving fire-colored regions. The flickering flames can be detected by finding zero crossings of the wavelet transform coefficients which are an indicator of activity within the region. In [@xu2007automatic] the flicker of smoke is detected computing the temporal high-frequency activity of foreground pixels by analysing the local wavelet energy. They also use frequency information to find smoke boundaries by detecting the decrease in high frequency content since the smoke is semi-transparent at its boundaries. In [@calderara2008smoke] the authors present an algorithm which uses a Discrete Wavelet Transform to analyse the blockwise energy of the image. The variance over time of this energy value was found to be an indicator of the presence or absence of smoke. In [@gonzalez2010wavelet] the authors perform a vertical, horizontal and diagonal frequency analysis using the Stationary Wavelet Transform and its inverse. A characterization of smoke behaviour in the wavelet domain was done in [@gubbi2009smoke] by decomposing the image into wavelets and analyzing its frequency information at various scales.
[ [**Optical flow:**]{} ]{} Optical flow estimates the motion of all the pixels of a frame with respect to the previous frame [@beauchemin1995computation; @lucas1981iterative]. The objective of algorithms based on optical flow is to determine the motion of objects in consecutive frames. Many techniques have been proposed in the literature to perform optical flow [@brox2004high; @brox2011large; @gautama2002phase], however the most widely used technique is the one proposed by Lucas-Kanade [@lucas1981iterative] with some extensions [@bouguet2001pyramidal].
Optical flow algorithms have been used in several computer vision systems for emergency management, such as for the detection of fire and smoke and anomalous situations on roads or in crowds. In the case of fire and smoke detection several approaches use the pyramidal Lucas-Kanade optical flow algorithm to determine the flow of candidate regions which likely to contain fire or smoke and to discard regions that do not present a high variation on the flow rate characteristic of the fire turbulence [@rinsurongkawong2012fire; @yu2013real]. The pyramidal Lucas-Kanade optical flow algorithm assumes that the objects in the video are rigid with a Lambertian surface and negligible changes in brightness but the drawback is that these assumptions do not hold for fire and smoke. Addressing this problem, Kolesov et al. [@kolesov2010natural] proposed a modification to Lucas-Kanade in which they assume brightness and mass conservation. Optical flow has been also used to analyse the motion of crowds in order to determine abnormal and therefore potentially dangerous situations [@andrade2006hidden; @ihaddadene2008real; @mehran2009abnormal]. In [@sultani2010abnormal] they use an optical flow technique to model the traffic dynamics of the road, however instead of doing a pixel by pixel matching they consider each individual car as a particle for the optical flow model.
[ [**Background modeling and subtraction:**]{} ]{} The separation of background from foreground is an important step in many of the applications for emergency management. Background modeling is an often applied technique for applications with static cameras (see Section \[minisec:static\]) and static background. The aim of this method is to compute a representation of the video scene with no moving objects, a comprehensive review can be seen in [@piccardi2004background].
In the context of emergency management, the method of Stauffer and Grimson [@stauffer2000learning] has been used to segment fire, smoke and persons from the background [@andrade2006modelling; @calderara2008smoke; @lu2004vision; @thome2008falling; @truong2012fire; @wang2013spatial]. Although not as popular, other background subtraction algorithms have been proposed in the literature [@collins2000system; @izadi2008robust; @kim2005real; @li2003foreground] and used in systems for emergency management [@ihaddadene2008real; @ko2010early; @liao2012slip; @shieh2012falling; @toreyin2006computer]. An interesting application of background modeling was described in [@kamijo2000traffic] in the context of abnormal event detection in emergency situations. They assume that vehicles involved in an accident or an abnormal situation remain still for a considerable period of time. Therefore an abnormal situation or accident can be detected in a surveillance video by comparing consecutive background models.
[ [**Tracking:**]{} ]{} Tracking in computer vision is the problem of locating moving objects of interest in consecutive frames of a video. The shape of tracked objects can be represented by points, primitive geometric shapes or the object silhouette [@yilmaz2006object]. Representing the object to be tracked as points is particularly useful when object boundaries are difficult to determine, for example when tracking crowds [@ihaddadene2008real] where detecting and tracking each individual person may be infeasible. It is also possible to represent the tracked objects as primitive objective shapes, for example in [@lu2004vision] the authors use ellipses to represent swimmers for the detection of drowning incidents. It is also common to represent the tracked objects as rectangles, e.g. in swimmer detection [@chen2011framework] and traffic accident detection [@kamijo2000traffic]. Finally, tracked objects can also be represented with the object’s silhouette or contour [@kam2002video].
Feature selection is crucial for tracking accuracy. The features chosen must be characteristic of the objects to be tracked and as unique as possible. In the context of fall detection, the authors of [@hazelhoff2008video] use skin color to detect the head of the tracked person. Many tracking algorithms use motion as the main feature to detect the object of interest, for example applications in traffic emergencies [@hu2006system; @kamijo2000traffic], fall detection [@jiang2013real; @liao2012slip], and drowning person detection [@lu2004vision]. Haar-like features [@viola2004robust] have been widely used in computer vision for object detection, and have been applied to swimmer detection [@chen2011framework]. Finally, descriptors such as Harris [@harris1988combined] or HOG [@dalal2005histograms] can also be useful for crowd analysis [@garate2009crowd; @ihaddadene2008real].
### Convolutional features
Recently, convolutional features have led to a very good performance in analyzing visual imagery. Convolutional Neural Networks (CNN) were first proposed in 1990 by LeCun et al. [@lecun1990handwritten]. However, due to the lack of training data and computational power, it was not until 2012 [@krizhevsky2012imagenet] that these networks started to be extensively used in the computer vision community.
CNNs have also been used in the recent years for some emergency challenges, mainly fire and smoke detection in images or video and anomalous event detection in videos. For the task of smoke detection, some researchers proposed novel networks to perform the task [@frizzi2016convolutional; @xu2017deep]. In [@frizzi2016convolutional] the authors propose a 9-layer convolutional network and they use the output of the last layer as a feature map of the regions containing fire, while in [@xu2017deep] the authors construct a CNN and use domain adaptation to alleviate the gap between their synthetic fire and smoke training data and their real test data.
Other researchers have concentrated on solutions built using existing networks [@lagerstrom2016image; @maksymiv2016deep]. In [@maksymiv2016deep] the authors retrain some existing architectures, while in [@lagerstrom2016image] the authors use already trained architectures to give labels to the images and then classify those labels into fire or not fire. For the task of anomaly detection, algorithms need to take temporal and spatial information into account. To handle this problem the authors in [@chong2017abnormal; @medel2016anomaly] propose to use a Long Short-Term Memory (LSTM) to predict the evolution of the video based on previous frames. If the reconstruction error between subsequent frames and the predictions is very high it is considered to be an anomaly. In [@zhou2016spatial] the authors consider only the regions of the video which have changed over the last period of time and send those regions through a spatial-temporal CNN. In [@sabokrou2016fully] the authors pass a pixel-wise average of every pair of frames through a 4-layered fully connected network and an autoencoder trained on normal regions.
Machine learning
----------------
In the vast majority of the applications in emergency management machine learning algorithms are applied. They are used for classification and regression to high-level semantic information, to impose spatial coherence, as well as to estimate uncertainty of the models. Here we briefly discuss the most used algorithms.
### Artificial Neural Networks
Artificial Neural Networks (ANN) have received considerable interest due to their wide range of applications and ability to handle problems involving imprecise and complex nonlinear data. Moreover, their learning and prediction capabilities combine with impressive accuracy values for a wide range of problems. A neural network is composed of a large number of interconnected nonlinear computing elements or neurons organized in an input layer, a variable number of hidden layers and an output layer. The design of a neural network model requires three steps: selection of the statistical variables or features, selection of the number of layers (really the hidden layers) and nodes and selection of transfer function. So, the different statistical variables or features, the number of nodes and the selected transfer function, make the difference between the papers found showing applications of the ANN to prevention and detection of emergencies using computer vision and image processing techniques.
Applications of ANNs to fire and smoke detection and prevention can be seen in papers [@kolesov2010natural; @xu2007smoke; @yu2013real; @yuan2011video]. They mainly differ in the characteristics of the network. In [@xu2007smoke] the authors extract features of the moving target which are normalized and fed to a single-layer artificial neural network with five inputs, one hidden layer with four processing elements, and one output layer with one processing element to recognize fire smoke. In [@yu2013real] they propose a neural network to classify smoke features. In [@kolesov2010natural] they implement an ANN fire/smoke pixel classifier. The classifier is implemented as a single-hidden-layer neural network with twenty hidden units, with a softmax non-linearity in the hidden layer. The method proposed in [@yuan2011video] describes a neural network classifier with a hidden layer is trained and used for discrimination of smoke and non-smoke objects. The output layer uses a sigmoid transfer function.
In the system presented in [@alhimale2014fallin] a neural network of perceptrons was used to classify falls against the set of predefined situations. Here, the neural network analyses the binary map image of the person and identifies which plausible situation the person is at any particular instant in time.
In general, the usage of ANNs provide a performance improvement compared to other learning methods [@chunyu2010smoke]. The results rely highly on the selected statistical values for training. If the selection of statistical values is not appropriate, the results on the accuracy might be lower and simpler machine learning algorithm may be preferred.
[ [**Deep Learning**]{} ]{} ANN algorithms that consist of more than one hidden layer are usually called *deep* learning algorithms. Adding more layers to the network increases the complexity of the algorithm making it more suitable to learn more complex tasks. However, increasing the layers of the network comes at a higher computational cost and a higher probability to overfit or converge to a local minima. Nevertheless, the advances in computers and the increase in training data has facilitated the research towards deeper models which have proven to improve the results.
In [@chunyu2010smoke] the authors propose a multi-layered ANN for smoke features classification. In the works [@alhimale2014fallin; @foroughi2008falling; @juang2007falling; @sixsmith2004smart] we find applications of ANNs to prevent and detect single-person emergencies. A neural network is used to classify falls in [@sixsmith2004smart] using vertical velocity estimates derived either directly from the sensor data or from the tracker. The network described in [@juang1998neural], with four hidden layers, is used in the classifier design to do posture classification that can be used to determine if a person has fallen. In [@foroughi2008falling] the authors monitor human activities using a multilayer perceptron with focus on detecting three types of fall.
### Support Vector Machines
Support vector machines (SVMs) are a set of supervised machine learning algorithms based on statistical learning and risk minimization which are related to classification and regression problems. Given a set of labeled samples a trained SVM creates an hyperplane or a set of them, in a high dimensional space, that optimally separates the samples between classes maximizing the separation margin among classes. Many computer vision systems related to emergency situations can be considered as a binary classification problems: fire or no fire, fall or no fall, crash or no crash. To solve this classification problem many researches make use of SVMs [@mirmahboub2013automatic; @suard2006pedestrian; @tian2011smoke; @truong2012fire] based on the classification performance that they can reach, their effectiveness to work on high dimensional spaces and their ability to achieve good performance with few data.
In more complex systems where it is difficult to predict the type of emergencies that may occur, for example in systems that supervise crowded areas or traffic, algorithms that consider abnormal behaviours as potential emergencies have been proposed [@piciarelli2008trajectory; @wang2012histograms]. A difficulty of these kind of systems is that they can only be trained with samples of one class (normal scenes). For these kind of problems one-class SVMs, which find a region of a high dimensional space that contains the majority of the samples, are well suited.
### Hidden Markov Models
Hidden Markov Models (HMMs), which are especially known for their application in temporal pattern recognition such as speech, handwriting, gesture and action recognition [@yamato1992recognition], are a type of stochastic state transition model [@rabiner1986introduction]. HMMs make it possible to deal with time-sequence data and can provide time-scale invariability in recognition. Therefore it is one of the preferred algorithms to analyze emergency situations subject to time variability. To detect smoke regions from video clips, a dynamic texture descriptor is proposed in [@ye2015dynamic]. The authors propose a texture descriptor with Surfacelet transformation ([@lu2007surfacelets]) and Hidden Markov Tree model (HMT). Examples of use of Markov models to study human-caused emergencies that affect a single person can be seen in the following works [@chen2010hidden; @eng2003automatic; @jiang2013real; @thome2008falling; @toreyin2006falling], covering fall detection and detection behavior of a swimmer and drownings. Hidden Markov Models have been applied on monitoring emergency situations in crowds by learning patterns of normal crowd behaviour in order to identify unusual or emergency events, as can be seen in [@andrade2006hidden; @andrade2006detection]. In these two works Andrade and his team presents similar algorithms in order to detect emergency events in crowded scenarios, and based on optical flow to extract information about the crowd behaviour and the Hidden markov Models to detect abnormal events in the crowd.
Töreyin and his team in [@toreyin2006falling] use Hidden Markov Models to recognize possible human motions in video. Also they use audio track with a three-state HMM to distinguish a person sitting on a floor from a person stumbling and falling. In [@thome2008falling] the motion analysis is performed by means of a layered hidden markov model (LHMM). A fall detection system based on motion and posture analysis with surveillance video is presented in [@jiang2013real]. The authors propose a context-based fall detection system by analyzing human motion and posture using a discrete hidden Markov model. The problem of recognition of several specific behaviours of a swimmer (including backstroke, breaststroke, freestyle and butterfly, and abnormal behaviour) in a swimming pool using Hidden Markov Models is addressed in [@chen2010hidden]. The algorithm presented in [@eng2003automatic] detects water crises within highly dynamic aquatic environments where partial occlusions are handled using a Markov Random Field framework.
Hidden Markov Models have been applied on monitoring emergency situations in crowds by learning patterns of normal crowd behavior in order to identify unusual or emergency events, as can be seen in [@andrade2006hidden; @andrade2006detection]. In these two works Andrade and his team presents similar algorithms in order to detect emergency events in crowded scenarios, and based on optical flow to extract information about the crowd behavior and the Hidden Markov Models to detect abnormal events in the crowd.
### Fuzzy logic
Fuzzy sets theory and fuzzy logic are powerful frameworks for performing automated reasoning, and provide a principled approach to address the inherent uncertainty related to modeling; an aspect which relevant in many emergency management application. Fuzzy set theory, originally introduced by Lofti A. Zadeh in 1965 [@Zad65], is an extension of classical set theory where to each element $x$ from a set $A$ is associated with a value between $0$ and $1$, representing its degree of belonging to the set $A$. One important branch of the fuzzy set theory is fuzzy logic [@Zad73]. An inference engine operators on rules that are structured in an IF-THEN rules, are presented. Rules are constructed from linguistic variables. A fuzzy approach has the advantage that the rules and linguistic variables are understandable and simplify addition, removal, and modification of the system’s knowledge.
In the papers [@anderson2009falling] and [@thome2008falling] the authors proposed a multi-camera video-based methods for monitoring human activities, with a particular interest to the problem of fall detection. They use methods and techniques from fuzzy logic and fuzzy sets theory. Thome et al. [@thome2008falling] apply a fuzzy logic fusion process that merges features of each camera to provide a multi-view pose classifier efficient in unspecified conditions (viewpoints). In [@anderson2009falling], they construct a three-dimensional representation of the human from silhouettes acquired from multiple cameras monitoring the same scene, called voxel person. Fuzzy logic is used to determine the membership degree of the person to a pre-determined number of states at each image. A method, based on temporal fuzzy inference curves representing the states of the voxel person, is presented for generating a significantly smaller number of rich linguistic summaries of the humans state over time with the aim to detect of fall detection.
Another problem addressed with methods and techniques from theory of fuzzy sets is fire prevention by smoke detection. It has been observed that the bottom part of a smoke region is less mobile than top part over time; this is called swaying motion of smoke. In [@wang2014smoke] a method to detect early smoke in video is presented, using swaying motion and a diffusion feature. In order to realized data classification based on information fusion, Choquet fuzzy integral (see [@grabisch2000fuzzy]) was adopted to extract dynamic regions from video frames, and then, a swaying identification algorithm based on centroid calculation was used to distinguish candidate smoke region from other dynamic regions. Finally, Truong et al. [@truong2012fire] presented a four-stage smoke detection algorithm, where they use a fuzzy c-means algorithm method to cluster candidate smoke regions from the moving regions.
Algorithmic trends in emergency management {#sec:trends}
------------------------------------------
-- -- --
-- -- --
To analyze the usage of the discussed algorithms across emergency types, we provide a more in depth analysis of the three most studied emergencies: fire, road accidents, and human fall. In Figure \[fig:graphs\](a-c) we break down the algorithms that have been used for these three types of emergencies. As can be seen in the figures, for the three types of emergencies, the most commonly applied algorithms are the ones that use temporal features. For fire and human fall management background subtraction is the most popular algorithm since most times the most discriminative feature of the objects of interest in this scenarios move on top of a static background. In the case of road accident management it is common to use non-static cameras (such as cameras on board a vehicle) and therefore the background is not static and background subtraction algorithms are not useful. However, temporal features are used to perform optical flow or tracking in order to determine the trajectory of the objects in the road or close by. For fire color features are very distinctive and therefore more researched.
Discussion
==========
There are many video monitoring systems whose aim is to monitor various types of emergency scenarios. Since continuous human vigilance of these recordings is infeasible, most of this data is used *a posteriori*. Although this information is very useful for the later understanding of the event or as an incriminatory proof, many researchers study the possibility of exploiting these huge amounts of data and analysing them in real time with the hope of preventing some of these emergencies or to facilitate a faster or more efficient response. In this survey we reviewed and analyzed the progress of computer vision covering many types of emergencies with the goal of unveiling the overlap across emergency research in computer vision, especially in the hardware and algorithms developed, making it easier to researchers to find relevant works. In order to restrict the scope of this survey we first gave a description of what we consider emergency situations. Taking into account the description provided, we restrict ourselves to emergency scenarios that have been studied in the context of computer vision. Taking all these considerations into account we identified eleven different emergency scenarios, studied in computer vision, which we classified into natural emergencies and emergencies caused by humans. After a thorough analysis of the state-of-the-art algorithms on these topics, we inferred that there are four main axes that have to be taken into account when creating computer vision systems to help in emergency scenarios: the type of emergency that is going to be addressed, the monitoring objective, the hardware and sensors needed, and the algorithms used.
When talking about the monitoring objective, we have seen that prevention through computer vision techniques can only be done when there is visual evidence of risk factors such as for fire detection using smoke features, car accidents by detecting pedestrians, driver drowsiness and unusual behaviours, and finally fall prevention falls by detecting slips. The most common objective is emergency detection, which has been studied in all the emergencies studied in this survey. It is also common to use computer vision to detect humans in disaster areas to assist emergency responders. Finally, computer vision has not played a role from the emergency understanding point of view.
From all the acquisition methods studied in Section \[sec:acquisition\] we have concluded that the most common sensors are monocular RGB or grayscale cameras – mainly for their low cost – and that they are normally situated in fixed locations. However, we see a trend in the use of UAVs which are becoming cheaper and provide great flexibility through their mobility.
Finally, as discussed in Section \[sec:trends\], it is very difficult to draw conclusions on algorithmic trends for every emergency since some of them have little impact in computer vision. However, there is a clear tendency towards using color and temporal features for fire emergencies, a clear preference for temporal features in road accidents, and tracking algorithms and background subtraction techniques are the most common when detecting falling person events.
This up-to-date survey will be important for helping readers obtain a global view. At the same time they can find novel ideas of how computer vision can be helpful to prevent, respond and recover from emergency scenarios, and learn about some required hardware and relevant algorithms that are developed to solve these problems.
It is extremely difficult to compare different algorithms proposed in the literature for solving similar problems due to the lack of unified benchmarks. From all the papers studied during this review, over 60% used non-publicly available datasets. In order to facilitate a common benchmark to compare algorithms, in Table \[tab:datasets\] we report some of the most important publicly available datasets used among the emergencies studied in this survey. Note that some pedestrian datasets are shared among emergencies since detecting humans is considered useful in more than one emergency.
[**Public datasets**]{} [**General comments**]{}
-- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------
[VISOR smoke dataset](http://imagelab.ing.unimore.it/visor/video_videosInCategory.asp?idcategory=-1) Videos with evidence of smoke.
[Fire dataset from the Bilkent University](http://signal.ee.bilkent.edu.tr/VisiFire/) Videos with evidence of smoke and fire.
[MESH database of news content](http://www.mesh-ip.eu) Instances of catastrophe related videos from the Deutsche Welle broadcaster with several news related to fires in a variety of conditions.
[FASTData](http://fire.nist.gov/fastdata) Collection of fire images and videos.
[CAVIAR data](http://homepages.inf.ed.ac.uk/rbf/CAVIAR/)
[VISOR smoke dataset](http://imagelab.ing.unimore.it/visor/)
[Fire and smoke dataset](http://signal.ee.bilkent.edu.tr/VisiFire/Demo/SampleClips.html) Clips containing evidence of fire and smoke in several scenarios.
[Fire and smoke dataset](http://signal.ee.bilkent.edu.tr/VisiFire/Demo/SmokeClips/) Fire and smoke clips.
[CIRL fall detection dataset](http://www.derektanderson.com/fallrecognition/datasets.html) Indoor videos for action recognition and fall detection, not available for comercial purposes.
[MMU fall detection dataset](http://foe.mmu.edu.my/digitalhome/FallVideo.zip)
[Multiple camera fall dataset](http://www.iro.umontreal.ca/~labimage/Dataset/) 24 videos of falls and fall confounding situations recorded with 8 cameras in different angles.
[NICTA dataset](https://research.csiro.au/data61/automap-datasets-and-code/) Contains a total of 25551 unique pedestrians.
[MIT pedestrian dataset](http://cbcl.mit.edu/software-datasets/PedestrianData.html) 924 images of 64x128 containing pedestrians.
[MIT pedestrian dataset](http://cbcl.mit.edu/software-datasets/PedestrianData.html) 924 images of 64x128 containing pedestrians.
[INRIA pedestrian dataset](http://pascal.inrialpes.fr/data/human/) Images of pedestrians and negative images.
[CAVIAR dataset](http://groups.inf.ed.ac.uk/vision/CAVIAR/CAVIARDATA1/) Sequences containing pedestrians, also suitable for action recognition.
[Trajectory based anomalous event detection](http://ieeexplore.ieee.org/document/4633642/) Synthetic and real-world data.
[The German Traffic Sign detection benchmark](http://benchmark.ini.rub.de/?section=gtsdb&subsection=dataset) 900 images of roads with traffic sign ground truth.
[Thermal pedestrian database](http://vcipl-okstate.org/pbvs/bench/) 10 sequences containing pedestrians recorded with a thermal camera.
[TUD Brussels and TUD Paris](https://www.mpi-inf.mpg.de/departments/computer-vision-and-multimodal-computing/research/people-detection-pose-estimation-and-tracking/multi-cue-onboard-pedestrian-detection/) Images of pedestrians taken by on-board cameras.
[Simulation of Crowd Problems for Computer Vision](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.59.8399&rep=rep1&type=pdf) Approach for generating video evidence of dangerous situations in crowded scenes.
[PETS dataset](http://www.cvg.reading.ac.uk/PETS2009/a.html) Sequences containing different crowd activities.
[Unusual crowd activity dataset](http://mha.cs.umn.edu/Movies/Crowd-Activity-All.avi) Clip with unusual crowd activities.
[USCD anomaly detection dataset](http://www.svcl.ucsd.edu/projects/anomaly/dataset.htm) Clips of the street with stationary camera.
[UMN monitoring human activity dataset](http://mha.cs.umn.edu/proj_recognition.shtml#crowd_count) Videos of crowded scenes.
[**Type of emergency**]{} [**Public datasets**]{} [**General comments**]{} [**Url**]{}
--------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------
[VISOR smoke dataset](http://imagelab.ing.unimore.it/visor/video_videosInCategory.asp?idcategory=-1) Videos with evidence of smoke. <http://imagelab.ing.unimore.it/visor>
[Fire dataset from the Bilkent University](http://signal.ee.bilkent.edu.tr/VisiFire/) Videos with evidence of smoke and fire. <http://signal.ee.bilkent.edu.tr/VisiFire/>
[MESH database of news content](http://www.mesh-ip.eu/?Page=Project) Instances of catastrophe related videos from the Deutsche Welle broadcaster with several news related to fires in a variety of conditions. <http://www.mesh-ip.eu/>
[FASTData](http://fire.nist.gov/fastdata) Collection of fire images and videos. <http://fire.nist.gov/fastdata>
[Fire and smoke dataset](http://signal.ee.bilkent.edu.tr/VisiFire/Demo/SampleClips.html) Clips containing evidence of fire and smoke in several scenarios. <http://signal.ee.bilkent.edu.tr/VisiFire/Demo>
[CIRL fall detection dataset](http://www.derektanderson.com/fallrecognition/datasets.html) Indoor videos for action recognition and fall detection, not available for comercial purposes. <http://www.derektanderson.com/fallrecognition/datasets.html>
[MMU fall detection dataset](http://foe.mmu.edu.my/digitalhome/FallVideo.zip) 20 indoor videos including 38 normal activities and 29 different falls <http://foe.mmu.edu.my/digitalhome/FallVideo.zip>
[Multiple camera fall dataset](http://www.iro.umontreal.ca/~labimage/Dataset/) 24 videos of falls and fall confounding situations recorded with 8 cameras in different angles. <http://www.iro.umontreal.ca/~labimage/Dataset/>
[NICTA dataset](https://research.csiro.au/data61/automap-datasets-and-code/) Contains a total of 25551 unique pedestrians. <https://research.csiro.au/data61/automap-datasets-and-code>
[MIT pedestrian dataset](http://cbcl.mit.edu/software-datasets/PedestrianData.html) 924 images of 64x128 containing pedestrians. <http://cbcl.mit.edu/software-datasets/PedestrianData.html>
[MIT pedestrian dataset](http://cbcl.mit.edu/software-datasets/PedestrianData.html) 924 images of 64x128 containing pedestrians. <http://cbcl.mit.edu/software-datasets/PedestrianData.html>
[INRIA pedestrian dataset](http://pascal.inrialpes.fr/data/human/) Images of pedestrians and negative images. <http://pascal.inrialpes.fr/data/human/>
[CAVIAR dataset](http://groups.inf.ed.ac.uk/vision/CAVIAR/CAVIARDATA1/) Sequences containing pedestrians, also suitable for action recognition. <http://groups.inf.ed.ac.uk/vision/CAVIAR>
[Trajectory based anomalous event detection](http://ieeexplore.ieee.org/document/4633642/) Synthetic and real-world data. <http://ieeexplore.ieee.org/document/4633642/>
[The German Traffic Sign detection benchmark](http://benchmark.ini.rub.de/?section=gtsdb&subsection=dataset) 900 images of roads with traffic sign ground truth. <http://benchmark.ini.rub.de>
[Thermal pedestrian database](http://vcipl-okstate.org/pbvs/bench/) 10 sequences containing pedestrians recorded with a thermal camera. <http://vcipl-okstate.org/pbvs/bench/>
[TUD Brussels and TUD Paris, Multi-Cue Onboard Pedestrian Detection](https://www.mpi-inf.mpg.de/departments/computer-vision-and-multimodal-computing/research/people-detection-pose-estimation-and-tracking/multi-cue-onboard-pedestrian-detection/) Images of pedestrians taken by on-board cameras. <ttps://www.mpi-inf.mpg.de/departments/computer-vision-and-multimodal-computing/research/people-detection-pose-estimation-and-tracking>
[Simulation of Crowd Problems for Computer Vision](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.59.8399) Approach for generating video evidence of dangerous situations in crowded scenes. <https://www.icsa.inf.ed.ac.uk/publications/online/0493.pdf>
[PETS dataset](http://www.cvg.reading.ac.uk/PETS2009/a.html) Sequences containing different crowd activities. <http://www.cvg.reading.ac.uk/PETS2009/a.html>
[Unusual crowd activity dataset](http://mha.cs.umn.edu/Movies/Crowd-Activity-All.avi) Clip with unusual crowd activities. <http://mha.cs.umn.edu/Movies/Crowd-Activity-All.avi>
[USCD anomaly detection dataset](http://www.svcl.ucsd.edu/projects/anomaly/dataset.htm) Clips of the street with stationary camera. <http://www.svcl.ucsd.edu/projects/anomaly/dataset.htm>
[UMN monitoring human activity dataset](http://mha.cs.umn.edu/proj_recognition.shtml#crowd_count) Videos of crowded scenes. <http://mha.cs.umn.edu/proj_recognition.shtml#crowd_count>
In general, we have seen a tendency to create more complex systems that combine different algorithms or algorithms of a higher computational complexity which in many cases could not have been feasible with the previous generations of computers. Among these computationally complex algorithms, the emerging techniques that stand out are those based on Convolutional Neural Networks (CNN) and Deep Learning [@gope2016deep; @kang2016deep; @zhang2016deep]. The field of computer vision is experiencing rapid change since the rediscovery of CNNs. These algorithms have already been introduced in some emergency scenarios studied in this survey and we believe that the impact of CNNs will grow over the years in the field of emergency management with computer vision. Moreover, with the decreasing cost of visual acquisition devices such as infrared cameras or stereo cameras, there is a trend of combining different information sources which can help to increase accuracy rates and decrease false alarms. We also believe that, with the help of this survey and further study of the overlap that exists between the algorithms developed to detect and study different types of emergency situations, it will be possible in the future to create system that can cover more than one type of emergency.
\[sec:discussion\]
[^1]: This work was partially supported by the Spanish Grants TIN2016-75404-P AEI/FEDER, UE, TIN2014-52072-P, TIN2013-42795-P and the European Commission H2020 I-REACT project no. 700256. Laura Lopez-Fuentes benefits from the NAERINGSPHD fellowship of the Norwegian Research Council under the collaboration agreement Ref.3114 with the UIB. We thank the NVIDIA Corporation for support in the form of GPU hardware.
|
---
abstract: 'The newly proposed panoptic segmentation task, which aims to encompass the tasks of instance segmentation (for things) and semantic segmentation (for stuff), is an essential step toward real-world vision systems and has attracted a lot of attention in the vision community. Recently, several works have been proposed for this task. Most of them focused on unifying two tasks by sharing the backbone but ignored to highlight the significance of fully interweaving features between tasks, such as providing the spatial context of objects to both semantic and instance segmentation. However, being well aware of locations of objects is fundamental to many vision tasks, e.g., object detection, instance segmentation, semantic segmentation. In this paper, we propose object spatial information flows, which can bridge all tasks together by delivering the spatial context from the box regression task to others. Based on these flows, we present a location-aware and unified framework for panoptic segmentation, [*SpatialFlow*]{}. The spatial information flows in [*SpatialFlow*]{} can provide clues for segmenting both things and stuff and help networks better understand the whole image. Moreover, instead of endowing Mask R-CNN with a stuff segmentation branch on a shared backbone, we design four parallel sub-networks for sub-tasks, which facilitate the feature integration among different tasks. We perform a detail ablation study on MS-COCO and Cityscapes panoptic benchmarks. Extensive experiments show that SpatialFlow achieves state-of-the-art results and can boost the performance of things and stuff segmentation at the same time.'
author:
- |
Qiang Chen^1,2^, Anda Cheng^1,2^, Xiangyu He^1,2^, Peisong Wang^1^, Jian Cheng^1,2^\
^1^NLPR, Institute of Automation, Chinese Academy of Sciences\
^2^University of Chinese Academy of Sciences\
{qiang.chen, xiangyu.he, peisong.wang, jcheng}@nlpr.ia.ac.cn\
chenganda2017@ia.ac.cn
bibliography:
- 'formatting-instructions-latex-2020.bib'
title: 'SpatialFlow: Bridging All Tasks for Panoptic Segmentation '
---
Introduction
============
![An illustration of the panoptic segmentation task. We also provide the box for each object.[]{data-label="fig0"}](panoptic){width="47.00000%"}
Real-world vision systems, such as autonomous driving or augmented reality, require a rich and complete understanding of the scene. Scene understanding is a classic vision task that has been studied in pre-deep learning era [@detectorcrfs; @wholescene]. As the results of instance and semantic segmentation have been rapidly improved by deep learning over a short period of time and to drive deep vision algorithms a step further toward real-world, [@panopticsegmentation] proposed the panoptic segmentation task to unify instance and semantic segmentation, as illustrated in Figure \[fig0\]. In this task, countable objects, such as people, animals, tools, are considered as [*things*]{}, and amorphous regions of similar texture or material, such as grass, sky, road, are referred to [*stuff*]{}. In [@panopticsegmentation], the authors used two independent models, Mask R-CNN [@maskrcnn] and PSPNet [@pspnet], for things and stuff segmentation[^1] respectively, then they used a heuristic post-processing method to merge the two outputs. Recently, several unified frameworks [@jsisnet; @tascnet; @panopticfpn; @aunet; @upsnet; @panopticranking] had been proposed for panoptic segmentation. Most of them focused on unifying two tasks by sharing the backbone but ignored to highlight the significance of interweaving features between tasks, such as providing the spatial context of objects to both things and stuff segmentation.
However, being well aware of locations of objects is fundamental to many vision tasks, e.g., object detection [@fasterrcnn; @rfcn; @retinanet], instance segmentation [@maskrcnn; @retinamask], semantic segmentation [@deeplab; @deeplabv3; @pspnet]. As a combination of these tasks, panoptic segmentation can benefit from interweaving spatial features between sub-tasks. To fully leverage the reciprocal relationship among detection, things segmentation, and stuff segmentation, we carefully consider two main aspects when designing a new unified framework for panoptic segmentation task.
[*First, utilizing the underlying relationship in spatial dimension among tasks*]{}. As shown in Figure \[fig0\] (b), all sub-tasks in panoptic segmentation are related to locations of objects: object detection aims to localize and recognize objects; things segmentation focuses on predicting a segmentation mask for each instance relying on the box location predicted by detectors; stuff segmentation assigns class labels to the pixels which are outside of objects in the image. Based on mentioned above, we can build a global view of image segmentation by considering locations of objects.
[*Second, integrating features of different tasks in pixel-level*]{}. Although things and stuff segmentation are complementary tasks, their activate features are inconsistent - things segmentation is dominated by instance-level features, while stuff segmentation is influenced by pixel-level features. There is a gap in directly integrating the features of two tasks. More importantly, instances can be overlapping [@panopticsegmentation], which makes it hard to map instance-level features back to pixel-level. Fortunately, the features are in the format of pixel-level before they are cropped by RoIAlign [@maskrcnn] or RoIPool [@fastrcnn] layer in things segmentation. It is intuitive to implement feature integration between two tasks in pixel-level.
To this end, we present a location-aware and unified framework for panoptic segmentation, named [*SpatialFlow*]{}. We propose object spatial information flows to bridge all tasks by delivering the spatial context from box regression task to others. Moreover, instead of endowing Mask R-CNN [@maskrcnn] with a stuff segmentation branch, our SpatialFlow extends RetinaNet [@retinanet]. To strengthen the information flow among tasks, we design four parallel sub-networks for sub-tasks, as demonstrated in Figure \[fig1\]. The overall design fully leverages spatial context and interweaves all tasks by integrating features among them, leading to better refinement of features, more robust representations for image segmentation, and better prediction results.
SpatialFlow proves its competitive performance through comparison with the state-of-the-art on MS-COCO [@coco] and Cityscapes [@cityscapes] benchmarks. Also, we perform a detail ablation study on various components; extensive experimental results show that the components of SpatialFlow can boost the performance of things and stuff segmentation at the same time.
Related Works
=============
#### Instance Segmentation.
Instance segmentation, also named as things segmentation in this paper, is a task that requires a pixel-level mask for each instance. Existing methods can be divided into two main categories, segmentation-based and region-based methods. Segmentation-based approaches, such as [@instancelevelseg; @categorylevelinstancelevelseg], first generate a pixel-level segmentation map over the image and then perform grouping to identify object instances. While region-based methods, such as [@maskrcnn; @panet; @htc], are closely related to object detection algorithms. They predict the instance masks within the bounding boxes generated by detectors. Region-based methods can achieve higher performance than their segmentation-based counterparts, which motivate us to resort to the region-based methods. In SpatialFlow, we add a mask branch on RetinaNet for instance segmentation.
#### Semantic Segmentation.
Fully convolutional networks are essential to semantic segmentation [@fcn], and its variants achieve state-of-the-art results on various benchmarks. It has been proven that contextual information plays a vital role in segmentation [@contextrole]. A bunch of works followed this idea: Dilated convolution [@dilatedconv] was invented to keep feature resolution and maintain contextual details; Deeplab series [@deeplab; @deeplabv3] proposed Atrous Spatial Pyramid Pooling (ASPP) to capture global and multi-scale contextual information; PSPNet [@pspnet] used spatial pyramid pooling to collect contextual priors; the encoder-decoder networks [@unet; @learningdeconv] are designed to capture contextual information in encoder and gradually recover the details in decoder. Our SpatialFlow, built upon FPN [@fpn], uses an encoder-decoder architecture for stuff segmentation to capture the contextual information. We take the spatial context of object detection into consideration and build a connection for things and stuff segmentation.
#### Panoptic Segmentation.
The panoptic segmentation task was proposed in [@panopticsegmentation], where the authors provided a baseline method with two separate networks, then used a heuristic post-processing method to merge two outputs. Later, Li et al. [@weaklypanoptic] followed this task and proposed a weakly- and semi-supervised panoptic segmentation method. Recently, a number of unified frameworks had been proposed: De Geus et al. [@jsisnet] used a shared backbone for both things and stuff segmentation; Li et al. [@tascnet] proposed a unified network named TASCNet by considering things and stuff consistency; Li et al. [@aunet] proposed AUNet which utilized the attention module to capture the complementary information delivered by things and stuff; Kirillov et al. [@panopticfpn] introduced PanopticFPN by endowing Mask R-CNN [@maskrcnn] with a stuff branch; Liu et al. [@panopticranking] designed a spatial ranking module to deal with the occlusion problem between the predicted instances; Xiong et al. [@upsnet] proposed a parameter-free panoptic head to resolve the conflicts between things and stuff segmentation; Yang et al. [@deeperlab] present a single-shot, bottom-up approach for panoptic segmentation. However, most of these methods ignored to highlight the significance of interweaving features between tasks. Our SpatiaFlow helps build a panoptic view for image segmentation by information flows.
{width="90.00000%"}
SpatialFlow
===========
We present a location-aware and unified framework for panoptic segmentation. The sketch of our network is illustrated in Figure \[fig1\]. The SpatialFlow can be divided into three parts: a backbone with FPN, four parallel sub-networks, and four task-specific heads. In this section, we describe the details of the major components in each part.
Backbone
--------
The backbone contains FPN, whose outputs are five levels of features named $\{P_3, P_4, P_5, P_6, P_7\}$ with a downsample rate of $8$, $16$, $32$, $64$, $128$ respectively. In FPN, all features have $256$ channels. The details are shown in the blue dashed rectangle of Figure \[fig2\], $P_6$ is obtained via a $3 \times 3$ stride-$2$ convolution on $C_5$, and $P_7$ is computed by applying ReLU followed by a $3 \times 3$ stride-$2$ conv on $P_6$. Following [@retinamask], we treat these features differently against various tasks: we use all the five levels to predict the bounding boxes in detection but only send $\{P_3, P_4, P_5\}$ to mask and stuff sub-networks.
Parallel sub-networks
---------------------
#### RetinaNet-based framework.
To deal with the panoptic segmentation task, we begin with RetinaNet. We first add a mask head for things segmentation and tackle stuff segmentation with a stuff head. Under this assumption, there are only two sub-networks - classification sub-network (cls sub-net for short) and regression sub-network (reg sub-net for short) - in this part. The operations in the sub-networks, which transform the features of FPN to the inputs of downstream heads, can be formulated as follows: $$\label{eq1}
\begin{split}
&P_{reg_{i, j}} = \zeta(P_{reg_{i, j-1}}), \quad P_{cls_{i, j}} = \zeta(P_{cls_{i, j-1}}); \\
&P_{reg_{i, 0}} = P_i, \quad P_{cls_{i, 0}} = P_i; \\
&P_{mask_i} = P_i, \quad P_{stuff_i} = P_i.
\end{split}$$ Here, $i$ represents the level index of feature, $P_i$ is the $i$-th level output of FPN, and $i \in \{3, 4, 5, 6, 7\}$ in reg sub-net and cls sub-net while $i \in \{3, 4, 5\}$ for mask and stuff; $j$ indicates the index of feature map in sub-networks, and $j \in \{1, 2, 3, 4\}$ in reg sub-net and cls sub-net; $\zeta()$ denotes a block that contains a convolution and a ReLU layer.
#### Stuff and Mask sub-networks.
In the architecture discussed above, there is no direct path for the spatial information to flow across tasks, which prevents further improvements on both things and stuff segmentation. In SpatialFlow, we consider the spatial information for all tasks. To improve the information flow, we design a stuff sub-network, which is parallel to reg sub-net and consists of four Conv-ReLU blocks. Moreover, to avoid the inconsistency between instance-level and pixel-level features, we propose to add a mask sub-network with one Conv-ReLU block. The integrations between features are in pixel-level. Until now, between the FPN and the task-specific heads, there are four parallel sub-networks. We present the modifications in architecture below: $$\label{eq2}
\begin{split}
&P_{reg_{i, j}} = \zeta(P_{reg_{i, j-1}}), \quad P_{cls_{i, j}} = \zeta(P_{cls_{i, j-1}}); \\
&P_{mask_{i, 1}} = \zeta(P_i), \quad P_{stuff_{i, j}} = \zeta(P_{stuff_{i, j-1}}); \\
&P_{reg_{i, 0}} = P_i, \quad P_{cls_{i, 0}} = P_i, \quad P_{stuff_{i, 0}} = P_i.
\end{split}$$ where $j \in \{1, 2, 3, 4\}$ in stuff sub-net. We directly use $P_{mask_{i, 1}}$ to represent the output feature of mask sub-net as there is only one Conv-ReLU block.
{width="100.00000%"}
#### Spatial information flow.
We carefully consider two main aspects to make all tasks location-aware: (1) Being well aware of object locations is fundamental to many vision tasks [@maskrcnn], such as object detection, instance segmentation, semantic segmentation. (2) All sub-tasks in panoptic segmentation are related to the locations of objects. With the help of the spatial context, the features can be more discriminative, which further boosts the performance. Furthermore, things and stuff segmentation are complementary tasks, which indicates that the semantic feature in the stuff sub-net will be a benefit to the mask segmentation by providing additional context.
Thus, we propose a four parallel sub-networks design, then add the spatial information flow to all tasks and deliver the stuff semantic feature to mask sub-net, as shown in the green dashed box of Figure \[fig2\]. We obtain a final version of our sub-networks, which can be implemented as follows: $$\label{eq3}
\begin{split}
&P_{reg_{i, j}} = \zeta(P_{reg_{i, j-1}}), \quad P_{cls_{i, j}} = \zeta(P_{cls_{i, j-1}} + \phi(P_{reg_{i, j}})); \\
&P_{stuff_{i, j}} = \zeta(P_{stuff_{i, j-1}} + \phi(P_{reg_{i, j}})); \\
&P_{mask_{i, 1}} = \zeta(P_i + \phi(P_{reg_{i, 4}}) + \psi(P_{stuff_{i, 4}})); \\
&P_{reg_{i, 0}} = P_i, \quad P_{cls_{i, 0}} = P_i, \quad P_{stuff_{i, 0}} = P_i.
\end{split}$$ Here, $\phi()$ denotes an adaptation convolution from box regression task to others; $\psi()$ denotes a adaption convolution from stuff sub-net to mask sub-net. We use a $3 \times 3$ convolution layer for both $\phi()$ and $\psi()$. To generate $P_{mask_{i, 1}}$, we deliver $P_{reg_{i, 4}}$ and $P_{stuff_{i, 4}}$, which are the last features of reg and stuff sub-net, to the feature $P_i$ in the mask sub-net. All features have 256 channels in this part.
Task-specific heads
-------------------
As illustrated in the orange dashed box of Figure \[fig1\], we use four heads for box classification, box regression, things segmentation, and stuff segmentation respectively. To generate the final output of SpatialFlow, we first obtain the detection results by considering the outputs of reg head and cls head jointly, then make segmentation mask predictions for all instances based on the predicted boxes; at the same time, we generate stuff segmentation map by applying stuff head; finally, we implement a heuristic post-processing method [@panopticfpn] to merge the things and stuff segmentation results.
In cls and reg head, we apply a $3 \times 3$ convolution to the outputs of cls and reg sub-nets. In mask head, we adopt the same design as Mask R-CNN [@maskrcnn], which consists of a RoIAlign layer, four consecutive $3 \times 3$ convolutions, a $3 \times 3$ transposed convolution and a $1 \times 1$ prediction convolution. The stuff head is shown in the orange dashed box of Figure \[fig2\]. After stuff sub-net, we obtain three feature maps with scales of $1/8$, $1/16$, $1/32$ of the original image. We perform upsampling on each feature map gradually by blocks, each of which contains a $3 \times 3$ convolution, a group norm [@groupnorm] layer, a ReLU layer and a $2\times$ bilinear upsampling operation. All the features are upsampled to the scale of $1/4$, which are then element-wise summed. A final $1 \times 1$ convolution, a $4\times$ bilinear upsampling operation and a softmax are applied to get the segmentation result.
Implementation Details
----------------------
#### Training.
As a unified framework for panoptic segmentation, there are four different losses for SpatialFlow to optimize during the training stage. The loss function can be formulated as follow: $$\label{eq4}
\mathcal{L} = (\mathcal{L}_{cls} + \mathcal{L}_{reg} + \mathcal{L}_{mask}) + \lambda \cdot \mathcal{L}_{stuff}$$ where $\mathcal{L}_{cls}$, $\mathcal{L}_{reg}$ and $\mathcal{L}_{mask}$ belong to the things segmentation task, and $\mathcal{L}_{stuff}$ is the loss of the stuff segmentation. We add a hyper-parameter $\lambda$ to balance the losses between things and stuff segmentation.We implement our SpatialFlow with a toolbox [@mmdet] based on PyTorch [@pytorch].
We inherit all the hyper-parameters from RetinaNet except that we set the threshold of NMS to $0.4$ when generating proposals during training. For mask prediction, we add the ground truth boxes to the proposals set and run the mask head for all proposals. For training strategies, we fix the batch norm layer in the backbone and train our model over $4$ GPUs with a total of 8 images per minibatch. On MS-COCO [@coco], all models are trained for $20$ epochs with an initial learning rate of $5 \times 10^{-3}$, which is decreased by $10$ after $16$ and $19$ epochs; on Cityscapes [@cityscapes], we set the initial learning rate as $1.25 \times 10^{-2}$ and the number of epochs as $176$, then drop the learning rate by 10 on the $112$th and the $152$th epoch. For image size, we resize the shorter edge of the image to $600$ pixels and $800$ pixels on COCO, while on Cityscapes, we adopt $512 \times 1024$ image crops after scaling each image by $0.5$ to $2.0\times$. As Kirillov et al. [@panopticfpn] did, we also predict a particular ‘other’ class for all things categories in stuff head on COCO benchmark.
#### Inference.
Our model follows a pipeline in the inference stage: (1) generate the detection results; (2) obtain the maps of things and stuff segmentation; (3) merge the two outputs to form a final panoptic segmentation map. In detection, we set the threshold of NMS to $0.4$ for each class separately, and choose the top-$100$ scoring bounding boxes to send to mask head. During merging, we first ignore the stuff regions labeled ‘other’; then we resolve the overlap problem between instances based on their scores; at last, we merge the things and stuff map in favor of things.
Experiments
===========
Dataset and Evaluation metric
-----------------------------
#### Dataset.
We evaluate our model on both COCO [@coco] and Cityscapes [@cityscapes]. COCO consists of 80 things and 53 stuff classes. We use the 2017 data splits with 118k/5k/20k [*train*]{}/[*val*]{}/[*test*]{} images. We use [*train*]{} split for training, and report leison and sensitive studies by evaluating on [*val*]{} split. For our main results, we report our panoptic performance on the [*test-dev*]{} split. Cityscapes has $5k$ high-resolution images with fine pixel-accurate annotations: $2975$ train, $500$ val, and $1525$ test. There are $19$ classes on Cityscapes, $8$ with instance-level masks. For all experiments, we report our performance on [*val*]{} split with $11$ stuff classes and $8$ things classes.
#### Evaluation metric.
We adopt the [*panoptic quality*]{} (PQ) as the metric. As proposed in [@panopticsegmentation], PQ can be formulated as follow: $$\label{eq5}
PQ = \underbrace{\frac {\sum_{(p,g) \in TP} IoU(p, g)}{|TP|}}_{\text{segmentation quality (SQ)}} \times \underbrace{\frac {|TP|}{|TP| + \frac 12 |FP| + \frac 12 |FN|}}_{\text{recognition quality (RQ)}}$$ where $p$ and $g$ are predicted and ground truth segments, TP (true positives), FP (false positives), and FN (false negatives) represent matched pairs of segments ($IoU(p, g) > 0.5$), unmatched predicted segments, and unmatched ground truth segments, respectively. Besides, PQ can be explained as the multiplication of a segmentation quality (SQ) and a recognition quality (RQ). We also use SQ and RQ to measure the performance in our experiments.
Ablation studies
----------------
In this section, we carefully design various experiments to reveal the contribution of each component in our unified framework. We adopt the Retina-based framework as our baseline model. All the results and discussions are presented below.
$\lambda$ PQ PQ$^{Th}$ PQ$^{St}$
----------- ---------- ----------- -----------
1.0 37.5 41.8 30.9
0.75 38.2 43.0 **31.1**
0.5 38.8 44.0 31.0
0.4 38.9 44.3 30.8
0.3 39.1 44.5 30.9
**0.25** **39.3** **45.1** 30.5
0.2 39.0 44.9 30.0
0.1 38.5 **45.1** 28.5
: The results of the baseline model on COCO [*val*]{} split with different values of $\lambda$ based on ResNet-50 with an image size of $600$px.[]{data-label="tab1"}
{width="100.00000%"}
model PQ PQ$^{Th}$ PQ$^{St}$
------------------------ ---------- ----------- -----------
baseline 39.7 46.0 30.2
+ stuff + mask sub-net 40.3 46.2 31.4
+ reg-cls flow 40.5 46.6 31.4
+ reg-stuff flow 40.7 46.3 **32.0**
+ reg-mask flow 40.7 46.4 31.8
+ stuff-mask flow **40.9** **46.8** 31.9
: The contribution of each component in SpatialFlow with ResNet-50 and an image size of $800$px on COCO [*val*]{} split. Each row adds an extra component to the above row.[]{data-label="tab2"}
#### Multi-Task training.
We perform multi-task training on panoptic segmentation by tackling things and stuff simultaneously. However, there is an inconsistency between the gradients of things and stuff segmentation [@panopticranking], which is caused by the different objectives of two tasks - things segmentation tries to optimize the instance-level masks while stuff segmentation aims to learn to predict the pixel-level segmentation maps. To balance the gradients between two tasks, we search the best value for the hyper-parameter $\lambda$ on COCO dataset with image shorter edge of $600$px. We show the results of the baseline model with various values of $\lambda$ in Table \[tab1\], which are 1.0, 0.75, 0.5, 0.4, 0.3, 0.25, 0.2, 0.1. According to the results, we found that directly adding the losses together ($\lambda=1.0$) leads to low performance, which is 37.5 PQ. We demonstrate the power of $\lambda$ and discover that the best value to balance the losses is $0.25$, with which the baseline model achieves 39.3 PQ and earns a 1.8 PQ gain compared with $\lambda=1.0$. For Cityscapes dataset, we follow [@panopticfpn] and set $\lambda=1.0$.
model PQ PQ$^{Th}$ PQ$^{St}$
------------------------ ---------- ----------- -----------
baseline 57.3 53.5 60.0
+ stuff + mask sub-net 57.5 53.6 60.3
+ reg-cls flow 58.0 **55.1** 60.1
+ reg-stuff flow 58.3 54.6 60.9
+ reg-mask flow 58.5 54.7 61.3
+ stuff-mask flow **58.6** 54.9 **61.4**
: The contribution of each component in SpatialFlow with ResNet-50 on Cityscapes [*val*]{} split. Each row adds an extra component to the above row. []{data-label="tab3"}
#### Stuff and Mask sub-networks.
In this section, we decompose the sub-nets to make clear their effects. Table \[tab2\] shows that the sub-nets bring considerable gain on stuff segmentation, which is $1.2$ PQ$^{St}$. In Table \[tab3\], we also discover that the sub-nets slightly bring consistency improvements on Cityscape dataset. Stuff and Mask sub-nets are designed to enlarge the region for integrating features and improve the information flow between tasks. These improvements demonstrate their additional function that can help the model learn better representations.
#### Spatial information flow.
Backbone size PQ PQ$^{Th}$ PQ$^{St}$
------------ ------- ---------- ----------- -----------
ResNet-50 600px 40.3 45.6 32.2
ResNet-101 600px **41.2** **46.6** **32.9**
ResNet-50 800px 40.9 46.8 31.9
ResNet-101 800px **42.2** **48.2** **33.1**
: The results of SpatialFlow on COCO [*val*]{} split with different backbones and various image sizes.
\[tab4\]
model backbone image size PQ PQ$^{Th}$ PQ$^{St}$ SQ RQ
-------------- ---------------- ------------------- ---------- ----------- ----------- ---------- ---------- -- --
JSIS-Net ResNet-50 400 $\times$ 400 27.2 29.6 23.4 71.9 35.9
DeeperLab Xception-71 321 $\times$ 321 34.3 37.5 29.6 77.1 43.1
PanopticFPN ResNet-101-FPN \[640, 800\] 40.9 48.3 29.7 - -
OANet ResNet-101-FPN 800 $\times$ 1333 41.3 **50.4** 27.7 - -
SpatialFlow ResNet-101-FPN 600 $\times$ 1000 41.8 47.6 33.0 **79.2** 51.0
SpatialFlow ResNet-101-FPN 800 $\times$ 1333 **42.8** 49.1 **33.1** 78.9 **52.1**
model backbone PQ PQ$^{Th}$ PQ$^{St}$ SQ RQ
----------------- ----------------- ---------- ----------- ----------- ---------- ---------- -- --
Megvii (Face++) ensemble model 53.2 62.2 39.5 83.2 62.9
Caribbean ensemble model 46.8 54.3 35.5 80.5 57.1
PKU 360 ResNeXt-152-DCN 46.3 58.6 27.6 79.6 56.1
AUNet ResNeXt-152-DCN 46.5 **55.9** 32.5 81.0 56.1
UPSNet ResNet-101-DCN 46.6 53.2 36.7 80.5 **56.9**
SpatialFlow ResNet-101-DCN **47.3** 53.5 **37.9** **81.8** **56.9**
We conduct experiments to highlight the significance of fully interweaving features between tasks in this section. We propose the spatial information flows to connect all the tasks. There are three paths to deliver the spatial context from box regression task to others: the path from the reg sub-net to the cls sub-net (reg-cls flow), the path to the stuff sub-net (reg-stuff flow), and the path to the mask sub-net (reg-mask flow). The results are reported in Table \[tab2\] and Table \[tab3\]. At first, we add the reg-cls path, and we obtain a $0.4$ PQ$^{Th}$ improvement on COCO and a $1.5$ PQ$^{Th}$ gain on Cityscapes, which is brought by better detection results. Adding spatial context helps cls sub-net to extract discriminative features, which is essential for detection. Then we build a spatial path for stuff sub-net, as shown in the $5$th row of Table \[tab2\] and Table \[tab3\], we earn a $0.6$ PQ$^{St}$ gain on COCO and a $0.8$ PQ$^{St}$ gain compared with the former model, which indicates that the spatial context helps segment stuff more accurately. The reg-mask path and the semantic path also show their effectiveness on both things and stuff segmentation. Comparing with the original model, SpatialFlow can achieve a consistent gain in both things and stuff. The results prove the significance of the spatial context in panoptic segmentation to some extent. It is worth noting that we only apply the element-wise sum operation to integrate the spatial context in this work. We believe that we can get a greater improvement by implementing attention modules. We also show the results of different image sizes and different backbones on COCO in Table \[tab4\]. We find that the stuff segmentation is robust to the image size, while things segmentation benefits from large image sizes. Our SpatialFlow can achieve 42.2 PQ with a single ResNet-101 backbone on the [*val*]{} split. Also, SpatialFlow can process 10 images per second during inference with an image size of 600 $\times$ 1000 on a single Tesla V100 GPU.
model PQ PQ$^{Th}$ PQ$^{St}$
------------------------ ---------- ----------- -----------
PanopticFPN-101 58.1 52.0 62.5
AUNet-101 59.0 54.8 62.1
TASCNet-101-COCO 59.2 56.0 61.5
UPSNet-101-COCO-M 61.8 **57.6** 64.8
SpatialFlow-101 59.6 55.0 63.1
SpatialFlow-101-COCO-M **62.5** 56.6 **66.8**
: Comparison to the state-of-the-art methods on Cityscapes [*val*]{} split. In this table, ‘-101’ represents that the backbone is ResNet-101; ‘-COCO’ means using COCO preatrin model; ‘-M’ is the multi-scale testing. The implementation details can be found in Supplementary File.[]{data-label="tab7"}
#### From quantitative improvements to qualitative difference.
In this section, we want to find out how the quantitative improvements translate to the qualitative difference in SpatialFlow. In Figure \[fig4\], we choose to study the models in Table \[tab2\] and visualize the last feature map in cls-head and stuff-head of both baseline and SpatialFlow via CAM [@cam]. The visualized heatmaps illustrate that spatial information can help things branch to focus on objects and make the stuff branch aware of the precise boundary of things and stuff. In the third row of Figure \[fig4\], by comparing the segmentation maps of SpatialFlow with those of the baseline, it is intuitive how the quantitative gains translate to visual qualitative improvements. As we can see, the spatial information flow helps the model retrieve missing stuff parts, correct the wrong prediction area, and split crowded instances.
Comparisons to the state-of-the-art
-----------------------------------
In this section, we compare our SpatialFlow to the existing methods. The results are shown in Table \[tab5\], Table \[tab6\], and Table \[tab7\]. On COCO, to make a fair comparison, we report the results of different image sizes. With a single ResNet-101-FPN backbone, our SpatialFlow can achieve **41.8** PQ and **42.8** PQ with the shorter image side of 600 pixels and 800 pixels respectively. As illustrate in Table \[tab5\], SpatialFlow outperforms PanopticFPN [@panopticfpn] by 1.9 PQ and OANet [@panopticranking] by 1.5 PQ. More importantly, SpatialFlow achieves a new state-of-the-art performance on PQ$^{St}$, **33.1** PQ, which outperform other models by a large margin (3.4 PQ and 5.4 PQ respectively). The results demonstrate the effectiveness of integrating features in pixel-level, which has a great impact on stuff segmentation. We also report the results with a stronger backbone in Table \[tab6\]. We achieve $\textbf{47.3}$ PQ, which is the state-of-the-art results on COCO panoptic benchmark. Moreover, we show our results on Cityscapes dataset in Table \[tab7\]. Without COCO pretrain model, SpatialFlow can achieve $\textbf{59.6}$ PQ on Cityscapes [*val*]{} split, which is $1.5$ PQ and $0.6$ PQ higher than PanopticFPN [@panopticfpn] and AUNet [@aunet] respectively. With COCO pretrain model, Our SpatialFlow can achieve $\textbf{62.5}$ PQ on Cityscapes with multi-scale testing, which is $0.7$ PQ higher than UPSNet [@upsnet]. SpatialFlow achieves the state-of-the-art results on both COCO and Cityscapes benchmarks.
Conclusion
==========
In this work, we propose a new location-aware and unified framework, SpatialFlow, for panoptic segmentation. We emphasize the importance of the spatial context and bridge all the tasks by building spatial information flow, then achieve state-of-the-art performance on both COCO panoptic benchmark [*test-dev*]{} split and Cityscapes panoptic benchmark [*val*]{} split, which prove the effectiveness of our model. Moreover, we only use the element-wise sum when doing feature fusion in this work, and we believe that we can achieve higher performance by introducing attention modules to SpatialFlow. In the future, we will do more investigation work on how to fully integrating features.
Supplementary file
==================
In this file, we first provide experimental details on both COCO and Cityscapes. Then we give some visualization examples on two datasets.
Experimental details
--------------------
#### COCO
To make a fair comparison in Table 6, we add several additional components based on SpatialFlow in Table 5. Similar to UPSNet, we adopt deformable convolutions and the multi-scale trick. For deformable convolution, we apply it in the last three stage of ResNet-101, while the implementation in sub-networks is little different. In sub-networks, we want to fully leverage the spatial context and task-specific context, thus we first combine the spatial information flow and the task-specific feature, then use the combined feature to generate the offsets for the deformable convolution on the task-specific sub-network. The processes in sub-networks can be formulated as follow: $$\label{eq6}
\begin{split}
&P_{reg_{i, j}} = \zeta(P_{reg_{i, j-1}}); \\
&P_{cls_{i, j}} = \zeta_{dcn}(P_{cls_{i, j-1}}, \phi_{offset}(P_{reg_{i, j}} +P_{cls_{i, j-1}})); \\
&P_{stuff_{i, j}} = \zeta_{dcn}(P_{stuff_{i, j-1}}, \phi_{offset}(P_{reg_{i, j}} + P_{stuff_{i, j-1}})); \\
&P_{mask_{i, 1}} = \zeta(P_i + \psi(P_{stuff_{i, 4}}), \phi_{offset}(P_{reg_{i, 4}} + P_i)); \\
&P_{reg_{i, 0}} = P_i, \quad P_{cls_{i, 0}} = P_i, \quad P_{stuff_{i, 0}} = P_i.
\end{split}$$ Here, $\zeta_{dcn}$ represents deformable convolution, $\phi_{offset}$ means an adaptation convolution, which generates offsets for deformable convolution. For multi-scale trick, we feed multi-scale images for SpatialFlow, and the scales are $(1500, 1000)$, $(1800, 1200)$, and $(2100, 1400)$ with horizontal flip. For the hyper-parameters of SpatialFlow in the inference stage, we set the confidence score threshold for instance mask as $0.37$, set the overlap threshold of instance masks as $0.37$, and set the area limit threshold of stuff regions as $4900$.
#### Cityscapes
In Table 7, we adopt COCO pre-trained model on Cityscapes. To obtain the result of $62.5$ PQ on Cityscapes [*val*]{} split, we first replace the convolution layers in stuff with deformable convolutions as UPSNet does, then we follow the steps below: (1) Finetune the COCO pre-trained model. As the number of things and stuff classes in Cityscapes is smaller than the number in COCO, $11/8$ vs. $80/53$, we have to finetune the layers that related to the number of classes. We freeze the rest layers and use a learning rate of $2.5 \times 10^{-3}$ to train for $2$ epochs. (2) Train the finetuned model as the standard SpatialFlow does. (3) Apply the multi-scale trick. The scales that we use in Cityscapes are $(2304, 1152)$, $(2432, 1216)$, $(2560, 1280)$, and $(2688, 1344)$ with horizontal flip. For the hyper-parameters on Cityscapes, we set the confidence score threshold for instance mask as $0.37$, set the overlap threshold of instance masks as $0.25$, and set the area limit threshold of stuff regions as $2048$.
Results visualization
---------------------
We show some visualization examples of SpatialFlow on COCO and Cityscapes in Figure \[fig5\] and Figure \[fig6\] respectively.
{width="100.00000%"} \[fig5\]
{width="100.00000%"} \[fig6\]
[^1]: Refer to instance segmentation and semantic segmentation, in this paper, we use the things and the stuff to emphasize the tasks in panoptic segmentation.
|
---
abstract: 'Camera and [lidar]{}are important sensor modalities for robotics in general and self-driving cars in particular. The sensors provide complementary information offering an opportunity for tight sensor-fusion. Surprisingly, lidar-only methods outperform fusion methods on the main benchmark datasets, suggesting a gap in the literature. In this work, we propose PointPainting: a sequential fusion method to fill this gap. PointPainting works by projecting lidar points into the output of an image-only semantic segmentation network and appending the class scores to each point. The appended (painted) point cloud can then be fed to any lidar-only method. Experiments show large improvements on three different state-of-the art methods, Point-RCNN, VoxelNet and PointPillars on the KITTI and nuScenes datasets. The painted version of PointRCNN represents a new state of the art on the KITTI leaderboard for the [bird’s-eye view]{}detection task. In ablation, we study how the effects of Painting depends on the quality and format of the semantic segmentation output, and demonstrate how latency can be minimized through pipelining.'
author:
- |
Sourabh Vora Alex H. Lang Bassam Helou Oscar Beijbom\
nuTonomy: an Aptiv Company\
[{sourabh, alex, bassam, oscar}@nutonomy.com]{}
bibliography:
- '../references.bib'
title: 'PointPainting: Sequential Fusion for 3D Object Detection'
---
|
---
author:
- 'Alain Connes[^1]'
title: An essay on the Riemann Hypothesis
---
Introduction {#intro}
============
Let $\pi(x):=\#\{p\mid p\in \cP, \, p<x\}$ be the number of primes less than $x$ with $\frac12$ added when $x$ is prime. Riemann [@Riemann] found for the counting function [^2] $$f(x):=\sum \frac 1n \pi(x^{\frac 1n}),$$ the following formula involving the integral logarithm function $\li(x)=\int_0^x\frac{dt}{\log t}$, $$\label{Riemann1}
f(x)=\li(x)-\sum_\rho \li(x^{\rho})+\int_x^\infty \frac{1}{t^2-1}\,\frac{dt}{t\log t}-\log 2$$ in terms[^3] of the non-trivial zeros $\rho$ of the analytic continuation (shown as well as two proofs of the functional equation by Riemann at the beginning of his paper) of the Euler zeta function $$\zeta(s)=\sum\frac{1}{n^s}$$ Reading Riemann’s original paper is surely still the best initiation to the subject. In his lecture given in Seattle in August 1996, on the occasion of the 100-th anniversary of the proof of the prime number theorem, Atle Selberg comments about Riemann’s paper: [@Selbergseattle]
> [*It is clearly a preliminary note and might not have been written if L. Kronecker had not urged him to write up something about this work (letter to Weierstrass, Oct. 26 1859). It is clear that there are holes that need to be filled in, but also clear that he had a lot more material than what is in the note[^4]. What also seems clear : Riemann is not interested in an asymptotic formula, not in the prime number theorem, what he is after is an exact formula!*]{}
The Riemann hypothesis (RH) states that all the non-trivial zeros of $\zeta$ are on the line $\frac 12+i\R$. This hypothesis has become over the years and the many unsuccessful attempts at proving it, a kind of “Holy Grail" of mathematics. Its validity is indeed one of the deepest conjectures and besides its clear inference on the distribution of prime numbers, it admits relations with many parts of pure mathematics as well as of quantum physics.
It is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. There are many excellent texts on RH, such as [@B2] which explain in great detail what is known about the problem, and the many implications of a positive answer to the conjecture. When asked by John Nash to write a text on RH[^5], I realized that writing one more encyclopedic text would just add another layer to the psychological barrier that surrounds RH. Thus I have chosen deliberately to adopt another point of view, which is to navigate between the many forms of the explicit formulas (of which is the prime example) and possible strategies to attack the problem, stressing the value of the elaboration of new concepts rather than “problem solving".
- [*RH and algebraic geometry*]{}
We first explain the Riemann-Weil explicit formulas in the framework of adeles and global fields in §\[subsecrw\]. We then sketch in §\[rrstrat\] the geometric proof of RH for function fields as done by Weil, Mattuck, Tate and Grothendieck. We then turn to the role of RH in generating new mathematics, its role in the evolution of algebraic geometry in the XX-th century through the Weil conjectures, proved by Deligne, and the elaboration by Grothendieck of the notions of scheme and of topos.
- [*Riemannian Geometry, Spectra and trace formulas*]{}
Besides the proof of analogues of RH such as the results of Weil and of Deligne, there is another family of results that come pretty close. They give another natural approach of RH using analysis, based on the pioneering work of Selberg on trace formulas. These will be reviewed in Section \[analysisattack\] where the difficulty arising from the minus sign in front of the oscillatory terms will be addressed.
- [*The Riemann-Roch strategy: A Geometric Framework*]{}
In Section \[algeomattack\], we shall describe a geometric framework, established in our joint work with C. Consani, allowing us to transpose several of the key ingredients of the geometric proof of RH for function fields recalled in §\[rrstrat\]. It is yet unclear if this is the right set-up for the final Riemann-Roch step, but it will illustrate the power of RH as an incentive to explore new parts of mathematics since it gives a clear motivation for developing algebraic geometry in characteristic $1$ along the line of tropical geometry. This will take us from the world of characteristic $p$ to the world of characteristic $1$, and give us an opportunity to describe its relation with semi-classical and idempotent analysis, optimization and game theory[^6], through the Riemann-Roch theorem in tropical geometry [@BN; @GK; @MZ].
- [*Absolute Algebra and the sphere spectrum*]{}
The arithmetic and scaling sites which are the geometric spaces underlying the Riemann-Roch strategy of Section \[algeomattack\] are only the semiclassical shadows of a more mysterious structure underlying the compactification of $\Spec\Z$ that should give a cohomological interpretation of the explicit formulas. We describe in this last section an essential tool coming from algebraic topology: Segal’s $\Gamma$-rings and the sphere spectrum, over which all previous attempts at developing an absolute algebra organize themselves. Moreover, thanks to the results of Hesselholt and Madsen in particular, topological cyclic homology gives a cohomology theory suitable to treat in a unified manner the local factors of $L$-functions.
RH and algebraic geometry
=========================
I will briefly sketch here the way RH, once transposed in finite characteristic, has played a determining role in the upheaval of the very notion of geometric space in algebraic geometry culminating with the notions of scheme and topos due to Grothendieck, with the notion of topos offering a frame of thoughts of incomparable generality and breadth. It is a quite remarkable testimony to the unity of mathematics that the origin of this discovery lies in the greatest problem of analysis and arithmetic.
The Riemann-Weil explicit formulas, Adeles and global fields {#subsecrw}
------------------------------------------------------------
Riemann’s formula is a special case of the “explicit formulas" which establish a duality between the primes and the zeros of zeta. This formula has been extended by Weil in the context of global fields which provides a perfect framework for a generalization of RH since it has been solved, by Weil, for all global fields except number fields.
### The case of $\zeta$
Let us start with the explicit formulas ([[*cf.*]{} ]{}[@weilpos0; @weilpos; @EB; @patter]). We start with a function $F(u)$ defined for $u\in [1,\infty)$, continuous and continuously differentiable except for finitely many points at which both $F(u)$ and $F'(u)$ have at most a discontinuity of the first kind, [^7] and such that, for some $\epsilon>0$, $F(u)=O(u^{-1/2-\epsilon})$. One then defines the Mellin transform of $F$ as $$\label{expl}
\Phi(s)=\int_1^\infty F(u)\,u^{s-1}du$$ The explicit formula then takes the form $$\label{explfor}
\Phi(\frac 12)+\Phi(-\frac 12)-\sum_{\rho\in \rmz}\Phi(\rho-\frac 12)=\sum_p\sum_{m=1}^\infty \log p \,\,p^{-m/2}F(p^m)+$$ $$+(\frac \gamma 2+\frac{\log\pi}{2})F(1)
+\int_1^\infty\frac{t^{3/2}F(t)-F(1)}{t(t^2-1)}dt$$ where $\gamma=-\Gamma'(1)$ is the Euler constant, and the zeros are counted with their multiplicities [[*i.e.*]{} ]{}$\sum_{\rho \in\rmz}\Phi(\rho-\frac 12)$ means $\sum_{\rho\in \rmz}{\rm order}(\rho)\Phi(\rho-\frac 12)$.
### Adeles and global fields
By a result of Iwasawa [@Iwasawa] a field $\K$ is a finite algebraic number field, or an algebraic function field of one variable over a finite constant field, if and only if there exists a semi-simple ([[*i.e.*]{} ]{}with trivial Jacobson radical [@Jac]) commutative ring $R$ containing $\K$ such that $R$ is locally compact, but neither compact nor discrete and $\K$ is discrete and cocompact in $R$. This result gives a conceptual definition of what is a “global field" and indicates that the arithmetic of such fields is intimately related to analysis on the parent ring $R$ which is called the ring of adeles of $\K$ [@Weil; @tate]. It is the opening door to a whole world which is that of automorphic forms and representations, starting in the case of $\GL_1$ with Tate’s thesis [@tate] and Weil’s book [@Weil]. Given a global field $\K$, the ring $\A_\K$ of adeles of $\K$ is the restricted product of the locally compact fields $\K_v$ obtained as completions of $\K$ for the different places $v$ of $\K$. The equality $dax=\vert a\vert dx$ for the additive Haar measure defines the module $\mmod:\K_v\to \R_+$, $\mmod(a):=\vert a\vert$ on the local fields $\K_v$ and also as a group homomorphism $\mmod:C_\K\to \R_+^*$ where $C_\K=\GL_1(\A_\K)/\K^\times$ is the idele class group. The kernel of the module is a compact subgroup $C_{\K,1}\subset C_\K$ and the range of the module is a cocompact subgroup $\mmod(\K)\subset \R_+^*$. On any locally compact modulated group, such as $C_\K$ or the multiplicative groups $\K_v^*$, one normalizes the Haar measure $d^*u$ uniquely so that the measure of $\{u\mid 1\leq \vert u\vert \leq \Lambda\}$ is equivalent to $\log \Lambda$ when $\Lambda\to \infty$.
### Weil’s explicit formulas {#sectweilexpl}
As shown by Weil, in [@weilpos], adeles and global fields give the natural framework for the explicit formulas. For each character $\chi\in \widehat{C_{\K,1}}$ one chooses an extension $\tilde \chi$ to $C_\K$ and one lets $Z_{\tilde\chi}$ be the set (with multiplicities and taken modulo the orthogonal of $\mmod(\K)$, [[*i.e.*]{} ]{}$\{s\in\C\mid q^s=1, \forall q\in \mmod(\K)\}$) of zeros of the $L$-function associated to $\tilde\chi$. Let then $ \alpha$ be a nontrivial character of $\A_\K/\K$ and $ \alpha = \prod \, \alpha_v$ its local factors. The explicit formulas take the following form, with $h \in \cS (C_\K)$ a Schwartz function with compact support: $$\label{weil4}
\hat h (0) + \hat h (1) - \sum_{\chi\in \widehat{C_{\K,1}}}\,
\sum_{Z_{\tilde\chi}} \hat h (\tilde\chi , \rho) = \sum_v
\int'_{\K_v^*} \frac{h(u^{-1}) }{ \vert 1-u \vert} \, d^* u$$ where the principal value $\int'_{\K_v^*}$ is normalized by the additive character $ \alpha_v$ ([[*cf.*]{} ]{}[@CMbook] Chapter II, 8.5, Theorem 2.44 for the precise notations and normalizations) and for any character $\omega$ of $C_\K$ one lets $$\label{fourier}
\hat h (\omega , z): = \int h(u) \,
\omega(u) \, \vert u \vert^z \, d^* u, \ \ \hat h (t):=\hat h (1,t)$$ For later use in §\[counting\] we compare with the Weil way of writing the explicit formulas. Let the function $h$ be the function on $C_\Q$ given by $h(u):=\vert u\vert^{-\frac 12}F(\vert u\vert)$ (with $F(v)=0$ for $v<1$). Then $\hat h (\omega , z)=0$ for characters with non-trivial restriction to $C_{\Q,1}=\hatz$, while $\hat h (1 , z)=\Phi(z-\frac 12)$. Moreover note that for the archimedean place $v$ of $\K=\Q$ one has, disregarding the principal values for simplicity, $$\int_{\K_v^*} \frac{h(u^{-1}) }{ \vert 1-u \vert} \, d^* u=\int_{\R^*} \frac{h(u) }{ \vert 1-u^{-1} \vert} \, d^* u$$ $$=\frac 12\int_1^\infty h(t)\left( \frac{1}{\vert 1-t^{-1}\vert}+\frac{1}{\vert 1+t^{-1}\vert}\right)\frac{dt}{t}=\int_1^\infty\frac{t^{3/2}F(t)}{t(t^2-1)}dt$$ where the $\frac 12$ comes from the normalization of the multiplicative Haar measure of $\R^*$ viewed as a modulated group. In a similar way, the normalization of the multiplicative Haar measure on $\Q_p^*$ shows that for the finite place associated to the prime $p$ one gets the term $\sum_{m=1}^\infty \log p \,\,p^{-m/2}F(p^m)$.
RH for function fields {#subsecff}
----------------------
When the module $\mmod(\K)$ of a global field is a discrete subgroup of $ \R_+^*$ it is of the form $\mmod(\K)=q^\Z$ where $q$ is a prime power, and the field $\K$ is the function field of a smooth projective curve $C$ over the finite field $\F_q$.
Already at the beginning of the XX-th century, Emil Artin and Friedrich Karl Schmidt have generalized RH to the case of function fields. We refer to the text of Cartier [@Cart] where he explains how Weil’ s definition of the zeta function associated to a variety over a finite field slowly emerged, starting with the thesis of E. Artin where this zeta function was defined for quadratic extensions of $\F_q[T]$, explaining F. K. Schmidt’ s generalization to arbitrary extensions of $\F_q[T]$ and the work of Hasse on the “Riemann hypothesis" for elliptic curves over finite fields.
When the global field $\K$ is a function field, geometry comes to the rescue. The problem becomes intimately related to the geometric one of estimating the number $N(q^r):=\#\,C(\F_{q^r})$ of points of $C$ rational over a finite extension $\F_{q^r}$ of the field of definition of $C$. The analogue of the Riemann zeta function is a generating function: the Hasse-Weil zeta function $$\label{HW}
\zeta_C(s):=Z(C,q^{-s}), \ \ Z(C,T) := \exp\left(\sum_{r\geq 1}N(q^r)\frac{T^r}{r}\right)$$ The analogue of RH for $\zeta_C$ was proved by Andr' e Weil in 1940. Pressed by the circumstances (he was detained in jail) he sent a Comptes-Rendus note to E. Cartan announcing his result. Friedrich Karl Schmidt and Helmut Hasse had previously been able to transpose the Riemann-Roch theorem in the framework of geometry over finite fields and shown its implications for the zeta function: it is a rational fraction (of the variable $T$) and it satisfies a functional equation. But it took André Weil several years to put on solid ground a general theory of algebraic geometry in finite characteristic that would justify his geometric arguments and allow him to transpose the Hodge index theorem in the form due to the Italian geometers Francesco Severi and Guido Castelnuovo at the beginning of the XX-th century.
The proof using Riemann-Roch on $\bar C\times \bar C$ {#rrstrat}
-----------------------------------------------------
Let $C$ be a smooth projective curve over the finite field $\F_q$. The first step is to extend the scalars from $\F_q$ to an algebraic closure $\bar\F_q$. Thus one lets $$\label{extscal}
\bar C:=C\otimes_{\F_q}\bar\F_q$$ This operation of extension of scalars does not change the points over $\bar\F_q$, [[*i.e.*]{} ]{}one has $\bar C(\bar\F_q)=C(\bar\F_q)$. The Galois action of the Frobenius automorphism of $\bar\F_q$ raises the coordinates of any point $x\in C(\bar\F_q)$ to the $q$-th power and this transformation of $C(\bar\F_q)$ coincides with the [*relative Frobenius*]{} $\fr_r:=\fr_C\times \id$ of $\bar C$, where $\fr_C$ is the [*absolute Frobenius*]{} of $C$ (which is the identity on points of the scheme and the $q$-th power map in the structure sheaf). The relative Frobenius $\fr_r$ is $\bar\F_q$-linear by construction and one can consider its graph in the surface $X=\bar C\times_{\bar\F_q} \bar C$ which is the square of $\bar C$. This graph is the Frobenius correspondence $\Psi$. It is important to work over an algebraically closed field in order to have a good intersection theory. This allows one to express the right hand side of the explicit formula for the zeta function $\zeta_C$ as an intersection number $D.\Delta$, where $\Delta$ is the diagonal in the square and $D=\sum a_k\Psi^k$ is the divisor given by a finite integral linear combination of powers of the Frobenius correspondence. The terms $\hat h (0)$, $\hat h (1)$ in the explicit formula are also given by intersection numbers $D.\xi_j$, where $$\label{xin}
\xi_0=e_0\times \bar C\,, \ \xi_1=\bar C\times e_1$$ where the $e_j$ are points of $\bar C$. One then considers divisors on $X$ up to the additive subgroup of principal divisors [[*i.e.*]{} ]{}those corresponding to an element $f\in \cK$ of the function field of $X$. The problem is then reduced to proving the negativity of $D.D$ (the self-intersection pairing) for divisors of degree zero. The Riemann-Roch theorem on the surface $X$ gives the answer. To each divisor $D$ on $X$ corresponds an index problem and one has a finite dimensional vector space of solutions $H^0(X,\cO(D))$ over $\bar\F_q$. Let $$\label{dim}
\ell(D)={\rm dim}\, H^0(X,\cO(D))$$ The best way to think of the sheaf $\cO(D)$ is in terms of Cartier divisors, [[*i.e.*]{} ]{}a global section of the quotient sheaf $\cK^\times/\cO_X^\times$, where $\cK$ is the constant sheaf corresponding to the function field of $X$ and $\cO_X$ is the structure sheaf. The sheaf $\cO(D)$ associated to a Cartier divisor is obtained by taking the sub-sheaf of $\cK$ whose sections on $U_i$ form the sub $\cO_X$-module generated by $f_i^{-1}\in \Gamma(U_i,\cK^\times)$ where the $f_i$ represent $D$ locally. One has a “canonical" divisor $K$ and Serre duality $$\label{Sdual}
{\rm dim}\, H^2(X,\cO(D))={\rm dim}\, H^0(X,\cO(K-D))$$ Moreover the following Riemann-Roch formula holds $$\label{RRform}
\sum_0^2(-1)^j {\rm dim}\, H^j(X,\cO(D))=\frac 12 D.(D-K)+\chi(X)$$ where $\chi(X)$ is the arithmetic genus. All this yields the Riemann-Roch inequality $$\label{rrine}
\ell(D)+ \ell(K-D)\geq \frac 12 D.(D-K)+\chi(X)$$ One then applies Lemma \[simplelem1\] to the quadratic form $\inter(D,D')=D.D'$ using the $\xi_j$ of . One needs three basic facts ([@grmt])
1. If $\ell(D)>1$ then $D$ is equivalent to a strictly positive divisor.
2. If $D$ is a strictly positive divisor then $$D.\xi_0+D. \xi_1>0$$
3. One has $\xi_0.\xi_1=1$ and $\xi_j.\xi_j=0$.
One then uses to show (see [@grmt]) that if $D.D>0$ then after a suitable rescaling by $n>0$ or $n<0$ one gets $\ell(nD)>1$ which shows that the hypothesis (2) of the following simple Lemma \[simplelem1\] is fulfilled, and hence that RH holds for $\zeta_C$,
\[simplelem1\] Let $\inter(x,y)$ be a symmetric bilinear form on a vector space $E$ (over $\Q$ or $\R$). Let $\xi_j \in E$, $j\in \{0,1\}$, be such that
1. $\inter(\xi_j,\xi_j)=0$ and $\inter(\xi_0,\xi_1)=1$.
2. For any $x\in E$ such that $\inter(x,x)>0$ one has $\inter(x,\xi_0)\neq 0$ or $\inter(x,\xi_1)\neq 0$.
Then one has the inequality $$\label{negative4}
\inter(x,x)\leq 2 \inter(x,\xi_0)\inter(x,\xi_1)\qqq x\in E$$
The proof takes one line but the meaning of this lemma is to reconcile the “naive positivity" of the right hand side of the explicit formula (which is positive when $h\geq 0$ vanishes near $u=1$) with the negativity of the left hand side needed to prove RH ([[*cf.*]{} ]{}§\[sectSelberg\] below).
Such a tentative framework will be explained in Section \[algeomattack\]. It involves in particular the refinement of the notion of geometric space which was uncovered by Grothendieck and to which we now briefly turn.
Grothendieck and the notion of topos {#subsecff}
------------------------------------
The essential ingredients of the proof explained in §\[rrstrat\] are the intersection theory for divisors on $\bar C\times \bar C$, sheaf cohomology and Serre duality, which give the formulation of the Riemann-Roch theorem. Both owe to the discovery of sheaf theory by J. Leray and the pioneering work of J. P. Serre on the use of sheaves for the Zariski topology in the algebraic context, with his fundamental theorem comparing the algebraic and analytic frameworks. The next revolution came from the elaboration by A. Grothendieck and M. Artin of etale $\ell$-adic cohomology. It allows one to express the Weil zeta function of a smooth projective variety $X$ defined over a finite field $\F_q$ [[*i.e.*]{} ]{}the function $Z(X,t)$ given by with $t=q^{-s}$ which continues to make sense in general, as an alternate product of the form $$\label{etalecohomol}
Z(X,t)=\prod_{j=0}^{2\, \dim X} \det(1-t F^*\mid H^j({\bar X}_\ett,\Q_\ell))^{(-1)^{j+1}}$$ where $F^*$ corresponds to the action of the Frobenius on the $\ell$-adic cohomology and $\ell$ is a prime which is prime to $q$. This equality follows from a Lefschetz formula for the number $N(q^r)$ of fixed points of the $r$-th power of the Frobenius and when $X=C$ is a curve the explicit formulas reduce to the Lefschetz formula. The construction of the cohomology groups $H^j(\bar X_{\ett},\Q_\ell)$ is indirect and they are defined as : $$H^j(\bar X_{\ett},\Q_\ell)=\varprojlim_n \left( H^j(\bar X_\ett,\Z/\ell^n\Z) \right)\otimes_{\Z_\ell}\Q_\ell$$ where $\bar X_{\ett}$ is the etale site of $\bar X$. Recently the etale site of a scheme has been refined [@BS] to the [*pro-etale*]{} site whose objects no longer satisfy any finiteness condition. The cohomology groups $H^j(\bar X_{\rm proet},\bar\Q_\ell)$ are then directly obtained using the naive interpretation (without torsion coefficients). One needs to pay attention in to the precise definition of $F$, it is either the relative Frobenius $\fr_r$ or the [*Geometric Frobenius*]{} $\fr_g$ which is the inverse of the [*Arithmetic Frobenius*]{} $\fr_a$. The product $\fr_a\circ \fr_r=\fr_r\circ \fr_a$ is the absolute Frobenius $\fr$ which acts trivially on the $\ell$-adic cohomology. To understand the four different incarnations of “the Frobenius" it is best to make them explicit in the simplest example of the scheme $\Spec R$ where $R=\bar\F_q[T]$ is the ring of polynomials $P(T)=\sum a_j T^j$, $a_j\in \bar\F_q$
- Geometric Frobenius: $\sum a_j T^j\mapsto \sum a_j^{1/q} T^j$
- Relative Frobenius: $P(T)\mapsto P(T^q)$
- Absolute Frobenius: $P(T)\mapsto P(T)^q$
- Arithmetic Frobenius: $\sum a_j T^j\mapsto \sum a_j^q T^j$
The motivation of Grothendieck for developing etale cohomology came from the search of a Weil cohomology and the Weil conjectures which were solved by Deligne in 1973 ([@Deligne]).
In his quest Grothendieck uncovered several key concepts such as those of schemes and above all that of topos, in his own words:
> [*C’est le thème du topos, et non celui des schémas, qui est ce “lit”, ou cette “rivière profonde”, où viennent s’épouser la géométrie et l’algèbre, la topologie et l’arithmétique, la logique mathématique et la théorie des catégories, le monde du continu et celui des structures “discontinues” ou “discrètes”. Si le thème des schémas est comme le [*cœur*]{} de la géométrie nouvelle, le thème du topos en est l’enveloppe, ou la [*demeure*]{}. Il est ce que j’ai conçu de plus vaste, pour saisir avec finesse, par un même langage riche en résonances géométriques, une “essence” commune à des situations des plus éloignées les unes des autres, provenant de telle région ou de telle autre du vaste univers des choses mathématiques.* ]{}
Riemannian Geometry, Spectra and trace formulas {#analysisattack}
===============================================
Riemannian Geometry gives a wealth of “spectra" of fundamental operators associated to a geometric space, such as the Laplacian and the Dirac operators.
The Selberg trace formula {#sectSelberg}
-------------------------
In the case of compact Riemann surfaces $X$ with constant negative curvature $-1$, the Selberg trace formula [@Selberg], takes the following form where the eigenvalues of the Laplacian are written in the form[^8] $\lambda_n=-(\frac 14+r_n^2)$. Let $\delta>0$, $h(r)$ be an analytic function in the strip $\vert \Im(r)\vert\leq \frac 12+\delta$ and such that $h(r)=h(-r)$ and with $(1+r^2)^{1+\delta}\vert h(r)\vert$ being bounded. Then [@Selberg; @[Se]; @Hejhal], with $A$ the area of $X$, $$\label{Selberg}
\sum h(r_n)=\frac{A}{4\pi}\int_{-\infty}^\infty {\rm tanh}(\pi r)h(r)rdr
+\sum_{\{T\}}\frac{\log N(T_0)}{N(T)^\frac 12-N(T)^{-\frac 12}}g(\log N(T))$$ where $g$ is the Fourier transform of $h$, [[*i.e.*]{} ]{}more precisely $g(s)=\frac{1}{2\pi}\int_{-\infty}^\infty h(r)e^{-irs}dr$. The $\log N(T)$ are the lengths of the periodic orbits of the geodesic flow with $\log N(T_0)$ being the length of the primitive one. Already in 1950-51, Selberg saw the striking similarity of his formula with which ([[*cf.*]{}]{} [@Hejhal]) can be rewritten in the following form, with $h$ and $g$ as above and the non-trivial zeros of zeta expressed in the form $\rho=\frac 12+i\gamma$, $$\label{Hejhal}
\sum_\gamma h(\gamma)=h(\frac i2)+h(-\frac i2)+\frac{1}{2\pi}\int_{-\infty}^\infty \omega(r)h(r)dr-2\sum \Lambda(n)n^{-\frac 12}g(\log n)$$ where $$\omega(r)=\frac{\Gamma'}{\Gamma}\left(\frac 14+i\frac r2\right)-\log \pi, \ \ \frac{\Gamma'}{\Gamma}\left(s\right)=\int_0^1\frac{1-t^{s-1}}{1-t}dt-\gamma\qqq s,\Re(s)>0$$ and $\Lambda(n)$ is the von-Mangoldt function with value $\log p$ for powers $p^\ell$ of primes and zero otherwise. Moreover Selberg found that there is a zeta function which corresponds to in the same way that $\zeta(s)$ corresponds to . The role of Hilbert space is crucial in the work of Selberg to ensure that the zeros of his zeta function satisfy the analogue of RH. This role of Hilbert space is implicit as well in RH which has been reformulated by Weil as the positivity of the functional $W(g)$ defined as both sides of . More precisely the equivalent formulation is that $W(g\star g^*)\geq 0$ on functions $g$ which correspond to Fourier transforms of analytic functions $h$ as above ([[*i.e.*]{} ]{}even and analytic in a strip $\vert\Im z\vert \leq \frac 12+\delta$) where for even functions one has $g^*(s):=\overline{ g(-s)}=\overline{ g(s)}$. Moreover by [@B3; @burnol1], it is enough, using Li’s criterion ([[*cf.*]{} ]{}[@Li; @B3]), to check the positivity on a small class of explicit real valued functions with compact support. In fact for later purposes it is better to write this criterion as $$\label{negcrit}
RH \iff \inter(f,f)\leq 0 \qqq f \mid \int f(u)d^*u=\int f(u)du=0$$ where for real compactly supported functions on $\R_+^*$, we let $\inter(f,g):=N(f\star \tilde g)$ where $\star$ is the convolution product on $\R_+^*$, $\tilde g(u):=u^{-1}g(u^{-1})$, and $$\label{negcrit1}
N(h):= \sum_{n=1}^\infty \Lambda(n)h(n)+ \int_1^\infty\frac{u^2h(u)-h(1)}{u^2-1}d^*u+c\, h(1)\,, \ c=\frac12(\log\pi+\gamma)$$
The Selberg trace formula has been considerably extended by J. Arthur and plays a key role in the Langland’s program. We refer to [@Arthur] for an introduction to this vast topic.
The minus sign and absorption spectra
-------------------------------------
The Selberg trace formula [@Selberg; @[Se]] for Riemann surfaces of finite area, acquires additional terms which make it look [[*e.g.*]{} ]{}in the case of $X=H/PSL(2,\Z)$ (where $H$ is the upper half plane with the Poincaré metric) even more similar to the explicit formulas, since the parabolic terms now involve explicitly the sum $$2\sum_{n=1}^\infty \frac{\Lambda(n)}{n}g(2\log n)$$ Besides the square root in the $\Lambda(n)$ terms in the explicit formulas $$-2\sum_{n=1}^\infty \frac{\Lambda(n)}{n^{\frac 12}}g(\log n)$$ there is however a striking difference which is that these terms occur with a positive sign instead of the negative sign in , as discussed in [@Hejhal] §12. This discussion of the minus sign was extended to the case of the semiclassical limit of Hamiltonian systems in physics in [@Berry]. In order to get some intuition of what this reveals, it is relevant to go back to the origin of spectra in physics, [[*i.e.*]{} ]{}to the very beginning of spectroscopy. It occurred when Joseph Von Fraunhofer (1787-1826) could identify, using self-designed instruments, about 500 dark lines in the light coming from the sun, decomposed using the dispersive power of a spectroscope such as a prism ([[*cf.*]{} ]{}Figure \[absspectrum\]). These dark lines constitute the “absorption spectrum" and it took about 45 years before Kirchhoff and Bunsen noticed that several of these Fraunhofer lines coincide ([[*i.e.*]{} ]{}have the same wave length) with the bright lines of the “emission" spectrum of heated elements, and showed that they could be reobtained by letting white light traverse a cold gas. In his work on the trace formula in the finite covolume case, Selberg had to take care of a superposed continuous spectrum due to the presence of the non-compact cusps of the Riemann surface.
![ []{data-label="absspectrum"}](spectra5.pdf){width="\textwidth"}
The adele class space and the explicit formulas {#adeleclass}
-----------------------------------------------
I had the chance to be invited at the Seattle meeting in 1996 for the celebration of the proof of the prime number theorem. The reason was the paper [@BC] (inspired from [@[J2]]) in which the Riemann zeta function appeared naturally as the partition function of a quantum mechanical system (BC system) exhibiting phase transitions. The RH had been at the center of discussions in the meeting and I knew the analogy between the BC-system and the set-up that V. Guillemin proposed in [@gui] to explain the Selberg trace formula using the action of the geodesic flow on the horocycle foliation. To a foliation is associated a von Neumann algebra [@Co-foliations], and the horocycle foliation on the sphere bundle of a compact Riemann surface gives a factor of type II$_\infty$ on which the geodesic flow acts by scaling the trace. An entirely similar situation comes canonically from the BC-system at critical temperature and after interpreting the dual system in terms of adeles, I was led by this analogy to consider the action of the idele class group of $\Q$ on the adele class space, [[*i.e.*]{} ]{}the quotient $\Q^\times\backslash \A_\Q$ of the adeles $\A_\Q$ of $\Q$ by the action of $\Q^\times$. I knew from the BC-system that the action of $\Q^\times$, which preserves the additive Haar measure, is ergodic for this measure and gives the same factor of type II$_\infty$ as the horocycle foliation. Moreover the dual action scales the trace in the same manner.
Let $\K$ be a global field and $C_\K=\GL_1(\A_\K)/\K^\times$ the idele class group. The module $\mmod:C_\K\to \R_+^*$ being proper with cocompact range, one sees that the Haar measure on the Pontrjagin dual group of $C_\K$ is diffuse. Since a point is of measure $0$ in a diffuse measure space there is no way one can see the absorption spectrum without introducing some smoothness on this dual which is done using a Sobolev space $L_{\delta }^2 ( C_\K )$ of functions on $C_\K$ which (for fixed $\delta>1$) is defined as $$||\xi||^2=\int_{C_\K} \vert \xi (x) \vert^2 \, \rho(x) \, d^* x, \ \, \ \ \rho(x):=(1 + \log \vert x
\vert^2)^{\delta / 2} \label{sobolevck}$$
\[adclassspace\] Let $\K$ be a global field, the adele class space of $\K$ is the quotient $X_\K=\A_\K/\K^\times$ of the adeles of $\K$ by the action of $\K^\times$ by multiplication.
We then consider the codimension 2 subspace $\cS
(\A_\K)_0$ of the Bruhat-Schwartz space $\cS (\A_\K)$ ([[*cf.*]{}]{} [@Bruhat]) given by the conditions $
f(0) = 0 \, , \ \int f \, dx = 0
$ The Sobolev space $L_{\delta }^2 (X_\K)_0$ is the separated completion of $\cS
(\A_\K)_0$ for the norm with square $$||f||^2=\int_{C_\K} \vert \sum_{q\in \K^*} f (qx) \vert^2 \, \rho(x) \, \vert x\vert d^* x \label{spec2}$$ Note that by construction all functions of the form $f(x)=g(x)-g(qx)$ for some $q\in \K^\times$ belong to the radical of the norm , which corresponds to the operation of quotient of Definition \[adclassspace\]. In particular the representation of ideles on $\cS (\A_\K)$ given by $$(\urep (\alpha) \xi) (x) = \xi (\alpha^{-1} x) \ \ \forall \,
\alpha \in \GL_1(\A_\K) \, , \ x \in \A_\K \, \label{spec(3)}$$ induces a representation $\urep_a$ of $C_\K$ on $L_{\delta }^2 (X_\K)_0$. One has by construction a natural isometry $\mapE:L_{\delta }^2 (X_\K)_0\to L_{\delta }^2 ( C_\K )$ which intertwines the representation $\urep_a$ with the regular representation of $C_\K$ in $L_{\delta }^2 ( C_\K )$ multiplied by the square root of the module. This representation restricts to the cokernel of the map $\mapE$, which splits as a direct sum of subspaces labeled by the characters of the compact group $C_{\K,1}=\Ker\, \mmod$ and its spectrum in each sector gives the zeros of $L$-functions with Grössencharakter. The shortcoming of this construction is in the artificial weight $\rho(x)$, which is needed to see this absorption spectrum but only sees the zeros which are on the critical line and where the value of $\delta$ artificially cuts the multiplicities of the zeros ([[*cf.*]{}]{} [@Co-zeta]).
This state of affairs is greatly improved if one gives up trying to prove RH but retreats to an interpretation of the explicit formulas as a trace formula. One simply replaces the above Hilbert space set-up by a softer one involving nuclear spaces [@Meyer]. The spectral side now involves all non-trivial zeros and, using the preliminary results of [@Co-zeta; @Co99; @burnol] one gets that the geometric side is given by: $$\label{geomside}
\Tr_{\rm distr}\left(\int h(w)\urep(w)d^*w\right )=\sum_v\int_{\K^\times_v}\,\frac{h(w^{-1})}{|1-w|}\,d^*w$$ We refer to [@Co-zeta; @Meyer; @CMbook] for a detailed treatment. The subgroups $\K^\times_v\subset C_\K=\GL_1(\A_\K)/\GL_1(\K)$ arise as isotropy groups. One can understand why the terms $\displaystyle \frac{h(w^{-1})}{|1-w|}$ occur in the trace formula by computing, formally as follows, the trace of the scaling operator $T=\urep_{w^{-1}}$ when working on the local field $\K_v$ completion of the global field $\K$ at the place $v$, one has $$T\xi(x)=\xi(w x)=\int k(x,y)\xi(y)dy\,$$ so that $T$ is given by the distribution kernel $k(x,y)=\delta(w x-y)$ and its trace is $$\Tr_{\rm distr}(T)=\int k(x,x)\,dx=\int \delta(w x-x)\,dx=\frac{1}{|w-1|}\int \delta(z)\,dz=\frac{1}{|w-1|}$$
When working at the level of adeles one treats all places on the same footing and thus there is an overall minus sign in front of the spectral contribution. Thus the Riemann spectrum appears naturally as an absorption spectrum from the adele class space. As such, it is difficult to show that it is “real". While this solves the problem of giving a trace formula interpretation of the explicit formulas, there is of course still room for an interpretation as an emission spectrum. However from the adelic point of view it is unnatural to separate the contribution of the archimedean place.
The Riemann-Roch strategy: A Geometric Framework {#algeomattack}
================================================
In this section we shall present a geometric framework which has emerged over the years in our joint work with C. Consani and seems suitable in order to transpose the geometric proof of Weil to the case of RH. The aim is to apply the Riemann-Roch strategy of §\[rrstrat\]. The geometry involved will be of elaborate nature inasmuch as it relies on the following three theories:
1. [Noncommutative Geometry.]{}
2. [Grothendieck topoi.]{}
3. [Tropical Geometry.]{}
The limit $q\to 1$ and the Hasse-Weil formula {#counting}
---------------------------------------------
In [@Soule] ([[*cf.*]{}]{} §6), C. Soulé, motivated by [@Man-zetas] ([[*cf.*]{}]{} §1.5) and [@Steinberg; @Tits; @Ku; @Den1; @Den2; @Kapranov], introduced the zeta function of a variety $X$ over $\F_1$ using the [*polynomial*]{} counting function $N(x)\in\Z[x]$ associated to $X$. The definition of the zeta function is as follows $$\label{zetadefn}
\zeta_X(s):=\lim_{q\to 1}Z(X,q^{-s}) (q-1)^{N(1)},\qquad s\in\R$$ where $Z(X,q^{-s})$ denotes the evaluation at $T=q^{-s}$ of the Hasse-Weil exponential series $$\label{zetadefn1}
Z(X,T) := \exp\left(\sum_{r\ge 1}N(q^r)\frac{T^r}{r}\right)$$ For instance, for a projective space $\P^n$ one has $N(q)=1+q+\ldots +q^n$ and $$\zeta_{\P^n(\F_1)}(s) =\lim_{q\to 1} (q-1)^{n+1} \zeta_{\P^n(\F_q)}(s) = \frac{1}{\prod_0^n(s-k)}$$ It is natural to wonder on the existence of a “curve” $C$ suitably defined over $\F_1$, whose zeta function $\zeta_C(s)$ is the complete Riemann zeta function $\zeta_\Q(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ ([[*cf.*]{}]{} also [@Man-zetas]). The first step is to find a counting function $N(q)$ defined for $q\in [1,\infty)$ and such that gives $\zeta_\Q(s)$. But there is an obvious difficulty since as $N(1)$ represents the Euler characteristic one should expect that $N(1)=-\infty$ (since the dimension of $H^1$ is infinite). This precludes the use of and also seems to contradict the expectation that $N(q)\geq 0$ for $q\in (1,\infty)$. As shown in [@CC0; @CC1] there is a simple way to solve the first difficulty by passing to the logarithmic derivatives of both terms in equation and observing that the Riemann sums of an integral appear from the right hand side. One then gets instead of the equation: $$\label{logzetabis}
\frac{\partial_s\zeta_N(s)}{\zeta_N(s)}=-\int_1^\infty N(u)\, u^{-s}d^*u$$ Thus the integral equation produces a precise equation for the counting function $N_C(q)=N(q)$ associated to $C$: $$\label{special}
\frac{\partial_s\zeta_\Q(s)}{\zeta_\Q(s)}=-\int_1^\infty N(u)\, u^{-s}d^*u$$ One finds that this equation admits a solution which is a [*distribution*]{} and is given with $ \varphi(u):=\sum_{n<u}n\,\Lambda(n)$, by the equality $$\label{Nu}
N(u)=\frac{d}{du}\varphi(u)+ \kappa(u)$$ where $\kappa(u)$ is the distribution which appears in the explicit formula , $$\int_1^\infty\kappa(u)f(u)d^*u=\int_1^\infty\frac{u^2f(u)-f(1)}{u^2-1}d^*u+cf(1)\,, \qquad c=\frac12(\log\pi+\gamma)$$ The conclusion is that the distribution $N(u)$ is positive on $(1,\infty)$ and is given by $$\label{fin2}
N(u)=u-\frac{d}{du}\left(\sum_{\rho\in Z}{\rm order}(\rho)\frac{u^{\rho+1}}{\rho+1}\right)+1$$ where the derivative is taken in the sense of distributions, and the value at $u=1$ of the term $\displaystyle{\omega(u)=\sum_{\rho\in Z}{\rm order}(\rho)\frac{u^{\rho+1}}{\rho+1}}$ is given by $\frac 12+ \frac \gamma 2+\frac{\log4\pi}{2}-\frac{\zeta'(-1)}{\zeta(-1)}
$.
![ []{data-label="figcounting"}](expl10.pdf){width="\textwidth"}
The primitive $J(u)=\frac{u^2}{2}-\omega(u)+u$ of $N(u)$ is an increasing function on $(1,\infty)$, but tends to $-\infty$ when $u\to 1+$ while its value $J(1)$ is finite. The tension between the positivity of the distribution $N(q)$ for $q>1$ and the expectation that its value $N(1)$ should be $N(1)=-\infty$ is resolved by the theory of distributions: $N$ is [*finite*]{} as a distribution, but when one looks at it as a function its value at $q=1$ is formally given by $$N(1)=2-\lim_{\epsilon\to 0}\frac{\omega(1+\epsilon)-\omega(1)}{\epsilon}\sim-\frac 12 E \log E,\qquad \ E=\frac 1\epsilon$$ which is $-\infty$ and in fact reflects, when $\epsilon\to 0$, the density of the zeros. Note that this holds independently of the choice of the principal value in the explicit formulas. This subtlety does not occur for function fields $\K$ since their module $\mmod(\K)$ is discrete so that distributions and functions are the same thing. There is one more crucial nuance between the case $\K=\Q$ and the function fields: the distribution $\kappa(u)$ which is the archimedean contribution to $N(u)$ in , does not fulfill the natural inequality $N(q)\leq N(q^r)$ expected of a counting function. This is due to the terms $\vert 1-u\vert^{-1}$ in the Weil explicit formula, which as explained in §\[sectweilexpl\] contribute non-trivially at the archimedean place, and indicate that the counting needs to take into account an ambient larger space and transversality factors as in [@gui]. In fact, we have seen in Section \[adeleclass\] that the noncommutative space of adele classes of a global field provides a framework to interpret the explicit formulas of Riemann-Weil in number theory as a trace formula, and that the geometric contributions give the right answer. In [@CC1], we showed that the quotient $$\label{doublequot}
X_\Q:=\Q^\times\backslash \A_\Q/\hatz$$ of the adele class space $\Q^\times\backslash \A_\Q$ of the rational numbers by the maximal compact subgroup $\hatz$ of the idele class group, gives by considering the induced action of $\R_+^\times$, the above counting distribution $N(u)$, $u\in [1,\infty)$, which determines, using the Hasse-Weil formula in the limit $q\to 1$, the complete Riemann zeta function. The next step is to understand that the action of $\R_+^\times$ on the space $X_\Q$ is in fact the action of the Frobenius automorphisms $\fr_\lambda$ on the points of the arithmetic site– an object of algebraic geometry–over $\rmax$. To explain this we first need to take an excursion in the exotic world of “characteristic one".
The world of characteristic $1$ {#sectchar1}
-------------------------------
The key words here are: Newton polygons, Thermodynamics, Legendre transform, Game theory, Optimization, Dequantization, Tropical geometry. One alters the basic operation of addition of positive real numbers, replacing $x+y$ by $x\vee y:=\max(x,y)$. When endowed with this operation as addition and with the usual multiplication, the positive real numbers become a semifield $\rmax$. It is of characteristic $1$, [[*i.e.*]{} ]{}$1\vee 1=1$ and contains the smallest semifield of characteristic $1$, namely the Boolean semifield $\B=\{0,1\}$. Moreover, $\rmax$ admits non-trivial automorphisms and one has $${\rm Gal}_\B(\rmax):=\Aut_\B(\rmax)=\R_+^*, \ \ \fr_\lambda(x)=x^\lambda \qqq x\in \rmax, \ \lambda \in \R_+^*$$ thus providing a first glimpse of an answer to Weil’s query in [@Weilcdc] of an algebraic framework in which the connected component of the idele class group would appear as a Galois group. More generally, for any abelian ordered group $H$ we let $H_{\rm max}=H\cup \{-\infty\}$ be the semifield obtained from $H$ by the max-plus construction, [[*i.e.*]{} ]{}the addition is given by the max, and the multiplication by $+$. In particular $\R_{\rm max}$ is isomorphic to $\rmax$ by the exponential map ([[*cf.*]{}]{} [@Gaubert]). Historically, and besides the uses of $\rma$ in idempotent analysis and tropical geometry which are discussed below, an early use of $\rma$ occurred in the late fifties in the work of R. Cuninghame-Green in Birmingham, who established the spectral theory of irreducible matrices with entries in $\rma$ ([[*cf.*]{} ]{}[@Cunni]) and in the sixties, in Leningrad, where Vorobyev used the $\rma$ formalism in his work motivated by combinatorial optimization, and proved a fundamental covering theorem. A systematic use of the $\rma$ algebra was developed by the INRIA group at the beginning of the 80’s in their work on the modelization of discrete event systems [@Gaubert2]. We refer to [@Gaubert; @Gaubert1] for a more detailed history of the subject, and for overwhelming evidence of its relevance in mathematics. We shall just give here a sample of this evidence starting by a really early occurrence in the work of C.G.J. Jacobi[^9] and hoping to convince the reader that it would be a mistake to dismiss this algebraic formalism and the analogy with ordinary algebra as trivial.
### Optimization, Jacobi
One of the early instances, around 1840, of the use of matrices over $\R_{\rm max}$ is the work of C.G.J. Jacobi [@Jacobi] on optimal assignment problems, where he states
> **
>
> Problema
>
> Disponantur nn quantitates $h_k^{(i)}$ quaecunque in schema Quadrati, ita ut habeantur n series horizontales et n series verticales, quarum quaeque est n terminorum. Ex illis quantitatibus eligantur n transversales, [[*i.e.*]{} ]{}in seriebus horizontalibus simul atque verticalibus diversis positae, quod fieri potest n! modis; ex omnibus illis modis quaerendus est is, qui summam n numerorum electorum suppeditet maximam.
In other words, given a square matrix $m_{ik}=h_k^{(i)}$ he looks for the maximum over all permutations $\sigma$ of the quantity $\sum m_{j\sigma(j)}$. Using the algebraic rules of $\rma$ one checks that he is in fact computing the analogue of the determinant for the matrix $m_{ik}$. In fact the perfect definition of the determinant is more subtle and was obtained in the work of Gondran-Minoux [@GM1], instead of $\max\sum m_{j\sigma(j)}$ where $\sigma$ runs over all permutations, one uses the signature of permutations and considers the pair $$({\rm det}_+(m_{ik}),{\rm det}_-(m_{ik})), \ \ {\rm det}_\pm(m_{ik})=\max \sum_{{\rm sign}(\sigma)=\pm} m_{j\sigma(j)}$$ The remarkable fact is that the Cayley-Hamilton theorem now holds, as the equality of two terms $P_+(m)=P_-(m)$ corresponding to the characteristic polynomial $P=(P_+,P_-)$. Each of the terms $P_\pm(m)\in M_n(\rma)$ is computed from the original matrix $m\in M_n(\rma)$ using the rules of matrices with entries in $\rma$ which turn $M_n(\rma)$ into a semiring.
### Idempotent analysis
The essence of the theory of semiclassical analysis in physics rests in the comparison of quantum systems with their semiclassical counterpart, [@GS; @Gutz; @Gutz1; @Maslov1; @Berry]. In the eighties V. P. Maslov and his collaborators developed a satisfactory algebraic framework which encodes the semiclassical limit of quantum mechanics. They called it idempotent analysis. We refer to [@Maslov; @Lit] for a detailed account and just mention briefly some salient features here. The source of the variational formulations of mechanics in the classical limit is the behavior of sums of exponentials $$\sum e^{-\frac{ S_j}{ \hbar}} \sim e^{-\frac{ \inf S_j}{ \hbar}}, \ \ \text{when}\ \ \hbar\to 0$$ which are, when $\hbar\to 0$, dominated by the contribution of the minimum of $S$. The starting observation is that one can encode this fundamental principle by simply conjugating the addition of numbers by the power operation $x\mapsto x^\epsilon$ and passing to the limit when $\epsilon\to 0$. The new addition of positive real numbers is $$\lim_{\epsilon \to 0}\left(x^{\frac 1\epsilon}+y^{\frac 1\epsilon}\right)^\epsilon=\max \{x,y\}=x\vee y$$ and one recovers $\rmax$ as the natural home for semiclassical analysis. The superposition principle of quantum mechanics, [[*i.e.*]{} ]{}addition of vectors in Hilbert space, now makes sense in the limit and moreover the “fixed point argument" proof of the Perron-Frobenius theorem works over $\rmax$ and shows that irreducible compact operators have one and only one eigenvalue[^10], thus reconciling classical determinism with the quantum variability. But the most striking discovery of this school of Maslov, Kolokolstov and Litvinov [@Maslov; @Lit] is that the Legendre transform which plays a fundamental role in all of physics and in particular in thermodynamics in the nineteenth century, is simply the Fourier transform in the framework of idempotent analysis!
The contact between the INRIA school and the Maslov school was established in 92 when Maslov was invited in the Seminar of Jacques Louis Lions in College de France. At the BRIMS HP-Labs workshop on Idempotency in Bristol (1994) organized by J. Gunawardena, several of the early groups of researchers in the field were there, and an animated discussion took place on how the field should be named. The names “max-plus", “exotic", “tropical", “idempotent" were considered, each one having its defaults.
### Tropical geometry, Riemann-Roch theorems and the chip firing game
The tropical semiring $\N_{\rm min}=\N\cup\{\infty\}$ with the operations $\min$ and $+$ was introduced by Imre Simon in [@Simon] to solve a decidability problem in rational language theory. His work is at the origin of the term “tropical" used in tropical geometry which is a vast subject, see [[*e.g.*]{} ]{}[@Gelfand; @Kap1; @Mik; @Sturm]. We refer to [@virotagaki] for an excellent introduction starting from the sixteenth Hilbert problem. In its simplest form ([[*cf.*]{} ]{}[@GK]) a tropical curve is given by a metric graph $\Gamma$ ([[*i.e.*]{} ]{}a graph with a usual line metric on its edges). The natural structure sheaf on $\Gamma$ is the sheaf $\cO$ of real valued functions which are continuous, convex, piecewise affine with integral slopes. The operations on such functions are given by the pointwise operations of $\rma$-valued functions, [[*i.e.*]{} ]{}$(f\vee g)(x)=f(x)\vee g(x)$ for all $x\in \Gamma$ and similar for the product which is given by pointwise addition. One also adjoins the constant $-\infty$ which plays the role of the zero element in the semirings of sections. One proceeds as in the classical case with the construction of the sheaf $\cK$ of semifields of quotients and finds the same type of functions as above but no longer convex. Cartier divisors make sense and one finds that the order of a section $f$ of $\cK$ at a point $x\in\Gamma$ is given by the sum of the (integer valued) outgoing slopes. The conceptual explanation of why the discontinuities of the derivative should be interpreted as zeros or poles is due to Viro, [@viro] who showed that it follows automatically if one understands that[^11] the sum $x\vee x$ of two equal terms in $\rma$ should be viewed as ambiguous with all values in the interval $[-\infty, x]$ on equal footing. In their work Baker and Norine [@BN] proved in the discrete set-up of graphs (where $g$ is the genus and $K$ the canonical divisor) the Riemann-Roch equality in the form $$\label{rr1}
r(D)-r(K-D)=\Deg(D)-g+1$$ where by definition $r(D):=\max \{k\mid H^0(D-\tau )\neq \{-\infty\}\qqq \tau \geq 0,\ \Deg(\tau)=k\}$ and $H^0(D)$ is the $\rma$-module of global sections $f$ of the associated sheaf $\cO_D$ [[*i.e.*]{} ]{}sections of $\cK$ such that $D+(f)\geq 0$. The essence of the proof of [@BN] is that the inequality ${\rm Deg}(D)\geq g$ for a divisor implies $H^0(D)\neq \{-\infty\}$. Once translated in the language of the chip firing game ([[*op.cit.*]{} ]{}), this fact is equivalent to the existence of a winning strategy if one assumes that the total sum of dollars attributed to the vertices of the graph is $\geq g$ where $g$ is the genus. We refer to [@GK; @MZ] for variants of the above Riemann-Roch theorem, and to [@Dhar; @Shor; @PostS] for early occurrences of these ideas in a different context (including sandpile models and parking functions!).
The arithmetic and scaling sites
--------------------------------
### The arithmetic site and Frobenius correspondences
The [*arithmetic site*]{} [@CC; @CCas1] is an object of algebraic geometry involving two elaborate mathematical concepts: the notion of topos and of (structures of) characteristic $1$ in algebra. A nice fact ([[*cf.*]{}]{} [@Golan]) in characteristic $1$ is that, provided the semiring $R$ is ([[*i.e.*]{} ]{}equivalently if it injects in its semifield of fractions) the map $x\mapsto x^n=\fr_n(x)$ is, for any integer $n\in \nt$, an injective endomorphism $\fr_n$ of $R$. One thus obtains a canonical action of the semigroup $\nt$ on any such $R$ and it is thus natural to work in the topos $\wnt$ of sets endowed with an action of $\nt$.
\[site\] The arithmetic site $\aarith=\arith$ is the topos $\wnt$ endowed with the [*structure sheaf*]{} $\cO:=\zmax$ viewed as a semiring in the topos using the action of $\nt$ by the Frobenius endomorphisms.
The topological space underlying the arithmetic site is the Grothendieck topos of sets endowed with an action of the multiplicative monoïd $\nt$ of non-zero positive integers. As we have seen above the semifield $\rmax$ of tropical real numbers admits a one parameter group of Frobenius automorphisms $\fr_\lambda$, $\lambda\in \R_+^\times$, given by $\fr_\lambda(x)=x^\lambda$ $\forall x\in \rmax$. Using a straightforward extension in the context of semi-ringed topos of the classical notion of algebraic geometry of a point over a ring, one then gets the following result which gives the bridge between the noncommutative geometry and topos points of view:
[@CC; @CCas1] \[structure3z\] The set of points of the arithmetic site $\aarith$ over $\rmax$ is canonically isomorphic with $X_\Q=\Q^\times\backslash \A_\Q/\hatz$. The action of the Frobenius automorphisms $\fr_\lambda$ of $\rmax$ on these points corresponds to the action of the idele class group on $X_\Q=\Q^\times\backslash \A_\Q/\hatz$.
The square of the arithmetic site is the topos $\wntb$ endowed with the structure sheaf defined globally by the multiplicatively cancellative semiring associated to the tensor square $\nbo$ over the smallest Boolean semifield of characteristic one. In this way one obtains the semiring whose elements are Newton polygons and whose operations are given by the convex hull of the union and the sum. The points of the square of the arithmetic site over $\rmax$ coincide with the product of the points of the arithmetic site over $\rmax$. Then, we describe the Frobenius correspondences $\Psi(\lambda)$ as congruences on the square parametrized by positive real numbers $\lambda\in \R_+^\times$.
In the context of semirings, the congruences [[*i.e.*]{} ]{}the equivalence relations compatible with addition and product, play the role of the ideals in ring theory. The Frobenius correspondences $\Psi(\lambda)$, for a rational value of $\lambda$, are deduced from the diagonal of the square, which is described by the product structure of the semiring, by composition with the Frobenius endomorphisms. We interpret these correspondences geometrically, in terms of the congruence relation on Newton polygons corresponding to their belonging to the same half planes with rational slope $\lambda$. These congruences continue to make sense also for irrational values of $\lambda$ and are described using the best rational approximations of $\lambda$, while different values of the parameter give rise to distinct congruences. The composition of the Frobenius correspondences is given for $\lambda, \lambda' \in \R_+^\times$ such that $\lambda\lambda'\notin \Q$ by the rule [@CC; @CCas1] $$\label{compideps}
\Psi(\lambda)\circ \Psi(\lambda')=\Psi(\lambda\lambda')$$ The same equality still holds if $\lambda$ and $\lambda'$ are rational numbers. When $\lambda, \lambda'$ are irrational and $\lambda\lambda'\in \Q$ one has $$\label{compideps1}
\Psi(\lambda)\circ \Psi(\lambda')=\id_\epsilon\circ \Psi(\lambda\lambda')$$ where $\id_\epsilon$ is the tangential deformation of the identity correspondence.
### The scaling site and Riemann-Roch theorems
The Scaling Site $\scal1$, [@CCss], is the algebraic geometric space obtained from the arithmetic site $\aarith$ of [@CC; @CCas1] by extension of scalars from the Boolean semifield $\B$ to the tropical semifield $\rmax$. The points of $\scal1$ are the same as the points $\aarith(\rmax)$ of the arithmetic site over $\rmax$. But $\scal1$ inherits from its structural sheaf a natural structure of tropical curve, in a generalized sense, allowing one to define the sheaf of rational functions and to investigate an adequate version of the Riemann-Roch theorem in characteristic $1$. In [@CCss], we tested this structure by restricting it to the periodic orbits of the scaling flow, [[*i.e.*]{} ]{}the points over the image of $\Spec\Z$ under the canonical morphism of toposes $\Theta:\spz\to \aarith$ ([[*cf.*]{}]{} [@CCas1], §5.1). We found that for each prime $p$ the corresponding circle of length $\log p$ is endowed with a quasi-tropical structure which turns this orbit into the analogue $C_p=\R_+^*/p^\Z$ of a classical elliptic curve $\C^*/q^\Z$. In particular rational functions, divisors, etc all make sense. A new feature is that the degree of a divisor can now be any real number. The Jacobian of $C_p$ ([[*i.e.*]{} ]{}the quotient $J(C_p)$ of the group of divisors of degree $0$ by principal divisors) is a cyclic group of order $p-1$. For each divisor $D$ there is a corresponding Riemann-Roch problem with solution space $H^0(D)$ and the continuous dimension $\cdim(H^0(D))$ of this $\rma$-module is defined as the limit $$\label{rr1}
\cdim(H^0(D)):=\lim_{n\to \infty} p^{-n}\tdim(H^0(D)^{p^n})$$ where $H^0(D)^{p^n}$ is a natural filtration and $\tdim(\cE)$ is the topological dimension of an $\rma$-module $\cE$. One has the following Riemann-Roch formula [@CCss],
\[RRperiodic\] $(i)$ Let $D\in \div(C_p)$ be a divisor with $\deg(D)\geq 0$. Then the limit in converges and one has $\cdim(H^0(D))=\deg(D)$.$(ii)$ The following Riemann-Roch formula holds $$\cdim(H^0(D))-\cdim(H^0(-D))=\deg(D)\qqq D\in \div(C_p)$$
The appearance of arbitrary positive real numbers as continuous dimensions in the Riemann-Roch formula is due to the density in $\R$ of the subgroup $H_p\subset \Q$ of fractions with denominators a power of $p$. This outcome is the analogue in characteristic $1$ of what happens for modules over matroid $C^*$-algebras and the type II normalized dimensions as in [@dix].
Here $f(\lambda)$ is a real valued function with compact support of the variable $\lambda\in \R_+^*$ and $\inter(f,f)$ is as in . More precisely $D.D$ should be obtained as the intersection number of $D\circ \tilde D$ (defined using composition of correspondences) with the diagonal $\Delta$ and hence as a suitably defined distributional trace as for the counting function $N(u)$ of §\[counting\] so that $\frac 12 D(f).D(f)=\inter(f,f)$ with the notations of . So far the Riemann-Roch formula in tropical geometry is limited to curves and there is no Serre duality or good cohomological version of $H^j$ for $j\neq 0$, but in the above context one can hope that a Riemann-Roch inequality of the type , [[*i.e.*]{} ]{}of the form $$\cdim(H^0(D))+\cdim(H^0(-D))\geq \frac 12 D.D$$ would suffice to apply the strategy of Section \[rrstrat\] to prove the key inequality .
---------------------------------------------- ----------------------------------------------------
$C$ curve over $\F_q$ Arithmetic Site $\aarith= ( \wnt,\zmax)$ over $\B$
Structure sheaf $\cO_{C}$ Structure sheaf $\zmax$
$\bar C=C\otimes_{\F_q} \bar\F_q$ Scaling Site $\scal1=(\rnt,\cO)$ over $\rmax$
$C( \bar\F_q)=\bar C(\bar\F_q)$ $\aarith(\rmax)=\scal1(\rmax)$
Galois action on $C( \bar\F_q)$ Galois action on $\aarith(\rmax)$
Structure sheaf $\cO_{\bar C}$ Structure sheaf $\cO=\zmax\hat\otimes_\B\rmax$
of $\bar C=C\otimes_{\F_q} \bar\F_q$ piecewise affine convex functions, integral slopes
Sheaf $\cK$ of rational functions $\bar C$ Sheaf $\cK$ of piecewise affine functions
on $\bar C=C\otimes_{\F_q} \bar\F_q$ with integral slopes
Cartier divisors $=$ sections of $\cK/\cO^*$ Sections of $\cK/\cO^*$
$X= \bar C\times \bar C$ $\scal1\times \scal1$
$D=\sum a_k\Psi^k$ $D=\int \Psi(\lambda) f(\lambda) d^*\lambda$
Frobenius correspondence $\Psi$ Correspondences $\Psi(\lambda)$
---------------------------------------------- ----------------------------------------------------
: Here are a few entries in the analogy:[]{data-label="tab:1"}
Absolute Algebra and the sphere spectrum
========================================
Even if the Riemann-Roch strategy of Section \[algeomattack\] happened to be successful, one should not view the arithmetic and scaling sites for more than what they are, namely a semiclassical shadow of a still mysterious structure dealing with compactifications of $\Spec \Z$. An essential role in the unveiling of this structure should be played, for the reasons briefly explained below, by the discovery made by algebraic topologists in the 80’s (see [@DGM]) that in their world of “spectra" (in their sense) the sphere spectrum is a generalized ring $\sss$ which is more fundamental than the ring $\Z$ of integers, while the latter becomes an $\sss$-algebra. Over the years the technical complications of dealing with spaces “up to homotopy" have greatly been simplified, in particular for the smash product of spectra. For the purpose of arithmetic applications, Segal’s $\Gamma$-rings provide a very simple algebraic framework which succeeds to unify several attempts pursued in recent times in order to define the meaning of “absolute algebra". In particular it contains the following three possible categories that had been considered previously to handle this unification: namely the category $\mathbf\cM$ of monoïds as in [@deit; @deit1; @CC1; @CC3], the category $\mathbf\cH$ of hyperrings of [@CC2; @CC4; @CC5] and finally the category $\mathbf\cS$ of semirings as in [@C; @CC; @CCas1; @CCss]. Thanks to the work of L. Hesselholt and I. Madsen briefly explained below in §\[topcyc\] one now has at disposal a candidate cohomology theory in the arithmetic context: topological cyclic homology.
Segal’s $\Gamma$-rings
----------------------
Let $\Gamma^{\rm op}$ be the small, full subcategory of the category of finite pointed sets whose objects are the the pointed finite sets[^12] $k_+:=\{0,\ldots ,k\}$, for $k\geq 0$. The object $0_+$ is both initial and final so that $\gop$ is a [*pointed category*]{}. The notion of a discrete $\Gamma$-space, [[*i.e.*]{} ]{}of a $\Gamma$-set is as follows:
\[defngamset\] A $\Gamma$-set $F$ is a functor $F:\gop\longrightarrow\Ses$ between pointed categories from $\gop$ to the category of pointed sets.
The morphisms $\Hom_\gop(M,N)$ between two $\Gamma$-sets are natural transformations of functors. The category $\gam$ of $\Gamma$-sets is a symmetric closed monoidal category ([[*cf.*]{}]{} [@DGM], Chapter II). The monoidal structure is given by the smash product (denoted $X\wedge Y$) of $\Gamma$-sets which is a Day product. The closed structure property is shown in [@Lyd] ([[*cf.*]{}]{} also [@DGM] Theorem 2.1.2.4). The specialization of Definition 2.1.4.1. of [@DGM] to the case of $\Gamma$-sets yields the following
\[defnsalg\] A $\Gamma$-ring $\mathcal A$ is a $\Gamma$-set $\mathcal A: \gop\longrightarrow\Ses$ endowed with an associative multiplication $\mu:\cA \wedge \cA\to \cA$ and a unit $1: \sss\to \cA$, where $
\sss:\gop\longrightarrow \Ses
$ is the inclusion functor.
Thus $\Gamma$-rings[^13] make sense and the sphere spectrum corresponds to the simplest possible $\Gamma$-ring: $\sss$. One can then easily identify the category $\gam$ of $\Gamma$-sets with the category $\catmo(\sss)$ of $\sss$-modules. In [@Durov], N. Durov developed a geometry over $\F_1$ intended for Arakelov theory applications by using monads as generalizations of classical rings. While in the context of [@Durov] the tensor product $\Z\otimes_{\F_1} \Z$ produces an uninteresting output isomorphic to $\Z$, we showed in [@CCsalg] that the same tensor square, re-understood in the theory of $\sss$-algebras, provides a highly non-trivial object. The Arakelov compactification of $\Spec\Z$ is endowed naturally with a structure sheaf of $\sss$-algebras and each Arakelov divisor provides a natural sheaf of modules over the structure sheaf. This new structure of $\overline{\Spec\Z}$ over $\sss$ endorses a one parameter group of weakly invertible sheaves whose tensor product rules are the same as the composition rules , of the Frobenius correspondences over the arithmetic site [@CC; @CCas1]. The category $\catmo(\sss)$ of $\sss$-modules is not an abelian category and thus the tools of homological algebra need to be replaced along the line of the Dold-Kan correspondence, which for an abelian category $\cA$ gives the correspondence between chain complexes in $\geq 0$ degrees and simplicial objects [[*i.e.*]{} ]{}objects of $\cA^\dop$.
We refer to Table \[tab:2\] for a short dictionary. The category of $\Gamma$-spaces is the central tool of [@DGM], while the relations between algebraic $K$-theory and topological cyclic homology is the main topic.
-------------------------------------- ------------------------------------
$X\in Ch_{\geq 0}(\cA)$ $M\in \catmo(\sss)^\dop$
$H_q(X)$ $\pi_q(M)$
$H_q(f):H_q(X)\simeq H_q(Y)$ $\pi_q(f):\pi_q(M)\simeq \pi_q(N)$
quasi-isomorphism weak equivalence
$f_n:X_n\stackrel{\subset}{\to} Y_n$ cofibration
+ projective cokernel (stable)
$f_n:X_n\to Y_n$ stable
surjective if $n>0$ fibration
-------------------------------------- ------------------------------------
: Short dictionary homology–homotopy[]{data-label="tab:2"}
Topological cyclic homology {#topcyc}
---------------------------
As shown in [@CCsalg] the various attempts done in recent times to develop “absolute algebra" are all unified by means of the well established concept of $\sss$-algebra, [[*i.e.*]{} ]{}of $\Gamma$-rings. Moreover ([[*cf.*]{} ]{}[@DGM]) this latter notion is at the root of the theory of topological cyclic homology which can be understood as cyclic homology over the absolute base $\sss$, provided one uses the appropriate Quillen model category. In particular, topological cyclic homology is now available to understand the new structure of $\overline{\Spec\Z}$ using its structure sheaf and modules. The use of cyclic homology in the arithmetic context is backed up by the following two results:
- At the archimedean places, and after the initial work of Deninger [@Den1; @Den2] to recast the archimedean local factors of arithmetic varieties [@Se3] as regularized determinants, we showed in [@CC6] that cyclic homology in fact gives the correct infinite dimensional (co)homological theory for arithmetic varieties. The key operator $\Theta$ in this context is the generator of the $\lambda$-operations $\Lambda(k)$ [@Loday; @weibel; @Weibelcris] in cyclic theory. More precisely, the action $u^\Theta$ of the multiplicative group $\R_+^\times$ generated by $\Theta$ on cyclic homology, is uniquely determined by its restriction to the dense subgroup $\Q_+^\times\subset \R_+^\times$ where it is given by the formula $$\label{actiontheta}
k^\Theta|_{HC_n}=\Lambda(k)\,k^{-n} \qqq n\geq 0, \, \ k\in \N^\times\subset \R_+^\times$$ Let $X$ be a smooth, projective variety of dimension $d$ over an algebraic number field $\K$ and let $\nu\vert\infty$ be an archimedean place of $\K$. Then, the action of the operator $\Theta$ on the archimedean cyclic homology ${HC^{\rm ar}}$ ([[*cf.*]{} ]{}[@CC6]) of $X_\nu$ satisfies $$\label{dettheta0}
\prod_{0\leq w \leq 2d} L_\nu(H^w(X),s)^{(-1)^{w}}=\frac{det_\infty(\frac{1}{2\pi}(s-\Theta)|_{{HC^{\rm ar}}_{\rm od}(X_\nu)})}{
det_\infty(\frac{1}{2\pi}(s-\Theta)|_{{HC^{\rm ar}}_{\rm ev}(X_\nu)})}$$ The left-hand side of is the product of Serre’s archimedean local factors of the complex $L$-function of $X$ ([[*cf.*]{}]{}[@Se3]). On the right-hand side, $det_\infty$ denotes the regularized determinant and one sets $${HC^{\rm ar}}_{\rm ev}(X_\nu)=\bigoplus_{n=2k\ge 0} {HC^{\rm ar}}_{n}(X_\nu), \ {HC^{\rm ar}}_{\rm od}(X_\nu)=\bigoplus_{n=2k+1\ge 1} {HC^{\rm ar}}_{n}(X_\nu)$$
- L. Hesselholt and I. Madsen have shown ([[*cf.*]{}]{} [[*e.g.*]{} ]{}[@HM; @H; @H1]) that the de Rham-Witt complex, an essential ingredient of crystalline cohomology ([[*cf.*]{} ]{}[@Berth; @Illusie]), arises naturally when one studies the topological cyclic homology of smooth algebras over a perfect field of finite characteristic. One of the remarkable features in their work is that the arithmetic ingredients such as the Frobenius and restriction maps are naturally present in the framework of topological cyclic homology. Moreover L. Hesselholt has shown [@Hessel1] how topological periodic cyclic homology with its inverse Frobenius operator may be used to give a cohomological interpretation of the Hasse-Weil zeta function of a scheme smooth and proper over a finite field in the form ([[*cf.*]{} ]{}[@Hessel1]): $$\label{detthetaH}
\zeta(X,s)=\frac{det_\infty(\frac{1}{2\pi}(s-\Theta)|_{{TP}_{\rm od}(X)})}{
det_\infty(\frac{1}{2\pi}(s-\Theta)|_{{TP}_{\rm ev}(X)})}$$
One of the stumbling blocks in order to reach a satisfactory cohomology theory is the problem of coefficients. Indeed, the natural coefficients at a prime $p$ for crystalline cohomology are an extension of $\Q_p$ and it is traditional to relate them with complex numbers by an embedding of fields. Similarly, uses an embedding of the Witt ring $W(\F_q)\to \C$. To an analyst it is clear that since such embeddings cannot be measurable[^14] they will never be effectively constructed. This begs for a better construction, along the lines of Quillen’s computation of the algebraic $K$-theory of finite fields, which instead would only involve the ingredient of the Brauer lifting, [[*i.e.*]{} ]{}a group injection of the multiplicative group of $\bar \F_p$ as roots of unity in $\C$.
Final remarks
-------------
The Riemann hypothesis has been extended far beyond its original formulation to the question of localization of the zeros of $L$-functions. There are a number of constructions of $L$-functions coming from three different sources, Galois representations, automorphic forms and arithmetic varieties. Andr' e Weil liked to compare ([[*cf.*]{} ]{}[@B2] §12 and also [@weilcomplete] vol. 1, p. 244–255 and vol. 2, p. 408–412), the puzzle of these three different writings to the task of deciphering hieroglyphics with the help of the Rosetta Stone. In some sense the $L$-functions play a role in modern mathematics similar to the role of polynomials in ancient mathematics, while the explicit formulas play the role of the expression of the symmetric functions of the roots in terms of the coefficients of the polynomial. If one follows this line of thought, the RH should be seen only as a first step since in the case of polynomials there is no way one should feel to have understood the zeros once one proves that they are, say, real numbers. In fact Galois formulated precisely the problem as that of finding all numerical relations between the roots of an equation, with the trivial ones being given by the symmetric functions, while the others, when determined, will reveal a complete understanding of the zeros as obtained, in the case of polynomials, by Galois theory. In a fragment, page 103, of the complete works of Galois [@Galois] concerning the memoir of February 1830, he delivers the essence of his theory:
> [ *Remarquons que tout ce qu’une équation numérique peut avoir de particulier, doit provenir de certaines relations entre les racines. Ces relations seront rationnelles c’est-à-dire qu’elles ne contiendront d’irrationnelles que les coefficients de l’équation et les quantités adjointes. De plus ces relations ne devront pas être invariables par toute substitution opérée sur les racines, sans quoi on n’aurait rien de plus que dans les équations littérales. Ce qu’il importe donc de connaître, c’est par quelles substitutions peuvent être invariables des relations entre les racines, ou ce qui revient au même, des fonctions des racines dont la valeur numérique est déterminable rationnellement.[^15]* ]{}
### Acknowledgement {#acknowledgement .unnumbered}
I am grateful to J. B. Bost for the reference [@Tatebbk], to J. B. Bost, P. Cartier, C. Consani, D. Goss, H. Moscovici, M. Th. Rassias, C. Skau and W. van Suijlekom for their detailed comments, to S. Gaubert for his help in Section 4.2 and to Lars Hesselholt for his comments and for allowing me to mention his forthcoming paper [@Hessel1].
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[^1]: Collège de France, 3 rue d’Ulm, Paris 75005 France. IHÉS, 35 Route de Chartres, Bures sur Yvette and Ohio State University, Columbus, Ohio; email : alain@connes.org
[^2]: Similar counting functions were already present in Chebyshev’s work
[^3]: More precisely Riemann writes $\sum_{\Re(\alpha)>0}\left(\li(x^{\frac 12+\alpha i})+\li(x^{\frac 12-\alpha i})\right)$ instead of $\sum_\rho \li(x^{\rho})$ using the symmetry $\rho\to 1-\rho$ provided by the functional equation, to perform the summation.
[^4]: See [@RZeta] Chapter VII for detailed support to Selberg’s comment
[^5]: My warmest thanks to Michael Th. Rassias for the communication
[^6]: one of the topics in which John Nash made fundamental contributions
[^7]: and at which the value of $F(u)$ is defined as the average of the right and left limits there
[^8]: where the argument of $r_n$ is either $0$ or $-\pi/2$
[^9]: I am grateful to S. Gaubert for pointing out this early occurrence
[^10]: as mentioned above, this result was obtained already for matrices in 1962 by R. Cuninghame-Green
[^11]: as seen when using $\rma$ as the target of a valuation
[^12]: where $0$ is the base point.
[^13]: equivalently $\sss$-algebras
[^14]: A measurable group homomorphism from $\Z_p^\times$ to $\C^\times$ cannot be injective
[^15]: In 2012 I had to give, in the French academy of Sciences, the talk devoted to the 200-th anniversary of the birth of Evariste Galois. On that occasion I read for the $n+1$-th time the book of his collected works and was struck by the pertinence of the above quote in the analogy with $L$-functions. In the case of function fields one is dealing with Weil numbers and one knows a lot on their Galois theory using results such as those of Honda and Tate [[*cf.*]{} ]{}[@Tatebbk].
|
---
abstract: 'We report radial-speed evolution of interplanetary coronal mass ejections (ICMEs) detected by the *Large Angle and Spectrometric Coronagraph* onboard the *Solar and heliospheric Observatory* (SOHO/LASCO), interplanetary scintillation (IPS) at 327 MHz,and *in-situ* observations. We analyzed solar-wind disturbance factor (*g*-value) data derived from IPS observations during 1997–2009 covering nearly whole period of Solar Cycle 23. By comparing observations from SOHO/LASCO, IPS, and *in situ*, we identified 39 ICMEs that could be analyzed carefully. Here, we defined two speeds \[${V_{\mathrm{SOHO}}}$ and ${V_{\mathrm{bg}}}$\], which are initial speed of the ICME and the speed of the background solar wind, respectively. Examination of these speeds yield the following results: i) Fast ICMEs (with $V_{\mathrm{SOHO}} - V_\mathrm{bg} > 500$ ${\mathrm{km~s^{-1}}}$) rapidly decelerate, moderate ICMEs (with $0$ $\mathrm{km~s^{-1}}$ $\le V_{\mathrm{SOHO}} - V_\mathrm{bg} \le 500$ ${\mathrm{km~s^{-1}}}$) show either gradually decelerating or uniform motion, and slow ICMEs (with $V_{\mathrm{SOHO}} - V_\mathrm{bg} <$ $0$ ${\mathrm{km~s^{-1}}}$) accelerate. The radial speeds converge on the speed of the background solar wind during their outward propagation. We subsequently find; ii) both the acceleration and deceleration are nearly complete by $0.79 \pm 0.04$ AU, and those are ended when the ICMEs reach a $489 \pm 21$ $\mathrm{km~s^{-1}}$. iii) For ICMEs with $V_{\mathrm{SOHO}} - V_\mathrm{bg} \ge$ $0$ ${\mathrm{km~s^{-1}}}$, *i.e.* fast and moderate ICMEs, a linear equation $a = -{\gamma}_{\mathrm{1}}(V - V_\mathrm{bg})$ with ${\gamma}_{\mathrm{1}} = 6.58 \pm 0.23 \times 10^{-6}$ ${\mathrm{s^{-1}}}$ is more appropriate than a quadratic equation $a = -{\gamma}_{\mathrm{2}}(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$ to describe their kinematics, where ${{\gamma}_\mathrm{1}}$ and ${{\gamma}_\mathrm{2}}$ are coefficients, and $a$ and $V$ are the acceleration and ICME speed, respectively, because the ${\chi^{\mathrm{2}}}$ for the linear equation satisfies the statistical significance level of 0.05, while the quadratic one does not. These results support the assumption that the radial motion of ICMEs is governed by a drag force due to interaction with the background solar wind. These findings also suggest that ICMEs propagating faster than the background solar wind are controlled mainly by the hydrodynamic Stokes drag.'
author:
- 'T. $^{1}$, M. $^{1}$, K. $^{1}$'
bibliography:
- 'fasticme.bib'
title: Radial Speed Evolution of Interplanetary Coronal Mass Ejections During Solar Cycle 23
---
\
**Abbreviations**\
ACE *Advanced Composition Explorer*\
AU Astronomical unit\
CC Correlation coefficient\
CDAW Coordinated Data Analysis Workshop\
CME Coronal mass ejection\
CPI *Comprehensive Plasma Instrumentation*\
ESA European Space Agency\
FOV Field-of-view\
GSFC Goddard Space Flight Center\
ICME Interplanetary coronal mass ejection\
IDED IPS disturbance event day\
IDEDs IPS disturbance event days\
IMP *Interplanetary Monitoring Platform*\
IPS Interplanetary Scintillation\
LASCO *Large Angle and Spectrometric Coronagraph*\
LOS Line-of-sight\
MIT *Massachusetts Institute of Technology Faraday Cup Experiment*\
NASA National Aeronautics and Space Administration\
OMNI Operating Missions as Nodes on the Internet\
SOHO *Solar and Heliospheric Observatory*\
STEL Solar-Terrestrial Environment Laboratory\
STEREO *Solar-Terrestrial Relations Observatory*\
SWE *Solar Wind Experiment*\
SWEPAM *Solar Wind Electron, Proton, and Alpha Monitor*
Introduction
============
Coronal mass ejections (CMEs) are transient events in which large amounts of plasma are ejected from the solar corona (*e.g.* ). Interplanetary counterparts of CMEs are called interplanetary coronal mass ejections (ICMEs). Since ICMEs seriously affect the space environment around the Earth, understanding of their fundamental physics, *e.g.* generation, propagation, and interaction with the Earth’s magnetosphere, is very important for space-weather forecasting (*e.g.* ; ). In particular, the dynamics of ICME propagation is one of the key pieces of information for predicting geomagnetic storms.
Propagation of ICMEs has been studied by various methods. Earlier studies combining space-borne coronagraphs with *in-situ* observations revealed that ICME speeds significantly evolve between near-Sun and 1 AU. reported the correlation between CMEs and interplanetary disturbances using the *P78-1/Solwind* coronagraph, the *Helios-1* and -2 solar probes, and a ground-based H$\alpha$ coronagraph. He showed that fast CMEs associated with flares exhibit no acceleration into interplanetary space, while slow CMEs related to prominence eruptions accelerate. examined the relation between propagation speeds of CMEs observed by the *Solwind* coronagraph and *Solar Maximum Mission* coronagraph/polarimeter and those of the ICMEs observed by the *Helios-1* and *Pioneer Venus Orbiter* for 31 CMEs and their associated ICMEs. They found a good correlation between the speeds of CMEs and those of ICMEs observed in interplanetary space between 0.7 and 1 AU. They also found that the speeds of most ICMEs range from 380 ${\mathrm{km~s^{-1}}}$ to 600 ${\mathrm{km~s^{-1}}}$, while CME speeds show a wider range of from $\approx 10$ ${\mathrm{km~s^{-1}}}$ to 1500 ${\mathrm{km~s^{-1}}}$. These findings suggest that the ICME speeds tend to converge to an average solar-wind speed as they propagate through interplanetary space. determined an effective acceleration for 28 CMEs observed by the *Large Angle and Spectrometric Coronagraph* (LASCO: ) onboard the *Solar and Heliospheric Observatory* (SOHO) spacecraft between 1996 and 1998. On the assumption that the acceleration is constant, they found a very good anti-correlation between the accelerations and initial speeds of CMEs, and a critical speed of 405 ${\mathrm{km~s^{-1}}}$; this value is close to the typical speed of the solar wind in the equatorial plane. Following this research, described an empirical model for predicting of arrival of the ICMEs at 1 AU; this model is based on their previous work [@Gopalswamy2000] and its accuracy is improved by allowing for cessation of the interplanetary acceleration before 1 AU. They showed that the acceleration cessation distance is 0.76 AU, and this result agrees reasonably well with observations by SOHO, *Advanced Composition Explorer* (ACE: ), and other spacecraft at 1 AU.
We expect that the acceleration or deceleration of ICMEs is controlled by a drag force caused by interaction between ICMEs and the solar wind. proposed an advanced model for the motion of ICMEs; this model considered the interaction with solar wind using a simple expression for the acceleration: $a = -{\gamma}_{\mathrm{1}}(V - V_\mathrm{bg})$, where ${\gamma}_{\mathrm{1}}$, $V$, and $V_\mathrm{bg}$ are the coefficient, ICME speed, and speed of the background solar wind, respectively. They also compared their model with a drag-acceleration model $a = -{\gamma}_{\mathrm{2}}(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$, where ${\gamma}_{\mathrm{2}}$ is the coefficient for this equation; this expression is known as the aerodynamic drag force (*e.g.* ; ). Both models have been tested by comparing with CME observations. studied the propagation of a CME that occurred on 5 April 2003 using observations by the SOHO/LASCO, the *Solar Mass Ejection Imager* onboard the *Coriolis* satellite, and the *Ulysses* spacecraft. derived the three-dimensional kinematics for three ICMEs detected between 2008 and 2009 using the *Solar-Terrestrial Relations Observatory-A* (STEREO-A) and -B spacecraft observations. examined the influence of the solar wind on the propagation of some ICMEs using the STEREO-A and -B spacecraft. Although the propagation of ICMEs has been studied by many investigators, their dynamics is still not well understood. This is mainly due to the lack of observational data about ICMEs between 0.1 and 1 AU. Almost all ICME observations are currently limited to the near-Earth area in the equatorial plane.
Remote sensing using radio waves is a suitable method for collecting global data on ICMEs. For example, derived kinematical parameters for 42 ICME/shocks from measurements of type-II radio emission. studied the shock propagation using Doppler-scintillation measurements of radio waves emitted from planetary spacecraft, and showed the speed profiles of shocks between 0.05 and 0.93 AU. In addition to these measurements, interplanetary scintillation (IPS) is a type of remote sensing. IPS is a phenomenon where signals from a point-like radio source, such as quasars and active galactic nuclei, fluctuate due to density irregularities in the solar wind [@Hewish1964]. IPS observations allow us to probe the inner heliosphere using many radio sources, and this is a useful means to study the global structure and propagation dynamics of ICMEs in the solar wind (*e.g.* ; ; ; , [-@Tokumaru2003]; ; ; ; ; ; see also ). For the kinematics of interplanetary disturbances, reported that the radial dependence of speed can be represented by a power-law function \[$V \approx R^{-\alpha}$\] with $\alpha$ in the range $0.25 < \alpha < 1$ from analysis of all-sky scintillationindices maps. examined radial evolution of 30 CMEs observed by SOHO/LASCO, ACE, and the Ooty radio-telescope between 1998 and 2004. He showed that most CMEs tend to attain the speed of the ambient flow at 1 AU and also reported a power-law form of radial-speed evolution for these events.
We take advantage of IPS observation to determine the ICME speed and acceleration. In the current study, we analyze the solar-wind disturbance factor (*g*-value) derived from IPS observations during 1997–2009 covering nearly the whole of Solar Cycle 23 and make a list of disturbance event days in the period. We define an “ICME” as a series of events including a near-Sun CME, an interplanetary disturbance, and a near-Earth ICME in this study. By comparing our list with that of CME/ICME pairs, we identify many events that are detected at three locations between the Sun and the Earth’s orbit, *i.e.* near-Sun, interplanetary space, and near-Earth, and derive their radial speed profiles. We then analyze the relationship between the acceleration and speed difference for the ICMEs. The outline of this article is as follows: Section \[observation\] describes the IPS observations made with the 327 MHz radio-telescope system of the Solar-Terrestrial Environment Laboratory (STEL), Nagoya University. Section \[method\] describes the criteria for ICME identification and the method for estimating ICME speeds and accelerations between the corona and 1 AU. Section \[results\] provides the radial-speed profiles of ICMEs and the analyses of the propagation properties. Section \[discussion\] discusses the results, while Section \[conclusion\] summarizes the main conclusions of our study.
STEL IPS Observation {#observation}
====================
STEL IPS observations have been carried out regularly since the early 1980s using multiple ground-based radio-telescope stations operated at 327 MHz (; ). The IPS observations at 327 MHz allow us to determine the solar-wind condition between 0.2 and 1 AU with a cadence of 24 hours. In our observations, nearly 30 radio sources within a solar elongation of $60^{\circ}$ are observed daily between April and December. The IPS observations on a given day are made when each radio source traverses the local meridian.
The solar-wind speed and disturbance factor, the so called “*g*-value” [@Gapper1982], are derived from IPS observations. A *g*-value is calculated for each source using the following equation: $$\label{eq.gvalue}
g = \frac{{\Delta}S}{{\Delta}S_{\mathrm{m}}(\varepsilon)},$$ where ${{\Delta}S}$ and ${{\Delta}S_{\mathrm{m}}(\varepsilon)}$ are the observed fluctuation level of radio signals and their yearly mean, respectively. ${{\Delta}S_{\mathrm{m}}(\varepsilon)}$ varies with the solar elongation angle \[$\varepsilon$\] for a line-of-sight (LOS) from an observed radio source to a telescope. When a radio signal is weakly scattered, the *g*-value is given by the following equation (, [-@Tokumaru2006]): $$\label{eq.gvalue2}
g^{2} = \frac{1}{K} \int_{0}^{\infty} \mathrm{d}z{\{}{\Delta}N_\mathrm{e}{\}}^{2}{\omega}(z),$$ here, $z$ is the distance along a LOS, ${N_\mathrm{e}}$ is the fluctuation level of solar-wind (electron) density, $K$ is the normalization factor based on the mean density fluctuation of the background solar wind, and ${\omega}(z)$ is the IPS weighting function [@Young1971]. We note that ${{\Delta}N_\mathrm{e}}$ is nearly proportional to the solar-wind density \[${N_\mathrm{e}}$\]; ${\Delta}N_\mathrm{e} \propto N_\mathrm{e}$ [@Coles1978], and the weak-scattering condition holds for $R > 0.2$ AU, where $R$ is the radial distance from the Sun.
****
A *g*-value represents the relative level of density fluctuation integrated along a LOS. For quiet solar-wind conditions, the *g*-value is around unity. With dense plasma or high turbulence as an ICME passes across a LOS, the *g*-value becomes greater than unity because of the ${{\Delta}N_\mathrm{e}}$ ($\propto N_\mathrm{e}$) increase. In contrast, a *g*-value less than unity indicates a rarefaction of the solar wind. Hence, detecting an abrupt increase in *g*-value is a useful means to detect an ICME.
The location of the LOS for a radio source exhibiting a *g*-value enhancement in the sky plane indicates a turbulent region is present. A sky-map of enhanced *g*-values for the sources observed in a day is called a “*g*-map” [@Gapper1982; @Hewish1986]. This map provides information on the spatial distribution of ICMEs. Figure 1 shows an example of a *g*-map for a CME event. A white-light difference image of a CME observed by the SOHO/LASCO-C2 coronagraph is shown in the left-hand panel of Figure \[fig1\]. As shown here, a bright balloon-like structure was observed on the northeast limb on 11 July 2000. This event was reported as an asymmetric halo CME in the SOHO/LASCO CME Catalog (; ; available at [cdaw.gsfc.nasa.gov/CME\_list/](cdaw.gsfc.nasa.gov/CME_list/)). The right-hand panel of Figure \[fig1\] is a *g*-map derived from our IPS observation on 12 July 2000. The center of the map corresponds to the location of the Sun, and the horizontal and vertical axes are parallel to the East–West and North–South directions, respectively. The concentric circles indicate the radial distances to the closest approach of the LOS of 0.3 AU, 0.6 AU, and 0.9 AU. The radial distance \[$r_\mathrm{IPS}$\] for each LOS is given by $r_\mathrm{IPS} = r_{\mathrm{E}} \sin{\varepsilon}$, where $r_\mathrm{E}$ is the distance between the Sun and the Earth, *i.e.* 1 AU and ${\varepsilon}$ is the solar elongation angle for the LOS. This calculation is based on the approximation that a large fraction of IPS is given by the wave scattering at the closest point to the Sun (the P-point) on a LOS [@Hewish1964]. Since ten LOSs between 0.4 and 0.7 AU in the eastern hemisphere (left-hand side of *g*-map) exhibit high *g*-values, a group of them is considered as the interplanetary counterpart of the 11 July 2000 CME event. This CME was also detected by *in-situ* observation at 1 AU on 13 July 2000 and reported as a near-Earth ICME [@Richardson2010]. In this way, a *g*-map can visualize an ICME between 0.2 and 1 AU. The *g*-value data have been available from our IPS observation since 1997 [@Tokumaru2000b]. To find the *g*-value enhancements due to ICMEs from the *g*-value data obtained between 1997 and 2009, we define criteria for the ICME identifications as mentioned in the next section.
Method
======
ICME Identification {#identification}
-------------------
First, we define disturbance days due to an ICME in the IPS data. In this determination, we consider a threshold *g*-value and the number of sources exhibiting the threshold or beyond. The average \[$a_{g}$\] and standard deviation \[${\sigma}_{g}$\] for the *g*-values obtained by STEL IPS observations between 1997 and 2009 are 1.07 and 0.47, respectively. From these, we regard a *g*-value for a disturbed condition on a given day to be $a_{g} + {\sigma}_{g}$ or more, and we decide to use 1.5 as this threshold. We also define an “observation day” as a day on which 15 or more sources are observed by our radio-telescope system; this minimum number is equal to half the mean number of sources observed in a day. In an observation day, when five or more sources showed a disturbed condition, we judge that a disturbance had occurred. Combining the above criteria, we define an “IPS disturbance event day” (IDED) as a day on which $g \ge 1.5$ sources numbered five or more on an observation day. Using this definition, we find 656 IDEDs in our period of research. From these, we eliminate periods with four or more consecutive IDEDs because they are likely related to co-rotating stream interaction regions [@Gapper1982]. However, we do not eliminate two periods including the 2000 Bastille Day (illustrated in Figure \[fig1\]) and 2003 Halloween events from among the IDEDs above, because consecutive disturbances in them are caused by successive CMEs (*e.g.* ; ). As a result, 159 out of 656 IDEDs are excluded, and the remaining 497 IDEDs are listed as candidates for ICME events.
Next, we examine the relationship between CME/ICME pairs and selected IDEDs. In this examination, we use the list of near-Earth ICMEs and associated CMEs compiled by . This includes 322 ICMEs associated with a halo or a partial halo or normal CMEs during Solar Cycle 23; here, “normal” means that the exterior of CME is neither a halo nor a partial halo. In the above study, CMEs were observed by the SOHO/LASCO coronagraphs, and ICMEs were detected by *in-situ* observation using spacecraft such as ACE and the *Interplanetary Monitoring Platform*-8 (IMP-8). We compare the list of IDEDs with that of ICMEs using the assumption that an ICME caused the IDED. When an IDED is between the appearance date of an associated CME and the detection date of a near-Earth ICME, we assume that the IDED was related to the ICME.
Using the above method, we find 66 IDEDs from our list that were probably related to ICMEs. However, we also find that 16 IDEDs of the 66 had multiple associated CMEs. For these 16 events, we identify the optimal one-to-one correspondence by comparing positions for LOS exhibiting high *g*-values in a *g*-map with the direction of the associated CME eruption in the LASCO field-of-view (FOV).
At the end of this selection, we identify 50 CMEs and their associated ICMEs that were detected by the SOHO/LASCO, IPS, and *in-situ* observations. For these, we estimate radial speeds and accelerations in interplanetary space using the method described in the next subsection.
Estimations of ICME Radial Speeds and Accelerations {#estimation}
---------------------------------------------------
The ICME radial speeds and accelerations are estimated in two interplanetary regions, *i.e.* the region between SOHO and IPS observations (the SOHO–IPS region, from 0.1 to ${\approx}$ 0.6 AU) and that between IPS and *in-situ* observations (the IPS–Earth region, from ${\approx}$ 0.6 to 1 AU). In these estimations, we assume that locations of LOS for disturbed sources in a *g*-map give the location of the ICME.
First, we calculate radial speeds at reference distances for each ICME. For each radio source of $g \ge 1.5$ in a *g*-map, distances \[${r_\mathrm{1}}$ and ${r_\mathrm{2}}$\] and radial speeds \[${v_\mathrm{1}}$ and ${v_\mathrm{2}}$\] are derived from the following equations: $$\label{eq.r1v1}
r_\mathrm{1} = \frac{r_\mathrm{S} + r_\mathrm{IPS}}{2},~
v_\mathrm{1} = \frac{r_\mathrm{IPS} - r_\mathrm{S}}{t_\mathrm{IPS} - T_\mathrm{SOHO}}~
\mbox{(for the SOHO--IPS region),}$$ and $$\label{eq.r2v2}
r_\mathrm{2} = \frac{r_\mathrm{IPS} + r_\mathrm{E}}{2},~
v_\mathrm{2} = \frac{r_\mathrm{E} - r_\mathrm{IPS}}{T_\mathrm{Earth} - t_\mathrm{IPS}}~
\mbox{(for the IPS--Earth region),}$$ respectively. Here, ${r_\mathrm{S}}$ is the minimum radius of SOHO/LASCO-C2 FOV, *i.e.* 0.009 AU, ${r_\mathrm{IPS}}$ is the radial distance of P-point on the LOS, ${r_\mathrm{E}}$ is the distance between the Sun and the Earth, *i.e.* 1 AU, ${T_\mathrm{SOHO}}$ is the appearance time of CME in the SOHO/LASCO-C2 FOV, ${t_\mathrm{IPS}}$ is the observation time for a $g \ge 1.5$ source, and ${T_\mathrm{Earth}}$ is the onset time of near-Earth ICME by *in-situ* observation. Using these values, the average reference distances \[${R_\mathrm{1}}$ and ${R_\mathrm{2}}$\] and the average radial speeds \[${V_\mathrm{1}}$ and ${V_\mathrm{2}}$\] for the ICME are found for values of ${r_\mathrm{1}}$, ${r_\mathrm{2}}$, ${v_\mathrm{1}}$, and ${v_\mathrm{2}}$ for all $g \ge 1.5$ sources, respectively on a given day.
Next, we calculate accelerations using the values above. In these calculations, we use the approximation that the accelerations are constant within each region. The average accelerations, *i.e.* ${a_\mathrm{1}}$ and ${a_\mathrm{2}}$, for ICMEs were given by $$\label{eq.accel1}
a_\mathrm{1} = \frac{1}{n} \sum_{k = 1}^{n} \frac{v_{\mathrm{IPS},k} - V_\mathrm{SOHO}}{t_{\mathrm{IPS},k} - T_\mathrm{SOHO}}~
\mbox{(for the SOHO--IPS region),}$$ and $$\label{eq.accel2}
a_\mathrm{2} = \frac{1}{n} \sum_{k = 1}^{n} \frac{V_\mathrm{Earth} - v_{\mathrm{IPS},k}}{T_\mathrm{Earth} - t_{\mathrm{IPS},k}}~
\mbox{(for the IPS--Earth region),}$$ respectively. Here, $$\label{eq.vips}
v_{\mathrm{IPS},k} = \frac{v_{\mathrm{1},k} + v_{\mathrm{2},k}}{2},$$ ${t_{\mathrm{IPS},k}}$ is the observation time for each $g \ge 1.5$ source, $n$ is the number of $g \ge 1.5$ sources, and ${V_\mathrm{SOHO}}$ and ${V_\mathrm{Earth}}$ are the radial speed of the CME and of the near-Earth ICME, respectively. For the value of ${V_\mathrm{SOHO}}$ in the halo or the partial halo CMEs, we use $$\label{eq.vsoho}
V_\mathrm{SOHO} = 1.20 \times V_\mathrm{POS},$$ where ${V_\mathrm{POS}}$ is the speed measured in the sky plane by the SOHO/LASCO, because the coronagraph measurement for them tends to underestimate the radial speed [@Michalek2003], while we use $V_\mathrm{SOHO} = V_\mathrm{POS}$ for the normal ones. In this study, we use the linear speeds reported in the SOHO/LASCO CME Catalog ([cdaw.gsfc.nasa.gov/CME\_list/index.html](cdaw.gsfc.nasa.gov/CME_list/index.html)) for those of ${V_\mathrm{POS}}$ with a 0.08 AU reference distance corresponding to half the LASCO FOV value. Those are derived from the bright leading edges of CME [@Yashiro2004], while the associated shocks show a faint structure ahead of them [@Ontiveros2009], and then indicate the speeds of CME itself in the sky plane [@Vourlidas2012]. For values of ${V_\mathrm{Earth}}$, we use the average ICME speeds listed by . We note that the values of ${V_\mathrm{SOHO}}$ and ${V_\mathrm{Earth}}$ represent an average in the near-Sun and near-Earth regions, respectively, and ${V_\mathrm{1}}$, ${V_\mathrm{2}}$, ${a_\mathrm{1}}$, and ${a_\mathrm{2}}$ are averages in the interplanetary space. The ICME speeds in the near-Earth region are measured when the spacecraft passes through them. Thus, those are equivalent to the plasma flow speed on the trajectory of the spacecraft during the passage of an ICME, indicated by the enhancement of the charge state and the rotation of magnetic-field direction [@Richardson2010]. The speed of the solar wind measured by *in-situ* observations is sometimes highly variable during the passage of an ICME. However, the majority of ICMEs listed by them have only $< 100$ ${\mathrm{km~s^{-1}}}$ difference between the peak and average speeds. Hence, we consider it justified that the average flow speed can be used as the propagation speed of ICMEs.
Classification of ICMEs {#classification}
-----------------------
Here, we introduce ${V_\mathrm{IPS}}$ which is given as the average value of ${v_\mathrm{IPS}}$ for each ICME; the ${v_\mathrm{IPS}}$ is derived from Equation \[eq.vips\]. In addition, we also introduce ${V_\mathrm{bg}}$ as the speed of the background solar wind. To determine the value of ${V_\mathrm{bg}}$ as the average background wind speed between ${T_\mathrm{SOHO}}$ and ${T_\mathrm{Earth}}$ for each ICME, we used plasma data obtained by space-borne instruments including *Solar Wind Electron, Proton, and Alpha Monitor* onboard ACE (ACE/SWEPAM: ), *Solar Wind Experiment* on *Wind* (Wind/SWE: ), *Massachusetts Institute of Technology Faraday cup experiment* on IMP-8 (IMP-8/MIT: ), and the *Comprehensive Plasma Instrumentation* on GEOTAIL (GEOTAIL/CPI: ); these are determined from the NASA/GSFC OMNI dataset through OMNIWeb Plus ([omniweb.gsfc.nasa.gov/](omniweb.gsfc.nasa.gov/)).
Using the values of ${V_\mathrm{SOHO}}$, ${V_\mathrm{IPS}}$, and ${V_\mathrm{bg}}$, we classify the 50 ICMEs into three types: fast ($V_{\mathrm{SOHO}} - V_\mathrm{bg} > 500$ ${\mathrm{km~s^{-1}}}$), moderate ($0$ $\mathrm{km~s^{-1}}$ $\le V_{\mathrm{SOHO}} - V_\mathrm{bg} \le 500$ ${\mathrm{km~s^{-1}}}$), and slow ($V_{\mathrm{SOHO}} - V_\mathrm{bg} < 0$ ${\mathrm{km~s^{-1}}}$). In our results, the numbers of fast, moderate, and slow ICMEs are 19, 25, and 6, respectively. Here, we eliminate 5 of the 19 fast ICMEs and a moderate ICME because they show an extreme zigzag profile of propagation speeds, *i.e.* $V_{\mathrm{1}} - V_{\mathrm{2}} > 1000$ ${\mathrm{km~s^{-1}}}$. The value of $V_{\mathrm{1}} - V_{\mathrm{2}} > 1000$ ${\mathrm{km~s^{-1}}}$ implies that the ICME has a strange acceleration, and then shows an unrealistic propagation. We also eliminate 4 of the 24 moderate ICMEs and one of the six slow ones because they exhibit the unusual values of ${V_\mathrm{IPS}}$ of $V_\mathrm{IPS} - V_\mathrm{bg} > 500$ ${\mathrm{km~s^{-1}}}$ and $V_\mathrm{IPS} - V_\mathrm{bg} > 100$ ${\mathrm{km~s^{-1}}}$, respectively. The values of $V_\mathrm{IPS} - V_\mathrm{bg} > 500$ ${\mathrm{km~s^{-1}}}$ for moderate and $V_\mathrm{IPS} - V_\mathrm{bg} > 100$ ${\mathrm{km~s^{-1}}}$ for slow ICMEs imply that the ICME has a strange acceleration since ${V_\mathrm{IPS}}$ is larger than ${V_\mathrm{SOHO}}$ and ${V_\mathrm{Earth}}$, and an unrealistic ICME propagation that indicates a higher speed in the region beyond coronagraph distances, and less at 1 AU.
Finally, we obtain physical properties for 39 ICMEs which consist of 14 fast, 20 moderate, and five slow ones.
Results
=======
Properties and Speed profiles of the 39 ICMEs {#icmeproperties}
---------------------------------------------
The properties of the 39 ICMEs identified from our analysis are listed in Tables \[table1\] and \[table2\] which including ${T_{\mathrm{IPS}}}$, ${R_\mathrm{0}}$, ${\alpha}$, ${\beta}$, and ${V_\mathrm{Tr}}$ in addition to ${T_{\mathrm{SOHO}}}$, ${V_{\mathrm{POS}}}$, ${V_{\mathrm{SOHO}}}$, ${R_\mathrm{1}}$, ${V_\mathrm{1}}$, ${a_\mathrm{1}}$, ${R_\mathrm{2}}$, ${V_\mathrm{2}}$, ${a_\mathrm{2}}$, ${T_{\mathrm{Earth}}}$, ${V_{\mathrm{Earth}}}$, and ${V_\mathrm{bg}}$ above. Here, ${T_{\mathrm{IPS}}}$ and ${R_\mathrm{0}}$ are the mean time and the average radial distance for an ICME detected by IPS observations; those are given as the averages of ${t_\mathrm{IPS}}$ and of ${r_\mathrm{IPS}}$ for the $g \ge 1.5$ sources, respectively. The ${\alpha}$ and ${\beta}$ are the index and coefficient for a power-law form of the radial speed evolution described as $$\label{eq.powerlaw}
V = {\beta}R^{\alpha},$$ where $R$ is the heliocentric distance. ${V_\mathrm{Tr}}$ is the transit speed: $$\label{eq.vtr}
V_\mathrm{Tr} = \frac{r_\mathrm{E}}{T_\mathrm{Earth} - T_\mathrm{SOHO}}.$$ This is equivalent to the average speed of ICMEs between the Sun and the Earth. In addition, we plot all of the speed profiles in order to show radial speed evolutions of ICMEs in Figure \[fig2\]. Here, data points for each ICME are connected by solid lines instead of fitting in Equation (\[eq.powerlaw\]). As shown here, ICME propagation speeds in the near-Sun region exhibit a wide range from 90 ${\mathrm{km~s^{-1}}}$ to ${\approx}~2100$ ${\mathrm{km~s^{-1}}}$, while those in the near-Earth region range from 310 ${\mathrm{km~s^{-1}}}$ to 790 ${\mathrm{km~s^{-1}}}$. Moreover, the range of ICME propagation speeds in interplanetary space decreases with increasing distance. In addition, speeds of the background solar wind also show a relatively narrow span from 286 ${\mathrm{km~s^{-1}}}$ to 662 ${\mathrm{km~s^{-1}}}$.
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Fast, Moderate, and Slow ICMEs, and Their Accelerations {#accelerations}
-------------------------------------------------------
For the fast, moderate, and slow ICMEs, we show representative examples of speed profiles in Figures \[fig3\], \[fig4\], and \[fig5\], respectively. These are plotted using the values of ${V_\mathrm{SOHO}}$, ${R_\mathrm{1}}$, ${V_\mathrm{1}}$, ${R_\mathrm{2}}$, ${V_\mathrm{2}}$, ${V_\mathrm{Earth}}$, and ${V_\mathrm{bg}}$. Figure \[fig3\] shows a speed profile for a fast ICME observed as a halo by SOHO/LASCO on 5 November 1998, a subsequent disturbance from the IPS observations on 7 November 1998, and the event detected at 1 AU by *in-situ* observations on 9 November 1998 (see No. 4 in Tables \[table1\] and \[table2\]). These data show that the ICME speed rapidly decreases to the value of ${V_\mathrm{bg}}$ with an increase in radial distance; the initial speed ${V_\mathrm{SOHO}}$ value is 1342 ${\mathrm{km~s^{-1}}}$, while $V_\mathrm{bg} = 385$ ${\mathrm{km~s^{-1}}}$ for this ICME. This speed profile is well fit by a power-law function; the fitting-line has a value of ${\alpha} = -0.478$ from Equation (\[eq.powerlaw\]). Figure \[fig4\] shows the speed profile for a moderate ICME; this ICME was observed as a normal event (neither a halo nor a partial halo) by SOHO/LASCO on 17 July 2000, on 19 July 2000 in IPS, and detected by *in-situ* observations on 20 July 2000 (see No. 14 in Tables \[table1\] and \[table2\]). As shown here, for this ICME, the 788 ${\mathrm{km~s^{-1}}}$ initial speed gradually decreases to $V_\mathrm{bg} = 574$ ${\mathrm{km~s^{-1}}}$ with an increase in radial distance; we have a value of ${\alpha} = -0.079$. Figure \[fig5\] exhibits a speed profile for a slow ICME observed as a normal event by SOHO/LASCO on 29 May 2009, on 1 June 2009 by IPS observations, and detected by *in-situ* observations on 4 June 2009 (see No. 39 in Tables \[table1\] and \[table2\]). For this event, we confirm that $V_\mathrm{SOHO} = 139$ ${\mathrm{km~s^{-1}}}$, and that the propagation speed increases to $V_\mathrm{bg} = 327$ ${\mathrm{km~s^{-1}}}$ with radial distance. This ICME shows acceleration, and the fit has a value of ${\alpha} = 0.276$.
Figure \[fig6\] shows the average radial acceleration for groups of fast, moderate, and slow ICMEs; the average acceleration in the two regions \[$a_\mathrm{1}$ and $a_\mathrm{2}$\] are calculated first using Equations (\[eq.accel1\]) and (\[eq.accel2\]) for each ICME, and each is subsequently averaged for respective groups. For all of them, the mean values of ${R_\mathrm{1}}$ and ${R_\mathrm{2}}$ with the standard errors are $0.33 \pm 0.04$ and $0.79 \pm 0.04$ AU, respectively. From this figure, we confirm that the acceleration levels vary toward zero with an increase in distance, and this trend is conspicuous for the group of fast ICMEs. We also confirm that the group of moderate ICMEs shows little acceleration.
Critical Speed for Zero Acceleration {#criticalspeed}
------------------------------------
If ICMEs accelerate or decelerate by interaction with the solar wind, we expect that the acceleration will become zero when the propagation speed of ICMEs reaches the speed of the background solar wind. Therefore, it is important to know the ICME propagation speed in this situation in order to verify our expectations. Here, we call this speed “the critical speed for zero acceleration”. In Figures \[fig7\] and \[fig8\], we give information on this critical speed for zero acceleration in two ways. In Figure \[fig7\], we show the relationship between initial ICME speeds \[${V_\mathrm{SOHO}}$\] and indices \[${\alpha}$\]. The ${\alpha}$ indicates the type of ICME motion, *i.e.* acceleration (${\alpha} > 0$), uniform (${\alpha} = 0$), and deceleration (${\alpha} < 0$). As shown here, ${\alpha}$ ranges from $0.486$ to $-0.596$ with an increase in ${V_\mathrm{SOHO}}$. Table \[table3\] gives the mean values of the critical speed for zero acceleration \[${V_\mathrm{c1}}$\], coefficients \[${k_\mathrm{1}}$, ${k_\mathrm{2}}$, and ${k_\mathrm{3}}$\] for the best-fit curve, and their standard errors. Figure \[fig8\] shows the relationship between ICME speeds \[${V_\mathrm{SOHO}}$ and ${V_\mathrm{IPS}}$\] and accelerations \[${a_\mathrm{1}}$ and ${a_\mathrm{2}}$\]. Table \[table4\] presents the mean values of the critical speed for zero acceleration \[${V_\mathrm{c2}}$\] slope, and intercept for the best-fit line and their standard errors, which are estimated using the from the IDL Astronomy User’s Library ([idlastro.gsfc.nasa.gov/homepage.html](idlastro.gsfc.nasa.gov/homepage.html)). From the above examinations, we find $V_\mathrm{c1} = 471 \pm 19$ ${\mathrm{km~s^{-1}}}$ and $V_\mathrm{c2} = 480 \pm 21$ ${\mathrm{km~s^{-1}}}$ as the critical speed for zero acceleration.
${k_\mathrm{1}}$ ${k_\mathrm{2}}$ ${k_\mathrm{3}}$ ${V_\mathrm{c1}}$ \[${\mathrm{km~s^{-1}}}$\]
---------------- ------------------------- -------------------------- ------------------------- ---------------------------------------------- --
Mean ${4.31 \times 10^{-1}}$ ${-1.06 \times 10^{-3}}$ ${3.04 \times 10^{-7}}$ 471
Standard error ${5.58 \times 10^{-2}}$ ${1.16 \times 10^{-4}}$ ${5.22 \times 10^{-8}}$ 19
: Mean values of coefficients \[${k_\mathrm{1}}$, ${k_\mathrm{2}}$, and ${k_\mathrm{3}}$\] for the best-fit quadratic curve ${\alpha} = k_\mathrm{1} + k_\mathrm{2}V_\mathrm{SOHO} + k_\mathrm{3}V_{\mathrm{SOHO}}^{2}$ and the critical speed for zero acceleration \[${V_\mathrm{c1}}$\], and their standard errors, which were derived from the relationship between ${V_\mathrm{SOHO}}$ and ${\alpha}$. []{data-label="table3"}
Slope \[${\mathrm{s^{-1}}}$\] Intercert \[${\mathrm{m~s^{-2}}}$\] ${V_\mathrm{c2}}$ \[${\mathrm{km~s^{-1}}}$\]
---------------- ------------------------------- ------------------------------------- ---------------------------------------------- --
Mean ${-7.38 \times 10^{-6}}$ 3.54 480
Standard error ${2.03 \times 10^{-7}}$ ${1.24 \times 10^{-1}}$ 21
: Mean values of slope and intercept for the best-fit line and the critical speed for zero acceleration \[${V_\mathrm{c2}}$\] and their standard errors, which were derived from the relationship between speeds and accelerations of ICMEs. []{data-label="table4"}
Relationship Between Acceleration and Difference in Speed {#speeddifference}
---------------------------------------------------------
We investigated how the ICME acceleration relates to the difference in speed between it and the background solar wind. In this investigation, we attempted to show which is more suitable to describe the relationship between acceleration and difference in speed: $a = -{\gamma}_{\mathrm{1}}(V - V_\mathrm{bg})$ or $a = -{\gamma}_{\mathrm{2}}(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$; these expressions were introduced and also tested in the earlier study by . Here, $a$, $V$, and ${V_\mathrm{bg}}$ denote the acceleration, ICME speed, and speed of the background solar wind, respectively. Although it was assumed that the coefficients \[${\gamma}_{\mathrm{1}}$ and ${\gamma}_{\mathrm{2}}$\] decrease with the heliocentric distance in the earlier study, for this analysis we assume that the values of coefficients are constants because we want as few variables as possible to describe the relationship. We also assume that the speed of the background solar wind \[${V_\mathrm{bg}}$\] is constant for heliocentric distances ranging from ${\approx}~0.1$ to 1 AU. This assumption has been verified approximately between 0.3 and 1 AU by and . In Figure \[fig9\], the top panel shows the relationship between $a$ and $(V - V_\mathrm{bg})$, and the bottom panel that between $a$ and $(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$ for ICMEs with $(V_\mathrm{SOHO} - V_\mathrm{bg}) \ge 0$ ${\mathrm{km~s^{-1}}}$, *i.e.* the fast and moderate ICMEs. Table \[table5\] exhibits the values of ${{\gamma}_{\mathrm{1}}}$ and ${{\gamma}_{\mathrm{2}}}$, correlation coefficients, and reduced ${\chi^{\mathrm{2}}}$ derived from this analysis. It is noted that the ${{\gamma}_{\mathrm{1}}}$, ${{\gamma}_{\mathrm{2}}}$, and ${\chi^{\mathrm{2}}}$ are calculated using the . Although we also examined the slow ICMEs in the same way, we did not obtain a conclusive result. We discuss interpretations of these results in the next section.
Equation Mean Standard error CC $\chi^{2}$
----------- -------------------------- -------------------------- ----------- ------------ --
Linear ${6.58 \times 10^{-6}}$ ${2.34 \times 10^{-7}}$ ${-0.93}$ 1.26
Quadratic ${6.10 \times 10^{-12}}$ ${2.25 \times 10^{-13}}$ ${-0.90}$ 2.90
: Coefficients \[${\gamma_\mathrm{1}}$ and ${\gamma_\mathrm{2}}$\], correlation coefficient \[CC\], and reduced $\chi^{2}$ for the linear and quadratic equations. []{data-label="table5"}
Discussion
==========
From Figures \[fig2\], \[fig3\], and \[fig4\], we confirm that fast and moderate ICMEs are rapidly and gradually decelerating during their outward propagation, respectively, while slow ICMEs are accelerating, and consequently all attain speeds close to those of the background solar wind. As shown in Figure \[fig5\], the distribution of ICME propagation speeds in the near-Sun region is wider than in the near-Earth region for all of the ICMEs identified in this study. This is consistent with the earlier study by . We also confirm that the distribution of ICME propagation speed in the near-Earth region is similar to that of the background solar-wind speed at 1 AU. We interpret these results as indicating that ICMEs accelerate or decelerate by interaction with the solar wind; the magnitude of the propelling or retarding force acting upon ICMEs depends on the difference between ICMEs and the solar wind. Thus, ICMEs attain final speeds close to the solar-wind speed as they move outward from the Sun. Figure \[fig5\] also shows the radial evolution of ICME propagation speeds between 0.08 and 1 AU. We show that ICME speeds reach their final value at $0.79 \pm 0.04$ AU or at a solar distance slightly less than 1 AU. In addition, we confirm from Figure \[fig6\] that the acceleration at $0.79 \pm 0.04$ AU is much lower than at $0.33 \pm 0.04$ AU; this is the clearest for the group of fast ICMEs. From this, we thus conclude that most of the ICME acceleration or deceleration ends by $0.79 \pm 0.04$ AU. This is consistent with an earlier result obtained by .
We expect that the critical speed of zero acceleration will be close to that of the background solar-wind speed on the basis of the above. We derive two different critical speeds of $V_\mathrm{c1} = 471 \pm 19$ ${\mathrm{km~s^{-1}}}$ and of $V_\mathrm{c2} = 480 \pm 21$ ${\mathrm{km~s^{-1}}}$ from the observational data. Although there is agreement between them, both are somewhat higher than the ${\approx}~380$ ${\mathrm{km~s^{-1}}}$ reported to be the threshold speed by and the 405 ${\mathrm{km~s^{-1}}}$ reported by . We suggest that this discrepancy is caused by the difference in our analysis methods and also the time interval chosen for the analysis. Because the properties of the background solar wind (*e.g.* speed and density) vary with the change in solar activity, we consider this discrepancy to be minor, and we note that both critical speeds in our result are within the typical speed of the solar wind: $V_\mathrm{bg} = 445 \pm 95$ ${\mathrm{km~s^{-1}}}$ from our sample. Here, we adopt the speed of 480 ${\mathrm{km~s^{-1}}}$ as the critical speed for zero acceleration as a mean that is derived from the relationship between propagation speeds and accelerations without the assumption of a power-law form for the motion of the ICME.
and point out that the radial evolution of ICME speeds can be represented by a power-law function. A power-law speed evolution also applies to the ICMEs identified in this study as shown in Figures \[fig2\], \[fig3\], and \[fig4\]. As indicated by Figure \[fig7\], the value of ${\alpha}$ varies from 0.499 (acceleration) to ${-0.596}$ (strongly deceleration) as ICME speeds increase. This result is consistent with that exhibited in Figure \[fig8\].
The relationship between acceleration and speed-difference for ICMEs is usually expressed by either of the following: a linear equation $a = -{\gamma}_{\mathrm{1}}(V - V_\mathrm{bg})$ or a quadratic equation $a = -{\gamma}_{\mathrm{2}}(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$. As shown in Figure \[fig9\], these equations are evaluated using the acceleration and speed-difference data derived from our observations. From this and Table \[table5\], we find that the reduced ${\chi^{2}}$ for the former relationship is smaller than for the latter. The assessment of the significance level shows that ${\chi}^{2} = 1.26$ for the linear equation is smaller than the reduced ${\chi^{2}}$ corresponding to the probability of 0.05 with 66 degrees of freedom, while ${\chi}^{2} = 2.90$ for the quadratic one is larger. We therefore conclude that the linear equation is more suitable than the quadratic one to describe the kinematics of ICMEs with $(V_\mathrm{SOHO} - V_\mathrm{bg}) \ge 0$ ${\mathrm{km~s^{-1}}}$. From the viewpoint of fluid dynamics, a linear equation suggests that the hydrodynamic Stokes drag force is operating, while the quadratic equation suggests the aerodynamic drag force. found that the acceleration of a fast ICME showed a linear dependence on the speed difference, while that of a slow ICME showed a quadratic dependence. Our conclusion is consistent with their finding only for the fast and moderate ICMEs. We could not verify their result for the slow ICMEs because we lack sufficient observational data for the slow ICMEs in our sample. We expect to make a more detailed examination for the motion of slow ICMEs in a future study.
We also obtained the mean value of $6.58 \times 10^{-6}~\mathrm{s^{-1}}$ for the coefficient ${{\gamma}_{\mathrm{1}}}$ in our analysis. Substituting our value of ${{\gamma}_{\mathrm{1}}}$ in our linear equation, we obtain the following simple expression: $$\label{eq.kinematics}
a = -6.58 \times 10^{-6}(V - V_\mathrm{bg}),$$ where $a$, $V$, and ${V_\mathrm{bg}}$ are the acceleration, ICME propagating speed, and speed of the background solar wind, respectively, as a useful way to determine the dynamics of ICMEs.
Last, we discuss why the linear equation with a constant ${{\gamma}_{\mathrm{1}}}$ can explain the observational result. Our IPS radio-telescope system observes fluctuations of radio signals. These fluctuations are proportional to the solar-wind (electron) density \[${N_\mathrm{e}}$\]. Therefore, low-density ICMEs may not be detected by our system. Moreover, we used a threshold *g*-value more severe than that used by or for identification of ICMEs. Hence, it is conceivable that almost all detected ICMEs are high-density events in this study. In addition, from a theoretical study, indicated that with dense ICMEs, the factor ${\gamma}$ and ${C_\mathrm{D}}$ (the dimensionless drag coefficient) become approximately constant for aerodynamic drag deceleration; here, ${\gamma}C_\mathrm{D} = {\gamma}_{\mathrm{2}}$ in our notation. From this, we surmise that a constant value of ${{\gamma}C_\mathrm{D}}$ indicates that both interplanetary-space conditions and the properties of dense ICMEs are unchanged in the range from the Sun to the Earth. Therefore, ${{\gamma}_{\mathrm{1}}}$ must also become approximately constant over the same range from the Sun to the Earth. Thus, to recapitulate, the events detected using our IPS radio-telescope system give results for dense ICMEs, and the dynamics of these are well explained by a linear equation with ${\gamma}_{\mathrm{1}} =$ constant.
Summary and Conclusions {#conclusion}
=======================
We investigate radial evolution of propagation speed for 39 ICMEs detected by SOHO/LASCO, IPS at 327 MHz, and *in-situ* observations during 1997–2009 covering nearly all of Solar Cycle 23. In this study, we first analyze *g*-values obtained by STEL IPS observations in the above period, and find 497 IPS disturbance event days (IDEDs) as candidates for ICME events. Next, we compare the list of these IDEDs with that of CME/ICME pairs observed by SOHO/LASCO and *in-situ* observations, and finally we are left with 50 ICMEs; those ICMEs that traveled from the Sun to the Earth, and were detected at three locations between the Sun and the Earth’s orbit, *i.e.* near-Sun, interplanetary space, and near-Earth. For these ICMEs, we determine reference distances and derive the propagation speeds and accelerations in the SOHO–IPS and IPS–Earth regions. Our examinations yield the following results.
1. Fast ICMEs (with $V_\mathrm{SOHO} - V_\mathrm{bg} > 500$ ${\mathrm{km~s^{-1}}}$) rapidly decelerate, moderate ICMEs (with $0$ ${\mathrm{km~s^{-1}}}$ $\le V_\mathrm{SOHO} - V_\mathrm{bg} \le 500$ ${\mathrm{km~s^{-1}}}$) show either gradually deceleration or uniform motion, while slow ICMEs (with $V_\mathrm{SOHO} - V_\mathrm{bg} < 0$ ${\mathrm{km~s^{-1}}}$) accelerate, where ${V_\mathrm{SOHO}}$ and ${V_\mathrm{bg}}$ are the initial speed of ICME and the speed of the background solar wind, respectively. Consequently, radial speeds converge to the speed of the background solar wind during their outward propagation. Thus, the distribution of ICME propagation speeds in the near-Earth region is narrower than in the near-Sun region, as shown in Figure \[fig5\]. This is consistent with the earlier study by .
2. Both the ICME accelerations and the decelerations are nearly complete by $0.79 \pm 0.04$ AU. This is consistent with an earlier result obtained by . Both critical speeds (where the speed of ICME acceleration becomes zero) derived from our analysis, *i.e.* $471 \pm 19$ ${\mathrm{km~s^{-1}}}$ and $480 \pm 21$ ${\mathrm{km~s^{-1}}}$, are somewhat higher than the values reported by and . However, this discrepancy is most likely explained because our analysis methods and data collection periods are different. Both critical speeds in our result do not differ much from the typical speed of the solar wind, and we adopt the mean value of 480 ${\mathrm{km~s^{-1}}}$ as the critical speed for zero acceleration. This is close to the speed of the background solar wind, $V_\mathrm{bg} = 445 \pm 95$ ${\mathrm{km~s^{-1}}}$, during this period of study.
3. For ICMEs with $(V_\mathrm{SOHO} - V_\mathrm{bg}) \ge 0$ ${\mathrm{km~s^{-1}}}$, a linear equation $a = -{\gamma}_{\mathrm{1}}(V - V_\mathrm{bg})$ with ${\gamma}_{\mathrm{1}} = 6.58 \pm 0.23 \times 10^{-6}$ ${\mathrm{s^{-1}}}$ is more appropriate than a quadratic equation $a = -{\gamma}_{\mathrm{2}}(V - V_\mathrm{bg})|V - V_\mathrm{bg}|$ to describe their kinematics, where ${\gamma}_{\mathrm{1}}$ and ${\gamma}_{\mathrm{2}}$ are coefficients, $a$, $V$, and ${V_\mathrm{bg}}$ are the acceleration and propagation speed of ICMEs, and the speed of the background solar wind, respectively, because the reduced ${\chi^{2}}$ for the linear equation satisfies the statistical significance level at 0.05, while the quadratic one does not.
These results support our assumption that ICMEs are accelerated or decelerated by a drag force caused by an interaction with the solar wind; the magnitude of the drag force acting upon ICMEs depends on the difference in speed, and, thus, ICMEs attain final speeds close to the solar-wind speed when the force becomes zero. In particular, our result iii) suggests that ICMEs propagating faster than the background solar wind are controlled mainly by the hydrodynamic Stokes drag force. Moreover, our result iii) confirms the finding by only for the fast and moderate ICMEs that we measure. From the characteristics of the IPS observations and the result of , we conclude that the ICMEs detected by the IPS observations in this study are probably high-density events. A combination of the space-borne coronagraph, ground-based IPS, and satellite *in-situ* observations serves to detect many ICMEs between the Sun and the Earth, and is a useful means to study their kinematics.
The IPS observations were carried out under the solar-wind program of the Solar-Terrestrial Environment Laboratory (STEL) of Nagoya University. We acknowledge use of the SOHO/LASCO CME catalog; this CME catalog is generated and maintained at the CDAW Data Center by NASA and the Catholic University of America in cooperation with the Naval Research Laboratory. SOHO is a project of international cooperation between ESA and NASA. We thank NASA/GSFC’s Space Physics Data Facility for use of the OMNIWeb service and OMNI data. We thank the IDL Astronomy User’s Library for the use of IDL software. We acknowledge use of the comprehensive ICME catalog compiled by I.G. Richardson and H.V. Cane. We also thank B.V. Jackson for useful help and comments.
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abstract: 'We consider a Higgs mechanism in scale-invariant theories of gravitation. It is shown that in spontaneous symmetry breakdown of scale invariance, gauge symmetries are also broken spontaneously even without the Higgs potential if the corresponding charged scalar fields couple to a scalar curvature in a non-minimal way. In this gravity-inspired new Higgs mechanism, the non-minimal coupling term, which is scale-invariant, plays a critical role. Various generalizations of this mechanism are possible and particularly the generalizations to non-abelian gauge groups and a scalar field with multi-components are presented in some detail. Moreover, we apply our finding to a scale-invariant extension of the standard model (SM) and calculate radiative corrections. In particular, we elucidate the coupling between the dilaton and the Higgs particle and show that the dilaton mass takes a value around the GeV scale owing to quantum effects even if the dilaton is massless at the classical level.'
---
=cmr5 \#1[\#1\^[\^]{}]{}
\#1
/\#1[\#1]{} \#1[\#1]{}
DPUR/TH/39\
August, 2013\
[**Higgs Mechanism in Scale-Invariant Gravity** ]{}
Ichiro Oda [^1]
Department of Physics, Faculty of Science, University of the Ryukyus,\
Nishihara, Okinawa 903-0213, Japan.\
Introduction
============
Current understanding of elementary particle physics is based on two celebrated fundamental principles, which are gauge symmetry and spontaneous symmetry breakdown of the gauge symmetry. In four interactions among elementary particles, strong, weak and electro-magnetic interactions are known to be described on the same footing in terms of a gauge theory which is the standard model (SM) on the basis of $SU(3) \times SU(2) \times U(1)$ gauge group, and gravitational interaction is believed to be also described by a gauge theory whose final formalism is still far from complete at present.
The gauge principle alone, however, cannot describe the known structure of elementary particles. The gauge principle requires elementary particles to be massless[^2], so in order to generate masses for elementary particles the $SU(2) \times U(1)$ gauge symmetry must be spontaneously broken at any rate. The idea of spontaneous symmetry breakdown itself is not new for elementary particle physics but has emerged as a universal phenomenon in physics, in particular, condensed matter physics. An alternative and indeed older description of super-conductivity, which was developed by Ginzburg and Landau, turned out to be a phenomenological representation of the BCS theory [@Ginzburg]. In this transcription, the complex “Ginzburg-Landau” scalar field is nothing but the Higgs boson representing a bound pair of electrons and holes, and its phase and amplitude components correspond to the massless Nambu-Goldstone boson and the massive Higgs type of excitations, respectively. As it happens, the collective excitations of both the types do exist in all phenomena of the super-fluidity type. This universality of the spontaneous symmetry breakdown, however, seems to have no implication in gravity so far. The concept of the mass is intimately connected with general relativity since the right-hand side of Einstein’s equations is constructed out of the energy-momentum tensor. In this article, we will investigate an idea such that a scale-invariant gravity induces the spontaneous symmetry breakdown of gauge symmetry without assuming the existence of the Higgs potential.
The SM based on $SU(3) \times SU(2) \times U(1)$ gauge group, together with classical general relativity, describes with amazing parsimony (only $19$ parameters) our world over scales that have been explored by experiments: from the Hubble radius of $10^{30} cm$ all the way down to scales of the order of $10^{-16} cm$. In other words, with the help of cosmological initial conditions when the universe was much smaller, the SM is believed to encode the information needed to deduce all the physical phenomena observed so far. There are, however, some obvious chinks in the armor of the SM. In particular, the origin of different scales in nature cannot be answered at all by the SM. One should recall that there is only one fundamental constant with the dimension of mass: Gravity comes with its own mass scale $M_p = 2.4 \times 10^{18} GeV$. All units of mass should be scaled to this fundamental scale. It is a source of great intellectual worry that the SM appears to be consistent at a scale which is so different from the Planck mass scale. Naive expectations are that all physical phenomena should occur at their natural scale which is of course the Planck scale.
Coleman-Mandula theorem [@Mandula] allows the Poincare group to be generalized to two global groups, one is the super-Poincare group and the other is the conformal group. It is remarkable to notice that these two groups might yield resolution of the gauge hierarchy problem by a completely different idea, and they also yield a natural generalization of local gauge group, the former gives rise to the local super-Poincare group leading to supergravity whereas the latter does the local conformal group leading to conformal gravity.
According to recent results by the LHC [@ATLAS; @CMS], supersymmetry on the basis of the super-Poincare group seems not to be taken by nature as resolution of the gauge hierarchy problem. Then, it is natural to ask ourselves if the conformal group, the other extension of the Poincare group, gives us resolution of the gauge hierarchy problem. Indeed, inspired by an interesting idea by Bardeen [@Bardeen], there has appeared to pursue the possibility of replacing the supersymmetry with the conformal symmetry near the TeV scale in an attempt to solve the hierarchy problem [@Meissner; @Iso]. It is worth noting that the principle of conformal invariance is more rigid than the supersymmetry in the sense that in many examples the conformal symmetry predicts the number of generations as well as a rich structure for the Yukawa couplings among various families. This inter-family rigidity is a welcome feature of the conformal approach to particle phenomenology [@Frampton].
In the conformal approach, it is thought that the electro-weak scale and the QCD scale as well as the masses of observed quarks and leptons are all so small compared to the Planck scale that it is reasonable to believe that in some approximation they are exactly massless. If so, then the quantum field theory which would be describing the massless fields should be a conformal theory as it has no mass scale. In this scenario, the fact that there are no large mass corrections follows from the condition of conformal invariance. In other words, the ’tHooft naturalness condition [@'tHooft] is satisfied in the conformal approach, namely in the absence of masses there is an enhanced symmetry which is the conformal symmetry. Of course, the breaking of conformal invariance should be soft in such a way that the idea of the conformal symmetry is relevant for solving the hierarchy problem.
In passing, in the present context, it seems to be of interest to consider the issue of renormalizability. Usually, in quantum field theories, the condition of renormalizability is imposed on a theory as if it were a basic principle to make the perturbation method to be meaningful, but its real meaning is unclear since there might exist a theory for which only the non-perturbative approach could be applied without relying on the perturbation method at all. To put differently, the concept of renormalizability means that even if one is unfamiliar with true physics beyond some higher energy scale, one can construct an effective theory by confining its ignorance to some parameters such as coupling constants and masses below the energy scale. Thus, from this point of view, it is unclear to require the renormalizability to theories holding at the highest energy scale, the Planck scale, such as quantum gravity and superstring theory. On the other hand, given a scale invariance in a theory, all the coupling constants must be dimensionless and operators in an action are marginal ones whose coefficient is independent of a certain scale, which ensures that the theory is manifestly renormalizable. In this world, all masses of particles must be then generated by spontaneous symmetry breakdown.
In previous works [@Oda1; @Oda2], we have shown that without resort to the Coleman-Weinberg mechanism [@Coleman], by coupling the non-minimal term of gravity, the U(1) B-L gauge symmetry in the model [@Iso] is spontaneously broken in the process of spontaneous symmetry breakdown of global or local scale symmetry at the tree level and as a result the U(1) B-L gauge field becomes massive via the Higgs mechanism. One of advantages in this mechanism is that we do not have to introduce the Higgs potential in a theory.
Then, we have the following questions of this mechanism of symmetry breaking of gauge symmetry:
1. Is it possible to generalize to the non-abelian gauge groups?
2. Is it possible to generalize many scalar fields?
3. What becomes of applying it to the standard model and what its radiative corrections are?
In this article, we would like to answer these questions in order. The structure of this article is the following: In Section 2, we present the simplest model which accomodates global scale symmetry and the abelian gauge symmetry, and explain our main idea. In Section 3, we generalize this simple model to a model with the non-abelian gauge symmetry. In Section 4, we extend our idea to a model of a scalar field with many of components. Moreover, we apply our finding to a scale-invariant extension of the standard model and calculate radiative corrections in Section 5. We conclude in Section 6. Two appendices are given, one of which is to explain a derivation of the dilatation current and the other is to put useful formulae for the calcualtion of radiative corrections.
Review of a globally scale-invariant Abelian model
==================================================
We start with a brief review of the simplest model showing a gravitational Higgs phenomenon which was previously discovered in case of a global scale invariance and the abelian gauge group [@Oda1].
With a background curved metric $g_{\mu\nu}$, a complex (singlet) scalar field $\Phi$ and the $U(1)$ gauge field $A_\mu$, the Lagrangian takes the form[^3]: $$\begin{aligned}
{\cal L} = \sqrt{-g} \left[ \xi \Phi^\dagger \Phi R - g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi)
- \frac{1}{4} g^{\mu\nu} g^{\rho\sigma} F_{\mu\rho} F_{\nu\sigma} \right],
\label{Lagr 1}\end{aligned}$$ where $\xi$ is a certain positive and dimensionless constant. The covariant derivative and field strength are respectively defined as $$\begin{aligned}
D_\mu \Phi = (\partial_\mu - i e A_\mu) \Phi, \quad
(D_\mu \Phi)^\dagger = (\partial_\mu + i e A_\mu) \Phi^\dagger, \quad
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,
\label{Def 1}\end{aligned}$$ with $e$ being a $U(1)$ real coupling constant.
Let us note that the Lagrangian (\[Lagr 1\]) is invariant under a global scale transformation. In fact, with a constant parameter $\Omega = e^\Lambda \approx 1 + \Lambda \ (|\Lambda| \ll 1)$ the scale transformation is defined as [@Fujii1][^4] $$\begin{aligned}
g_{\mu\nu} &\rightarrow& \tilde g_{\mu\nu} = \Omega^2 g_{\mu\nu}, \quad
g^{\mu\nu} \rightarrow \tilde g^{\mu\nu} = \Omega^{-2} g^{\mu\nu}, \quad
\nonumber\\
\Phi &\rightarrow& \tilde \Phi = \Omega^{-1} \Phi, \quad
A_\mu \rightarrow \tilde A_\mu = A_\mu.
\label{Scale transf}\end{aligned}$$ Then, using the formulae $\sqrt{-g} = \Omega^{-4} \sqrt{- \tilde g}, R = \Omega^2 \tilde R$, it is straightforward to show that ${\cal L}$ is invariant under the scale transformation (\[Scale transf\]). Following the Noether procedure $\Lambda J^\mu = \sum \frac{\partial {\cal L}}{\partial \partial_\mu \phi} \delta \phi$ where $\phi = \{g_{\mu\nu}, \Phi, \Phi^\dagger \}$, as shown in the Appendix A, the current for the scale transformation, what we call the dilatation current, takes the form[^5] $$\begin{aligned}
J^\mu = ( 6 \xi + 1 ) \sqrt{-g} g^{\mu\nu} \partial_\nu \left(\Phi^\dagger \Phi \right).
\label{Current}\end{aligned}$$ To prove that this current is conserved on-shell, it is necessary to derive a set of equations of motion from the Lagrangian (\[Lagr 1\]). The variation of (\[Lagr 1\]) with respect to the metric tensor produces Einstein’s equations $$\begin{aligned}
2 \xi \Phi^\dagger \Phi G_{\mu\nu} = T^{(A)}_{\mu\nu} + T^{(\Phi)}_{\mu\nu} - 2 \xi ( g_{\mu\nu} \Box - \nabla_\mu \nabla_\nu )
(\Phi^\dagger \Phi),
\label{Einstein eq}\end{aligned}$$ where d’Alembert operator $\Box$ is as usual defined as $\Box (\Phi^\dagger \Phi) = \frac{1}{\sqrt{-g}} \partial_\mu
(\sqrt{-g} g^{\mu\nu} \partial_\nu (\Phi^\dagger \Phi)) = g^{\mu\nu} \nabla_\mu \nabla_\nu (\Phi^\dagger \Phi)$ and the Einstein tensor is $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R$. Here the energy-momentum tensors $T^{(A)}_{\mu\nu}$ for the gauge field and $T^{(\Phi)}_{\mu\nu}$ for the scalar field are defined as, respectively $$\begin{aligned}
T^{(A)}_{\mu\nu} &=& - \frac{2}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu\nu}}
[ - \frac{1}{4} \sqrt{-g} g^{\alpha\beta} g^{\rho\sigma} F_{\alpha\rho} F_{\beta\sigma} ]
\nonumber\\
&=& g^{\rho\sigma} F_{\mu\rho} F_{\nu\sigma} - \frac{1}{4} g_{\mu\nu} F_{\rho\sigma}^2, \nonumber\\
T^{(\Phi)}_{\mu\nu} &=& - \frac{2}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu\nu}} [ - \sqrt{-g} g^{\rho\sigma} (D_\rho \Phi)^\dagger
(D_\sigma \Phi) ] \nonumber\\
&=& 2 (D_{(\mu} \Phi)^\dagger (D_{\nu)} \Phi) - g_{\mu\nu} (D_{\rho} \Phi)^\dagger (D^{\rho} \Phi),
\label{Energy-momentum}\end{aligned}$$ where we have used the notation of symmetrization $A_{(\mu} B_{\mu)} = \frac{1}{2} (A_\mu B_\nu + A_\nu B_\mu)$.
Next, the equation of motion for $\Phi^\dagger$ is of form $$\begin{aligned}
\xi \Phi R + \frac{1}{\sqrt{-g}} D_\mu (\sqrt{-g} g^{\mu\nu} D_\nu \Phi) = 0.
\label{Phi eq}\end{aligned}$$ Finally, taking the variation with respect to the gauge fields $A^{(i)}_\mu$ produces “Maxwell” equations $$\begin{aligned}
\nabla_\rho F^{\mu\rho} = - i e \left[ \Phi^\dagger (D^\mu \Phi) - \Phi (D^\mu \Phi)^\dagger \right].
\label{Maxwell eq}\end{aligned}$$ Now we wish to prove that the current (\[Current\]) for the scale transformation is indeed conserved on-shell by using these equations of motion. Before doing so, let us first take the divergence of the current, whose result is given by $$\begin{aligned}
\partial_\mu J^\mu = ( 6 \xi + 1 ) \sqrt{-g} \Box (\Phi^\dagger \Phi).
\label{Div-Current}\end{aligned}$$ In order to show that the expression in the right-hand side of Eq. (\[Div-Current\]) vanishes on-shell, let us take the trace of Einstein’s equations (\[Einstein eq\]) $$\begin{aligned}
\xi \Phi^\dagger \Phi R = 3 \xi \Box (\Phi^\dagger \Phi) + (D_\mu \Phi)^\dagger (D^\mu \Phi).
\label{Trace-Einstein eq}\end{aligned}$$ Next, multiplying Eq. (\[Phi eq\]) by $\Phi^\dagger$, and then eliminating the term involving the scalar curvature, i.e., $\xi \Phi^\dagger \Phi R$, with the help of Eq. (\[Trace-Einstein eq\]), we obtain $$\begin{aligned}
3 \xi \Box (\Phi^\dagger \Phi) + (D_\mu \Phi)^\dagger (D^\mu \Phi)
+ \frac{1}{\sqrt{-g}} \Phi^\dagger D_\mu (\sqrt{-g} g^{\mu\nu} D_\nu \Phi) = 0.
\label{Combined eq}\end{aligned}$$ At this stage, it is useful to introduce a generalized covariant derivative defined as ${\cal D}_\mu = D_\mu + \Gamma_\mu$ where $\Gamma_\mu$ is the usual affine connection. Using this derivative, Eq. (\[Combined eq\]) can be rewritten as $$\begin{aligned}
3 \xi {\cal D}_\mu {\cal D}^\mu (\Phi^\dagger \Phi) + ({\cal D}_\mu \Phi)^\dagger ({\cal D}^\mu \Phi)
+ \Phi^\dagger {\cal D}_\mu {\cal D}^\mu \Phi = 0.
\label{Re-Combined eq}\end{aligned}$$ Then, adding its Hermitian conjugation to Eq. (\[Re-Combined eq\]), we arrive at $$\begin{aligned}
(6 \xi + 1) {\cal D}_\mu {\cal D}^\mu (\Phi^\dagger \Phi) = 0.
\label{Re-Combined eq2}\end{aligned}$$ The quantity $\Phi^\dagger \Phi$ is a scalar and neutral under the U(1) charge, we obtain $$\begin{aligned}
(6 \xi + 1) \Box (\Phi^\dagger \Phi) = 0.
\label{Re-Combined eq3}\end{aligned}$$ Using this equation, the right-hand side in Eq. (\[Div-Current\]) is certainly vanishing, by which we can prove that the current of the scale transformation is conserved on-shell as promised.
Now we are willing to explain our finding about spontaneous symmetry breakdown of gauge symmetry in our model where the coexistence of both scale invariance and gauge symmetry plays a pivotal role. Incidentally, it might be worthwhile to comment that in ordinary examples of spontaneous symmetry breakdown in the framework of quantum field theories, one is accustomed to dealing with a potential which has the shape of the Mexican hat type and therefore induces the symmetry breaking in a natural way, but the same recipe cannot be applied to general relativity because of the lack of such a potential.[^6]
Let us note that a very interesting recipe which induces spontaneous symmetry breakdown of $\it{scale}$ invariance via local scale transformation has been already known [@Fujii1]. This recipe can be explained as follows: Suppose that we started with a scale-invariant theory with only dimensionless coupling constants. But in the process of local scale transformation, one cannot refrain from introducing the quantity with mass dimension, which is the Planck mass $M_p$ in the present context, to match the dimensions of an equation and consequently scale invariance is spontaneously broken.
Of course, the absence of a potential which induces symmetry breaking makes it impossible to investigate a stability of the selected solution, but the very existence of the solution including the Planck mass with mass dimension justifies the claim that this phenomenon is nothing but a sort of spontaneous symmetry breakdown. This fact can be also understood by using a dilatation charge as seen shortly.
The first technique for obtaining spontaneous symmetry breakdown of both scale and gauge invariances is to find a suitable local scale transformation which transforms dilaton gravity in the Jordan frame to general relativity with matters in the Einstein frame. Of course, note that our starting Lagrangian is invariant under not the local scale transformation but the global transformation, so the change of form of the Lagrangian after the local scale transformation is reasonable. Here it is useful to parametrize the complex scalar field $\Phi$ in terms of two real fields, $\Omega$ (or $\sigma$) and $\theta$ in polar form, defined as $$\begin{aligned}
\Phi(x) = \frac{1}{\sqrt{2 \xi}} \Omega(x) e^{i \alpha \theta(x)}
= \frac{1}{\sqrt{2 \xi}} e^{\zeta \sigma(x) + i \alpha \theta(x)},
\label{Parametrization}\end{aligned}$$ where $\Omega(x) = e^{\zeta \sigma(x)}$ is a local parameter field and the constants $\zeta, \alpha$ will be determined later.
Let us then consider the following local scale transformation: $$\begin{aligned}
g_{\mu\nu} \rightarrow \tilde g_{\mu\nu} = \Omega^2(x) g_{\mu\nu}, \quad
g^{\mu\nu} \rightarrow \tilde g^{\mu\nu} = \Omega^{-2}(x) g^{\mu\nu}, \quad
A_\mu \rightarrow \tilde A_\mu = A_\mu.
\label{L-scale transf}\end{aligned}$$ Note that apart from the local property of $\Omega(x)$, this local scale transformation is different from the scale transformation (\[Scale transf\]) in that the complex scalar field $\Phi$ is not transformed at all. Under the local scale transformation (\[L-scale transf\]), the scalar curvature is transformed as $$\begin{aligned}
R = \Omega^2 ( \tilde R + 6 \tilde \Box f - 6 \tilde g^{\mu\nu} \partial_\mu f \partial_\nu f ),
\label{Curvature}\end{aligned}$$ where we have defined as $f = \log \Omega = \zeta \sigma$ and $\tilde \Box f = \frac{1}{\sqrt{- \tilde g}}
\partial_\mu (\sqrt{- \tilde g} \tilde g^{\mu\nu} \partial_\nu f) = \tilde g^{\mu\nu}
\tilde \nabla_\mu \tilde \nabla_\nu f$.
With the critical choice $$\begin{aligned}
\xi \Phi^\dagger \Phi = \frac{1}{2} \Omega^2 = \frac{1}{2} e^{2 \zeta \sigma},
\label{Choice}\end{aligned}$$ the non-minimal term in (\[Lagr 1\]) reads the Einstein-Hilbert term (plus part of the kinetic term of the scalar field $\sigma$) up to a surface term as follows: $$\begin{aligned}
\sqrt{-g} \xi \Phi^\dagger \Phi R &=& \Omega^{-4} \sqrt{- \tilde g} \frac{1}{2} \Omega^2
\Omega^2 ( \tilde R + 6 \tilde \Box f - 6 \tilde g^{\mu\nu} \partial_\mu f \partial_\nu f ) \nonumber\\
&=& \sqrt{- \tilde g} \left( \frac{1}{2} \tilde R - 3 \zeta^2 \tilde g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right).
\label{1st term}\end{aligned}$$ Then, the second term in (\[Lagr 1\]) is cast to the form $$\begin{aligned}
- \sqrt{-g} g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi)
= - \frac{1}{2 \xi} \sqrt{- \tilde g} \tilde g^{\mu\nu} \left( \zeta^2
\partial_\mu \sigma \partial_\nu \sigma + e^2 B_\mu B_\nu \right),
\label{2nd term}\end{aligned}$$ where we have chosen $\alpha = e$ for convenience, and defined a new massive gauge field $B_\mu$ as $$\begin{aligned}
B_\mu = A_\mu + \partial_\mu \theta.
\label{B-field}\end{aligned}$$ In terms of this new gauge field $B_\mu$, the Maxwell’s Lagrangian in (\[Lagr 1\]) is described in the Einstein frame as $$\begin{aligned}
- \frac{1}{4} \sqrt{-g} g^{\mu\nu} g^{\rho\sigma} F_{\mu\rho} F_{\nu\sigma}
= - \frac{1}{4} \sqrt{- \tilde g} \tilde g^{\mu\nu} \tilde g^{\rho\sigma} \tilde F_{\mu\rho} \tilde F_{\nu\sigma},
\label{Maxwell Lagr}\end{aligned}$$ where $\tilde F_{\mu\nu} \equiv \partial_\mu B_\nu - \partial_\nu B_\mu$.
It is worthwhile to stress again that in the process of local scale transformation we have had to introduce the mass scale into a theory having no dimensional constants, thereby inducing the breaking of the scale invariance. More concretely, to match the dimensions in the both sides of the equation, the Planck mass $M_p$ must be introduced in the ciritical choice (\[Choice\]) (recovering the Planck mass) $$\begin{aligned}
\xi \Phi^\dagger \Phi = \frac{1}{2} \Omega^2 M_p^2 = \frac{1}{2} e^{2 \zeta \sigma} M_p^2.
\label{Choice2}\end{aligned}$$ It is also remarkable to notice that in the process of spontaneous symmetry breakdown of the scale invariance, the Nambu-Goldstone boson $\theta$ is absorbed into the $U(1)$ gauge field $A_\mu$ as a longitudinal mode and as a result $B_\mu$ acquires a mass, which is nothing but the Higgs mechanism! In other words, the $U(1)$ gauge symmetry is broken at the same time and on the same energy scale that the scale symmetry is spontaneously broken. The size of the mass $M_B$ of $B_\mu$ can be read off from (\[2nd term\]) as $M_B = \frac{e}{\sqrt{\xi}} M_p$ which is also equal to the energy scale on which the scale invariance is broken.
Putting (\[1st term\]), (\[2nd term\]) and (\[Maxwell Lagr\]) together, and defining $\zeta^{-2} = 6 + \frac{1}{\xi}$ (by which the kinetic term for the $\sigma$ field becomes a canonical form), the Lagrangian (\[Lagr 1\]) is reduced to the form $$\begin{aligned}
{\cal L} = \sqrt{- \tilde g} \left[ \frac{1}{2} M_p^2 \tilde R - \frac{1}{2} \tilde g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{1}{4} \tilde F_{\mu\nu}^2 - \frac{e^2}{2 \xi} M_p^2 B_\mu B^\mu \right],
\label{Lagr 2}\end{aligned}$$ where we have recovered the Planck mass $M_p$ for clarity. Let us note that the first term coincides with the Einstein-Hilbert term in general relativity, the second term implies that the dilaton $\sigma$ is massless at the classical level, and the last two terms means that the gauge field becomes massive via the new Higgs mechanism.
As an interesting application of our finding to phenomenology, we can propose two scenarios at the different energy scales. One scenario, which was adopted in case of the classically scale-invariant B-L model [@Iso; @Oda1; @Oda2], is the spontaneous symmetry breakdown at the TeV scale where $\frac{e}{\sqrt{\xi}} \approx 10^{-15}$, so the gravity is in the strong coupling phase. The other scenario is to trigger the spontaneous symmetry breakdown of both scale and gauge symmetry at the Planck scale, for which we take $\frac{e}{\sqrt{\xi}} \approx 1$ and the gravity is in the weak coupling phase.
Finally, let us comment on the physical meaning of the dilaton $\sigma$. The dilaton is a massless particle and interact with the other fields only through the covariant derivative $\tilde D_\mu = D_\mu + \zeta (\partial_\mu \sigma)$, but owing to its nature of the derivative coupling, at the low energy this coupling is so small that it is difficult to detect the dilaton experimentally.
To understand the physical meaning of the dilaton more clearly, it is useful to evaluate the dilatation current $J^\mu$ in (\[Current\]) in the Einstein frame. The result reads $$\begin{aligned}
J^\mu = \frac{1}{\zeta} \sqrt{- \tilde g} \tilde g^{\mu\nu} \partial_\nu \sigma.
\label{Current2}\end{aligned}$$ This is exactly the expected form of the current seen in the case of the conventional spontaneous symmetry breakdown, with $\frac{1}{\zeta}$ playing the role of the vacuum value of the order parameter, and the dilaton $\sigma$ doing of the Nambu-Goldstone boson associated with the spontaneous symmetry breakdown of the scale invariance. This result can be also reached by constructing the corresponding charge which is defined as $Q_D = \int d^3 x J^0$. Note that this charge does not annihilate the vacuum because of the linear form in $\sigma$ $$\begin{aligned}
Q_D | 0 > \neq 0.
\label{Vacuum}\end{aligned}$$ Of course, it is also possible to show $\partial_\mu J^\mu = 0$ in terms of equations of motion in the Einstein frame as proved in the Jordan frame before. It therefore turns out that the dilaton $\sigma$ is indeed the Nambu-Goldstone boson associated with spontaneous symmetry breakdown of the scale invariance. We will see later that although the dilaton is massless at the classical level, the trace anomaly makes the dilaton be massive at the quantum level.
The generalization to non-Abelian groups
========================================
In this section, we wish to extend the present formalism to arbitrary non-Abelian gauge groups. For clarity, we shall consider only the $SU(2)$ gauge group with a complex $SU(2)$-doublet of scalar field $\Phi^T = (\Phi_1, \Phi_2)$ since the generalization to a general non-Abelian gauge group is straightforward.
Let us start with the $SU(2)$ generalization of the Lagrangian (\[Lagr 1\]) $$\begin{aligned}
{\cal L} = \sqrt{-g} \left[ \xi \Phi^\dagger \Phi R - g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi)
- \frac{1}{4} g^{\mu\nu} g^{\rho\sigma} F_{\mu\rho}^a F_{\nu\sigma}^a \right],
\label{NA-Lagr 1}\end{aligned}$$ where $a$ is an $SU(2)$ index running over $1, 2, 3$, and the covariant derivative and field strength are respectively defined as $$\begin{aligned}
D_\mu \Phi &=& (\partial_\mu - i g \tau^a A_\mu^a) \Phi, \quad
(D_\mu \Phi)^\dagger = (\partial_\mu + i g \tau^a A_\mu^a) \Phi^\dagger, \quad \nonumber\\
F_{\mu\nu}^a &=& \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g \varepsilon^{abc} A_\mu^b A_\nu^c.
\label{NA-Def 1}\end{aligned}$$ Here $g$ is an $SU(2)$ coupling constant (Do not confuse with the determinant of the metric tensor since we use the same letter of the alphabet). Furthermore, the matrices $\tau^a$ are defined as half of the Pauli ones, i.e., $\tau^a = \frac{1}{2} \sigma^a$, so the following relations are satisfied: $$\begin{aligned}
\{ \tau^a, \tau^b \} = \frac{1}{2} \delta^{ab}, \quad
[ \tau^a, \tau^b ] = i \varepsilon^{abc} \tau^c.
\label{tau-matrix}\end{aligned}$$ In order to see the Higgs mechanism discussed in the previous section explicitly, it is convenient to go to the unitary gauge. To do that, we first parametrize the scalar doublet as $$\begin{aligned}
{\Phi(x)} = U^{-1}(x) \frac{1}{\sqrt{2 \xi}} \ e^{\zeta \sigma(x)} \left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right),
\label{U-gauge}\end{aligned}$$ where a unitary matrix $U(x)$ is defined as $U(x) = e^{ -i \alpha \tau^a \theta^a(x)}$ with $\alpha$ being a real number. Then, we will define new fields in the unitary gauge by $$\begin{aligned}
\Phi^u(x) &=& U(x) \Phi(x) = \frac{1}{\sqrt{2 \xi}} \ e^{\zeta \sigma(x)} \left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right)
= \frac{1}{\sqrt{2 \xi}} \ \Omega(x) \left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right), \nonumber\\
\tau^a B_\mu^a &=& U(x) \tau^a A_\mu^a U^{-1}(x) - \frac{i}{g} \partial_\mu U(x) U^{-1}(x).
\label{NA-new fields}\end{aligned}$$ Using these new fields, after an easy calculation, we find the following relations $$\begin{aligned}
D_\mu \Phi = U^{-1}(x) D_\mu \Phi^u, \quad
F_{\mu\nu}^a F^{a \mu\nu} = F_{\mu\nu}^a(B) F^{a \mu\nu}(B),
\label{NA-Rel 1}\end{aligned}$$ where $D_\mu \Phi^u$ and $F_{\mu\nu}^a(B)$ are respectively defined as $$\begin{aligned}
D_\mu \Phi^u = (\partial_\mu - i g \tau^a B_\mu^a) \Phi^u, \quad
F_{\mu\nu}^a(B) = \partial_\mu B_\nu^a - \partial_\nu B_\mu^a + g \varepsilon^{abc} B_\mu^b B_\nu^c.
\label{NA-Def 2}\end{aligned}$$ To reach the desired Lagrangian, we can follow a perfectly similar path of argument to the case of the Abelian gauge group in the previous section. In other words, we will take a critical choice $$\begin{aligned}
\xi \Phi^{u \dagger} \Phi^u = \frac{1}{2} \Omega^2 M_p^2 = \frac{1}{2} e^{2 \zeta \sigma} M_p^2,
\label{NA-Choice}\end{aligned}$$ and make use of the local scale transformation by the local parameter $\Omega(x)$ to move from the Jordan frame to the Einstein frame. After performing this procedure, the final Lagrangian reads $$\begin{aligned}
{\cal L} = \sqrt{- \tilde g} \left[ \frac{1}{2} M_p^2 \tilde R - \frac{1}{2} \tilde g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{1}{4} (\tilde F_{\mu\nu}^a)^2 - \frac{g^2}{8 \xi} M_p^2 B_\mu^a B^{a \mu} \right],
\label{NA-Lagr 2}\end{aligned}$$ where $\tilde F_{\mu\nu}^a \equiv F_{\mu\nu}^a(B)$. The mass of the massive gauge field $B_{\mu\nu}^a$ is easily read off to be $M_B = \frac{g}{2 \sqrt{\xi}} M_p$. As in the Abelian group, we can see that the massless dilaton $\sigma$ is the Nambu-Goldstone boson of spontaneous symmetry breakdown of scale symmetry by making the conserved dilatation current and its charge.
The generalization to scalar field with many components
=======================================================
Let us recall that the quantum field theory of a scalar field with many components goes in much the same way as that of a single component except that new interesting internal symmetry arises. This general fact is also valid even in the present formalism if we take a common “radial” field in all the components. When we consider a general “radial” field, we must face an annoying issue of getting the canonical kinetic term for the dilaton. The procedure of obtaining the canonical kinetic term is just a problem of matrix diagonalization and is not in principle a problem, but the general treatment makes our formalism very complicated. We will therefore focus on the case of the common “radial” field in this section. In the next section, we will meet the same situation since we consider two scalar fields coupling to a curvature scalar in the non-minimal manner, but a reasonable approximation can serve to avoid this annoying issue.
The starting Lagrangian is just a generalization of (\[Lagr 1\]) to $n$ complex scalar fields, or equivalently a complex scalar field with $n$ independent components $\Phi_i ( i = 1, 2, \cdots, n )$ $$\begin{aligned}
{\cal L} = \sum_{i=1}^{n} \sqrt{-g} \left[ \xi_i \Phi_i^\dagger \Phi_i R - g^{\mu\nu} (D_\mu \Phi_i)^\dagger (D_\nu \Phi_i)
- \frac{1}{4} g^{\mu\nu} g^{\rho\sigma} F_{\mu\rho}^{(i)} F_{\nu\sigma}^{(i)} \right],
\label{M-Lagr 1}\end{aligned}$$ where $\xi_i$ are positive and dimensionless constants. The covariant derivative and field strength are respectively defined as $$\begin{aligned}
D_\mu \Phi_i = (\partial_\mu - i e_i A_\mu^{(i)}) \Phi_i, \quad
(D_\mu \Phi_i)^\dagger = (\partial_\mu + i e_i A_\mu^{(i)}) \Phi_i^\dagger, \quad
F_{\mu\nu}^{(i)} = \partial_\mu A_\nu^{(i)} - \partial_\nu A_\mu^{(i)}.
\label{M-Def 1}\end{aligned}$$ Since the Lagrangian (\[M-Lagr 1\]) includes only dimensionless coupling constants, it is manifestly invariant under a global scale transformation. Following the Noether theorem, the current for the scale transformation reads $$\begin{aligned}
J^\mu = \sum_{i=1}^{n} ( 6 \xi_i + 1 ) \sqrt{-g} g^{\mu\nu} \partial_\nu \left(\Phi_i^\dagger \Phi_i \right).
\label{M-Current}\end{aligned}$$ In a similar way to the cases of both Abelian and non-Abelian gauge groups, we can show that this current is conserved on-shell.
As mentioned in the above, the point is to take a common “radial” (real) field $\Omega(x)$ such that $$\begin{aligned}
\Phi_i (x) = \frac{1}{\sqrt{2 n \xi_i}} \Omega(x) e^{i \alpha_i \theta_i (x)}
= \frac{1}{\sqrt{2 n \xi_i}} e^{\zeta \sigma(x) + i \alpha_i \theta_i (x)}.
\label{M-Parametrization}\end{aligned}$$ This is a great simplification in the sense that $n$ real component fields in $\Phi_i (x)$ is reduced to a single one, but this restriction is needed to obtain the canonical kinetic term for the dilaton in a rather simple way.
Now we would like to show that the Lagrangian (\[M-Lagr 1\]) has the property of spontaneous symmetry breakdown of gauge symmetry when scale symmetry is spontaneously broken. To do that, we proceed similar steps to the case of the single scalar field with the Abelian gauge group in Section 2. With the critical choice $$\begin{aligned}
\xi_i \Phi_i^\dagger \Phi_i = \frac{1}{2 n} \Omega^2 = \frac{1}{2 n} e^{2 \zeta \sigma},
\label{M-Choice}\end{aligned}$$ the non-minimal term in (\[M-Lagr 1\]) yields the Einstein-Hilbert term and part of the kinetic term of the scalar field $\sigma$) up to a surface term $$\begin{aligned}
\sum_{i=1}^{n} \sqrt{-g} \xi_i \Phi_i^\dagger \Phi_i R
= \sqrt{- \tilde g} \left( \frac{1}{2} \tilde R - 3 \zeta^2 \tilde g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right).
\label{M-1st term}\end{aligned}$$ Moreover, the second term in (\[M-Lagr 1\]) is reduced to $$\begin{aligned}
- \sum_{i=1}^{n} \sqrt{-g} g^{\mu\nu} (D_\mu \Phi_i)^\dagger (D_\nu \Phi_i)
= - \sum_{i=1}^{n} \frac{1}{2 n \xi_i} \sqrt{- \tilde g} \tilde g^{\mu\nu} \left( \zeta^2
\partial_\mu \sigma \partial_\nu \sigma + e_i^2 B_\mu^{(i)} B_\nu^{(i)} \right),
\label{M-2nd term}\end{aligned}$$ where we have selected $\alpha_i = e_i$ and defined new massive gauge fields $B_\mu^{(i)}$ by $$\begin{aligned}
B_\mu^{(i)} = A_\mu^{(i)} + \partial_\mu \theta_i.
\label{M-B-field}\end{aligned}$$ In terms of the new gauge fields $B_\mu^{(i)}$, the Maxwell’s Lagrangian in (\[M-Lagr 1\]) is cast to the form $$\begin{aligned}
- \frac{1}{4} \sum_{i=1}^{n} \sqrt{-g} g^{\mu\nu} g^{\rho\sigma} F_{\mu\rho}^{(i)} F_{\nu\sigma}^{(i)}
= - \frac{1}{4} \sum_{i=1}^{n} \sqrt{- \tilde g} \tilde g^{\mu\nu} \tilde g^{\rho\sigma} \tilde F_{\mu\rho}^{(i)}
\tilde F_{\nu\sigma}^{(i)},
\label{M-Maxwell Lagr}\end{aligned}$$ where $\tilde F_{\mu\nu}^{(i)} \equiv \partial_\mu B_\nu^{(i)} - \partial_\nu B_\mu^{(i)}$.
To summarize, the Lagrangian (\[M-Lagr 1\]) is given by $$\begin{aligned}
{\cal L} = \sqrt{- \tilde g} \left\{ \frac{1}{2} M_p^2 \tilde R - \frac{1}{2} \tilde g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
+ \sum_{i=1}^{n} \left[ - \frac{1}{4} (\tilde F_{\mu\nu}^{(i)})^2 - \frac{e_i^2}{2 n \xi_i} M_p^2 B_\mu^{(i)} B^{(i) \mu} \right] \right\},
\label{M-Lagr 2}\end{aligned}$$ where the definition $\zeta^{-2} = 6 + \frac{1}{n} \sum_{i=1}^{n} \frac{1}{\xi_i}$ is used. It is then obvious that this system also exhibits the new Higgs mechanism triggered by spontaneous symmetry breakdown of scale symmetry.
Radiative corrections and dilaton mass
======================================
In this section, as an example, we wish to apply our idea discussed so far to the standard model and evaluate the quantum effects. Since the standard model is known to not be classically scale-invariant because of the presence of the (negative) mass term of the Higgs field, we must replace the mass term with a new scalar field. It is then natural to identify this new scalar with the $\Phi$ field which couples to a scalar curvature in a non-minimal manner.
Let us first recall that in our previous work [@Oda1] we have already considered one-loop effects of a classically scale invariant B-L model [@Iso]. However, our finding of gravitational spontaneous symmetry breakdown of gauge symmetry as a result of spontaneous symmetry breakdown of scale symmetry is very universal in the sense that our ideas can be generalized not only to local scale symmetry as clarified in [@Oda2] but also to non-Abelian gauge groups and even a scalar field with many of components as discussed in this article. In fact, it is obvious that our ideas can be applied to any model which is scale-invariant and involves the non-minimal coupling terms between the curvature scalar and charged scalars associated with local gauge symmetries.
In our calculation, we are not ambitious enough to quantize the metric tensor field and take a fixed Minkowski background $g_{\mu\nu} = \eta_{\mu\nu}$. Moreover, we restrict ourselves to the calculation of radiative corrections between dilaton and matter fields in the weak-field approximation.
One of the motivations behind this study is to calculate the size of the mass of dilaton. As shown above, the dilaton is exactly massless at the classical level owing to scale symmetry, but it is well-known that radiative corrections violate the scale invariance thereby leading to the trace anomaly. Consequently, the dilaton becomes massive in the quantum regime. Since the dilaton is a scalar field like the Higgs particle, one might expect that there could be quadratic divergence for the self-energy diagram. On the other hand, since the dilaton is the Nambu-Goldstone boson resulting from the scale symmetry, the dilaton mass would be much lower in the such a way that the pion masses are very smaller as the pions can be understood as the Nambu-Goldstone boson coming from $SU(2)_L \times SU(2)_R \rightarrow SU(2)_V$ flavor symmetry breaking. As long as we know, nobody has calculated the dilaton mass in a reliable manner, so we wish to calculate the dilaton mass within the framework of the present formalism and determine which scenario, quadratic divergence and huge radiative corrections like the Higgs particle or very lower mass like the pions, is realized. Remarkably enough, it will be shown that although the dilaton mass is quadratic divergent, the cutoff scale, which is the Planck mass in the formalism at hand, is exactly cancelled by the induced coupling constant, by which the dilaton mass is kept to be around the GeV scale.
Whenever we evaluate anomalies, the key point is to adopt a suitable regularization method respecting classical symmetries existing in the action as much as possible. In this article, as a regularization method, we make use of the method of continuous space-time dimensions, for which we rewrite previous results in arbitrary $D$ dimensions [@Fujii2]. Like the dimensional regularization, the divergences will appear as poles $\frac{1}{D-4}$, which are cancelled by the factor $D-4$ that multiplies the dilaton coupling, thereby producing a finite result leading to an effective interaction term.
Basic formalism
---------------
Our starting Lagrangian, which is a scale-invariant extension of the standard model coupled to the non-minimal terms, is of form $$\begin{aligned}
{\cal L} &=& \sqrt{-g} \Big[ ( \xi_1 \Phi^\dagger \Phi + \xi_2 H^\dagger H ) R
- g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi) - g^{\mu\nu} (D_\mu H)^\dagger (D_\nu H)
\nonumber\\
&-& \frac{1}{4} g^{\mu\nu} g^{\rho\sigma} ( F_{\mu\rho}^{(1)} F_{\nu\sigma}^{(1)}
+ F_{\mu\rho}^{(2)} F_{\nu\sigma}^{(2)} + F_{\mu\rho}^a F_{\nu\sigma}^a )
- V (H, \Phi) + L_m \Big],
\label{Q-Lagr 1}\end{aligned}$$ where $L_m$ denotes the remaining Lagrangian part of the standard-model sector such as the Yukawa couplings and various definitions are given by the following expressions: $$\begin{aligned}
D_\mu \Phi &=& (\partial_\mu - i e_1 A_\mu^{(1)}) \Phi, \quad
D_\mu H = (\partial_\mu - i g \tau^a A_\mu^a - i \frac{e_2}{2} A_\mu^{(2)}) H, \nonumber\\
F_{\mu\nu}^{(i)} &=& \partial_\mu A_\nu^{(i)} - \partial_\nu A_\mu^{(i)}, \quad
F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g \varepsilon^{abc} A_\mu^b A_\nu^c,
\nonumber\\
V (H, \Phi) &=& \lambda_\Phi (\Phi^\dagger \Phi)^2 + \lambda_{H \Phi} (H^\dagger H) (\Phi^\dagger \Phi)
+ \lambda_H (H^\dagger H)^2.
\label{Qb 1}\end{aligned}$$ with $e_i (i = 1, 2)$ being $U(1)$ coupling constants and $g$ being an $SU(2)$ coupling constant. As in the Appendix A, in this model, we can also calculate the Noether current for scale transformation $$\begin{aligned}
J^\mu = \sqrt{-g} g^{\mu\nu} \partial_\nu [ ( 6 \xi_1 + 1 ) \Phi^\dagger \Phi
+ ( 6 \xi_2 + 1 ) H^\dagger H ].
\label{Q-Current}\end{aligned}$$ It turns out that this dilatation current is conserved on-shell as well.
For simplicity of presentation, we take the vanishing $SU(2)$ gauge field, $A_\mu^a = 0$ since this assumption does not change the essential conclusion for our purpose.
In general $D$ space-time dimensions, as a generalization of Eq. (\[L-scale transf\]), the local scale transformation is defined as $$\begin{aligned}
\hat g_{\mu\nu} &=& \Omega^2(x) g_{\mu\nu}, \quad \hat g^{\mu\nu} = \Omega^{-2}(x) g^{\mu\nu}, \quad
\hat \Phi = \Omega^{- \frac{D-2}{2}}(x) \Phi, \nonumber\\
\hat H &=& \Omega^{- \frac{D-2}{2}}(x) H, \quad \hat A^{(i)}_\mu = \Omega^{- \frac{D-4}{2}}(x) A^{(i)}_\mu.
\label{Q-L-scale transf}\end{aligned}$$ Under this local scale transformation (\[Q-L-scale transf\]), with the definition $f = \log \Omega$, the scalar curvature is transformed as $$\begin{aligned}
R = \Omega^2 \left[ \hat R + 2 (D-1) \hat \Box f - (D-1) (D-2) \hat g^{\mu\nu} \partial_\mu f \partial_\nu f \right],
\label{Q-Curvature}\end{aligned}$$ for which we set $D=4$ in what follows since we do not quantize the metric tensor and therefore do not have poles from the curvature.
In a physically more realistic situation, the scale symmetry must be broken spontaneously in the higher energy region before spontaneous symmetry breaking of the electro-weak symmetry since all quantum field theories must in principle contain the gravity from the beginning although contributions from the gravity can be usually ignored when dealing with particle physics processes.
Therefore, let us first break the scale invariance by taking the following value for the charged scalar field $\Phi$: $$\begin{aligned}
\Phi = \frac{1}{\sqrt{2 \xi_1}} \Omega^{\frac{D-2}{2}} e^{i \alpha \theta}
= \frac{1}{\sqrt{2 \xi_1}} e^{\zeta \sigma + i \alpha \theta},
\label{Q-Choice}\end{aligned}$$ where we have defined $\Omega(x) = e^{\frac{2}{D-2} \zeta \sigma}$ and $\zeta^{-2} \equiv 4 \frac{D-1}{D-2} + \frac{1}{\xi_1} = 6 + \frac{1}{\xi_1}$. Then, the first term in (\[Q-Lagr 1\]) takes the form $$\begin{aligned}
\sqrt{-g} \xi_1 \Phi^\dagger \Phi R =
\sqrt{- \hat g} \left( \frac{1}{2} \hat R - 3 \zeta^2 \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right).
\label{Q-1st term}\end{aligned}$$ Similarly, the third term in (\[Q-Lagr 1\]) is changed to the form $$\begin{aligned}
- \sqrt{-g} g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi)
= - \frac{1}{2 \xi_1} \sqrt{- \hat g} \hat g^{\mu\nu} \left( \zeta^2
\partial_\mu \sigma \partial_\nu \sigma + \hat e_1^2 \hat B_\mu^{(1)} \hat B_\nu^{(1)} \right),
\label{Q-3rd term}\end{aligned}$$ where we have defined a new coupling constant and massive gauge field as $$\begin{aligned}
\hat e_1 = \Omega^{\frac{D-4}{2}} e_1, \quad \hat B_\mu^{(1)} = \hat A_\mu^{(1)} - \partial_\mu \theta,
\label{Q-B-field}\end{aligned}$$ and chosen $\alpha = \hat e_1$ for convenience. Adding (\[Q-1st term\]) and (\[Q-3rd term\]) together yields the expression $$\begin{aligned}
\sqrt{-g} \left[ \xi_1 \Phi^\dagger \Phi R - g^{\mu\nu} (D_\mu \Phi)^\dagger (D_\nu \Phi) \right]
= \sqrt{- \hat g} \left( \frac{1}{2} \hat R - \frac{1}{2} \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{\hat e_1^2}{2 \xi_1} \hat g^{\mu\nu} \hat B_\mu^{(1)} \hat B_\nu^{(1)} \right).
\label{Q-1st+3rd term}\end{aligned}$$ On the other hand, the Lagrangian of matter fields turns out to depend on the dilaton field $\sigma$ in a non-trivial manner in general $D$ space-time dimensions. First, the non-minimal term for $H$ field becomes $$\begin{aligned}
\sqrt{-g} \xi_2 H^\dagger H R
= \sqrt{- \hat g} \ \xi_2 \hat H^\dagger \hat H
\left( \hat R + 6 \zeta^2 \hat \Box \sigma
- 6 \zeta^2 \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right).
\label{Q-H}\end{aligned}$$ Second, the kinetic term for $H$ is cast to $$\begin{aligned}
- \sqrt{-g} g^{\mu\nu} (D_\mu H)^\dagger (D_\nu H)
= - \sqrt{- \hat g} \hat g^{\mu\nu} (\hat D_\mu \hat H)^\dagger (\hat D_\nu \hat H),
\label{Q-H2}\end{aligned}$$ where the new covariant derivative is defined as $$\begin{aligned}
\hat D_\mu \hat H = ( \partial_\mu + \zeta \partial_\mu \sigma
- \frac{i}{2} \hat e_2 \hat A_\mu^{(2)} ) \hat H,
\label{Q-DH}\end{aligned}$$ with being $\hat e_2 = \Omega^{\frac{D-4}{2}} e_2$. Third, the electro-magnetic terms are reduced to the form $$\begin{aligned}
- \frac{1}{4} \sqrt{-g} \sum_{i=1}^2 g^{\mu\nu} g^{\rho\sigma} F^{(i)}_{\mu\rho} F^{(i)}_{\nu\sigma}
= - \frac{1}{4} \sqrt{- \hat g} \sum_{i=1}^2 \hat g^{\mu\nu} \hat g^{\rho\sigma}
\hat F^{(i)}_{\mu\rho} \hat F^{(i)}_{\nu\sigma},
\label{Q-EM}\end{aligned}$$ where the new field strengths are defined as[^7] $$\begin{aligned}
\hat F^{(1)}_{\mu\nu} &=& \Omega^{2- \frac{D}{2}} F^{(1)}_{\mu\nu}
= \partial_\mu \hat B^{(1)}_\nu + \frac{D-4}{2} \zeta \partial_\mu \sigma (\hat B^{(1)}_\nu
+ \partial_\nu \theta) - (\mu \leftrightarrow \nu), \nonumber\\
\hat F^{(2)}_{\mu\nu} &=& \Omega^{2- \frac{D}{2}} F^{(2)}_{\mu\nu}
= \partial_\mu \hat A^{(2)}_\nu + \frac{D-4}{2} \zeta \partial_\mu \sigma \hat A^{(2)}_\nu
- (\mu \leftrightarrow \nu).
\label{Q-F}\end{aligned}$$ Finally, the potential term can be rewritten as $$\begin{aligned}
\sqrt{-g} V (H, \Phi) &=& \sqrt{- \hat g} V(\hat H) \nonumber\\
&=& \sqrt{- \hat g} e^{\frac{2(D-4)}{D-2} \zeta \sigma} \left[ \frac{1}{4 \xi_1^2} \lambda_\Phi M_p^4
+ \frac{1}{2 \xi_1} \lambda_{H \Phi} M_p^2 (\hat H^\dagger \hat H)
+ \lambda_H (\hat H^\dagger \hat H)^2 \right].
\label{Q-P}\end{aligned}$$ To summarize, the starting Lagrangian is now of the form $$\begin{aligned}
{\cal L} &=& \sqrt{- \hat g} \Big[ \frac{1}{2} \hat R
- \frac{1}{2} \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{\hat e_1^2}{2 \xi_1} \hat g^{\mu\nu} \hat B_\mu^{(1)} \hat B_\nu^{(1)}
\nonumber\\
&+& \xi_2 \hat H^\dagger \hat H \left( \hat R + 6 \zeta^2 \hat \Box \sigma
- 6 \zeta^2 \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right)
- \hat g^{\mu\nu} (\hat D_\mu \hat H)^\dagger (\hat D_\nu \hat H)
\nonumber\\
&-& \frac{1}{4} \sum_{i=1}^2 \hat g^{\mu\nu} \hat g^{\rho\sigma}
\hat F^{(i)}_{\mu\rho} \hat F^{(i)}_{\nu\sigma}
- V (\hat H) + L_m \Big].
\label{Q-Lagr 2}\end{aligned}$$ Next, we are ready to deal with spontaneous symmetry breakdown of the electro-weak symmetry, which is assumed to occur at the lower energy, GeV scale, than breaking of scale symmetry. To realize the spontaneous symmetry breakdown of the electro-weak symmetry, we assume the conventional ansatz $$\begin{aligned}
\lambda_{H \Phi} < 0, \quad \lambda_{H} > 0.
\label{Q-Ansatz}\end{aligned}$$ With the parametrization $\hat H^T = (0, \frac{v + h}{\sqrt{2}}) e^{i \varphi}$, after the spontaneous symmetry breakdown of the electro-weak symmetry, the potential term can be described as $$\begin{aligned}
V(\hat H)
= e^{\frac{2(D-4)}{D-2} \zeta \sigma} \left[ \frac{1}{2} m_h^2 h^2 + \sqrt{\frac{\lambda_H}{2}} m_h h^3
+ \frac{\lambda_H}{4} h^4 \right],
\label{Q-P2}\end{aligned}$$ where a constant in the square bracket is discarded, and the vacuum expectation value $v$ and the Higgs mass $m_h$ are respectively defined as $$\begin{aligned}
v^2 = \frac{1}{2 \xi_1} \frac{|\lambda_{H \Phi}|}{\lambda_H} M_p^2, \quad
m_h = \sqrt{2 \lambda_H} v = \sqrt{\frac{|\lambda_{H \Phi}|}{\xi_1}} M_p,
\label{Q-P3}\end{aligned}$$ where the Planck mass $M_p$ is explicitly written.
Now we wish to consider couplings between the dilaton field $\sigma$ and matter fields which vanish at the classical level ($D=4$) but provide a finite contribution at the quantum level, interpreted as the trace anomaly. In the weak field approximation, let us extract terms linear in the dilaton $\sigma$ in $V(\hat H)$ as $$\begin{aligned}
e^{\frac{2(D-4)}{D-2} \zeta \sigma} \approx 1 + (D-4) \zeta \sigma.
\label{Q-EXP}\end{aligned}$$ Then, the potential $V(\hat H)$ is devided into two parts $$\begin{aligned}
V(\hat H) = V^{(0)} (\hat H) + V^{(1)} (\hat H),
\label{Q-P4}\end{aligned}$$ where we have defined as $$\begin{aligned}
V^{(0)} (\hat H) &=& \frac{1}{2} m_h^2 h^2 + \sqrt{\frac{\lambda_H}{2}} m_h h^3
+ \frac{\lambda_H}{4} h^4,
\nonumber\\
V^{(1)} (\hat H) &=& (D-4) \zeta V^{(0)} (\hat H) \sigma.
\label{Q-P5}\end{aligned}$$ Using the parametrization $\hat H^T = (0, \frac{v + h}{\sqrt{2}}) e^{i \varphi}$, the remaining part including the field $H$ except the Higgs potential is also rewritten, and consequently the whole Lagrangian (\[Q-Lagr 2\]) takes a little longer expression $$\begin{aligned}
{\cal L} &=& \sqrt{- \hat g} \ \Big\{ \frac{1}{2} M_p^2 \hat R
- \frac{1}{2} \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{\hat e_1^2}{2 \xi_1} M_p^2 \hat g^{\mu\nu} \hat B_\mu^{(1)} \hat B_\nu^{(1)}
\nonumber\\
&+& \frac{1}{2} \xi_2 v^2 \hat R + \xi_2 (v h + \frac{1}{2} h^2) ( \hat R - 6 \zeta^2 \frac{1}{M_p^2}
\hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma )
\nonumber\\
&-& 3 \xi_2 \zeta^2 \frac{v^2}{M_p^2} \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
-6 \xi_2 \zeta^2 \frac{v}{M_p} \hat g^{\mu\nu} \partial_\mu h \partial_\nu \sigma
+ 3 \xi_2 \zeta^2 \frac{1}{M_p} h^2 \hat \Box \sigma
\nonumber\\
&-& \hat g^{\mu\nu} \Big[ \frac{1}{2} \partial_\mu h \partial_\nu h
+ \zeta \frac{1}{M_p} \partial_\mu h \partial_\nu \sigma ( v + h )
+ \frac{1}{2} \zeta^2 \frac{1}{M_p^2} \partial_\mu \sigma \partial_\nu \sigma ( v + h )^2
\nonumber\\
&+& \frac{\hat e_2^2}{8} \hat A_\mu^{(2)} \hat A_\nu^{(2)} ( v + h )^2 \Big]
- \frac{1}{4} \sum_{i=1}^2 \hat g^{\mu\nu} \hat g^{\rho\sigma} \hat F^{(i)}_{\mu\rho} \hat F^{(i)}_{\nu\sigma}
- V (\hat H) + L_m \Big\},
\label{Q-Lagr 3}\end{aligned}$$ where we have recovered the Planck mass scale $M_p$. As mentioned in Section 4, given two non-minimal terms, we need to diagonalize the kinetic terms for the dilaton $\sigma$ and the Higgs field $h$ to get the canonical form. However, in the present context, the energy scale $v$ of the electro-weak symmetry breaking is much lower compared to that of the scale symmetry one, so it is reasonable to take the approximation $$\begin{aligned}
\frac{v}{M_p} \ll 1, \quad \xi_2 v^2 \ll M_p^2.
\label{Q-Approx}\end{aligned}$$ With this approximation, the Lagrangian is rather simplified to $$\begin{aligned}
{\cal L} &=& \sqrt{- \hat g} \ \Big[ \frac{1}{2} \hat R
- \frac{1}{2} \hat g^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma
- \frac{\hat e_1^2}{2 \xi_1} \hat g^{\mu\nu} \hat B_\mu^{(1)} \hat B_\nu^{(1)}
\nonumber\\
&+& \xi_2 (v h + \frac{1}{2} h^2) \hat R
- \frac{1}{2} \hat g^{\mu\nu} \partial_\mu h \partial_\nu h
- \frac{\hat e_2^2}{8} \hat g^{\mu\nu} \hat A_\mu^{(2)} \hat A_\nu^{(2)} ( v + h )^2
\nonumber\\
&-& \frac{1}{4} \sum_{i=1}^2 \hat g^{\mu\nu} \hat g^{\rho\sigma} \hat F^{(i)}_{\mu\rho} \hat F^{(i)}_{\nu\sigma}
- V (\hat H) + L_m \Big].
\label{Q-Lagr 4}\end{aligned}$$ Based on this Lagrangian, we wish to calculate quantum effects, in particular, on the dilaton coupling below. Since we are interested in the low energy region, the derivative coupling of the dilaton appearing in $\hat F_{\mu\nu}^{(i)}$ and $\hat D_\mu \hat H$ will be ignored in the calculation.
The coupling between dilaton and Higgs field
--------------------------------------------
In this subsection, we first switch off the $U(1)$ fields and calculate the coupling between the dilaton $\sigma$ and the Higgs particle $h$ and derive an effective Lagrangian at the one-loop level. The contribution from the $U(1)$ fields will be discussed in the later subsection. We will see that the $\sigma h^n (2 \le n \le 4)$ $(n+1)$-point diagrams are non-vanishing whereas the $\sigma h^n (n \ge 5)$ diagrams are vanishing.
First, let us consider three-point (with two Higgs $h$ and one dilaton $\sigma$ as the external particles), one-loop diagrams. Inspection of the vertices reveals that we have three types of one-loop divergent diagrams in which the Higgs field is circulating in the loop and one dilaton field, whose momentum is assumed to be vanishing, couples. Note that the divergences stemming from the Higgs one-loop diagrams provide us with poles $\frac{1}{D-4}$, which cancel the factor $D-4$ multiplying the dilaton coupling in $V^{(1)} (\hat H)$, thereby yielding a finite contribution.
One type of one-loop divergent diagram, which we call the diagram (A1), is a tadpole type and is given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) 3! \zeta \lambda_H$ in $V^{(1)} (\hat H)$. The corresponding amplitude ${\cal{T}}_{A1}$ is of form $$\begin{aligned}
{\cal{T}}_{A1} &=& - i (D-4) 3! \zeta \lambda_H \int \frac{d^D k}{(2 \pi)^D} \frac{1}{k^2 + m_h^2}
\nonumber\\
&=& - i (D-4) 3! \zeta \lambda_H \frac{i \pi^2}{(2 \pi)^4} (m_h^2)^{\frac{D}{2} - 1} \Gamma(1 - \frac{D}{2})
\nonumber\\
&=& \frac{3}{4 \pi^2} \zeta \lambda_H m_h^2,
\label{T-A1-1}\end{aligned}$$ where we have used the familiar formula in the dimensional regularization which corresponds to a specific case of the general formula in Appendix B $$\begin{aligned}
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{k^2 + m_h^2}
= \frac{i \pi^2}{(2 \pi)^4} (m_h^2)^{\frac{D}{2} -1} \Gamma(1- \frac{D}{2}),
\label{Q-F1}\end{aligned}$$ and the property of the gamma function $\Gamma(m+1) = m \Gamma(m)$.
The second type of one-loop divergent diagram, which we call the diagram (A2), is given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) \zeta m_h^2$ in $V^{(1)} (\hat H)$ and with the Higgs self-coupling vertex $- 3! \lambda_H$ in $V^{(0)} (\hat H)$. The amplitude ${\cal{T}}_{A2}$ is calculated as $$\begin{aligned}
{\cal{T}}_{A2} &=& i (D-4) 3! \zeta \lambda_H m_h^2 \int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_h^2)^2}
\nonumber\\
&=& i (D-4) 3! \zeta \lambda_H m_h^2 \frac{i}{16 \pi^2} \Gamma(2 - \frac{D}{2})
\nonumber\\
&=& \frac{3}{4 \pi^2} \zeta \lambda_H m_h^2,
\label{T-A2-1}\end{aligned}$$ where we have used the equation $$\begin{aligned}
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_h^2)^2}
&=& - \frac{\partial}{\partial m_h^2} \int \frac{d^D k}{(2 \pi)^D} \frac{1}{k^2 + m_h^2}
\nonumber\\
&=& \frac{i \pi^2}{(2 \pi)^4} (m_h^2)^{\frac{D}{2} - 2} (1 - \frac{D}{2}) \Gamma(1- \frac{D}{2})
\nonumber\\
&=& \frac{i}{16\pi^2} \Gamma(2- \frac{D}{2}).
\label{Q-F2}\end{aligned}$$ The final type of one-loop diagram, which we call the diagram (A3), is a little more involved and given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) 3! \zeta \sqrt{\frac{\lambda_H}{2}} m_h$ in $V^{(1)} (\hat H)$ and with the Higgs self-coupling vertex $- 3! \sqrt{\frac{\lambda_H}{2}} m_h$ in $V^{(0)} (\hat H)$. The amplitude ${\cal{T}}_{A3}$ reads $$\begin{aligned}
{\cal{T}}_{A3} &=& 2 i (D-4) \zeta \left(- 3! \sqrt{\frac{\lambda_H}{2}} m_h \right)^2 \int \frac{d^D k}{(2 \pi)^D}
\frac{1}{(k^2 + m_h^2)\left[(k+q)^2 + m_h^2 \right]}
\nonumber\\
&=& 36 i \zeta \lambda_H m_h^2 (D -4) \frac{i}{16 \pi^2} \Gamma(2 - \frac{D}{2})
\nonumber\\
&=& \frac{9}{2 \pi^2} \zeta \lambda_H m_h^2,
\label{T-A3-1}\end{aligned}$$ where $q$ is the external momentum of the Higgs field. In order to reach the final result in Eq. (\[T-A3-1\]), we have evaluated the integral as follows: $$\begin{aligned}
I &=& \int d^D k \frac{1}{(k^2 + m_h^2)\left[(k+q)^2 + m_h^2 \right]}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{1}{\left[ (k^2 + m_h^2) (1-x) + ( (k+q)^2 + m_h^2) x \right]^2}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{1}{\left[ (k+ xq)^2 + m_h^2 + x (1-x) q^2 \right]^2}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{1}{\left[ k^2 + m_h^2 + x (1-x) q^2 \right]^2}
\nonumber\\
&=& \int_0^1 d x \ i \pi^2 \Gamma(2-\frac{D}{2}) (m_h^2)^{\frac{D}{2} -2} (1 - x + x^2)^{\frac{D}{2} -2}
\nonumber\\
&=& i \pi^2 \Gamma(2-\frac{D}{2}).
\label{Q-A3-1}\end{aligned}$$ Here at the second equality, we have used the Feynman parameter formula (\[App-B-Feynman2\]) and at the fourth equality, we have shifted the momentum $k + x q \rightarrow k$, which is allowed since the integral is now finite owing to the regularization, and at the fifth equality we have used the on-mass-shell condition $q^2 = - m_h^2$ and Eq. (\[Q-F2\]).
Thus, adding three types of contributions, we have $$\begin{aligned}
{\cal{T}}_A = {\cal{T}}_{A1} + {\cal{T}}_{A2} + {\cal{T}}_{A3} = \frac{6}{\pi^2} \zeta \lambda_H m_h^2.
\label{All-A}\end{aligned}$$ From this result, we can construct an effective Lagrangian at the one-loop level $$\begin{aligned}
L_{\sigma h^2} = - \frac{3}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p} \sigma h^2,
\label{Eff-A}\end{aligned}$$ where we have explicitly written down the Planck mass dependence in such a way that we can recognize dimensions clearly.
Next, let us take account of four-point (with three Higgs and one dilaton as the external particles), one-loop diagrams. In this case, inspection of the vertices reveals again that there are two types of one-loop divergent diagrams where the Higgs field is circulating in the loop. One type of one-loop divergent diagram, which we call the diagram (B1), is given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) 3! \zeta \lambda_H$ in $V^{(1)} (\hat H)$ and with the Higgs self-coupling $- 3! \sqrt{\frac{\lambda_H}{2}} m_h$ in $V^{(0)} (\hat H)$. The corresponding amplitude ${\cal{T}}_{B1}$ reads $$\begin{aligned}
{\cal{T}}_{B1} &=& i (D-4) 3! \zeta \lambda_H \sqrt{\frac{\lambda_H}{2}} m_h 3! \int \frac{d^D k}{(2 \pi)^D}
\frac{1}{(k^2 + m_h^2)\left[(k + q)^2 + m_h^2 \right]}
\nonumber\\
&=& 36 i \zeta \lambda_H \sqrt{\frac{\lambda_H}{2}} m_h (D-4) \frac{i}{16 \pi^2} \Gamma(2 - \frac{D}{2})
\nonumber\\
&=& \frac{9}{2 \sqrt{2} \pi^2} \zeta \lambda_H \sqrt{\lambda_H} m_h,
\label{T-B1-1}\end{aligned}$$ where we have used Eq. (\[Q-A3-1\]).
The other type of one-loop divergent diagram, which is called the diagram (B2), is given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) 3! \zeta \sqrt{\frac{\lambda_H}{2}} m_h$ in $V^{(1)} (\hat H)$ and with the Higgs self-coupling $- 3! \lambda_H$ in $V^{(0)} (\hat H)$. The amplitude ${\cal{T}}_{B2}$ takes the form $$\begin{aligned}
{\cal{T}}_{B2} &=& i (D-4) 3! \zeta \sqrt{\frac{\lambda_H}{2}} m_h 3! \lambda_H \int \frac{d^D k}{(2 \pi)^D}
\frac{1}{(k^2 + m_h^2)\left[(k + q)^2 + m_h^2 \right]}
\nonumber\\
&=& \frac{9}{2 \sqrt{2} \pi^2} \zeta \lambda_H \sqrt{\lambda_H} m_h,
\label{T-B2-1}\end{aligned}$$ Putting the two types of contributions together, we obtain $$\begin{aligned}
{\cal{T}}_B = {\cal{T}}_{B1} + {\cal{T}}_{B2} = \frac{9}{\sqrt{2} \pi^2} \zeta \lambda_H \sqrt{\lambda_H} m_h.
\label{All-B}\end{aligned}$$ This result gives rise to an effective Lagrangian $$\begin{aligned}
L_{\sigma h^3} = - \frac{3}{2 \sqrt{2} \pi^2} \zeta \lambda_H \sqrt{\lambda_H} \frac{m_h}{M_p} \sigma h^3,
\label{Eff-B}\end{aligned}$$ where we have recovered the Planck mass again.
Now we turn our attention to five-point (with four Higgs and one dilaton as the external particles), one-loop diagrams. In this case, we find that there is only one type of one-loop divergent diagram where the Higgs field is circulating in the loop. This type of one-loop divergent diagram, which we call the diagram (C), is given by the Higgs loop to which the dilaton couples by the vertex $- (D-4) 3! \zeta \lambda_H$ in $V^{(1)} (\hat H)$ and with the Higgs self-coupling $- 3! \lambda_H$ in $V^{(0)} (\hat H)$. The corresponding amplitude ${\cal{T}}_{C}$ reads $$\begin{aligned}
{\cal{T}}_{C} &=& 2 i (D-4) 3! \zeta \lambda_H 3! \lambda_H \int \frac{d^D k}{(2 \pi)^D}
\frac{1}{(k^2 + m_h^2)\left[(k + p + q)^2 + m_h^2 \right]}
\nonumber\\
&=& \frac{1}{\pi^2} \zeta \lambda_H^2,
\label{T-C-1}\end{aligned}$$ where $p$ and $q$ are external momenta of the two Higgs fields. This quantum effect gives us an effective Lagrangian $$\begin{aligned}
L_{\sigma h^4} = - \frac{1}{24 \pi^2} \zeta \lambda_H^2 \frac{1}{M_p} \sigma h^4.
\label{Eff-C}\end{aligned}$$ Finally, it is straightforward to evaluate $(n+1)$-point (with $n \ge 5$ Higgs and one dilaton as external particles), one-loop diagrams in a similar manner. It turns out that these higher-point, one-loop diagrams do not yield any divergence, thereby leading to the vanishing effective Lagrangian. Moreover, we find that there are no divergences for the $\sigma^n h^m (n \ge 2)$-type of amplitudes at the one-loop level.
After all, we have a total effective Lagrangian at the one-loop level $$\begin{aligned}
L^{1-loop} = \left[ - \frac{3}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p} h^2
- \frac{3}{2 \sqrt{2} \pi^2} \zeta \lambda_H \sqrt{\lambda_H} \frac{m_h}{M_p} h^3
- \frac{1}{24 \pi^2} \zeta \lambda_H^2 \frac{1}{M_p} h^4 \right] \sigma.
\label{Eff-All}\end{aligned}$$ Note that this effective Lagrangian has the similar form to the potential as $V^{(1)} (\hat H)$ but each coefficient is suppressed by the Planck mass, which means that effects of radiative corrections are very tiny in the low energy region.
Yukawa coupling
---------------
In the standard model, the fermion masses arise from the Yukawa coupling between the fermions and the Higgs field. It is therefore of interest to evaluate radiative corrections of the Yukawa coupling in the present model.
It is easy to see that there are no radiative corrections to the coupling between the dilaton and the fermions at the one-loop level, but it turns out that the one-loop induced vertex produces radiative corrections to this coupling, so we are willing to calcuclate this quantum effect in this subsection.
Before delving into the calculation, let us go back to the basics of the Yukawa coupling. The Yukawa coupling between the fermions and the Higgs field is generically given by the following Lagrangian $$\begin{aligned}
{\cal{L}}_{H \bar \psi \psi} = - \sqrt{-g} g_Y \bar \psi_L H \psi_R,
\label{Yukawa1}\end{aligned}$$ where $g_Y$ is the Yukawa coupling constant, $\psi_L$ and $\psi_R$ are respectively a left-handed, $SU(2)$-doublet spinor and a right-handed singlet spinor.
To move the Jordan frame to the Einstein frame, we use the local scale transformation (\[Q-L-scale transf\]) and its fermionic one $$\begin{aligned}
\hat \psi_L = \Omega^{- \frac{D+2}{4}}(x) \psi_L, \quad
\hat \psi_R = \Omega^{- \frac{D+2}{4}}(x) \psi_R.
\label{Q-L-scale transf 2}\end{aligned}$$ Under this local scale transformation, the Lagrangian (\[Yukawa1\]) takes the same form $$\begin{aligned}
{\cal{L}}_{H \bar \psi \psi} = - \sqrt{- \hat g} g_Y \bar {\hat \psi_L} \hat H \hat \psi_R.
\label{Yukawa2}\end{aligned}$$ With the following definitions of spinors and the unitary gauge for the Higgs field $\hat H$, $$\begin{aligned}
\hat \psi_L^T = (\hat \chi, \hat \psi), \quad
\hat \psi_R = \hat \psi, \quad
\hat H^T = ( 0, \frac{v + h(x)}{\sqrt{2}} ),
\label{Yukawa3}\end{aligned}$$ the Lagrangian is reduced to $$\begin{aligned}
{\cal{L}}_{H \bar \psi \psi} = - \sqrt{- \hat g} ( M_\psi \bar {\hat \psi} \hat \psi
+ \frac{g_Y}{\sqrt{2}} \bar {\hat \psi} \hat \psi h ),
\label{Yukawa4}\end{aligned}$$ where we have defined $M_\psi = \frac{g_Y}{\sqrt{2}} v$.
We are now in a position to calculate the one-loop amplitude where two fermions and one dilaton appear as the external particles. In this case, there is no divergent diagram but we have a finite diagram where the fermion and the Higgs field propagate in the loop, which we call the diagram (D). In this diagram, the dilaton couples to the Higgs by the vertex $- \frac{6}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p}$ in (\[Eff-All\]), which is a one-loop effect[^8], and two fermions couple to the Higgs by the vertex $- \frac{g_Y}{\sqrt{2}}$ in (\[Yukawa4\]). Thus, this diagram is essentially a two-loop effect. The correponding amplitude is given by $$\begin{aligned}
{\cal{T}}_D &=& - i \frac{6}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p} (\frac{g_Y}{\sqrt{2}})^2
\int \frac{d^D k}{(2 \pi)^D}
\frac{\slash{q} - \slash{k} - M_\psi}{(k^2 + m_h^2)^2 \left[(q-k)^2 + M_\psi^2 \right]}
\nonumber\\
&=& \frac{3}{16 \pi^4} \zeta \lambda_H g_Y^2 \frac{m_h^2}{M_\psi M_p} f(\frac{M_\psi}{m_h}),
\label{T-D-1}\end{aligned}$$ where $q$ is the external momentum of the fermion field, which satisfies the on-mass-shell condition $q^2 = - M_\psi^2$ and the function $f(x)$ is defined as $$\begin{aligned}
f(x) = \log x + \frac{1 - 2 x^2}{\sqrt{1 - 4 x^2}} \log \left[ \frac{1}{2 x} ( 1 + \sqrt{1 - 4 x^2} ) \right].
\label{f(x)}\end{aligned}$$ To obtain the result in Eq. (\[T-D-1\]), we have calculated the integral as follows: $$\begin{aligned}
J &=& \int d^D k \frac{\slash{q} - \slash{k} - M_\psi}{(k^2 + m_h^2)^2 \left[(q-k)^2 + M_\psi^2 \right]}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{2 (1 - x) (\slash{q} - \slash{k} - M_\psi) }
{\left[ (k^2 + m_h^2) (1-x) + ( (q-k)^2 + M_\psi^2) x \right]^3}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{2 (1 - x) (\slash{q} - \slash{k} - M_\psi) }
{\left[ (k- xq)^2 + M_\psi^2 x^2 - m_h^2 x + m_h^2 \right]^3}
\nonumber\\
&=& \int_0^1 d x \int d^D k \frac{2 (1 - x) [ (1 - x ) \slash{q} - \slash{k} - M_\psi) }
{\left[ k^2 + M_\psi^2 x^2 - m_h^2 x + m_h^2 \right]^3}
\nonumber\\
&=& - 2 M_\psi \int_0^1 d x \int d^D k \frac{1 - x}
{\left[ k^2 + M_\psi^2 x^2 - m_h^2 x + m_h^2 \right]^3}
\nonumber\\
&=& \frac{i \pi^2}{M_\psi} f(\frac{M_\psi}{m_h}).
\label{Q-D-2}\end{aligned}$$ Here at the second equality, we have used the Feynman parameter formula (\[App-B-Feynman3\]). At the fourth equality, we have shifted the momentum $k - x q \rightarrow k$, and used that $\int d^D k \ k^\mu F(k^2) = 0$ for a general function $F$ in addition to $q^\mu \approx 0$ at the low energy. Furthermore, at the final equality, we have made use of the integral formula $$\begin{aligned}
\int d^D k \frac{1}{( k^2 + \Delta )^3} = \frac{i \pi^2}{2 \Delta},
\label{Q-D-3}\end{aligned}$$ which is a specific case of a general formula (\[App-B-Integral3\]).
From the above result, an effective Lagrangian for the interaction between the dilaton and fermions can be derived to $$\begin{aligned}
L_{\sigma \bar \psi \psi} = - g_\sigma \bar{\hat \psi} \hat \psi \sigma,
\label{Q-D-4}\end{aligned}$$ where the effective coupling $g_\sigma$ is defined by the absolute value of ${\cal{T}}_D$, i.e., $g_\sigma = | {\cal{T}}_D |$.
Dilaton mass
------------
As seen in the Lagrangian (\[Q-Lagr 4\]), the dilaton is exactly massless at the classical level since it is the Nambu-Goldstone boson stemming from spontaneous symmetry breakdown of scale symmetry. However, it is well-known that the scale symmetry is violated by the trace anomaly at the quantum-mechanical level, and as a result, the dilaton becomes massive.
It is very interesting to evaluate the size of the dilaton mass within the present formalism. It is in general expected that if any, the Nambu-Goldstone boson would not be so heavy as in the pions. Of course, the size of the dilaton mass would be closely related to an energy scale where the scale symmetry is broken spontaneously. On the other hand, because the dilaton is a representative example of scalar particle as well as the Higg particle, it is of interest to investigate if the dilaton would receive the quadratic divergence like the Higgs particle or not.
It turns out that at the one-loop effect, there is no quantum correction for the self-energy of the dilaton and it is at the two-loop effect that radiative corrections appear for it in the formalism at hand. Actually, we have a one-loop divergent diagram for the self-energy of the dilaton, which we call the diagram (E), where two external dilatons couple to the Higgs loop by the vertex $- (D-4) \zeta m_h^2$ in $V^{(1)} (\hat H)$ and the vertex $-\frac{6}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p}$ in (\[Eff-All\]) which is already a one-loop effect. Therefore, this one-loop diagram is essentially a two-loop contribution. The amplitude takes the form $$\begin{aligned}
{\cal{T}}_E &=& 2 i (D-4) \zeta m_h^2 \frac{6}{\pi^2} \zeta \lambda_H \frac{m_h^2}{M_p}
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_h^2)^2}
\nonumber\\
&=& \frac{3}{2 \pi^4} \zeta^2 \lambda_H \frac{m_h^4}{M_p}.
\label{Q-E-1}\end{aligned}$$ This amplitude directly gives rise to an effective action for the mass term of the dilaton $$\begin{aligned}
L_{\sigma^2}^{(2)} = - \frac{3}{4 \pi^4} \zeta^2 \lambda_H \frac{m_h^4}{M_p^2} \sigma^2
\equiv - \frac{1}{2} m_\sigma^2 \sigma^2,
\label{Q-E-2}\end{aligned}$$ where we have defined the induced dilaton mass $m_\sigma$ as $$\begin{aligned}
m_\sigma^2 = \frac{3}{2 \pi^4} \zeta^2 \lambda_H \frac{m_h^4}{M_p^2}.
\label{Q-E-3}\end{aligned}$$ As expected, it has turned out that the dilaton, which is massless classically, becomes massive because of radiative corrections.
From the result (\[Q-E-3\]), one might be tempted to conclude that the dilaton mass induced by radiative corrections is very small since the size of the mass is suppressed by the Planck mass and $\zeta \approx \lambda_H \approx {\cal {O}}(1)$. But the story has not ended yet because we have to take the quadratic divergence, which is the root of the hiearchy problem in case of the Higgs particle, into consideration. Since there is no interaction vertex $\sigma^2 h^2$ in the present formalism, the most severe quadratic divergence appears when the fermion is circulating in the loop via the vertex in the Lagrangian (\[Q-D-4\]). The amplitude ${\cal{T}}_F$, which is essentially a five-loop effect, is certainly quadratically divergent by power counting $$\begin{aligned}
{\cal{T}}_F = i g_\sigma^2 \int \frac{d^D k}{(2 \pi)^D}
\frac{1}{(\slash{k} + M_\psi)^2}
\approx i g_\sigma^2 \int \frac{d^D k}{(2 \pi)^D} \frac{1}{k^2}
\approx - g_\sigma^2 \Lambda^2,
\label{T-F-1}\end{aligned}$$ where $\Lambda$ is the ultra-violet cutoff. Then, with the reasonable choice $\Lambda = M_p$, the mass of the dilaton is approximately given by $$\begin{aligned}
m_\sigma \approx g_\sigma \Lambda = \frac{3}{16 \pi^4} \zeta \lambda_H g_Y^2
\frac{m_h^2}{M_\psi} | f(\frac{M_\psi}{m_h}) |,
\label{T-F-2}\end{aligned}$$ which is around the GeV scale since $| f(\frac{M_\psi}{m_h}) | \approx 1$ for $m_h \approx M_\psi$, which holds approximately for $\psi =$ top-quark. (Here it is reasonable to take $\zeta \approx
\lambda_H \approx g_Y \approx {\cal{O}}(1)$ at the low energy.) Note that the factor $\frac{1}{M_p}$ in $g_\sigma$ is cancelled by the cutoff $M_p$. It is remarkable that the quadratic divergence, which leads to the burdensome hierarchy problem in case of the Higgs particle, gives the GeV scale mass to the dilaton! At first sight, it appears that the GeV scale mass of the dilaton is against the results of the LHC owing to null results in searches for new scalar particles except the Higgs particle below a few TeV scale. However, as seen in the relation $g_\sigma = | {\cal{T}}_D |$ and Eq. (\[T-D-1\]), the coupling between the dilaton and the Higgs particle is so tiny that it is extremely difficult to detect the dilaton in the LHC.
Contributions from gauge fields
-------------------------------
In the previous subsections, we have switched off the $U(1)$ gauge fields. In this final subsection, we switch on the $U(1)$ gauge fields, and wish to calculate the coupling between the dilaton and the Higgs particle by using propagators and vertices from the sector of the gauge fields in the Lagrangian (\[Q-Lagr 4\]). The result is very simple and illuminating in the sense that we can obtain the similar form of the effective Lagrangian to (\[Eff-All\]) at the one-loop level, but the coefficient of each term is multiplied by the square of the “fine structure constant”.
For convenience, let us pick up part of the Lagrangian (\[Q-Lagr 4\]) which contains the gauge fields $$\begin{aligned}
{\cal L}_{EM} = \sqrt{- \hat g} \ \Big[ -\frac{1}{4} \sum_{i=1}^2 \hat g^{\mu\nu} \hat g^{\rho\sigma}
\hat F^{(i)}_{\mu\rho} \hat F^{(i)}_{\nu\sigma}
- \frac{\hat e_1^2}{2 \xi_1} \hat g^{\mu\nu} \hat B_\mu^{(1)} \hat B_\nu^{(1)}
- \frac{\hat e_2^2}{8} \hat g^{\mu\nu} \hat A_\mu^{(2)} \hat A_\nu^{(2)} ( v + h )^2 \Big].
\label{Q-EM-Lagr}\end{aligned}$$ With the following definitions of the mass of the gauge fields $$\begin{aligned}
\hat{m}_A^2 = \frac{1}{4} \hat{e}_2^2 v^2, \quad
\hat{m}_B^2 = \frac{\hat{e}_1^2}{\xi_1},
\label{Q-EM-Mass}\end{aligned}$$ the Lagrangian (\[Q-EM-Lagr\]) can be rewritten as $$\begin{aligned}
{\cal L}_{EM} = \sqrt{- \hat g} \ \Big[ -\frac{1}{4} \sum_{i=1}^2 ( \hat F^{(i)}_{\mu\nu} )^2
- \frac{\hat{m}_B^2}{2} ( \hat B_\mu^{(1)} )^2
- \frac{\hat{m}_A^2}{2} ( \hat A_\mu^{(2)} )^2
- \hat{m}_A^2 ( \frac{1}{v} h + \frac{1}{2 v^2} h^2 ) ( \hat A_\mu^{(2)} )^2 \Big].
\label{Q-EM-Lagr2}\end{aligned}$$ Because of the relations $\hat{e}_i = \Omega^{\frac{D-4}{2}} e_i = e^{\frac{D-4}{D-2} \zeta \sigma} e_i \ (i = 1, 2)$, we have $$\begin{aligned}
\hat{m}_A^2 \approx m_A^2 [ 1 + (D-4) \zeta \sigma ], \quad
\hat{m}_B^2 \approx m_B^2 [ 1 + (D-4) \zeta \sigma ],
\label{Q-EM-Mass2}\end{aligned}$$ where $m_A^2, m_B^2$ are defined as in (\[Q-EM-Mass\]) but without the hat on $e_i$. Thus, in the sector of the gauge fields, we have six different vertices $\sigma-B^2, \sigma-A^2,
h-A^2, h^2-A^2, \sigma-h-A^2, \sigma-h^2-A^2$, and two propagators of massive gauge fields $\hat A_\mu^{(2)}, \hat B_\mu^{(1)}$ for which we take the Feynman gauge.
Now, on the basis of these vertices and propagators, we would like to consider $(n+1)$-point (with $n$ Higgs particles and one dilaton as external particles), one-loop diagrams. For $n=2$, we have two types of one-loop divergent diagrams in which the gauge field $A_\mu^{(2)}$ is circulating in the loop. One type of the diagram, which we call the diagram (G1), is a tadpole type in which the dilaton couples to the loop composed of the gauge field $A_\mu^{(2)}$ by the vertex $- 2 \frac{m_A^2}{v^2} (D-4) \zeta \eta_{\mu\nu} = - \frac{1}{2} e_2^2 (D-4) \zeta \eta_{\mu\nu}$. The corresponding amplitude ${\cal{T}}_{G1}$ is given by $$\begin{aligned}
{\cal{T}}_{G1} &=& - 2i e_2^2 (D-4) \zeta
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{k^2 + m_A^2}
\nonumber\\
&=& \frac{1}{4 \pi^2} e_2^2 \zeta m_A^2
\nonumber\\
&=& \alpha_2^2 \zeta v^2,
\label{Q-G-1}\end{aligned}$$ where we have introduced the “fine structure constant” $\alpha_2 = \frac{e_2^2}{4 \pi}$.
The other type of one-loop diveregent diagram which we call the diagram (G2), is the self-energy type of the dilaton where the dilaton couples to the loop by the vertex $- \frac{1}{4} e_2^2 v
(D-4) \zeta \eta_{\mu\nu}$ and with the gauge-Higgs coupling vertex $- \frac{1}{4} e_2^2 v \eta_{\mu\nu}$. The amplitude ${\cal{T}}_{G2}$ is given by $$\begin{aligned}
{\cal{T}}_{G2} &=& i \frac{1}{2} (D-4) \zeta (e_2^2 v)^2
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_A^2)[ (k+q)^2 + m_A^2 ]}
\nonumber\\
&=& \alpha_2^2 \zeta v^2,
\label{Q-G-2}\end{aligned}$$ where $q$ is the external momentum of the Higgs particle.
Hence, adding the two results, we have $$\begin{aligned}
{\cal{T}}_G = {\cal{T}}_{G1} + {\cal{T}}_{G2} = 2 \alpha_2^2 \zeta v^2,
\label{Q-G}\end{aligned}$$ from which, we obtain an effective Lagrangian $$\begin{aligned}
L_{\sigma h^2} = - \alpha_2^2 \zeta \frac{v^2}{M_p} \sigma h^2.
\label{Q-G-L}\end{aligned}$$ Next, let us move to the evaluation of $4$-point (with $3$ Higgs particles and one dilaton as external particles), one-loop diagrams. In this case, there are two kinds of divergent diagrams. The one diagram, which we call the diagram (H1), has the $h-A^2$ vertex with the coefficient $- \frac{1}{4} e_2^2 v \eta_{\mu\nu}$ and $\sigma-h^2-A^2$ vertex with the coefficient $- \frac{1}{2} e_2^2 (D-4) \zeta \eta_{\mu\nu}$, so the amplitude takes the form $$\begin{aligned}
{\cal{T}}_{H1} &=& i \frac{1}{2} (D-4) \zeta e_2^4 v
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_A^2)[ (k+q)^2 + m_A^2 ]}
\nonumber\\
&=& \alpha_2^2 \zeta v,
\label{Q-H-1}\end{aligned}$$ where $q$ is the momentum carried by the external Higgs particle coupled to the loop via the vertex $h-A^2$.
The other divergent diagram, which is called (H2), has the vertex $h^2-A^2$ with the coefficient $- \frac{1}{2} e_2^2 \eta_{\mu\nu}$ and the vertex $\sigma-h-A^2$ with the coefficient $- \frac{1}{4} e_2^2 v (D-4) \zeta \eta_{\mu\nu}$. It turns out the corresponding amplitude is the same as ${\cal{T}}_{H1}$, so we have $$\begin{aligned}
{\cal{T}}_{H2} = \alpha_2^2 \zeta v.
\label{Q-H-2}\end{aligned}$$ Putting Eqs. (\[Q-H-1\]) and (\[Q-H-2\]) together, we obtain the result $$\begin{aligned}
{\cal{T}}_H = {\cal{T}}_{H1} + {\cal{T}}_{H2} = 2 \alpha_2^4 \zeta v,
\label{Q-H-3}\end{aligned}$$ from which an effective Lagrangian becomes $$\begin{aligned}
L_{\sigma h^3} = - \frac{1}{3} \alpha_2^2 \zeta \frac{v}{M_p} \sigma h^3.
\label{Q-H-L}\end{aligned}$$ Finally, as $5$-point (with $4$ Higgs particles and one dilaton as external particles), one-loop divergent diagrams, there is only one diagram constructed out of the vertex $h^2-A^2$ and the vertex $\sigma-h^2-A^2$. The amplitude ${\cal{T}}_{G3}$ is of form $$\begin{aligned}
{\cal{T}}_I &=& 2 i (D-4) \zeta e_2^4
\int \frac{d^D k}{(2 \pi)^D} \frac{1}{(k^2 + m_A^2)[ (k+p-q)^2 + m_A^2 ]}
\nonumber\\
&=& 4 \alpha_2^2 \zeta,
\label{Q-I-1}\end{aligned}$$ from which an effective Lagrangian can be derived to $$\begin{aligned}
L_{\sigma h^4} = - \frac{1}{6} \alpha_2^2 \zeta \frac{1}{M_p} \sigma h^4.
\label{Q-I-2}\end{aligned}$$ Incidentally, it is easy to check that there are no divergent, one-loop diagrams for the case of $(n+1)$-point (with $n \ge 5$ Higgs and one dilaton as external particles) as before.
Putting these results together, we have the following effective Lagrangian from the sector of the gauge fields: $$\begin{aligned}
L_{EM}^{1-loop} &=& \alpha_2^2 \left[ - \zeta \frac{v^2}{M_p} h^2
- \frac{1}{3} \zeta \frac{v}{M_p} h^3
- \frac{1}{6} \zeta \frac{1}{M_p} h^4 \right] \sigma
\nonumber\\
&=& \left(\frac{\alpha_2}{\lambda_H} \right)^2 \left[ - \frac{1}{2} \zeta \lambda_H \frac{m_h^2}{M_p} h^2
- \frac{1}{3 \sqrt{2}} \zeta \lambda_H \sqrt{\lambda_H} \frac{m_h}{M_p} h^3
- \frac{1}{6} \zeta \lambda_H^2 \frac{1}{M_p} h^4 \right] \sigma,
\label{Eff-All2}\end{aligned}$$ where we have used the relation $v = \frac{m_h}{\sqrt{2 \lambda_H}}$ in (\[Q-P3\]). The last equality shows that the Lagrangian (\[Eff-All\]) is more dominant than the Lagrangian (\[Eff-All2\]) because of $\lambda_H \gg \alpha_2$. Thus, the results about the dilaton mass obtained in the previous subsection in essence remain unchanged even if the contribution from the gauge fields is taken into consideration.
Conclusion
==========
In this article, we have investigated a Higgs mechanism in scale-invariant theories of gravitation in detail. After reviewing this new Higgs mechanism found in our previous articles [@Oda1; @Oda2] in terms of the simplest model, we have extended the Higgs mechanism to non-Abelian gauge groups and a scalar field with many components. Since we have already considered the Higgs mechanism in a locally scale-invariant theory of gravitation, i.e., conformal gravity, the validity of this mechanism in the scale-invariant gravitational theories is very universal and therefore would have some phenomenological applications in future.
Moreover, we have spelled out quantum effects of a scale-invariant extension of the standard model in a flat Minkowski background, and examined the coupling between the dilaton and the Higgs particle. An intriguing observation done in our analysis is that although the mass of the dilaton is exactly zero at the classical level owing to the Nambu-Goldstone theorem, it becomes non-zero and takes a finite size around the GeV scale because of radiative corrections. It is worthwhile to mention that we have succeeded in deriving the size of the dilaton mass deductively by starting with a fundamental theory and without any specific assumption. As long as we know, the dilaton mass has not thus far been obtained in such a priori manner, so we think our derivation of the dilaton mass to be very interesting. As mentioned in the article, the GeV scale mass of the dilaton is consistent with the recent null results of new scalar particles except the Higgs particle in the LHC since the coupling constant of the dilaton is too small to detect the dilaton in the LHC. However, the dilaton with the GeV scale mass would have some implication in cosmology, e.g., the dilaton could become one of candidates of dark matter if it is somehow stable because of some unknown mechanism.
Our consideration in this article is confined to the quantum analysis in a fixed Minkowski background. In other words, quantum effects coming from the gravity are completely ignored because of non-renormalizability of quantum gravity. Since quantum gravity effects are of course not so dominant as quantum effects from matter fields in the low energy region, it is physically reasonable to neglect quantum effects of the gravity as the first approximation of the calculation. Nevertheless, it is of interest to take into consideration the quantum effects from the gravity. In the future, we wish to study the quantum effects from the gravitational sector in the present formalism.
Another interesting study for an application of our finding is the Higgs inflation [@Bezrukov]. We wish to return this problem as well in near future.
[**Acknowledgements**]{}
This work is supported in part by the Grant-in-Aid for Scientific Research (C) Nos. 22540287 and 25400262 from the Japan Ministry of Education, Culture, Sports, Science and Technology.
Derivation of current for scale transformation {#App:AppendixA}
==============================================
In Appendix A, we will present a derivation of the dilatation current (\[Current\]) via the Noether theorem. It is easy to show that the Lagrangian (\[Lagr 1\]) is invariant under the scale transformation (\[Scale transf\]) without surface terms. Therefore, the expression of the Noether current is of form $$\begin{aligned}
\Lambda J^\mu = \sum \frac{\partial {\cal L}}{\partial \partial_\mu \phi} \delta \phi,
\label{App-A-Current}\end{aligned}$$ where $\phi = \{g_{\mu\nu}, \Phi, \Phi^\dagger \}$. Under the scale transformation (\[Scale transf\]) with a global parameter $\Omega = e^\Lambda \approx 1 + \Lambda \ (|\Lambda| \ll 1)$, the current reads $$\begin{aligned}
J^\mu = \frac{\partial {\cal L}}{\partial \partial_\mu g_{\rho\sigma}} 2 g_{\rho\sigma}
- \frac{\partial {\cal L}}{\partial \partial_\mu \Phi} \Phi
- \frac{\partial {\cal L}}{\partial \partial_\mu \Phi^\dagger} \Phi^\dagger,
\label{App-A-J}\end{aligned}$$ so we have to calculate three objects $\frac{\partial {\cal L}}{\partial \partial_\mu g_{\rho\sigma}},
\frac{\partial {\cal L}}{\partial \partial_\mu \Phi}, \frac{\partial {\cal L}}{\partial \partial_\mu \Phi^\dagger}$ to obtain the expression of the dilatation current $J^\mu$. In particular, calculating the first object $\frac{\partial {\cal L}}{\partial \partial_\mu g_{\rho\sigma}}$ is so complicated that we will present its derivation in detail.
First, with the definition $\varphi = \xi \Phi^\dagger \Phi$, let us consider the non-minimal term $$\begin{aligned}
{\cal L}_{NM} = \sqrt{-g} \ \varphi R = {\cal L}_1 + {\cal L}_2,
\label{App-A-NM}\end{aligned}$$ where we have defined $$\begin{aligned}
{\cal L}_1 &=& \sqrt{-g} \ \varphi g^{\mu\nu} ( \partial_\alpha \Gamma^\alpha_{\mu\nu}
- \partial_\nu \Gamma^\alpha_{\mu\alpha} ), \nonumber\\
{\cal L}_2 &=& \sqrt{-g} \ \varphi g^{\mu\nu} ( \Gamma^\alpha_{\sigma\alpha} \Gamma^\sigma_{\mu\nu}
- \Gamma^\alpha_{\sigma\nu} \Gamma^\sigma_{\mu\alpha} ),
\label{App-A-NM2}\end{aligned}$$ where as usual the affine connection and its contraction are defined as $$\begin{aligned}
\Gamma^\alpha_{\mu\nu} = \frac{1}{2} g^{\alpha\beta} ( \partial_\mu g_{\beta\nu} + \partial_\nu g_{\beta\mu}
- \partial_\beta g_{\mu\nu} ), \quad
\Gamma^\alpha_{\mu\alpha} = \frac{1}{2} g^{\alpha\beta} \partial_\mu g_{\alpha\beta}
= \frac{\partial_\mu \sqrt{-g}} {\sqrt{-g}}.
\label{App-A-Affine}\end{aligned}$$ ${\cal L}_1$ includes terms with second derivative of the metric, i.e., $\partial^2 g$, so we need to perform the integration by parts to transform them to terms with first derivative, i.e., $\partial g$. After the integration by parts, ${\cal L}_1$ is devided in two parts, one of which contains terms proportional to $\partial \varphi$ and the other part does terms proportional to $\varphi$ itself $$\begin{aligned}
{\cal L}_1 = - \sqrt{-g} \ \partial_\alpha \varphi ( g^{\mu\nu} \Gamma^\alpha_{\mu\nu}
- g^{\alpha\mu} \Gamma^\beta_{\mu\beta} ) - \varphi [ \partial_\alpha ( \sqrt{-g} g^{\mu\nu} ) \Gamma^\alpha_{\mu\nu}
- \partial_\nu ( \sqrt{-g} g^{\mu\nu} ) \Gamma^\alpha_{\mu\alpha} ].
\label{App-A-L1-1}\end{aligned}$$ Now let us focus on the second term, which we call $A$, and show that $A$ is equal to $-2 {\cal L}_2$. $$\begin{aligned}
A &\equiv& - \varphi [ \partial_\alpha ( \sqrt{-g} g^{\mu\nu} ) \Gamma^\alpha_{\mu\nu}
- \partial_\nu ( \sqrt{-g} g^{\mu\nu} ) \Gamma^\alpha_{\mu\alpha} ] \nonumber\\
&=& - \varphi [ \sqrt{-g} ( \Gamma^\beta_{\alpha\beta} g^{\mu\nu} + \partial_\alpha g^{\mu\nu} ) \Gamma^\alpha_{\mu\nu}
- \sqrt{-g} ( \Gamma^\beta_{\nu\beta} g^{\mu\nu} + \partial_\nu g^{\mu\nu} ) \Gamma^\alpha_{\mu\alpha} ].
\label{App-A-L1-2}\end{aligned}$$ In terms of the definition of the affine connection (\[App-A-Affine\]), we can prove the following relations: $$\begin{aligned}
\partial_\alpha g^{\mu\nu} \Gamma^\alpha_{\mu\nu} &=& -2 g^{\mu\nu} \Gamma^\alpha_{\sigma\nu}
\Gamma^\sigma_{\mu\alpha}, \nonumber\\
\partial_\nu g^{\mu\nu} &=& - g^{\alpha\beta} \Gamma^\mu_{\alpha\beta}
- g^{\mu\nu} \Gamma^\alpha_{\nu\alpha}.
\label{App-A-L1-3}\end{aligned}$$ Inserting Eq. (\[App-A-L1-3\]) to Eq. (\[App-A-L1-2\]), we reach the result that $A$ is equal to $-2 {\cal L}_2$: $$\begin{aligned}
A = - 2 \varphi \ \sqrt{-g} g^{\mu\nu} ( \Gamma^\alpha_{\sigma\alpha} \Gamma^\sigma_{\mu\nu}
- \Gamma^\alpha_{\sigma\nu} \Gamma^\sigma_{\mu\alpha} ) = -2 {\cal L}_2.
\label{App-A-L1-4}\end{aligned}$$ Next, plugging this result into Eq. (\[App-A-L1-1\]) leads to $$\begin{aligned}
{\cal L}_1 = - \sqrt{-g} \ \partial_\alpha \varphi ( g^{\mu\nu} \Gamma^\alpha_{\mu\nu}
- g^{\alpha\mu} \Gamma^\beta_{\mu\beta} ) -2 {\cal L}_2.
\label{App-A-L1-5}\end{aligned}$$ Moreover, substituting Eq. (\[App-A-L1-5\]) into Eq. (\[App-A-NM\]), we have $$\begin{aligned}
{\cal L}_{NM} &=& - \sqrt{-g} \ \partial_\alpha \varphi ( g^{\mu\nu} \Gamma^\alpha_{\mu\nu}
- g^{\alpha\mu} \Gamma^\beta_{\mu\beta} ) - {\cal L}_2 \nonumber\\
&\equiv& {\cal L}_K - {\cal L}_2.
\label{App-A-L1-6}\end{aligned}$$ Here we have defined $$\begin{aligned}
{\cal L}_K &=& - \sqrt{-g} \ \partial_\alpha \varphi ( g^{\mu\nu} \Gamma^\alpha_{\mu\nu}
- g^{\alpha\mu} \Gamma^\beta_{\mu\beta} ) \nonumber\\
&\equiv& - \sqrt{-g} \ \partial_\alpha \varphi K^\alpha,
\label{App-A-K1}\end{aligned}$$ where $K^\alpha$ is defined as $$\begin{aligned}
K^\alpha &=& g^{\mu\nu} \Gamma^\alpha_{\mu\nu}
- g^{\alpha\mu} \Gamma^\beta_{\mu\beta} \nonumber\\
&=& ( g^{\alpha\rho} g^{\mu\sigma} - g^{\alpha\mu} g^{\rho\sigma} ) \partial_\mu g_{\rho\sigma},
\label{App-A-K2}\end{aligned}$$ where at the second equality we have used Eqs. (\[App-A-Affine\]) and (\[App-A-L1-3\]). With this expression (\[App-A-K2\]), it is straightforward to take the variation of ${\cal L}_K$ with respect to $\partial_\mu g_{\rho\sigma}$ whose result is given by $$\begin{aligned}
\frac{\partial {\cal L}_K}{\partial \partial_\mu g_{\rho\sigma}}
= - \sqrt{-g} \ \partial_\alpha \varphi ( g^{\alpha(\rho} g^{\sigma)\mu} - g^{\alpha\mu} g^{\rho\sigma} ).
\label{App-A-K3}\end{aligned}$$ Thus, we have $$\begin{aligned}
\frac{\partial {\cal L}_K}{\partial \partial_\mu g_{\rho\sigma}} 2 g_{\rho\sigma}
= 6 \sqrt{-g} \ \partial^\mu \varphi.
\label{App-A-K4}\end{aligned}$$ Taking the variation of ${\cal L}_2$ with respect to $\partial_\mu g_{\rho\sigma}$ is a bit tedious but straightforward since the whole calculation can be performed by using the formula $$\begin{aligned}
\frac{\partial \Gamma^\lambda_{\alpha\beta}}{\partial \partial_\mu g_{\rho\sigma}}
= \frac{1}{2} [ g^{\lambda\rho} \delta^\mu_{(\alpha} \delta^\sigma_{\beta)}
+ g^{\lambda\sigma} \delta^\mu_{(\alpha} \delta^\rho_{\beta)}
- g^{\lambda\mu} \delta^\rho_{(\alpha} \delta^\sigma_{\beta)} ].
\label{App-A-L2-1}\end{aligned}$$ After a straightforward calculation using Eq. (\[App-A-L2-1\]), we have the result $$\begin{aligned}
\frac{\partial {\cal L}_2}{\partial \partial_\mu g_{\rho\sigma}}
= \sqrt{-g} \ \varphi [ \frac{1}{2} g^{\rho\sigma} g^{\alpha\beta} \Gamma^\mu_{\alpha\beta}
- g^{\rho\alpha} g^{\sigma\beta} \Gamma^\mu_{\alpha\beta}
+ \frac{1}{2} ( g^{\mu\rho} g^{\nu\sigma} + g^{\mu\sigma} g^{\nu\rho}
- g^{\mu\nu} g^{\rho\sigma} ) \Gamma^\alpha_{\nu\alpha} ].
\label{App-A-L2-2}\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\frac{\partial {\cal L}_2}{\partial \partial_\mu g_{\rho\sigma}} 2 g_{\rho\sigma}
= 2 \sqrt{-g} \ \varphi K^\mu.
\label{App-A-L2-3}\end{aligned}$$ Accordingly, Eqs. (\[App-A-K4\]) and (\[App-A-L2-3\]) give us $$\begin{aligned}
\frac{\partial {\cal L}_{NM}}{\partial \partial_\mu g_{\rho\sigma}} 2 g_{\rho\sigma}
= 2 \sqrt{-g} ( 3 \partial^\mu \varphi - \varphi K^\mu).
\label{App-A-NM-2}\end{aligned}$$ Since $\partial_\mu g_{\rho\sigma}$ is only included in $ {\cal L}_{NM}$, from the definition $\varphi = \xi \Phi^\dagger \Phi$, we have $$\begin{aligned}
\frac{\partial {\cal L}}{\partial \partial_\mu g_{\rho\sigma}} 2 g_{\rho\sigma}
= 2 \xi \sqrt{-g} [ 3 g^{\mu\nu} \partial_\nu (\Phi^\dagger \Phi) - \Phi^\dagger \Phi K^\mu ].
\label{App-A-g}\end{aligned}$$ Furthermore, it is easy to calculate the variation of the Lagrangian with respect to $\partial_\mu \Phi,
\partial_\mu \Phi^\dagger$. The results read $$\begin{aligned}
\frac{\partial {\cal L}}{\partial \partial_\mu \Phi} \Phi
&=& - \sqrt{-g} [ \xi \Phi^\dagger \Phi K^\mu + g^{\mu\nu} (D_\nu \Phi)^\dagger \Phi ],
\nonumber\\
\frac{\partial {\cal L}}{\partial \partial_\mu \Phi^\dagger} \Phi^\dagger
&=& - \sqrt{-g} [ \xi \Phi^\dagger \Phi K^\mu + g^{\mu\nu} (D_\nu \Phi) \Phi^\dagger ].
\label{App-A-Phi}\end{aligned}$$ Putting together Eqs. (\[App-A-g\]) and (\[App-A-Phi\]), the dilatation current (\[App-A-J\]) is calculated to be $$\begin{aligned}
J^\mu = ( 6 \xi + 1 ) \sqrt{-g} g^{\mu\nu} \partial_\nu \left(\Phi^\dagger \Phi \right).
\label{App-A-Current2}\end{aligned}$$
Useful formulae in the loop calculation {#App:AppendixB}
=======================================
In Appendix B, we summarize useful formulae in calculating radiative corrections in Section 5. Following Ref. [@Fujii1], as a regularization method, we adopt the method of continuous space-time dimensions $D$ in a flat Minkowski space-time. In this regularization method, all the quantities are extended from four dimensions to $D$ dimensions. Let us therefore focus on the following loop integral: $$\begin{aligned}
I (m, n) = \int d^D k \frac{(k^2)^{m-2}}{(k^2 + \Delta)^n},
\label{App-B-Integral1}\end{aligned}$$ where $m, n$ are integers and $\Delta$ is a constant. By power counting, this integral is convergent as long as $D < 2n - 2m + 4$.
With a Wick rotation $k^0 = i k^D$ and the ansatz of spherical symmetry, the integral (\[App-B-Integral1\]) can be rewritten as $$\begin{aligned}
I (m, n) = i V(D) \int_0^\infty d k \frac{k^{2m + D - 5}}{(k^2 + \Delta)^n},
\label{App-B-Integral2}\end{aligned}$$ where $V(D) = \frac{2 \pi^{\frac{D}{2} } } {\Gamma(\frac{D}{2})}$ is a D-dimensional volume form, e.g., $V(4) = 2 \pi^2$. Via the change of variables from $k$ to $t = \frac{k^2}{\Delta}$, the integral is reduced to $$\begin{aligned}
I (m, n) &=& i V(D) \frac{1}{2} \Delta^{m-n+\frac{D}{2} -2} \int_0^\infty d t \frac{t^{m + \frac{D}{2} - 3}}{(1 + t)^n}
\nonumber\\
&=& i V(D) \frac{1}{2} \Delta^{m-n+\frac{D}{2} -2} B(m + \frac{D}{2} -2, n - m - \frac{D}{2} + 2),
\label{App-B-Integral3}\end{aligned}$$ where the definition of the beta function is used: $$\begin{aligned}
B(\alpha, \beta) = \int_0^1 d x \ x^{\alpha - 1} (1 - x)^{\beta-1}
= \int_0^\infty d t \ t^{\alpha - 1} (1 + t)^{-\alpha-\beta}
= \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}.
\label{App-B-Beta}\end{aligned}$$ Since $\Gamma(z)$ has isolated poles at $z = 0, -1, -2, \cdots$, the integral (\[App-B-Integral3\]) has isolated poles at $D = 2 (n - m + 2), 2 (n - m + 3), \cdots$. We often make use of a relation for the gamma function, which holds for positive real numbers $x > 0$ $$\begin{aligned}
\Gamma (x+1) = x \Gamma(x),
\label{App-B-Gamma}\end{aligned}$$ and $\Gamma(1) = 1$.
In Section 5, to combine propagator denominators we utilize the Feyman parameter formula $$\begin{aligned}
\frac{1}{A_1 A_2 \cdots A_n} = \int_0^1 d x_1 \cdots d x_n \delta(\sum x_i -1)
\frac{( n - 1 )! } {(x_1 A_1 + x_2 A_2 + \cdots + x_n A_n)^n}.
\label{App-B-Feynman1}\end{aligned}$$ In the case of only two denominator factors, this formula reduces to $$\begin{aligned}
\frac{1}{A B} = \int_0^1 d x \frac{1} {[x A + (1-x) B]^2}.
\label{App-B-Feynman2}\end{aligned}$$ Taking differentiation of Eq. (\[App-B-Feynman2\]) with respect to $B$, we can derive another formula $$\begin{aligned}
\frac{1}{A B^2} = \int_0^1 d x \frac{2 (1-x)} {[x A + (1-x) B]^3}.
\label{App-B-Feynman3}\end{aligned}$$
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[^1]: E-mail address: ioda@phys.u-ryukyu.ac.jp
[^2]: To tell this statement more precisely, gauge invariance with the conventional form forbids the presence of a mass term of gauge field, but such a mass term can exist if we change the gauge transformation to a more complicated form. This situation happens when we consider a theory with spontaneous symmetry beakdown where a translation of the field by a constant must be accompanied [@Iliopoulos].
[^3]: We follow notation and conventions by Misner et al.’s textbook [@MTW], for instance, the flat Minkowski metric $\eta_{\mu\nu} = diag(-, +, +, +)$, the Riemann curvature tensor $R^\mu \ _{\nu\alpha\beta} =
\partial_\alpha \Gamma^\mu_{\nu\beta} - \partial_\beta \Gamma^\mu_{\nu\alpha} + \Gamma^\mu_{\sigma\alpha}
\Gamma^\sigma_{\nu\beta} - \Gamma^\mu_{\sigma\beta} \Gamma^\sigma_{\nu\alpha}$, and the Ricci tensor $R_{\mu\nu} = R^\alpha \ _{\mu\alpha\nu}$. The reduced Planck mass is defined as $M_p = \sqrt{\frac{c \hbar}{8 \pi G}} = 2.4 \times 10^{18} GeV$. Through this article, we adopt the reduced Planck units where we set $c = \hbar = M_p = 1$ though we sometimes recover the Planck mass $M_p$ for the clarification of explanation. In this units, all quantities become dimensionless. Finally, note that in the reduced Planck units, the Einstein-Hilbert Lagrangian density takes the form ${\cal L}_{EH} = \frac{1}{2} \sqrt{-g} R$.
[^4]: In this article, we use the terminology such that scale or conformal transformation means global transformation whereas its local version is called local scale transformation or local conformal transformation.
[^5]: The case $\xi = - \frac{1}{6}$ corresponds to conformal gravity, for which there is no dilatation current.
[^6]: In the case of massive gravity, a similar situation occurs in breaking the general coordinate invariance spontaneously [@Oda].
[^7]: The presence of the Nambu-Goldstone mode $\theta$ in $\hat F^{(1)}_{\mu\nu}$ merely shows that scale invariance of the theory under consideration is violated in any space-time dimension except four dimensions.
[^8]: As will seen later, we also have the similar contribution from the $U(1)$ gauge sector at the one-loop level, but we will now neglect it since the contribution from the gauge sector is smaller than that from (\[Eff-All\]).
|
---
abstract: 'Motivated by the studies of the superconducting pairing states in the iron-based superconductors, we analyze the effects of Brillouin zone folding procedure from a space group symmetry perspective for a general class of materials with the $P4/nmm$ space group. The Brillouin zone folding amounts to working with an effective one-Fe unit cell, instead of the crystallographic two-Fe unit cell. We show that the folding procedure can be justified by the validity of a glide reflection symmetry throughout the crystallographic Brillouin zone and by the existence of a minimal double degeneracy along the edges of the latter. We also demonstrate how the folding procedure fails when a local spin-orbit coupling is included although the latter does not break any of the space group symmetries of the bare Hamiltonian. In light of these general symmetry considerations, we further discuss the implications of the glide reflection symmetry for the superconducting pairing in an effective multi-orbital $t-J_{1}-J_{2}$ model. We find that the $P4/nmm$ space group symmetry allows only pairing states with even parity under the glide reflection and zero total momentum.'
author:
- 'Emilian M. Nica'
- Rong Yu
- Qimiao Si
title: 'Glide reflection symmetry, Brillouin zone folding and superconducting pairing for the $P4/nmm$ space group'
---
Introduction {#Sec:Space_group_no_SOC }
=============
The iron-based superconductors form a large family of materials which exhibits considerable diversity in their lattice structures. Examples include the 1111, 111, and 122 iron pnictides, as well as the 11 iron chalcogenides. The structural unit cell of most of these superconductors consists of two Fe and two pnictogen/chalcogen atoms and it is typically labeled as a 2-Fe unit cell. In momentum space, the corresponding physical Brillouin zone (BZ) is usually referred to as the folded BZ (FBZ). However, many theoretical studies have been based on tight-binding models defined on an effective “unfolded BZ” (UBZ) of 1-Fe unit cell with an implicit equivalence between the former and the FBZ. The mapping between the effective 1-Fe UBZ and the physical 2-Fe FBZ is via a BZ folding procedure. In view of the widespread use of this mapping we believe it is important to better understand the necessary conditions for its employment.
An early discussion of such a BZ folding procedure for 1111 iron-pnictide superconductors was provided by Lee and Wen [@Lee_Wen:2008]. They noticed that a single Fe-As plane contains a glide reflection symmetry which consists of a fractional unit cell translation followed by a reflection about the Fe-plane (see below). This symmetry can be used to define a pseudo-crystal momentum which can label the single-particle wave-functions with different parities under the operation. The last step in turn allows the definition on an unfolding procedure from the 2-Fe FBZ of the physical momentum space to the 1-Fe UBZ of the pseudo-crystal momentum space. More recently, a thorough group theoretical description of the electronic structure in the iron-based superconductors with a $P4/nmm$ space group symmetry has been given in Ref. . There, it was found that the glide-reflection symmetry classifies the Bloch states near the Fermi level and puts strong constraints on the low-energy effective model of the system.
The immediate motivation for the current work has come from the strong-coupling approach to superconductivity in the multi-orbital models for the iron-based materials. Superconducting paring in this approach has been studied by using $t-J_1-J_2$ models with three or more $3d$ orbitals, involving at least the $3d_{xz}$, $3d_{yz}$ and $3d_{xy}$ set, in an effective 1-Fe unit cell [@Goswami:2009; @Yu_Nat_Comm:2013; @Yu_PRB_2014]; such studies have been motivated by both the multi-orbital nature of the electronic structure and the bad-metal behavior of the parent compounds [@Yu_Nat_Comm:2013]. As we will discuss, the Bloch states formed directly from $3d_{xz}$ and $3d_{yz}$ Wannier states are odd under the reflection about the Fe plane, whereas those associated with the $3d_{xy}$ orbital are even under this symmetry operation. In the former case, the unfolding procedure becomes trivial as both types of Bloch states map onto the same quasi-crystal momentum in an unfolded BZ. By contrast, states derived directly from the $3d_{xy}$ orbital are mapped onto a quasi-crystal momentum which is shifted w.r.t. the other two orbital states. Given this, the validity of calculations done directly in a 1-Fe unit cell comes into question and a careful examination is required to establish whether a more involved 2-Fe unit cell basis should be used instead. It appears that this potential issue is not restricted to the particular $t-J_{1}-J_{2}$ case. Indeed, within a more general but related context, different ways of taking into account the two inequivalent pnictogen/chalcogen atoms have recently been proposed [@Ong_Coleman:2013; @Hu_Hao:2012]. A consensus does not appear to be reached since even the lattice symmetries considered in these studies are quite different: Ref. considered a $C_{4v}$ point-group symmetry about each As atom, while in Ref. a local $S_4$ symmetry about each Fe atom was taken into account. The proposed effective tight-binding models and the superconducting pairing symmetries in these two works are also quite distinct. Indeed, while Ref. proposed a spin singlet, orbital triplet $d+id$ $A_{1g}$ pairing (denoted as “TAO pairing") as a consequence of the two inequivalent As atoms, Refs. discussed an $s$-wave odd-parity pairing with nonzero total momentum, labeled $\eta$-pairing. These proposals not only question the validity of using the an effective 1-Fe UBZ, they also point to the need for a clear link between the superconducting pairing and lattice (space group) symmetries. This is particularly the case for the $\eta$-pairing, where it is still uncertain whether such pairing is allowed by the space group symmetry [@Hu:2013; @Hu_Hao_Wu:2013; @Lin_Ku:2014; @Wang_Maier:2014].
We believe that part of the confusion regarding the pairing symmetry is related to the issue we mentioned in the beginning of our paper: To what extent can a model for iron-based superconductors defined on the 1-Fe UBZ reproduce results consistent with the one defined on a 2-Fe FBZ? Here we study this problem for systems with a $P4/nmm$ space group (which include the 111 and 1111 iron pnictides and the 11 iron chalcogenides). The stringent conditions imposed by the space group constitute our starting points in the analysis of the validity of the 1-Fe unit cell formulation. To our knowledge there have been only a few attempts [@Lee_Wen:2008; @Andersen_Boeri:2011; @Cvetkovic_Vafek:2013; @Tomic_Jeschke_Valenti:2014] at placing this procedure on a firmer footing. More specifically, we wish to give a better description of the notion of the glide reflection symmetry within a more formal group-theoretical context and provide some justification for it’s implicit use as a general symmetry which can be used to label Bloch states of arbitrary momenta.
The glide reflection operation is a part of the $P4/nmm$ space group, and is characteristic of the non-symmorphic nature of this group. As we will see, it plays an important role in establishing, in the absence of any spin-orbit coupling, the validity of using the 1-Fe UBZ for both the single-particle dispersion and the pairing states. From analyzing the effect of the glide reflection, we are also able to show that treating the local spin-orbit coupling would require working with the 2-Fe FBZ. We should stress that we will consider the effect of the glide reflection in the context of the entire space group symmetry.
The remainder of the paper is organized as follows. In Section \[Subsection:Folding\_wo\_SOC\], we aim to give a rigorous analysis of the mapping from the physical 2-Fe BZ to an effective 1-Fe BZ for a class of materials with $P4/nmm$ space group symmetry. Our considerations apply to the ideally 2D case for which the conduction electrons do not disperse along the c-axis. We show that in this case, the existence of a glide reflection symmetry for all momenta is *guaranteed* by this particular space group. Without a spin-orbit coupling, the classification of all Bloch states under the glide reflection and the particular degeneracies along the BZ ensure that the folding procedure does not violate any space group symmetries of the system. In Section \[Sec:Effects\_pairing\] we discuss some consequences of the space group symmetry on the superconducting pairing in the multi-orbital $t-J_{1}-J_{2}$ model. In Section \[Sec:SOC\], we discuss the effects of an atomic spin-orbit coupling term on the electronic properties of both normal and superconducting states. We first show in Section \[Sec:Space\_group\_with\_SOC\] that despite the glide reflection still being a valid symmetry for all momenta, the lack of degeneracies along part of the BZ edge nullifies the usual unfolding procedure. In Sec. \[Sec:SOC\_numerics\] we present and compare numerical results for the normal state bandstructure with and without a spin-orbit coupling to confirm the preceding symmetry-based arguments. The direct consequences of the spin-orbit coupling on the pairing are then discussed in Sec. \[Sec:Pairing\_SOC\]. In Sec. \[Sec:Discussion\] we show how the symmetry arguments on the BZ folding survive for a 2D dispersion even when the interlayer couplings are taken into account. We also discuss the constraint imposed by the space group symmetry on the glide-reflection parity of the pairing channels, and subsequently examine the compatibility of several proposed pairings with the previously described space group symmetry. Concluding remarks are given in Sec. \[Sec:Conclusion\]. Appendix \[Sec:Appendix\_A\] contains a derivation of the important ansatz used in the main sections while related details are given in Appendix \[Sec:Appendix\_B\].
The $P4/nmm$ space group symmetry and the Brillouin zone folding {#Sec:BZ_folding}
================================================================
In a large group of Fe-based materials which have the $P4/nmm$ space group symmetry, the identical Fe-pnictogen/chalcogen layers are stacked on top of each other along the c-axis. The nontrivial spatial symmetry properties can be traced back to the structure of a single layer which is composed of a square Fe lattice in between two square As lattices shifted horizontally w.r.t. the Fe lattice and each other. The projection of the layer onto the Fe plane is shown in Fig. \[Fig:Structure\_BZ\] (a). Two adjacent Fe sites have different nearest-neighbor As configurations defining the $A$ and $B$ sublattices.
The space group of the 1111 Fe-based superconductors is $P4/nmm$. The latter is non-symmorphic such that under any choice of unit cell one cannot decompose the set of symmetry operations into a point subgroup and its coset made up of proper lattice translations. In particular, for the conventional 2-Fe unit cell choice [@Int_Union:1969], the crystal is invariant under a glide reflection symmetry $ \{ \sigma_z|\frac{1}{2} \frac{1}{2} 0 \}$ composed of a fractional unit cell translation $T_{ \pmb { \tau } } = \{ E | \frac{1}{2} \frac{1}{2} 0 \}$ in units of the sublattice translation, followed by a reflection about the Fe-plane $P_z = \{ \sigma_z| 0 0 0 \} $. The notation was chosen to be consistent with Ref. .
The crucial point to consider is that the glide reflection $ T_{ \pmb { \tau } } P_z $ can be used to classify Bloch states of arbitrary momentum $\pmb{k}$. As observed in Ref. , for a general $ \pmb{k}=( k_{x}, k_{y}, 0) $ in the Folded Brillouin Zone (FBZ) corresponding to the 2-Fe unit cell, the group of the wave-vector is isomorphic to the $C_{1h} $ point group. The latter has two irreducible representations which have even/odd parity under the simple reflection $P_{z}$. Consequently, states belonging to the irreducible representations of the space group $P4/nmm$ for general $ \pmb{k} $ are also either Even (E) or Odd (O) under the glide reflection. In our view, this provides a connection between the particular glide reflection symmetry as part of the space group of the Hamiltonian and it’s use in classifying states of arbitrary 2D momentum $\pmb{k}$, an argument we feel lacks an explicit exposition in the literature. By general group theoretical arguments [@Hammermesh:1964], states belonging to different irreducible representations cannot mix and thus the even/odd Bloch states do not hybridize. As we show in the following, this, together with the particular degeneracies along the entire folded BZ edge (see Fig. \[Fig:Structure\_BZ\]) allows the reduction to the 1-Fe unit cell.
![ (a) Projection onto the Fe plane of the crystal structure of 1111 systems. The thick lines define the crystallographic 2-Fe unit cell while the dashed lines denote the effective 1-Fe unit cell. As throughout the text, the unit distance is defined by the NNN Fe translation (same sublattice). (b) The Folded Brillouin Zone (FBZ-thick line) corresponding to the true 2-Fe unit cell and the unfolded BZ (UBZ-dashed line) of the effective 1-Fe unit cell (for definiteness, in the $k_{z}=0$ plane). The FBZ is defined by the translation in reciprocal space by $\pmb{Q}= \left( \pm 2\pi, 0, 0 \right) $ or $\left( 0, \pm 2\pi, 0 \right)$, which correspond to $ \left( \pm \pi, \pm \pi , 0 \right) $ or $\left( \pm \pi, \pm \pi, 0 \right)$ in the notation for the effective 1-Fe unit cell. Here $\Gamma$, $X$ and $M$ label the points in the FBZ, while $\Delta$, $\Sigma$ and $Y$ mark the corresponding segments. []{data-label="Fig:Structure_BZ"}](Figure_unit_cell_BZ_re_size){width="1.0\columnwidth"}
The glide reflection symmetry and the Brillouin Zone folding {#Subsection:Folding_wo_SOC}
------------------------------------------------------------
The advantage of using an unfolded 1-Fe BZ consists in effectively reducing the number of bands in the calculation. In practice this means introducing a set of trial wavefunctions defined on an UBZ, carrying on the calculation, then folding back to the physical FBZ. In doing this, one must check that the Bloch functions defined in the UBZ do not introduce additional hybridization terms in the Hamiltonian, and that they do not violate any of the original symmetries of the space group. Here, we analyze the same procedure in reverse. We start with a set of trial wave functions which are in accord with a minimum space group symmetry, namely the glide reflection, then examine the unfolding process.
As mentioned in the previous section, the glide-reflection can be used to classify the irreducible representations of the space group for general $\pmb{k}$ in the folded BZ when there is no dispersion along the $z$-axis i.e. $k_{z}=0$. This ensures that eigenstates of this operator will be eigenstates of the Hamiltonian within a transformation on the orbital indices alone. As a consequence, the Hamiltonian can be written in block diagonal form at these general $\pmb{k}$. The detailed representation theory of this space group shows that for a number of higher-symmetry momenta in the FBZ, the irreducible representations are not required to be eigenstates of the glide [@Cvetkovic_Vafek:2013]. This does not invalidate our arguments since by continuity any off-diagonal terms must vanish here as well. We will show that this implies a one-to-one correspondence between the eigenstates of the glide-reflection and Bloch states defined on the UBZ, with no hybridization between different momenta.
We proceed to build Bloch eigenstates of the glide operation with $\pmb{k}$ in the folded zone (See Appendix \[Sec:Appendix\_A\]). The electron annihilation operator in the physical 2-Fe BZ on sublattice A(B) is defined as
$$\begin{aligned}
\label{Eq:sublattice}
C_{\pmb{k},A/B, \alpha}=\frac{1}{\sqrt{N_{s}}} \sum_{i} e^{i \pmb{k} \cdot \pmb{R}_{i,A/B}} C_{\pmb{R}_{i,A/B}, \alpha},\end{aligned}$$
where $\alpha$ is an orbital index, $N_s=N/2$ refers to the number of (2-Fe) unit cells and $i$ is the index of the unit cell. The two sublattice vectors are related by $\pmb{R}_{i,A} + \pmb{\tau} = \pmb{R}_{i,B}$ with $\tau$ the nearest-neighbour distance used in the translation of the glide operation. The anihilation operators do not have definite parity under the glide reflection $T_{ \pmb { \tau } } P_z$:
$$\begin{aligned}
(T_{ \pmb { \tau } } P_z) C_{\pmb{k},A/B, \alpha} = (-1)^{\alpha}
C_{\pmb{k},B/A, \alpha},\end{aligned}$$
where $(-1)^{\alpha} =\pm 1$ depending on the parity of the local orbital under a pure reflection. Without loss of generality we can define operators with definite parity :
$$\begin{aligned}
C_{\pmb{k},E,\alpha} = & \frac{1}{\sqrt{2}} \left[ C_{\pmb{k},A,\alpha} + (-1)^{\alpha} C_{\pmb{k},B,\alpha} \right] \label{Eq:Eigen_1_a} \\
C_{\pmb{k},O,\alpha} = & \frac{1}{\sqrt{2}} \left[ C_{\pmb{k},A,\alpha} - (-1)^{\alpha} C_{\pmb{k},B,\alpha} \right]. \label{Eq:Eigen_1_b}\end{aligned}$$
Note that the even ($E$) and odd ($O$) parity states refer to parity under the glide and so we can build both types of operators for arbitrary orbital parity $\alpha$. We also refer the reader to Appendix \[Sec:Appendix\_A\] for more details on the states defined above.
Since $C_{\pmb{k},E,\alpha}$ and $C_{\pmb{k},O,\alpha}$ have different parity under the glide for arbitrary “2-Fe” crystal momentum , they do not mix in a one-particle Hamiltonian. An arbitrary 2D tight-binding Hamiltonian consistent with the space group symmetries can then be expressed in the terms of the $E/O$ states as
$$\begin{aligned}
\label{Eq:2_Fe_Hamiltonian}
H_{TB} = \sum_{\pmb{k} \in FBZ } \left( \epsilon_{E}^{\alpha \beta} (\pmb{k}) C_{\pmb{k},E,\alpha}^{\dagger} C_{\pmb{k},E,\beta} + \epsilon_{O}^{\alpha \beta} (\pmb{k}) C_{\pmb{k},O,\alpha}^{\dagger} C_{\pmb{k},O,\beta}\right),\end{aligned}$$
where $FBZ$ stands for the 2-Fe Folded Brillouin Zone and $\epsilon_{E}^{\alpha \beta}(\pmb{k}), \epsilon_{O}^{\alpha \beta} (\pmb{k})$ are matrices in orbital space. We omitted the spin index for simplicity.
We now turn to the unfolding procedure and start by noting that the eigenstates of the glide reflection have (see Eqs. \[Eq:App\_state\_a\] and \[Eq:App\_state\_b\])
$$\begin{aligned}
\label{Eq:Irr_switch}
C_{\pmb{k}+\pmb{Q},E/O , \alpha} = C_{\pmb{k},O/E, \alpha},\end{aligned}$$
where $\pmb{Q}=(\pm 2 \pi, 0)$ or $\pmb{Q}=(0, \pm 2\pi)$ in units of the 2-Fe unit cell, and correspond to $ \left( \pm \pi, \pm \pi , 0 \right) $ or $\left( \pm \pi, \pm \pi, 0 \right)$ in units of the effective 1-Fe unit cell. That is, our states apparently violate the symmetry under the pure translation by a 2-Fe unit cell sublattice. In the simplest Bravais lattice case a state of arbitrary momentum is only labeled by the irreducible representation of the pure translation subgroup i.e by $\pmb{k}$ itself. Here, in addition to $\pmb{k}$ one can also label all the states according to the irreducible representations of the glide reflection as well, that is, by both $\pmb{k}$ and $\lambda$. The resolution to our apparent conundrum lies in the fact that states displaced by a reciprocal lattice vector $\pmb{Q}$ can switch their representation with respect to the glide reflections: $$\begin{aligned}
\label{Eq:E_to_O}
E \rightarrow O, \epsilon_{E}^{\alpha \beta} (\pmb{k}+ \pmb{Q}) = \epsilon_{O}^{\alpha \beta} (\pmb{k}).\end{aligned}$$ This ensures that we are not violating the original pure translation symmetry but it also introduces a constraint: Eigenstates of the system must be at least doubly degenerate all along the edge of the first (folded) BZ. Only in such case can the states switch representations as one goes beyond the folded BZ. Indeed, this is precisely what happens in Si and Ge at the X point along the edge of the BZ [@Slater_1954], [@Dresselhaus_1968]. The lattice in this case has a diamond structure and the two sublattices can be connected by a glide reflection. Due to the non-symmorphic nature of the space group, the states are degenerate at X and must exchange representations as one goes into the extended BZ. In the present case for $P4/nmm$, there is a minimal degeneracy throughout the edge of the folded BZ [@Cvetkovic_Vafek:2013], [@Cracknell_book:1972] which allows for the switching of the representations as one crosses into the extended zone.
We can connect with one of the initial folding procedures [@Lee_Wen:2008] which relied on a definition of a quasi-crystal-momentum dependent on the parity of the orbital state under the pure reflection. Using the equivalent form of the glide states in (\[Eq:App\_state\_a\]), (\[Eq:App\_state\_b\]), we can absorb the negative sign from $(-1)^{(\alpha=1)}=e^{\pm i \pmb{Q \cdot \tau }}=-1$ for states with odd parity under a pure reflection. Here $\pmb{Q}=(\pm 2 \pi, 0)$ or $\pmb{Q}=(0, \pm 2\pi)$ and $\tau=( \pm \frac{1}{2},\pm \frac{1}{2} )$ in units of the 2-Fe unit cell. Explicitly, the eigenstates of the glide become
$$\begin{aligned}
C_{\pmb{k},E,\alpha ~\text{even}}& = & C_{\pmb{k}, \alpha} & = & \frac{1}{\sqrt{N}} \sum_{\pmb{R}_i} \left[ e^{i \pmb{k} \cdot \pmb{R}_{i} } C_{\pmb{R}_i, \alpha } \right] \label{Eq:Orb_Fourier_a} \\
C_{\pmb{k},E,\alpha~\text{odd}}& = & C_{\pmb{k}+ \pmb{Q}, \alpha} & = & \frac{1}{\sqrt{N}} \sum_{\pmb{R}_i} \left[ e^{i ( \pmb{k} + \pmb{Q}) \cdot \pmb{R}_{i} } C_{\pmb{R}_i, \alpha} \right] \label{Eq:Orb_Fourier_b} \\
C_{\pmb{k},O,\alpha~\text{even}}& = & C_{\pmb{k}+ \pmb{Q}, \alpha} & = & \frac{1}{\sqrt{N}} \sum_{\pmb{R}_i} \left[ e^{i (\pmb{k} + \pmb{Q} ) \cdot \pmb{R}_{i} } C_{\pmb{R}_i, \alpha} \right] \label{Eq:Orb_Fourier_c} \\
C_{\pmb{k},O,\alpha~\text{odd}}& = & C_{\pmb{k}, \alpha} & = & \frac{1}{\sqrt{N}} \sum_{\pmb{R}_i} \left[ e^{i \pmb{k} \cdot \pmb{R}_{i} } C_{\pmb{R}_i, \alpha} \right] \label{Eq:Orb_Fourier_d}.\end{aligned}$$
The expressions above can be formally subtituted into the Hamiltonian (\[Eq:2\_Fe\_Hamiltonian\]). Under this mapping, the initial summation over $\pmb{k} \in $ FBZ for both $ E $ and $ O $ glide parity sectors can be re-written as a summation over $\slashed{\pmb{k}} \in $ UBZ (unfolded BZ) for the $E$ only sector (recall that $ \epsilon_{\alpha \beta}^{E} (\pmb{k}+ \pmb{Q}) = \epsilon_{\alpha \beta}^{O} (\pmb{k}) $). By taking into account the fact that different *orbital* parity states must be displaced by $\pmb{Q}$ w.r.t. to each other we can recast the tight-binding Hamiltonian as
$$\begin{aligned}
H_{TB} = & \sum_{\pmb{\slashed{k}} \in UBZ} \Bigg[ \sum_{ee} \left( \epsilon^{E}_{\alpha \beta}(\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}, \alpha} C_{\pmb{\slashed{k}}, \beta} \right) + \sum_{oo} \left( \epsilon^{E}_{\alpha \beta}(\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}+\pmb{Q}, \alpha} C_{\pmb{\slashed{k}}+\pmb{Q}, \beta} \right)
+ \sum_{eo} \left( \epsilon^{E}_{\alpha \beta}(\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}, \alpha} C_{\pmb{\slashed{k}}+\pmb{Q}, \beta} \right) + \sum_{oe} \left( \epsilon^{E}_{\alpha \beta}(\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}+\pmb{Q}, \alpha} C_{\pmb{\slashed{k}}, \beta} \right) \Bigg] \label{Eq:Conventioanal_1}\\
= & \sum_{\pmb{\tilde{k}} \in UBZ} \sum_{\alpha \beta} \left( \epsilon^{E}_{\alpha \beta}(\pmb{\tilde{k}}) C^{\dagger}_{\pmb{\tilde{k}}, \alpha} C_{\pmb{\tilde{k}}, \beta} \right),\end{aligned}$$
where $e,o$ refer to the parity of the orbitals $\alpha$ and $\beta$. The details of the above derivation are presented in Appendix \[Sec:Appendix\_B\]. In going to the second line we explicitly identified along with Ref. the dependence of $\slashed{\pmb{k}}$ on the orbital parity such that $\pmb{\tilde{k}}=\pmb{\slashed{k}}$ for even orbital states and $\pmb{\tilde{k}}= \pmb{\slashed{k}}+ \pmb{Q}$ for odd states. What allows us to equate the two forms is the fact that $\pmb{\slashed{k}}$ is effectively a dummy variable in (\[Eq:Conventioanal\_1\]) and the matrix structure is determined by the orbital parities alone. For the purpose of notational clarity, we stress the following: a) $\pmb{\slashed{k}}$ is defined in the larger Brillouin zone (UBZ) for the 1-Fe unit cell, and is associated with an orbital-dependent procedure in unfolding from the smaller Brillouin zone (FBZ). It has the same meaning as that used in Ref. ; b) $\pmb{\tilde{k}}$, also defined in the UBZ for the 1-Fe unit cell, arises in an orbital-independent unfolding procedure. The distinction between these two wave vectors will be important for our later considerations.
We can illustrate a different widely used unfolding procedure if we define two types of local states. Specifically, we can set $C_{\pmb{R_{A}}, \alpha}= C_{\pmb{R_{A}}+ \pmb{\tau}, \alpha}$ for even parity orbital $\alpha$ and $C_{\pmb{R_{A}}, \alpha}= (-1)C_{\pmb{R_{A}}+ \pmb{\tau}, \alpha}$ for odd parity orbital $\alpha$ in Eqs. \[Eq:Orb\_Fourier\_a\]-\[Eq:Orb\_Fourier\_d\] such that the local orbital state changes sign under the simple 1-Fe unit cell translation $T_{\tau}$. The additional minus sign for odd orbital states will generate and additional $e^{i\pmb{Q}\cdot \pmb{\tau}}$ factor for odd orbital states. This is equivalent to the orbital parity dependent gauge transformation encountered in the literature. Using the same notation as above the eigenstates become
$$\begin{aligned}
C_{\pmb{k},E,\alpha} =& C_{\pmb{k}, \alpha} \label{Eq:Gauge_a} \\
C_{\pmb{k},O, \alpha} = & C_{\pmb{k}+\pmb{Q}, \alpha} \label{Eq:Gauge_b}\end{aligned}$$
for arbitrary parity $\alpha$. If we identify $\epsilon^{O}_{\alpha \beta}(\pmb{k})= \epsilon^{E}_{\alpha \beta}(\pmb{k}+\pmb{Q})$ and use the above definitions in the Hamiltonian (\[Eq:2\_Fe\_Hamiltonian\]) the latter can be re-written as
$$\begin{aligned}
H_{TB} =& \sum_{\pmb{k} \in FBZ } \bigg( \epsilon_{E}^{\alpha \beta} (\pmb{k}) C_{\pmb{k},\alpha}^{\dagger} C_{\pmb{k},\beta} \notag \\
& + \epsilon_{E}^{\alpha \beta} (\pmb{k}+\pmb{Q}) C_{\pmb{k}+\pmb{Q},\alpha}^{\dagger} C_{\pmb{k}+\pmb{Q},\beta}\bigg) \\
= & \sum_{\pmb{\tilde{k}} \in UBZ }\epsilon_{E}^{\alpha \beta} (\pmb{\tilde{k}})C_{\pmb{\tilde{k}},\alpha}^{\dagger} C_{\pmb{\tilde{k}},\beta}\end{aligned}$$
where $\pmb{\tilde{k}}$ is in the Unfolded Brillouin Zone (UBZ).
We stress that both unfolding procedures discussed above were guaranteed by the absence of mixing between the $E,O$ sectors in the Hamiltonian for general $\pmb{k}$ together with the switching of the glide parity for the ansatz states beyond the first folded BZ (Eq. \[Eq:Irr\_switch\]). These two conditions are a direct consequence of the irreducible representations of the glide symmetry for arbitrary $\pmb{k}\in $ FBZ. None of the folding procedures discussed above and used in the literature would work if the Hamiltonian contained $E,O$ mixing terms since we could not define $\pmb{Q}$ shifted states related to the 1-Fe unit cell Fourier transforms for either even and odd parity orbitals.
Although the above arguments seem to be implicit in the literature [@Andersen_Boeri:2011; @Lee_Wen:2008], we have encountered few observations of the crucial connection to the irreducible representations of the $P4/nmm$ space group, with a notable exception of Ref. in the context of the electronic structure. Here, by constructing the electronic states which are even or odd under glide reflection, we are in position to discuss the the constraints imposed by the glide-reflection symmetry on the various types of pairing, which will be discussed in the next subsection, as well as the effects of a local spin-orbit coupling on the unfolding procedure, which we turn to in the following section. As we mentioned in Sec. \[Sec:BZ\_folding\], we believe that it is insufficient to argue that the presence of a glide reflection operation in the space group of the Hamiltonian guarantees the success of a folding procedure. Along with the glide reflection, the space group contains fifteen other symmetry elements [@Int_Union:1969] different from pure sublattice translations. Of these, only the glide reflection can be used to classify states according to the irreducible representations of the space group for *general two-dimensional crystal momenta*, since only this operation leaves an arbitrary $\pmb{ k } $ invariant. But as we will show in Sec. \[Sec:SOC\], the same folding procedure fails when a spin-orbit coupling is turned on, even though the additional interactions do not violate the glide reflection symmetry or any space symmetry of the original Hamiltonian.
The effects of the glide symmetry on the superconducting pairing interactions {#Sec:Effects_pairing}
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We now discuss some of the effects of the glide symmetry for the superconducting pairing interaction. We assume that the symmetry outlined in the previous sections is not violated such that without the pairing interactions one can unfold the BZ of the 2-Fe unit cell to the effective 1-Fe UBZ. We also assume that the original symmetry of the 2-Fe unit cell is not broken by the appearance of magnetic or “nematic" order. The analysis presented here also assumes a trivial spatial dependence of the bare pairing interactions such as would arise in a nearest neighbor (NN), next-nearest neighbor (NNN) $t-J_{1}-J_{2}$ Hamiltonian. Lastly, we consider singlet pairing, as evidenced by experiments for various iron-based superconductors. In this case, the antisymmetric nature of the pairing wavefunction requires the pairing to be even parity under inversion [@Sigrist_Ueda:1991].
Before proceeding in a manner analogous to that of Sec. \[Subsection:Folding\_wo\_SOC\], some additional remarks are in order. The tight-binding part of a Hamiltonian with pairing interactions at mean-field level is typically chosen as the identity representation of the space group. In general, the pairing functions are determined self-consistently and can lower the symmetry of the Hamiltonian to a subgroup of the full space group (as [*e.g.*]{}, what happens to the rotational invariance of the Hamiltonian under the $C_4$ operation of the $D_{4h}$ point group of the full $P4/nmm$ space group in the case of a d-wave pairing). The problem simplifies for the spin singlet pairing considered in this section. For Cooper pairs of equal and opposite momenta in the FBZ ( $\pmb{k}, -\pmb{k}$) associated with the 2-Fe unit cell, the symmetry properties of the pairing functions are completely determined by the $D_{4h}$ point group associated with the $P4/nmm$ space group. More precisely, the tensor irreducible representations of the space group used to classify the pairing are the irreducible representations of the group of the wave-vector labeled by total momentum $\pmb{k}+(-\pmb{k})=(0,0)$ (For a general argument, see Ref. ). Given this, we know that the irreducible representations of the $D_{4h}$ point group are either even or odd under inversion. In addition, for zero total momentum representations, the glide reflection is equivalent to a simple reflection about the z-plane. This means that the parity under inversion coincides with the parity under the glide-reflection for one dimensional irreducible tensor representations and is opposite for the two-dimensional ones [@Cvetkovic_Vafek:2013]. Restricting ourselves to the former 1D representations, a Hamiltonian containing any linear combination of inversion-even pairing terms cannot break the glide reflection symmetry, regardless of whether the pairing preserves the rotational invariance or not. The glide symmetry can be broken if and only if the system spontaneously breaks the inversion symmetry, which consequently results in triplet pairing. The remaining part of the section seeks to illustrate that no finite momentum pairing terms can be present in either the folded or unfolded BZ provided that the Hamiltonian is invariant under the glide-reflection. This also guarantees that the common folding procedure is still valid in this case.
We consider for illustration purposes the following NN pairing interaction:
$$\begin{aligned}
\label{Eq:Pairing_interaction}
H_{int}=\sum_{\alpha \beta}\sum_{ij,NN} J_{\alpha \beta} & \left[ C^{\dagger}_{i \alpha \uparrow}C^{\dagger}_{j \beta \downarrow} - C^{\dagger}_{i \alpha \downarrow}C^{\dagger}_{j \beta \uparrow} \right] \times \notag \\
& \left[ C_{i \alpha \uparrow}^{\phantom{\dagger}} C_{j \beta \downarrow} - C_{i \alpha \downarrow}C_{j \beta \uparrow} \right]+H.C.\end{aligned}$$
Based on Eqs. \[Eq:Eigen\_1\_a\], \[Eq:Eigen\_1\_b\] we can define Fourier transforms on each sublattice as
$$\begin{aligned}
C_{\pmb{R}_{A}, \alpha } & = \frac{1}{ \sqrt{ N_s } } \sum_{\pmb{k } \in FBZ} e^{-i \pmb{k} \cdot \pmb{ R_{A} } } \left[ C_{\pmb{ k },E,\alpha} + C_{\pmb{ k },O,\alpha} \right], \label{FT_A} \\
C_{\pmb{R}_{B}, \alpha} & = (-1)^{\alpha} \frac{1}{ \sqrt{ N_s } } \sum_{\pmb{k } \in FBZ} e^{-i \pmb{k} \cdot \pmb{ R_{B} } } \left[ C_{\pmb{ k },E,\alpha} - C_{\pmb{ k },O,\alpha} \right]. \label{FT_B}\end{aligned}$$
It is straightforward to see that the pairing interaction in Eq.\[Eq:Pairing\_interaction\] will contain equal numbers of $E$ and $O$ states. Indeed, this can be understood from the general discussion in the previous sections. Although formally at some special points of the FBZ the eigenstates are not required to also be eigenstates of the glide reflection, in practice, continuity must enforce this as was mentioned in the preceding section. All terms in the Hamiltonian must effectively be invariant under the glide reflection symmetry for arbitrary $\pmb{k}$, ensuring that all interactions must be even under the glide reflection symmetry. This constrains all pairing terms to have equal numbers of even and odd states by the space group symmetry arguments. The general form of the interactions can in principle generate pairing terms mixing $E$ and $O$ states which might result in a finite $\pmb{Q}$ momentum Cooper pairs.
To make progress, we can consider (\[Eq:Pairing\_interaction\]) in a mean-field (MF) approach. In view of the connection between the MF Hamiltonian and the equation of motion approach we anticipate the same results beyond the simplest level. As an illustration we analyze the term
$$\begin{aligned}
\label{Eq:MF_term}
& H'_{int, MF} = \sum_{\alpha\beta} \sum_{<ij>}\sum_{ e } \notag \\
& \left< J_{\alpha \beta} \left[ C^{\dagger}_{i \alpha \uparrow}C^{\dagger}_{j \beta \downarrow} - C^{\dagger}_{i \alpha \downarrow}C^{\dagger}_{j \beta \uparrow} \right] \right > \times C_{i \alpha \uparrow}C_{j \beta \downarrow}.\end{aligned}$$
The remaining terms in Eq.\[Eq:Pairing\_interaction\] can be obtained by flipping the spin indices of the last pair and adding the Hermitian conjugate terms. Decomposing the NN summation as $\sum_{<ij>}\equiv \sum_{\pmb{R_{A}}}\sum_{\pmb{e}=\pm \pmb{\hat{x}, \hat{y}}}+\sum_{\pmb{R_{B}}}\sum_{\pmb{e}=\pm \pmb{\hat{x}, \hat{y}}}$ and taking the sublattice specific Fourier transformation (F.T.) as in (\[FT\_A\]),(\[FT\_B\]) we obtain
$$\begin{aligned}
\label{Eq:Pairing_MF}
H'_{int, MF} =& \sum_{\alpha\beta} \sum_{ \pmb{e} } \sum_{\pmb{k} \in FBZ} e^{ i\pmb{ke} } \times \notag \\
\Bigg[ & (-1)^{\beta} \Delta^{A}_{\pmb{e},\alpha\beta} ( C_{\pmb{k},E, \alpha \uparrow}C_{\pmb{-k},E, \beta \downarrow} - C_{\pmb{k},E, \alpha \uparrow}C_{\pmb{-k},O, \beta \downarrow} + C_{\pmb{k},O, \alpha \uparrow }C_{\pmb{-k},E, \beta \downarrow} - C_{ \pmb{k},O, \alpha \uparrow}C_{ \pmb{-k},O,\beta \downarrow} ) + \notag \\
+ & (-1)^{\alpha} \Delta_{\pmb{e},B,\alpha\beta}^{B} (C_{\pmb{k},E, \alpha \uparrow}C_{\pmb{-k},E, \beta \downarrow} + C_{\pmb{k},E, \alpha \uparrow }C_{-k \beta \downarrow O} - C_{\pmb{ k},O, \alpha \uparrow}C_{ \pmb{-k},E, \beta \downarrow } - C_{ \pmb{k},O, \alpha \uparrow}C_{\pmb{-k},O, \beta \downarrow } ) \Bigg]\end{aligned}$$
where
$$\begin{aligned}
\label{Eq:Delta_A}
\Delta^{A}_{\pmb{e},\alpha\beta}=& \left< J_{\alpha \beta} \left[ C^{\dagger}_{\pmb{ R_{A}} \alpha \uparrow}C^{\dagger}_{\pmb{ R_{A}}+\pmb{e} \beta \downarrow} - C^{\dagger}_{\pmb{ R_{A}} \alpha \downarrow}C^{\dagger}_{\pmb{ R_{A}}+\pmb{e} \beta \uparrow} \right] \right >\end{aligned}$$
$$\begin{aligned}
\label{Eq:Delta_B}
\Delta^{B}_{\pmb{e},\alpha\beta}=& \left< J_{\alpha \beta} \left[ C^{\dagger}_{\pmb{ R_{B}} \alpha \uparrow}C^{\dagger}_{\pmb{ R_{B}}+\pmb{e} \beta \downarrow} - C^{\dagger}_{\pmb{ R_{B}} \alpha \downarrow}C^{\dagger}_{\pmb{ R_{B}}+\pmb{e} \beta \uparrow} \right] \right >\end{aligned}$$
The crucial assumption made in the above equations is that both $\Delta^{A}_{e}$, $\Delta^{B}_{e}$ are independent of their sublattice space indices $\pmb{R_{A}}$ and $\pmb{R_{B}}$, but $\Delta^{A}_{e} \neq \Delta^{B}_{e}$ in general. This amounts to having the pairing order parameters which are constant on the respective sublattices.
We distinguish two cases: i) $\alpha$ and $\beta$ corresponding to orbitals of the same parity under the reflection, ii) $\alpha$ and $\beta$ have different parities.
For case i) of same parity we can look at the real space expression (\[Eq:MF\_term\]) and demand invariance under a glide reflection. Since both $\alpha$, $\beta$ terms acquire the same orbital parity factor and each state gets shifted by one unit of the 1-Fe unit cell we have $\Delta^{A}_{\pmb{e},\alpha\beta}=\Delta^{B}_{\pmb{e},\alpha\beta}$. Plugging this into (\[Eq:Pairing\_MF\]) and setting $(-1)^{\alpha}=(-1)^{\beta}$, we see that all $EO$ and $OE$ mixed terms cancel and we get only same glide parity terms.
For case ii) where $\alpha$ and $\beta$ have different orbital parities we can do the same as above and impose the glide reflection symmetry in real space. We get $\Delta^{A}_{\pmb{e},\alpha\beta}=-\Delta^{B}_{\pmb{e},\alpha\beta}$. With $(-1)^{\alpha}=-(-1)^{\beta}$ we get the same cancellation as for case i).
Identical results are obtained for the next-nearest coupling (NNN, $J_2$) case.
In both cases of same and different orbital parity pairing, the results are completely analogous to those considered in Section \[Subsection:Folding\_wo\_SOC\] for a tight-binding model without spin-orbit coupling. Namely, the pairing part of the Hamiltonian splits into $EE$ and $OO$ glide parity sectors. The exact same unfolding arguments can be trivially extended to the pairing part. Calculations can be done in an UBZ using the $\pmb{\tilde{k}}$ F.T. and then fold the results to the 2-Fe BZ. In this scheme there is no finite momentum pairing.
The effects of spin-orbit coupling {#Sec:SOC}
==================================
Folding in the presence of spin-orbit coupling {#Sec:Space_group_with_SOC}
----------------------------------------------
A local (atomic) spin-orbit coupling (SOC) term $$\label{Eq:SOC_term}
I\pmb{L} \cdot \pmb{S} = \frac{I}{2} \left ( \pmb{J}^{2} - \pmb{L}^{2} - \pmb{S}^{2} \right )$$ preserves the $P4/nmm$ space group symmetry since the latter is a scalar under all point group operations and is local in space ($I$ is a constant). It does however lock the orbital and spin parts of the conduction electrons together such that the two do not transform independently of each other under space group operations. This forces us to change the irreducible representations of the space group to double valued representations [@Cracknell_book:1972]. In general, the reflection $ P_{z} = \{ \sigma_{z} | 0 0 0\} $ is equivalent to
$$\label{Eq:Sigma_z}
\begin{matrix}
\sigma_z & = & C_{ 2z } \otimes I \\
\end{matrix}$$ where $C_{2z}$ corresponds to a rotation by $ \pi $ along a chosen $z$-axis and $I$ is the inversion. Since spinor states are invariant under the inversion [@Cracknell_book:1972], the effect of $P_{z}$ is given by the Pauli matrix term $$\label{Eq:Sigma_z_spinor}
D^{ ( j= \frac {1} {2} ) }( \sigma_z )=
\begin{pmatrix}
-i & & 0 \\
& & \\
0 & & i
\end{pmatrix}
\times
\begin{pmatrix}
1 & & 0 \\
& & \\
0 & & 1
\end{pmatrix}
=
\begin{pmatrix}
-i & & 0 \\
& & \\
0 & & i
\end{pmatrix}
= - i \sigma_{3}$$
For arbitrary momentum $\pmb{k}$, one constructs the irreducible representations of the space group by determining the so-called group of the wave vector. That is the subgroup of the space group, with elements which either leave $\pmb{k}$ invariant or translate it by a reciprocal vector. By examining the effect of these elements on a set of properly chosen states, one can determine the group isomorphic to the group of the wave vector. In the case of double valued representations one must also consider, in addition to the operations in the single-valued case, those obtained by changing in sign of the states [@Cracknell_book:1972].
Even with SOC terms, the glide reflection is still part of the group of the wave-vector. Because of the locking of the spin and orbital states, the glide reflection generates factors of $i$, consistent with Eq. \[Eq:Sigma\_z\_spinor\]. By considering local states with and without spin and applying the operations that keep an arbitrary $\pmb{k}$ invariant one obtains a group isomorphic to the double valued group of $C_{1h}$. The irreducible representations of the latter are illustrated in Table \[Table:Double\_rep\].
\[Table:Double\_rep\]
$ C_{1h} $ $ E $ $ \sigma_{z} $ $ \bar{E} $ $ \bar{ \sigma}_{ z } $
---------------- ------- ---------------- ------------- --------------------------
$ \Gamma_{1} $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ \Gamma_{2} $ $ 1 $ $ -1 $ $ 1 $ $ -1 $
$ \Gamma_{3} $ $ 1 $ $ i $ $ -1 $ $ -i $
$ \Gamma_{4} $ $ 1 $ $ - i $ $ -1 $ $ i $
: The double-valued irreducible representations of the $ C_{1h} $ point group \[ C.J. Bradley and A.P. Cracknell, *Mathematical Theory of Symmetry in Solids* (Clarendon Press, Oxford 1972) \].
The inclusion of the spin in the glide reflection will always generate the pure phases $\pm i$. We then conclude that the physical irreducible representations correspond to $\Gamma_{3}$ and $\Gamma_{4}$. This indicates that, as for the $P4/nmm$ case without SOC, for arbitrary momentum in an unfolded BZ, eigenstates of a general tight-binding Hamiltonian will also be eigenstates of the glide reflection symmetry. We can connect with the ansatz states in Eqs. \[Eq:Eigen\_1\_a\], \[Eq:Eigen\_1\_b\] which coincide with Bloch states in a 1-Fe BZ. We remark that these are still eigenstates of the glide reflection operator but in the presence of SOC, we need to account for the transformation of the spins as well. We thus relabel
$$\begin{aligned}
C_{\pmb{k},E,\alpha, \uparrow} & \rightarrow C_{\pmb{k},\tilde{O},\alpha, \uparrow} \label{Eq:Corres_E_up}\\
C_{\pmb{k},E,\alpha, \downarrow} & \rightarrow C_{\pmb{k},\tilde{E},\alpha, \downarrow} \label{Eq:Corres_E_down} \\
C_{\pmb{k},O,\alpha, \uparrow} & \rightarrow C_{\pmb{k},\tilde{E},\alpha, \uparrow} \label{Eq:Corres_O_up} \\
C_{\pmb{k},O,\alpha, \downarrow} & \rightarrow C_{\pmb{k},\tilde{O},\alpha, \downarrow} \label{Eq:Corres_O_down}\end{aligned}$$
where the $\tilde{E}, \tilde{O}$ refer to the sign in front of the $i e^{i \pmb{k} \cdot \pmb{\tau}}$ term under the glide transformation. Since the irreducble representations are also eigenstates of the glide, the Hamiltonian excluding the pairing terms can only connect $\tilde{E}, \tilde{E}$ or $\tilde{O}, \tilde{O}$ states. Note however that states $E,O$ states in the original (no SOC) labeling such as $C_{\pmb{k},E,\alpha, \uparrow}, C_{\pmb{k},O,\alpha, \uparrow}$ both belong to the $\tilde{E}$ irreducible representation of the space group with SOC. So in general, the space group symmetry allows the mixture of the $E,O$ states invalidating the unfolding procedure since the two will always correspond to states with shifted momenta $\pmb{k}$ and $\pmb{k}+ \pmb{Q}.$
Also note that the eigenstates of the Hamiltonian in this case are not degenerate along the Y line of the unfolded BZ [@Cvetkovic_Vafek:2013], [@Cracknell_book:1972] (not counting Kramers degeneracy which is irrelevant here). In general, there cannot be an analogous switching between the $\Gamma_3$ and $\Gamma_4$ representations at the edge of the BZ and one must conserve the “parity" under the glide reflection as one crosses into adjacent zones. Therefore for finite SOC and for a general choice of tight-binding parameters,
$$\label{Eq:_NOT_BZ_translation}
C_{\pmb{k},E/O,\alpha}=C_{\pmb{k}+\pmb{Q}, O/E,\alpha}~~,$$
with $\pm i$ eigenstates under the glide-reflection for $\Gamma_3$ and $\Gamma_4$ respectively. This forces us to accept other eigenstates of the glide-reflection such as those in Eqs. \[Eq:SOC\_eigen\_a\], \[Eq:SOC\_eigen\_b\].
The relation above also signals that we cannot choose the simple sublattice superposition states in (\[Eq:Eigen\_1\_a\]),(\[Eq:Eigen\_1\_b\]) and so we cannot connect with the effective extended momentum Fourier transforms in Eqs. \[Eq:Orb\_Fourier\_a\]-\[Eq:Orb\_Fourier\_d\] or the gauge transformed states in (\[Eq:Gauge\_a\]),(\[Eq:Gauge\_b\]). In a practical calculation where the 1-Fe unit cell assumption is of any use we implicitly carry on calculations using the familiar 1-Fe unit Fourier transform defined on $\pmb{\tilde{k}} \in UBZ$ and then fold to the physical 2-Fe unit BZ. But our arguments show that in doing so we are violating space group symmetry and as a consequence we have no guarantee that the results thus obtained coincide with those done directly in the unfolded zone.
As in the zero SOC case, we argue that it is insufficient to justify the validity or invalidity of the 1-Fe unit cell by invoking a general symmetry of the Hamiltonian, namely, the glide reflection. In the SOC case, the space group of the Hamiltonian does not change since the extra coupling does not violate any symmetry of the latter. Our arguments, through which we attempt to treat the glide reflection within its natural space group symmetry perspective, can be used to give a more precise prescription to its use and to illustrate how the formulation of the problem in an 1-Fe unit cell can fail in spite of the validity of the glide reflection as a symmetry of the Hamiltonian.
The effects of the spin-orbit coupling on the normal-state bandstructure {#Sec:SOC_numerics}
------------------------------------------------------------------------
![ (a) Bandstructure of the five-orbital tight-binding model in the folded Brillouin zone (FBZ) without the spin-orbit coupling. (b) The corresponding Fermi surface in two quadrants of the FBZ.[]{data-label="Fig:Bands1"}](SOC000U000J000A5n600 "fig:"){width="80mm"} ![ (a) Bandstructure of the five-orbital tight-binding model in the folded Brillouin zone (FBZ) without the spin-orbit coupling. (b) The corresponding Fermi surface in two quadrants of the FBZ.[]{data-label="Fig:Bands1"}](FSSOC000U000J000A5N600 "fig:"){width="80mm"}
![(a) Bandstructure of the five-orbital tight-binding model in the folded Brillouin zone (FBZ) with a local spin-orbit coupling $\lambda_{\rm{SO}}=0.05$ eV. (b) The corresponding Fermi surface in two quadrants of the FBZ.[]{data-label="Fig:Bands2"}](SOC005U000J000A5n600.pdf "fig:"){width="80mm"} ![(a) Bandstructure of the five-orbital tight-binding model in the folded Brillouin zone (FBZ) with a local spin-orbit coupling $\lambda_{\rm{SO}}=0.05$ eV. (b) The corresponding Fermi surface in two quadrants of the FBZ.[]{data-label="Fig:Bands2"}](FSSOC005U000J000A5N600.pdf "fig:"){width="80mm"}
In light of the discussion on the folding procedure, in this section we study the effects of the spin-orbit coupling on the bandstructure of the normal state. We consider the following Hamiltonian: $H=H_{\rm{TB}} + H_{\rm{SO}}$, where $$\begin{aligned}
H_{\rm{TB}} = \sum_{\mathbf{k}\alpha\beta,\sigma} \epsilon_{\alpha\beta} (\mathbf{k}) C^\dagger_{\mathbf{k}\alpha\sigma} C_{\mathbf{k}\beta\sigma},\end{aligned}$$ is a five-orbital tight-binding model for the parent compound of iron pnictides. Here $\epsilon_{\alpha\beta} (\mathbf{k})$ are tight-binding parameters, which we adopted from Ref. . $H_{\rm{SO}} = \lambda_{\rm{SO}} \sum_{i} \mathbf{L}_i\cdot\mathbf{S}_i$, refers to a local spin-orbit coupling term. As we discussed, this term does not break any symmetry of the lattice, but couples the spatial and spin part of the single-particle wave function.
According to Sec. \[Subsection:Folding\_wo\_SOC\], the tight-binding Hamiltonian defined by the pseudo-crystal momentum $\pmb{\tilde{k}}$ in the UBZ can be transformed to the physical momentum in FBZ via the folding procedure. This allows us to study the effects of the spin-orbit coupling by comparing the bandstructures without and with a spin-orbit coupling in the FBZ. In the absence of the spin-orbit coupling, the bandstructure of the tight-binding model is shown in panels (a) and (b) of Fig. \[Fig:Bands1\]; it is in agreement with that from [*ab initio*]{} calculations using the density functional theory [@Graser:2009]. It is clearly seen that the bands are doubly degenerate along the boundary of the FBZ (Y line from the M to the X point), and as a consequence, the two elliptical electron pockets cross at a point along this direction. As we have emphasized in Sec. \[Subsection:Folding\_wo\_SOC\], this double degeneracy guarantees the successful folding procedure: the wave function can switch representations under the glide when crossing the FBZ boundary, and hence one can define a pseudo-crystal momentum according to the parity of the wave function under the glide. This ensures the equivalence between the models defined in the FBZ and UBZ. When a spin-orbit coupling is turned on, as shown in Fig. \[Fig:Bands2\] (a) and (b), the double degeneracy along the FBZ boundary (M-X direction) is lifted. As we discussed, this invalidates the unfolding procedure since the wave functions have to conserve the parity under glide across the FBZ boundary. As a result of the lifted degeneracy, the two electron pockets no longer cross, but a gap opens along the X-M direction.
Another feature of the bandstructure in the presence of spin-orbit coupling is the opening of hybridization gaps between the $d_{xy}$ and $d_{xz/yz}$ bands along the $\Gamma$-M direction as shown within the dashed ellipses in Fig. \[Fig:Bands2\]. In absence of spin-orbit coupling, these bands simply cross, without opening a gap because they have different parity (pseudo-crystal momenta) and can not mix. But with a finite spin-orbit coupling, the $C_{\mathbf{k},xz/yz,\uparrow}$ and $C_{\mathbf{k+Q},xy,\downarrow}$ have the same parity and they can hybridize via opening a gap. Recently, a hybridization gap between the $d_{xz/yz}$ and $d_{xy}$ bands along the $\Gamma$-M direction has been observed in ARPES measurements on several iron-based compounds [@Yi:2011; @Liu:2015]. It would be interesting to compare the experimental results with theoretical ones as this may provide valuable information about the strength of spin-orbit coupling in these systems. The spin-orbit couplin g also mixes the $d_{xz}$ and $d_{yz}$ orbitals, and lifts the double degeneracy between the $d_{xz}$ and $d_{yz}$ orbitals at $\Gamma$ point. But a local spin-orbit coupling does not lift the degeneracy between the $d_{xz}$ and $d_{xy}$ orbitals at the bottom of the electron bands at M point since it does not break the four-fold rotational symmetry of the P4/nmm group.
The pairing interactions in the presence of spin-orbit coupling {#Sec:Pairing_SOC}
---------------------------------------------------------------
In Sec. \[Sec:Space\_group\_with\_SOC\] we showed that the correspondence between states defined on a 1-Fe BZ and those belonging to the irreducible representations of the space group in the presence of a SOC breaks down even when there are no pairing interactions present. For consistency, here we analyze the direct effect of a local spin-orbit coupling term (Eq. \[Eq:SOC\_term\]) on the pairing interactions of a $t-J$ model.
The projective nature of this model [@Goswami:2009] excludes non-singlet pairing terms. However, the SOC does not conserve the electronic spin quantum number and in this case a $t-J$ model must include ad-hoc triplet pairing terms. In general, the resulting pairing must include terms which are odd under inversion. Based on the discussion at the beginning of Sec. \[Sec:Effects\_pairing\], we expect that the inversion-odd pairing, in the spin triplet case, will also be odd under the glide-reflection and thus correspond to finite-momentum pairs in the UBZ. We note that, as in the case for pairing without SOC, we restrict ourselves to one-dimensional representations of the point group, for which the above is correct. In the case of two-dimensional representations, the opposite holds, with inversion-even functions actually corresponding to glide-reflection odd states.
At mean-field level, the pairing can still be written in the form of Eq. \[Eq:MF\_term\] except for the spin structure allowing both same-spin and opposite spin pairing. More specifically, we use the correspondence in Eqs. \[Eq:Corres\_E\_up\]-\[Eq:Corres\_O\_down\] between the $E,O$ labels without SOC and the $\tilde{E}, \tilde{O}$ ones with SOC turned on to re-write the pairing in Eq. \[Eq:Pairing\_MF\] in terms of the latter. As a consequence, the real-space pairing functions $\Delta^{A}_{\pmb{e},\alpha\beta}, \Delta^{B}_{\pmb{e},\alpha\beta}$ (Eqs. \[Eq:Delta\_A\], \[Eq:Delta\_B\]) acquire a spin-index dependence. The previous argument (without SOC) relied on the transformation properties of the latter term under the glide reflection. We can apply the same procedure in real-space allowing the transformation of the spins. For singlet pairing the $\uparrow, \downarrow$ combination always generate products of $i, -i$, ensuring that $\Delta^{A,B}_{\alpha, \beta}$ have exactly the same properties as before. Applying the transformation (\[Eq:Corres\_E\_up\])-(\[Eq:Corres\_O\_down\]) in reverse, we note that singlet pairing can only have diagonal $E,E$ and $O,O$ pairing (original labeling) and so does not introduce any finite momentum Cooper pairs.
The situation is different when we allow triplet pairing. The same-spin terms always generate a minus sign under the glide reflection. When $\alpha, \beta$ have the same parity this means $\Delta^{A}_{\pmb{e},\alpha\beta}=-\Delta^{B}_{\pmb{e},\alpha\beta}$. In the original labeling, the only terms that survive are the off-diagonal $E,O$. Similarly, when $\alpha, \beta$ have different parity $\Delta^{A}_{\pmb{e},\alpha\beta}=\Delta^{B}_{\pmb{e},\alpha\beta}$ but the $(-1)$ terms coming from the orbital parity guarantee that we again obtain only off diagonal $E,O$ pairs. Now however, the $E,O$ terms will map to $\pmb{\tilde{k}},\pmb{\tilde{k}+ Q}$ in the UBZ producing finite-momentum Cooper pairs and invalidating the unfolding. The spin-symmetric combinations allow forms like those for singlet pairing and thus do not introduce any new terms.
Discussions {#Sec:Discussion}
===========
In this section we elaborate on some further issues related to the space group symmetry and BZ folding we detailed above.
Effects of three-dimensionality vs. spin-orbit coupling
-------------------------------------------------------
Real materials with a $P4/nmm$ space group have a 3D structure ensuring the bands are always dispersive, albeit weakly, along the $k_z$ direction. The group of the wave-vector for arbitrary $k_{z}$ cannot contain the glide reflection since the latter connects the generally inequivalent $k_{z}, -k_{z}$ components. For $k_{z}=0$, the folding can still work as we illustrate below. The dispersion along the $z$-direction can be accounted for by generalizing the 2D ansatze (\[Eq:Eigen\_1\_a\]), (\[Eq:Eigen\_1\_b\]) to
$$\begin{aligned}
C_{\pmb{k},k_{z},E,\alpha} = & \frac{1}{\sqrt{2}} \sum_{R_{z}} e^{ik_{z}R_{z}} \left[ C_{\pmb{k},A,\alpha, R_{z}} + (-1)^{\alpha} C_{\pmb{k},B,\alpha, R_{z}} \right], \label{Eq:Eigen_1_a_z} \\
C_{\pmb{k},k_{z},O,\alpha} = & \frac{1}{\sqrt{2}} \sum_{R_{z}} e^{ik_{z}R_{z}} \left[ C_{\pmb{k},A,\alpha, R_{z}} - (-1)^{\alpha} C_{\pmb{k},B,\alpha, R_{z}} \right], \label{Eq:Eigen_1_b_z}\end{aligned}$$
which for general $k_{z}$ are not eigenstates of $T_{\tau} P_{z}$ and we keep the $E,O$ indices for labeling purposes. Here, $\pmb{k}$ still refers to a purely 2D wave vector. Since for finite $k_{z}$, these states do not necessarily coincide with the irreducible representations of the space group, the Hamiltonian will contain terms mixing $E,O$ indices. For $k_{z}=0$ however, the glide does not affect the Bloch momentum label but it maps $R_{z} \rightarrow -R_{z}$. Due to the equivalence of the different planes, $C_{\pmb{k},A/B,\alpha, R_{z}}=C_{\pmb{k},A/B,\alpha, -R_{z}}$ since the two Wannier states are related by a proper translation. This means in the $k_{z}=0$ plane the states (\[Eq:Eigen\_1\_a\_z\]), (\[Eq:Eigen\_1\_b\_z\]) still coincide with the irreducible representations of the space group and the previous arguments still apply. Therefore, states with two-dimensional crystal momenta can still be mapped onto an effective 1-Fe BZ even when the dispersion along the z-axis is turned on.
Space group symmetry and parity of pairing
------------------------------------------
Recent studies have paid particular attention to the consequences specific to the 2-Fe unit cell such as the use of an additional isospin quantum number [@Ong_Coleman:2013] on the superconducting pairing. But some important issues are still unclear. For example, pairing channels with very different symmetries have been proposed theoretically, [@Ong_Coleman:2013; @Hu:2013] although the target compounds have the same spatial symmetry. As we discussed earlier in our paper, the space group already imposes strong constraints on the parity of symmetry-compatible pairing. This provides a means to check whether a pairing channel is allowed by the space group symmetry of the system. Here we examine this issue for two recently proposed pairing channels, the TAO pairing [@Ong_Coleman:2013] and the $\eta$-pairing [@Hu:2013]. As before, we focus on spin-singlet pairings in one-dimensional representations henceforth.
### TAO pairing
Discussed in Ref. , the TAO pairing refers to a spin singlet, orbital triplet $A_{1g}$ $d_{x^2-y^2}+id_{xy}$ pairing in the 2-Fe BZ. It is easy to check that this pairing channel has an even parity and is compatible with the $P4/nmm$ space symmetry. From Sec. \[Sec:Effects\_pairing\] we know that for a 2D dispersion of the conduction electrons and in the absence of spin-orbit coupling, any pairing defined on the 2-Fe unit cell FBZ which respects the space group symmetry must have an equivalent, albeit given by a different linear combination of channels, $(\pmb{\tilde{k}},-\pmb{\tilde{k}})$ pairing in the 1-Fe unit cell UBZ. The aforementioned TAO pairing defined in the 2-Fe BZ $(\pmb{k},-\pmb{k})$ is equivalent to one defined in the 1-Fe UBZ and does not incorporate any particular properties of the 2-Fe BZ not captured by the former. A possible advantage of the direct 2-Fe unit cell formulation might consist in expounding physical features which might be harder to illustrate in an equivalent 1-Fe unit cell picture.
### $\eta$-pairing
Recent discussions have also considered the $\eta$-pairing in the iron-based superconductors [@Hu:2013; @Hu_Hao_Wu:2013; @Lin_Ku:2014; @Wang_Maier:2014]. In its original proposal, [@Hu:2013] the $\eta$-pairing refers to a singlet pairing of two electrons with pseudo-crystal momenta $\pmb{\tilde{k}}$ and $-\pmb{\tilde{k}+Q}$, respectively. This pairing has nonzero total pseudo-crystal momentum in the 1-Fe UBZ, and the momentum dependent part of the wavefunction has odd parity under inversion. This proposal was based on the observation that the inversion center in an Fe plane lies half-way in between two (inequivalent) sites. In a real space basis, the inversion operation interchanges the two positions on the different sublattices. In a simpler case where the two lattices are equivalent, the antisymmetry under exchange of the pairing wavefunction forces the spatial ($\pmb{k}$-dependent) part to be even under inversion. The existence of the two inequivalent sites opens up the possibility of odd parity momentum dependence since the overall spatial part could be described as a direct product of a purely $\pmb{k}$-dependent part (in the FBZ, 2-Fe unit cell description) and a pseudo-spin matrix which captures the effect of the different sublattices. The purely $\pmb{k}$-dependent part can have odd parity under inversion as long as the remaining degrees of freedom compensate with a minus sign such that the total inversion parity is still even. Although not explicitly stated in Ref. , such a state could correspond to a $E,O$ pseudo-spin singlet. Upon unfolding, the combination will generate the finite momentum $(\pmb{\tilde{k}},-\pmb{\tilde{k}+Q})$ pairing. As already shown in Sec. \[Sec:Effects\_pairing\], for states of definite parity under inversion, or equivalently, of definite parity under the glide-reflection, the pairing can be written in general as a linear combination of terms with and without $E,O$ mixing together with their respective momentum-dependent parts. In addition, if we do not break the glide-reflection symmetry, the total pairing must be even. In a purely two dimensional BZ, the glide-reflection cannot change any of the momenta of the pair, such that terms with $E,O$ mixing change sign under the operation while those without do not. Indeed, as detailed in our Sec. \[Sec:Effects\_pairing\] , all mixing terms must vanish, ensuring that upon unfolding no finite momentum pairs can be generated. We stress that this conclusion holds whenever inversion is not spontaneously broken for a two dimensional BZ.
In recent studies on the $\eta$-pairing, there also seems to be confusion in the definition for this type of pairing in the literature which, we believe, is associated with different subsequent definitions of the 1-Fe BZ. While in the original work the $\eta$-pairing referred to a $(\pmb{\tilde{k}},-\pmb{\tilde{k}+Q})$ state, in more recent studies, it is associated with a $(\pmb{\slashed{k}},-\pmb{\slashed{k}+Q})$ pairing in the so-called “physical extended BZ“ [@Lin_Ku:2014; @Wang_Maier:2014] given by a mapping similar to the one in Eqs. \[Eq:Orb\_Fourier\_a\]-\[Eq:Conventioanal\_1\]. As in the original proposal [@Hu:2013], Ref. alludes to the inequivalence of the two sublattice sites but does not attempt to explicitly consider eigenstates of the glide reflection operation. Rather, the authors classify 2D states both in the ”physical“ ($\pmb{\slashed{k}}$ in our convention) and unfolded ($\pmb{\tilde{k}}$) representations according to the parity under the reflection about the z-plane. In the physical representation, an alternating minus sign in the hopping between different orbital parity Wannier states is ”absorbed" into the definition of the quasi-crystal momentum, resulting in a shift by $\pmb{Q}$ between the even and odd reflection parity states, as detailed in our Sec. \[Subsection:Folding\_wo\_SOC\]. The authors argue that a zero mometum pair in the UBZ ($\pmb{\tilde{k}}, - \pmb{\tilde{k}})$ or arbitrary parity under *the z-reflection alone* must decompose into a linear combination of even/even, odd/odd and even/odd terms which correspond to $(\pmb{\slashed{k}}+ \pmb{Q},
- \pmb{\slashed{k}}-\pmb{Q})$, $(\pmb{\slashed{k}}, - \pmb{\slashed{k}})$ and $(\pmb{\slashed{k}}+ \pmb{Q}, - \pmb{\slashed{k}} )$ respectively in the physical BZ. From this, they seem to explain inconsistencies in the spectral weights in the superconducting state between calculations done on a 1-Fe/UBZ ($\pmb{\tilde{k}}$) and subsequently folded down and ARPES experiments among others whose results are naturally obtained in a 2-Fe/FBZ.
In the context of a glide reflection symmetry, the two constructions referring to a finite momentum $(\pmb{\tilde{k}},-\pmb{\tilde{k}+Q})$ in the UBZ [@Hu:2013] and in the “physical” $(\pmb{\slashed{k}},-\pmb{\slashed{k}+Q})$ [@Lin_Ku:2014; @Wang_Maier:2014] are different. The $(\pmb{\tilde{k}},-\pmb{\tilde{k}+Q})$ pairing has an odd parity under glide reflection, while the $(\pmb{\slashed{k}},-\pmb{\slashed{k}+Q})$ -pairing is parity even under the glide. To see this, consider the latter (“physical” representation) case which corresponds to an odd/even parity combinations under *the pure reflection*. In our language, this corresponds to states given in Eqs. \[Eq:Orb\_Fourier\_a\]-\[Eq:Orb\_Fourier\_d\] with $\alpha$ odd and even respectively. It is not difficult to see that upon converting to the “true” 2-Fe/FBZ ($\pmb{k}$) this term corresponds to ($\pmb{k}, -\pmb{k}), O/O$ under the *glide-reflection* pairs. To the best of our knowledge, this point has not been clarified before. According to our analysis, the only symmetry-allowed pairing is the even glide parity one, *i.e.*, the $(\pmb{\tilde{k}},-\pmb{\tilde{k}})$ pairing (or equivalently, the $(\pmb{\slashed{k}},-\pmb{\slashed{k}+Q})$ pairing). It corresponds to normal zero-momentum pairing in both the 2-Fe FBZ ($\pmb{k}$) or the 1-Fe UBZ in pseudo-crystal momentum space $ (\pmb{\tilde{k}}) $. More precisely, the three formulations alluded to above are equivalent and we see no reason why 1-Fe unit cell/UBZ calculations which are folded down could not *a priori* capture experimental results. This correspondence naturally explains why the superconducting gap functions obtained from calculations in the 2-Fe FBZ are identical to the previous results in the 1-Fe UBZ [@Wang_Maier:2014]. Within our approach, we have also shown that the odd glide parity $(\pmb{\tilde{k}},-\pmb{\tilde{k}+Q})$ pairing is not allowed by symmetry.
Conclusions {#Sec:Conclusion}
===========
The glide reflection symmetry is valid for states of arbitrary momentum and without spin-orbit coupling there is a minimal double degeneracy all along the 2-Fe unit cell BZ edge. This ensures that a tight-binding Hamiltonian can be determined using an unfolded BZ corresponding to a 1-Fe unit cell. By contrast, although the glide symmetry still holds for arbitrary momentum when spin-orbit coupling is turned on, the latter mixes states corresponding to different pseudo-momenta in the unfolded BZ and lifts the degeneracy along the $Y$ line, forbidding the general use of the same unfolding procedure. These conclusions are consistent with bandstructure calculations with and without spin-orbit coupling which show the lack of and the presence of this hybridization.
We also conclude that for a $t-J$ type Hamiltonian without spin-orbit coupling the results obtained directly from a 1-Fe unit cell should coincide with those from 2-Fe unit cell, the two being related by the validity of the unfolding procedure. This applies to the TAO pairing [@Ong_Coleman:2013]: Though proposed in the 2-Fe BZ, it is equivalent to a $d_{x^2-y^2}+id_{xy}$ pairing with both intra- and inter-orbital contributions. One more remark is that this equivalence does not hold when a spin-orbit coupling term is included.
Another conclusion from our symmetry analysis is that the pairing channel compatible to the $P4/nmm$ space group symmetry must have an even parity (once again,we focus on spin-singlet pairings in one-dimensional representations). With this criterion, the $\eta$-pairing with $(\pmb{\tilde{k}},-\pmb{\tilde{k}}+Q)$, originally proposed in Ref. [@Hu:2013] is not symmetry-allowed since it is parity odd. The $\eta$-pairing discussed in most recent works [@Lin_Ku:2014; @Wang_Maier:2014], on the other hand, refers to a $(\pmb{\slashed{k}},-\pmb{\slashed{k}+Q})$ pairing, which corresponds to a total momentum zero $(\pmb{\tilde{k}},-\pmb{\tilde{k}})$ pairing with an even parity, and is thus compatible with the $P4/nmm$ space group symmetry.
Our analysis for the folding within a $P4/nmm$ space group symmetry can serve as a comparison point for a similar discussion in the more involved $I4/mmm$ case. As we stressed throughout the text, the validity of the glide-symmetry, together with the fortuitous double degeneracy along the edge of the FBZ guarantee the success of the folding for $k_{z}=0$ and no SOC. Since none of these appear to be valid for $I4/mmm$, the folding will probably not work. A rigorous analysis of this latter case is reserved for a future publication.
[*Acknowledgements. *]{} This work has been supported by the NSF Grant No. DMR-1309531 and the Robert A. Welch Foundation Grant No. C-1411 (E.M.N. & Q.S.). R.Y. was partially supported by the National Science Foundation of China Grant number 11374361, and the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China. All of us acknowledge the support provided in part by the NSF Grant No. NSF PHY11-25915 at KITP, UCSB, for our participation in the Fall 2014 program on “Magnetism, Bad Metals and Superconductivity: Iron Pnictides and Beyond". Q.S. also acknowledges the hospitality of the Institute of Physics of Chinese Academy of Sciences.
Eigenstates of the glide operation {#Sec:Appendix_A}
==================================
The irreducible representations of the $P_{4/nmm}$ space group for general $\pmb{k} \in FBZ$ and for the special loci $\Gamma$, $\Delta$, $\Sigma$ and $M$ [@Cvetkovic_Vafek:2013] transform as
$$\begin{aligned}
\label{Eq:Glide}
(T_{ \pmb { \tau } } P_z) C_{\pmb{k}\alpha} = e^{ i \pmb{k} \pmb{\tau}} \lambda C_{\pmb{k}\alpha},\end{aligned}$$
where $\lambda= \pm 1$. The above form is not the case for $X$ and $Y$. However, since the Hamiltonian must evolve continously with $\pmb{k}$ it is clear that we can form irreducible representations at the two above mentioned loci by taking linear combinations of even ($E$) only or odd only ($O$) such that our arguments are not affected.
We can derive the general form of states which transform according to (\[Eq:Glide\]) from a general superposition of operators defined on each sublattice
$$\begin{aligned}
\label{Eq:general_form}
C_{\pmb{k}\alpha}= & \frac{1}{\sqrt{N}} \left[\sum_{\pmb{R_A}}e^{i\pmb{k} \cdot \pmb{R_A}} C^{(A)}_{\pmb{R_A}\alpha} + \sum_{\pmb{R_B}} e^{i\theta_{\pmb{k}}}e^{i\pmb{k}\pmb{R_{B}}}C^{(B)}_{\pmb{R_B} \alpha} \right] \notag \\
= & \frac{1}{\sqrt{N}} \sum_{\pmb{R_A}} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ C^{(A)}_{\pmb{R_A}\alpha} + e^{i\theta_{\pmb{k}}}e^{i\pmb{k} \pmb{\tau} }C^{(B)}_{\pmb{R_A}+\pmb{\tau} \alpha} \right]\end{aligned}$$
where $\pmb{R}_{A}$, $\pmb{R}_{B}$ are summations over the position vectors of the respective sublattices, and $\alpha$ stands for the parity of the local degrees of freedom under a pure reflection $\sigma_z$. We stress that, in the most general case, the completely local states can be different ($C^{(A)}_{\pmb{r} \alpha} \neq C^{(B)}_{\pmb{r}\alpha}$) due to the physical inequivalence of the two sublattice sites. Although in principle the linear combinations of the two sublattice Bloch states can have arbitrary complex coefficients, the form chosen above is sufficient for our purposes.
Applying the glide reflection to the trial state (\[Eq:general\_form\]) and imposing (\[Eq:general\_form\]) we get
$$\begin{aligned}
\label{Eq:Transformed_state}
(T_{ \pmb { \tau } } P_z) C_{\pmb{k}\alpha}= \lambda e^{ i \pmb{k} \pmb{\tau}} \frac{1}{\sqrt{N}} \sum_{R_A} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ (-1)^\alpha C^{(A)}_{\pmb{R_A}+\pmb{\tau}\alpha} + e^{i\tilde{\theta_{\pmb{k}}}}e^{i\pmb{k}\pmb{\tau}} (-1)^\alpha C^{(B)}_{\pmb{R_A} +2\pmb{\tau} \alpha} \right]\end{aligned}$$
such that all position vectors get shifted by the fractional translation $\pmb{\tau}$, the local states generate the parity term $\alpha$, and the phase $\theta_{ \pmb{ k } } \rightarrow \tilde{ \theta_{\pmb{k}} } $ in general. Comparing (\[Eq:general\_form\]) and (\[Eq:Transformed\_state\]) we see there are a number of possibilities. In general, condition (\[Eq:Transformed\_state\]) cannot determine all the unknowns i.e. the phase factor and the relation between the the two displaced local states. We can connect with a folding procedure by making some assumptions regarding the phase and letting the above condition determine the local states.
A first possibility corresponds to taking $\lambda=1$ in (\[Eq:Transformed\_state\])
$$\begin{aligned}
\label{Eq:1_poss_1}
C^{(A)}_{\pmb{R_A}\alpha}= & (-1)^\alpha C^{(A)}_{\pmb{R_A}+\pmb{\tau}\alpha} \\
C^{(B)}_{\pmb{R_A} + \pmb{\tau}\alpha}= & (-1)^\alpha C^{(B)}_{\pmb{R_A}+2\pmb{\tau}\alpha} \\
\theta_{ \pmb{ k } } = & \tilde{ \theta_{\pmb{k}} }.\end{aligned}$$
For $\lambda=-1$ we can have
$$\begin{aligned}
\label{Eq:1_poss_2}
C^{(A)}_{\pmb{R_A}\alpha}= & - (-1)^{\alpha} C^{(A)}_{\pmb{R_A}+\pmb{\tau}\alpha} \\
C^{(B)}_{\pmb{R_A} + \pmb{\tau}\alpha}= & - (-1)^{\alpha} C^{(B)}_{\pmb{R_A}+2\pmb{\tau}\alpha} \\
\theta_{ \pmb{ k } } = & \tilde{ \theta_{\pmb{k}} }.\end{aligned}$$
and we set $\theta_{ \pmb{ k } }=0$. The conditions above are simply the transformation properties of the local states under a simple 1-Fe unit cell translation. If we choose the states at corresponding to $C_{A}$ and $C_{B}$ to have the same functional form, we can construct two distinct eigenstates of the glide by for arbitrary $\alpha$ :
$$\begin{aligned}
C_{\pmb{k},E,\alpha}= & \frac{1}{\sqrt{N}} \sum_{\pmb{R_A}} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ C_{\pmb{R_A}\alpha 1} + (-1)^{\alpha} e^{i\pmb{k} \pmb{\tau} }C_{\pmb{R_A}+\pmb{\tau} \alpha } \right] \label{Eq:App_state_a} \\
C_{\pmb{k},O,\alpha2}= & \frac{1}{\sqrt{N}} \sum_{\pmb{R_A}} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ C_{\pmb{R_A}\alpha} -(-1)^{\alpha} e^{i\pmb{k} \pmb{\tau} }C_{\pmb{R_A}+\pmb{\tau} \alpha} \right] \label{Eq:App_state_b},\end{aligned}$$
where we omitted the $A$,$B$ superscripts which are now irrelevant. Eqs. \[Eq:App\_state\_a\] and \[Eq:App\_state\_b\] are the operators in (\[Eq:Eigen\_1\_a\]) and (\[Eq:Eigen\_1\_b\]).
Note that in addition to the above states which allowed the unfolding we can also choose eigenstates of the glide reflection by imposing
$$\begin{aligned}
C^{(A)}_{\pmb{R_A}\alpha}= & \pm (-1)^{\alpha} e^{i\tilde{\theta_{\pmb{k}}}}e^{i\pmb{k}\pmb{\tau}} C^{(B)}_{\pmb{R_A}+2\pmb{\tau}\alpha} \\
e^{i\theta_{\pmb{k}}}e^{i\pmb{k} \pmb{\tau} }C^{(B)}_{\pmb{R_A}+\pmb{\tau} \alpha}= & \pm (-1)^{\alpha} C^{(A)}_{\pmb{R_A}+\pmb{\tau}\alpha}.\end{aligned}$$
We can choose the phase factor $e^{i \theta_{ \pmb{ k } }} = e^{i \tilde{ \theta_{\pmb{k}} }}= e^{-i\pmb{k\tau}}$ and disregard the $A,B$ distinction as in the first choice. The two sublattices are now related by a $\pmb{k}$-dependent phase rather than a simple sign. The initial ansatz (\[Eq:general\_form\]) becomes
$$\begin{aligned}
C_{\pmb{k},E, \alpha}= & \sqrt{\frac{1}{N}} \sum_{\pmb{R_A}} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ C_{\pmb{R_A}\alpha} +(-1)^{\alpha} C_{\pmb{R_A}+\pmb{\tau} \alpha} \right] \label{Eq:SOC_eigen_a} \\
C_{\pmb{k},O,\alpha}= & \sqrt{\frac{1}{N}} \sum_{\pmb{R_A}} e^{i\pmb{k} \cdot \pmb{R_A}} \left[ C_{\pmb{R_A}\alpha} - (-1)^{\alpha} C_{\pmb{R_A}+\pmb{\tau} \alpha} \right] \label{Eq:SOC_eigen_b}\end{aligned}$$
which clearly do not correspond to a 1-Fe unit cell.
The Hamiltonian in the physical extended momentum basis {#Sec:Appendix_B}
=======================================================
Direct substitution of the definitions (\[Eq:Orb\_Fourier\_a\])-(\[Eq:Orb\_Fourier\_d\]) into the 2-Fe BZ Hamiltonian (\[Eq:2\_Fe\_Hamiltonian\]) gives
$$\begin{aligned}
H_{TB}= & \sum_{\pmb{k} \in FBZ} \Bigg[ \sum_{ee} \left( \epsilon^{E}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}, \alpha} C_{\pmb{k}, \beta} + \epsilon^{O}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}+ \pmb{Q}, \alpha} C_{\pmb{k}+\pmb{Q}, \beta} \right) + \sum_{oo} \left( \epsilon^{E}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}+\pmb{Q}, \alpha} C_{\pmb{k}+\pmb{Q}, \beta} + \epsilon^{O}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}, \alpha} C_{\pmb{k}, \beta} \right) \nonumber \\
& + \sum_{eo} \left( \epsilon^{E}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}, \alpha} C_{\pmb{k}+\pmb{Q}, \beta} + \epsilon^{O}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}+\pmb{Q}, \alpha} C_{\pmb{k}, \beta} \right) + \sum_{oe} \left( \epsilon^{E}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}+\pmb{Q}, \alpha} C_{\pmb{k}, \beta} + \epsilon^{O}_{\alpha \beta}(\pmb{k}) C^{\dagger}_{\pmb{k}, \alpha} C_{\pmb{k}+\pmb{Q}, \beta} \right) \Bigg].\end{aligned}$$
We can use the fact that $\epsilon^{E(O)}_{\alpha \beta}(\pmb{k}+\pmb{Q})= \epsilon^{O(E)}_{\alpha \beta}(\pmb{k})$ to rewrite the first two terms as a total sum over $\pmb{\slashed{k}}$ over the unfolded BZ. We can also show that the third term can be expressed as a sum over $\pmb{\slashed{k}}$. Specifically,
$$\begin{aligned}
\sum_{\pmb{\slashed{k}} \in UBZ} \epsilon^{E}_{\alpha \beta} (\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}, \alpha \beta}C_{\pmb{\slashed{k}}+\pmb{Q}, \alpha \beta}= & \sum_{\pmb{\slashed{k}} \in FBZ} \epsilon^{E}_{\alpha \beta} (\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}, \alpha \beta}C_{\pmb{\slashed{k}}+\pmb{Q}, \alpha \beta} + \sum_{\pmb{\slashed{k}} \notin FBZ} \epsilon^{E}_{\alpha \beta} (\pmb{\slashed{k}}) C^{\dagger}_{\pmb{\slashed{k}}, \alpha \beta}C_{\pmb{\slashed{k}}+\pmb{Q}, \alpha \beta} \\
= & \sum_{\pmb{k} \in FBZ} \epsilon^{E}_{\alpha \beta} (\pmb{k}) C^{\dagger}_{\pmb{k}, \alpha \beta}C_{\pmb{k}+\pmb{Q}, \alpha \beta} + \sum_{\pmb{k} \in FBZ} \epsilon^{E}_{\alpha \beta} (\pmb{k+\pmb{Q}}) C^{\dagger}_{\pmb{k}+ \pmb{Q}, \alpha \beta}C_{\pmb{k}+2\pmb{Q}, \alpha \beta}\end{aligned}$$
which after recognizing that $C_{\pmb{k}+2\pmb{Q}}= C_{\pmb{k}}$ gives the form in Eq. \[Eq:Conventioanal\_1\].
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|
---
author:
- |
Matthias Neubert[^1]\
Theory Division, CERN, CH-1211 Geneva 23, Switzerland\
E-mail:
title: ' Model-Independent Analysis of $B\to\pi K$ Decays and Bounds on the Weak Phase $\gamma$ '
---
Introduction
============
The CLEO Collaboration has recently reported the observation of some rare two-body decays of the type $B\to\pi K$, as well as interesting upper bounds for the decays $B\to\pi\pi$ and $B\to
K\bar K$ [@CLEO]. In particular, they find the CP-averaged branching ratios $$\begin{aligned}
\frac 12 \Big[ \mbox{Br}(B^0\to\pi^- K^+)
+ \mbox{Br}(\bar B^0\to\pi^+ K^-) \Big]
&=& (1.4\pm 0.3\pm 0.1)\times 10^{-5} \,, \nonumber\\
\frac 12 \Big[ \mbox{Br}(B^+\to\pi^+ K^0)
+ \mbox{Br}(B^-\to\pi^-\bar K^0) \Big]
&=& (1.4\pm 0.5\pm 0.2)\times 10^{-5} \,, \nonumber\\
\frac 12 \Big[ \mbox{Br}(B^+\to\pi^0 K^+)
+ \mbox{Br}(B^-\to\pi^0 K^-) \Big]
&=& (1.5\pm 0.4\pm 0.3)\times 10^{-5} \,.
\label{CLEOvals}\end{aligned}$$ This observation caused a lot of excitement, because these decays offer interesting insights into the relative strength of various contributions to the decay amplitudes, whose interference can lead to CP asymmetries in the decay rates. It indeed appears that there may be potentially large interference effects, depending on the magnitude of some strong interaction phases (see, e.g., [@GrRo]). Thus, although at present only measurements of CP-averaged branching ratios have been reported, the prospects are good for observing direct CP violation in some of the $B\to\pi K$ or $B\to K\bar K$ decay modes in the near future.
It is fascinating that some information on CP-violating parameters can be extracted even without observing a single CP asymmetry, from measurements of CP-averaged branching ratios alone. This information concerns the angle $\gamma$ of the so-called unitarity triangle, defined as $\gamma=\mbox{arg}[(V_{ub}^* V_{ud})/(V_{cb}^* V_{cd})]$. With the standard phase conventions for the Cabibbo–Kobayashi–Maskawa (CKM) matrix, $\gamma=
\mbox{arg}(V_{ub}^*)$ to excellent accuracy. There have been proposals for deriving bounds on $\gamma$ from measurements of the ratios $$\begin{aligned}
R &=& \frac{\tau(B^+)}{\tau(B^0)}\,
\frac{\mbox{Br}(B^0\to\pi^- K^+)+\mbox{Br}(\bar B^0\to\pi^+ K^-)}
{\mbox{Br}(B^+\to\pi^+ K^0)+\mbox{Br}(B^-\to\pi^-\bar K^0)}
\,, \nonumber\\
R_* &=&
\frac{\mbox{Br}(B^+\to\pi^+ K^0)+\mbox{Br}(B^-\to\pi^-\bar K^0)}
{2[\mbox{Br}(B^+\to\pi^0 K^+)+\mbox{Br}(B^-\to\pi^0 K^-)]} \,,
\label{Rdef}\end{aligned}$$ whose current experimental values are $R=1.07\pm 0.45$ (we use $\tau(B^+)/\tau(B^0)=1.07\pm 0.03$) and $R_*=0.47\pm 0.24$. The Fleischer–Mannel bound $R\ge\sin^2\!\gamma$ [@FM] excludes values around $|\gamma|=90^\circ$ provided that $R<1$. However, this bound is subject to theoretical uncertainties arising from electroweak penguin contributions and strong rescattering effects, which are difficult to quantify [@BFM98]–[@Robert]. The bound $$1-\sqrt{R_*} \le \bar\varepsilon_{3/2}\,
|\delta_{\rm EW} - \cos\gamma| + O(\bar\varepsilon_{3/2}^2)
\label{ourb}$$ derived by Rosner and the present author [@us], where $\delta_{\rm EW}=0.64\pm 0.15$ accounts for electroweak penguin contributions, is less affected by such uncertainties; however, it relies on an expansion in the small parameter $$\bar\varepsilon_{3/2} = \sqrt 2\,R_{\rm SU(3)} \tan\theta_C \left[
\frac{\mbox{Br}(B^+\to\pi^+\pi^0) + \mbox{Br}(B^-\to\pi^-\pi^0)}
{\mbox{Br}(B^+\to\pi^+ K^0) + \mbox{Br}(B^-\to\pi^-\bar K^0)}
\right]^{1/2} \,,
\label{epsexp}$$ whose value has been estimated to be $\bar\varepsilon_{3/2}
=0.24\pm 0.06$. Here $\theta_C$ is the Cabibbo angle, and the factor $R_{\rm SU(3)}\simeq f_K/f_\pi$ accounts for SU(3)-breaking corrections. Assuming the smallness of certain rescattering effects, higher-order terms in the expansion in $\bar\varepsilon_{3/2}$ can be shown to strengthen the bound (\[ourb\]) provided that the value of $R_*$ is not much larger than indicated by current data, i.e., if $R_*<(1-\bar\varepsilon_{3/2}/\sqrt 2)^2\approx 0.7$ [@us].
Our main goal in the present work is to address the question to what extent these bounds can be affected by hadronic uncertainties such as final-state rescattering effects, and whether the theoretical assumptions underlying them are justified. To this end, we perform a general analysis of the various $B\to\pi K$ decay modes, pointing out where theoretical information from isopsin and SU(3) flavour symmetries can be used to eliminate hadronic uncertainties. Our approach will be to vary parameters not constrained by theory (strong-interaction phases, in particular) within conservative ranges so as to obtain a model-independent description of the decay amplitudes. An analysis pursuing a similar goal has recently been presented by Buras and Fleischer [@BFnew]. Where appropriate, we will point out the relations of our work with theirs and provide a translation of notations. We stress, however, that although we take a similar starting point, some of our conclusions will be rather different from the ones reached in their work.
In Section \[sec:2\], we present a general parametrization of the various isospin amplitudes relevant to $B\to\pi K$ decays and discuss theoretical constraints resulting from flavour symmetries of the strong interactions and the structure of the low-energy effective weak Hamiltonian. We summarize model-independent results derived recently for the electroweak penguin contributions to the isovector part of the effective Hamiltonian [@Ne97; @Robert] and point out constraints on certain rescattering contributions resulting from $B\to K\bar K$ decays [@Fa97; @Robert; @GRrescat; @He]. The main results of this analysis are presented in Section \[subsec:numerics\], which contains numerical predictions for the various parameters entering our parametrization of the decay amplitudes. The remainder of the paper deals with phenomenological applications of these results. In Section \[sec:3\], we discuss corrections to the Fleischer–Mannel bound resulting from final-state rescattering and electroweak penguin contributions. In Section \[sec:4\], we show how to include rescattering effects to the bound (\[ourb\]) at higher orders in the expansion in $\bar\varepsilon_{3/2}$. Detailed predictions for the direct CP asymmetries in the various $B\to\pi K$ decay modes are presented in Section \[sec:5\], where we also present a prediction for the CP-averaged $B^0\to\pi^0 K^0$ branching ratio, for which at present only an upper limit exists. In Section \[sec:6\], we discuss how the weak phase $\gamma$, along with a strong-interaction phase difference $\phi$, can be determined from measurements of the ratio $R_*$ and of the direct CP asymmetries in the decays $B^\pm\to\pi^0 K^\pm$ and $B^\pm\to\pi^\pm K^0$ (here $K^0$ means $K^0$ or $\bar K^0$, as appropriate). This generalizes a method proposed in [@us2] to include rescattering corrections to the $B^\pm\to\pi^\pm K^0$ decay amplitudes. Section \[sec:7\] contains a summary of our result and the conclusions.
Isospin decomposition {#sec:2}
=====================
Preliminaries
-------------
The effective weak Hamiltonian relevant to the decays $B\to\pi K$ is [@Heff] $${\mathcal H} = \frac{G_F}{\sqrt 2}\,\bigg\{ \sum_{i=1,2} C_i \Big(
\lambda_u\,Q_i^u + \lambda_c\,Q_i^c \Big) - \lambda_t
\sum_{i=3}^{10} C_i\,Q_i \bigg\} + \mbox{h.c.} \,,$$ where $\lambda_q=V_{qb}^* V_{qs}$ are products of CKM matrix elements, $C_i$ are Wilson coefficients, and $Q_i$ are local four-quark operators. Relevant to our discussion are the isospin quantum numbers of these operators. The current–current operators $Q_{1,2}^u\sim\bar b s\bar u u$ have components with $\Delta I=0$ and $\Delta I=1$; the current–current operators $Q_{1,2}^c\sim\bar b
s\bar c c$ and the QCD penguin operators $Q_{3,\dots,6} \sim \bar b
s\sum\bar q q$ have $\Delta I=0$; the electroweak penguin operators $Q_{7,\dots,10}\sim\bar b s \sum e_q\bar q q$, where $e_q$ are the electric charges of the quarks, have $\Delta I=0$ and $\Delta
I=1$. Since the initial $B$ meson has $I=\frac 12$ and the final states $(\pi K)$ can be decomposed into components with $I=\frac 12$ and $I=\frac 32$, the physical $B\to\pi K$ decay amplitudes can be described in terms of three isospin amplitudes. They are called $B_{1/2}$, $A_{1/2}$, and $A_{3/2}$ referring, respectively, to $\Delta I=0$ with $I_{\pi K}=\frac 12$, $\Delta I=1$ with $I_{\pi
K}=\frac 12$, and $\Delta I=1$ with $I_{\pi K}=\frac 32$ [@Ne97; @Gron; @NQ]. The resulting expressions for the decay amplitudes are $$\begin{aligned}
{\mathcal A}(B^+\to\pi^+ K^0)
&=& B_{1/2} + A_{1/2} + A_{3/2} \,, \nonumber\\
- \sqrt 2\,{\mathcal A}(B^+\to\pi^0 K^+)
&=& B_{1/2} + A_{1/2} - 2 A_{3/2} \,, \nonumber\\
- {\mathcal A}(B^0\to\pi^- K^+)
&=& B_{1/2} - A_{1/2} - A_{3/2} \,, \nonumber\\
\sqrt 2\,{\mathcal A}(B^0\to\pi^0 K^0)
&=& B_{1/2} - A_{1/2} + 2 A_{3/2} \,.
\label{isodec}\end{aligned}$$ From the isospin decomposition of the effective Hamiltonian it is obvious which operator matrix elements and weak phases enter the various isospin amplitudes. Experimental data as well as theoretical expectations indicate that the amplitude $B_{1/2}$, which includes the contributions of the QCD penguin operators, is significantly larger than the amplitudes $A_{1/2}$ and $A_{3/2}$ [@GrRo; @Ne97]. Yet, the fact that $A_{1/2}$ and $A_{3/2}$ are different from zero is responsible for the deviations of the ratios $R$ and $R_*$ in (\[Rdef\]) from 1.
Because of the unitarity relation $\lambda_u+\lambda_c+\lambda_t=0$ there are two independent CKM parameters entering the decay amplitudes, which we choose to be[^2] $-\lambda_c=e^{i\pi}|\lambda_c|$ and $\lambda_u=e^{i\gamma}|\lambda_u|$. Each of the three isospin amplitudes receives contributions proportional to both weak phases. In total, there are thus five independent strong-interaction phase differences (an overall phase is irrelevant) and six independent real amplitudes, leaving as many as eleven hadronic parameters. Even perfect measurements of the eight branching ratios for the various $B\to\pi K$ decay modes and their CP conjugates would not suffice to determine these parameters. Facing this problem, previous authors have often relied on some theoretical prejudice about the relative importance of various parameters. For instance, in the invariant SU(3)-amplitude approach based on flavour-flow topologies [@Chau; @ampl], the isospin amplitudes are expressed as linear combinations of a QCD penguin amplitude $P$, a tree amplitude $T$, a colour-suppressed tree amplitude $C$, an annihilation amplitide $A$, an electroweak penguin amplitude $P_{\rm EW}$, and a colour-suppressed electroweak penguin amplitude $P_{\rm EW}^C$, which are expected to obey the following hierarchy: $|P|\gg |T|\sim |P_{\rm EW}|\gg |C|\sim |P_{\rm EW}^C|
>|A|$. These naive expectations could be upset, however, if strong final-state rescattering effects would turn out to be important [@Ge97]–[@At97], a possibility which at present is still under debate. Whereas the colour-transparency argument [@Bj] suggests that final-state interactions are small in $B$ decays into a pair of light mesons, the opposite behaviour is exhibited in a model based on Regge phenomenology [@Dono]. For comparison, we note that in the decays $B\to D^{(*)} h$, with $h=\pi$ or $\rho$, the final-state phase differences between the $I=\frac 12$ and $I=\frac 32$ isospin amplitudes are found to be smaller than $30^\circ$–$50^\circ$ [@Stech].
Here we follow a different strategy, making maximal use of theoretical constraints derived using flavour symmetries and the knowledge of the effective weak Hamiltonian in the Standard Model. These constraints help simplifying the isospin amplitude $A_{3/2}$, for which the two contributions with different weak phases turn out to have the same strong-interaction phase (to an excellent approximation) and magnitudes that can be determined without encountering large hadronic uncertainties [@us]. Theoretical uncertainties enter only at the level of SU(3)-breaking corrections, which can be accounted for using the generalized factorization approximation [@Stech]. Effectively, these simplifications remove three parameters (one phase and two magnitudes) from the list of unknown hadronic quantities. There is at present no other clean theoretical information about the remaining parameters, although some constraints can be derived using measurements of the branching ratios for the decays $B^\pm\to K^\pm\bar K^0$ and invoking SU(3) symmetry [@Fa97; @Robert; @GRrescat; @He]. Nevertheless, interesting insights can be gained by fully exploiting the available information on $A_{3/2}$.
Before discussing this in more detail, it is instructive to introduce certain linear combinations of the isospin amplitudes, which we define as $$\begin{aligned}
B_{1/2} + A_{1/2} + A_{3/2} &=& P + A - \frac 13 P_{\rm EW}^C
\,, \nonumber\\
-3 A_{3/2} &=& T + C + P_{\rm EW} + P_{\rm EW}^C \,, \nonumber\\
-2(A_{1/2} + A_{3/2}) &=& T - A + P_{\rm EW}^C \,.
\label{combo}\end{aligned}$$ In the latter two relations, the amplitudes $T$, $C$ and $A$ carry the weak phase $e^{i\gamma}$, whereas the electroweak penguin amplitudes $P_{\rm EW}$ and $P_{\rm EW}^C$ carry the weak phase[^3] $e^{i\pi}$. Decomposing the QCD penguin amplitude as $P=\sum_q
\lambda_q P_q$, and similarly writing $A=\lambda_u A_u$ and $P_{\rm EW}^C=\lambda_t P_{{\rm EW},t}^C$, we rewrite the first relation in the form $$\begin{aligned}
B_{1/2} + A_{1/2} + A_{3/2}
&=& - \lambda_c (P_t - P_c - \textstyle\frac 13 P_{{\rm EW},t}^C)
+ \lambda_u (A_u - P_t + P_u) \nonumber\\
&\equiv& |P|\,e^{i\phi_P} \Big( e^{i\pi}
+ \varepsilon_a\,e^{i\gamma} e^{i\eta} \Big) \,.
\label{ampl1}\end{aligned}$$ By definition, the term $|P|\,e^{i\phi_P} e^{i\pi}$ contains all contributions to the $B^+\to\pi^+ K^0$ decay amplitude not proportional to the weak phase $e^{i\gamma}$. We will return to a discussion of the remaining terms below. It is convenient to adopt a parametrization of the other two amplitude combinations in (\[combo\]) in units of $|P|$, so that this parameter cancels in predictions for ratios of branching ratios. We define $$\begin{aligned}
\frac{-3 A_{3/2}}{|P|} &=& \varepsilon_{3/2}\,e^{i\phi_{3/2}}
(e^{i\gamma} - q\,e^{i\omega}) \,, \nonumber\\
\frac{-2(A_{1/2}+A_{3/2})}{|P|} &=& \varepsilon_T\,
e^{i\phi_T} (e^{i\gamma} - q_C\,e^{i\omega_C}) \,,
\label{ampl2}\end{aligned}$$ where the terms with $q$ and $q_C$ arise from electroweak penguin contributions. In the above relations, the parameters $\eta$, $\phi_{3/2}$, $\phi_T$, $\omega$, and $\omega_C$ are strong-interaction phases. For the benefit of the reader, it may be convenient to relate our definitions in (\[ampl1\]) and (\[ampl2\]) with those adopted by Buras and Fleischer [@BFnew]. The identificantions are: $|P|\,e^{i\phi_P}\leftrightarrow
\lambda_c|P_{tc}|\,e^{i\delta_{tc}}$, $\varepsilon_a\,e^{i\eta}\leftrightarrow -\rho\,e^{i\theta}$, $\phi_{3/2}\leftrightarrow\delta_{T+C}$, and $\phi_T\leftrightarrow\delta_T$. The notations for the electroweak penguin contributions conincide. Moreover, if we define $$\bar\varepsilon_{3/2}\equiv \frac{\varepsilon_{3/2}}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}} \,,
\label{bar32}$$ then $\bar\varepsilon_{3/2}\leftrightarrow r_{\rm c}$ and $\varepsilon_T/\varepsilon_{3/2}\leftrightarrow r/r_{\rm c}$. With this definition, the parameter $\bar\varepsilon_{3/2}$ is precisely the quantity that can be determined experimentally using the relation (\[epsexp\]).
Isovector part of the effective weak Hamiltonian
------------------------------------------------
The two amplitude combinations in (\[ampl2\]) involve isospin amplitudes defined in terms of the strong-interaction matrix elements of the $\Delta I=1$ part of the effective weak Hamiltonian.[^4] This part contains current–current as well as electroweak penguin operators. A trivial but relevant observation is that the electroweak penguin operators $Q_9$ and $Q_{10}$, whose Wilson coefficients are enhanced by the large mass of the top quark, are Fierz-equivalent to the current–current operators $Q_1$ and $Q_2$ [@Ne97; @Robert; @Fl96]. As a result, the $\Delta I=1$ part of the effective weak Hamiltonian for $B\to\pi K$ decays can be written as $${\mathcal H}_{\Delta I=1} = \frac{G_F}{\sqrt 2} \left\{ \left(
\lambda_u C_1 - \frac 32\lambda_t C_9 \right) \bar Q_1 + \left(
\lambda_u C_2 - \frac 32\lambda_t C_{10} \right) \bar Q_2 + \dots
\right\} + \mbox{h.c.} \,,
\label{newH}$$ where $\bar Q_i=\frac 12(Q_i^u-Q_i^d)$ are isovector combinations of four-quark operators. The dots represent the contributions from the electroweak penguin operators $Q_7$ and $Q_8$, which have a different Dirac structure. In the Standard Model, the Wilson coefficients of these operators are so small that their contributions can be safely neglected. It is important in this context that for heavy mesons the matrix elements of four-quark operators with Dirac structure $(V-A)\otimes (V+A)$ are not enhanced with respect to those of operators with the usual $(V-A)\otimes(V-A)$ structure. To an excellent approximation, the net effect of electroweak penguin contributions to the $\Delta I=1$ isospin amplitudes in $B\to\pi K$ decays thus consists of the replacements of the Wilson coefficients $C_1$ and $C_2$ of the current–current operators with the combinations shown in (\[newH\]). Introducing the linear combinations $C_\pm=(C_2\pm C_1)$ and $\bar Q_\pm=\frac 12
(\bar Q_2\pm\bar Q_1)$, which have the advantage of being renormalized multiplicatively, we obtain $${\mathcal H}_{\Delta I=1}\simeq \frac{G_F}{\sqrt 2}
|V_{ub}^* V_{us}| \left\{ C_+ (e^{i\gamma} - \delta_+)\,\bar Q_+
+ C_- (e^{i\gamma} - \delta_-)\,\bar Q_- \right\} + \mbox{h.c.} \,,
\label{Qpl}$$ where $$\delta_\pm = - \frac{3\cot\theta_C}{2\,|V_{ub}/V_{cb}|}\,
\frac{C_{10}\pm C_9}{C_2\pm C_1} \,.
\label{delpm}$$ We have used $\lambda_u/\lambda_t\simeq -\lambda_u/\lambda_c
\simeq -\tan\theta_C\,|V_{ub}/V_{cb}|\,e^{i\gamma}$, with the ratio $|V_{ub}/V_{cb}|=0.089\pm 0.015$ determined from semileptonic $B$ decays [@Rosnet].
From the fact that the products $C_\pm\,\bar Q_\pm$ are renormalization-group invariant, it follows that the quantities $\delta_\pm$ themselves must be scheme- and scale-independent (in a certain approximation). Indeed, the ratios of Wilson coefficients entering in (\[delpm\]) are, to a good approximation, independent of the choice of the renormalization scale. Taking the values $C_1=-0.308$, $C_2=1.144$, $C_9=-1.280\alpha$ and $C_{10}=0.328\alpha$, which correspond to the leading-order coefficients at the scale $\mu=m_b$ [@Heff], we find $(C_{10}+C_9)/(C_2+C_1)\approx -1.14\alpha$ and $(C_{10}-C_9)/(C_2-C_1)\approx 1.11\alpha$, implying that $\delta_-\approx -\delta_+$ to a good approximation. The statement of the approximate renormalization-group invariance of the ratios $\delta_\pm$ can be made more precise by noting that the large values of the Wilson coefficients $C_9$ and $C_{10}$ at the scale $\mu=m_b$ predominantly result from large matching contributions to the coefficient $C_9(m_W)$ arising from box and $Z$-penguin diagrams, whereas the $O(\alpha)$ contributions to the anomalous dimension matrix governing the mixing of the local operators $Q_i$ lead to very small effects. If these are neglected, then to next-to-leading order in the QCD evolution the coefficients $(C_{10}\pm C_9)$ are renormalized multiplicatively and in precisely the same way as the coefficients $(C_2\pm C_1)$. We have derived this result using the explicit expressions for the anomalous dimension matrices compiled in [@Heff].[^5] Hence, in this approximation the ratios of coefficients entering the quantities $\delta_\pm$ are renormalization-scale independent and can be evaluated at the scale $m_W$, so that $$\frac{C_{10}\pm C_9}{C_2\pm C_1} \simeq \pm C_9(m_W)
= \mp \frac{\alpha}{12\pi}\,\frac{x_t}{\sin^2\!\theta_W}
\left( 1 + \frac{3\ln x_t}{x_t-1} \right) + \dots
\approx \mp 1.18\alpha \,,
\label{niceeq}$$ where $\theta_W$ is the Weinberg angle, and $x_t=(m_t/m_W)^2$. This result agrees with an equivalent expression derived by Fleischer [@Fl96]. The dots in (\[niceeq\]) represent renormalization-scheme dependent terms, which are not enhanced by the factor $1/\sin^2\!\theta_W$. These terms are numerically very small and of the same order as the coefficients $C_7$ and $C_8$, whose values have been neglected in our derivation. The leading terms given above are precisely the ones that must be kept to get a consistent, renormalization-group invariant result. We thus obtain $$\delta_+ = - \delta_- = \frac{\alpha}{8\pi}\,
\frac{\cot\theta_C}{|V_{ub}/V_{cb}|}\,\frac{x_t}{\sin^2\!\theta_W}
\left( 1 + \frac{3\ln x_t}{x_t-1} \right)
= 0.68\pm 0.11 \,,$$ where we have taken $\alpha=1/129$ for the electromagnetic coupling renormalized at the scale $m_b$, and $m_t=\overline{m}_t(m_t)=170$GeV for the running top-quark mass in the $\overline{{\rm MS}}$ renormalization scheme. Assuming that there are no large $O(\alpha_s)$ corrections with this choice, the main uncertainty in the estimate of $\delta_+$ in the Standard Model results from the present error on $|V_{ub}|$, which is likely to be reduced in the near future.
We stress that the sensitivity of the $B\to\pi K$ decay amplitudes to the value of $\delta_+$ provides a window to New Physics, which could alter the value of this parameter significantly. A generic example are extensions of the Standard Model with new charged Higgs bosons such as supersymmetry, for which there are additional matching contributions to $C_9(m_W)$. We will come back to this point in Section \[sec:4\].
Structure of the isospin amplitude $A_{3/2}$ {#subsec:A32}
---------------------------------------------
$U$-spin invariance of the strong interactions, which is a subgroup of flavour SU(3) symmetry corresponding to transformations exchanging $d$ and $s$ quarks, implies that the isospin amplitude $A_{3/2}$ receives a contribution only from the operator $\bar Q_+$ in (\[Qpl\]), but not from $\bar Q_-$ [@us]. In order to investigate the corrections to this limit, we parametrize the matrix elements of the local operators $C_\pm \bar Q_\pm$ between a $B$ meson and the $(\pi K)$ isospin state with $I=\frac 32$ by hadronic parameters $K_{3/2}^\pm\,e^{i\phi_{3/2}^\pm}$, so that $$\begin{aligned}
-3 A_{3/2} &=& K_{3/2}^+\,e^{i\phi_{3/2}^+} (e^{i\gamma}-\delta_+)
+ K_{3/2}^-\,e^{i\phi_{3/2}^-} (e^{i\gamma}+\delta_+) \nonumber\\
&\equiv& \Big( K_{3/2}^+\,e^{i\phi_{3/2}^+} + K_{3/2}^-\,
e^{i\phi_{3/2}^-} \Big) (e^{i\gamma} - q\,e^{i\omega}) \,.
\label{Kdef}\end{aligned}$$ In the SU(3) limit $K_{3/2}^-=0$, and hence SU(3)-breaking corrections can be parametrized by the quantity $$\kappa\,e^{i\Delta\varphi_{3/2}}
\equiv \frac{2 K_{3/2}^-\,e^{i\phi_{3/2}^-}}
{K_{3/2}^+\,e^{i\phi_{3/2}^+} + K_{3/2}^-\,e^{i\phi_{3/2}^-}}
= 2 \left[ \frac{K_{3/2}^+}{K_{3/2}^-}\,
e^{i(\phi_{3/2}^+ -\phi_{3/2}^-)} + 1 \right]^{-1} \,,
\label{SU3br}$$ in terms of which $$q\,e^{i\omega} = \left( 1 - \kappa\,e^{i\Delta\varphi_{3/2}}
\right) \delta_+ \,.
\label{dEW}$$ This relation generalizes an approximate result derived in [@us].
The magnitude of the SU(3)-breaking effects can be estimated by using the generalized factorization hypothesis to calculate the matrix elements of the current–current operators [@Stech]. This gives $$\kappa\simeq 2 \left[ \frac{a_1+a_2}{a_1-a_2}\,
\frac{A_K + A_\pi}{A_K - A_\pi} + 1 \right]^{-1}
= (6\pm 6)\% \,,\qquad \Delta\varphi_{3/2}\simeq 0 \,,
\label{est}$$ where $A_K=f_K (m_B^2-m_\pi^2) F_0^{B\to\pi}(m_K^2)$ and $A_\pi=f_\pi
(m_B^2-m_K^2) F_0^{B\to K}(m_\pi^2)$ are combinations of hadronic matrix elements, and $a_1$ and $a_2$ are phenomenological parameters defined such that they contain the leading corrections to naive factorization. For a numerical estimate we take $a_2/a_1=0.21\pm 0.05$ as determined from a global analysis of nonleptonic two-body decays of $B$ mesons [@Stech], and $A_\pi/A_K=0.9\pm 0.1$, which is consistent with form factor models (see, e.g., [@BSW]–[@Casa]) as well as the most recent predictions obtained using light-cone QCD sum rules [@Ball]. Despite the fact that nonfactorizable corrections are not fully controlled theoretically, the estimate (\[est\]) suggests that the SU(3)-breaking corrections in (\[dEW\]) are small. More importantly, such effects cannot induce a sizable strong-interaction phase $\omega$. Since $\bar Q_+$ and $\bar Q_-$ are local operators whose matrix elements are taken between the same isospin eigenstates, it is very unlikely that the strong-interaction phases $\phi_{3/2}^+$ and $\phi_{3/2}^-$ could differ by a large amount. If we assume that these phases differ by at most $20^\circ$, and that the magnitude of $\kappa$ is as large as 12% (corresponding to twice the central value obtained using factorization), we find that $|\omega|<2.7^\circ$. Even for a phase difference $\Delta\varphi_{3/2}\simeq |\phi_{3/2}^+ - \phi_{3/2}^-|=90^\circ$, which seems totally unrealistic, the phase $|\omega|$ would not exceed $7^\circ$. It is therefore a safe approximation to work with the real value [@us] $$\delta_{\rm EW}\equiv (1-\kappa)\,\delta_+ = 0.64\pm 0.15 \,,$$ where to be conservative we have added linearly the uncertainties in the values of $\kappa$ and $\delta_+$. We believe the error quoted above is large enough to cover possible small contributions from a nonzero phase difference $\Delta\varphi_{3/2}$ or deviations from the factorization approximation. For completeness, we note that our general results for the structure of the electroweak penguin contributions to the isospin amplitude $A_{3/2}$, including the pattern of SU(3)-breaking effects, are in full accord with model estimates by Deshpande and He [@DeHe]. Generalizations of our results to the case of $B\to\pi\pi$, $K\bar K$ decays and the corresponding $B_s$ decays are possible using SU(3) symmetry, as discussed in [@Pirjol; @Agas].
In the last step, we define $K_{3/2}^+\,e^{i\phi_{3/2}^+}
+ K_{3/2}^-\,e^{i\phi_{3/2}^-}\equiv |P|\,\varepsilon_{3/2}\,
e^{i\phi_{3/2}}$, so that [@us] $$\frac{-3 A_{3/2}}{|P|} = \varepsilon_{3/2}\,e^{i\phi_{3/2}}
(e^{i\gamma}-\delta_{\rm EW}) \,.
\label{A32simple}$$ The complex quantity $q\,e^{i\omega}$ in our general parametrization in (\[ampl2\]) is now replaced with the real parameter $\delta_{\rm EW}$, whose numerical value is known with reasonable accuracy. The fact that the strong-interaction phase $\omega$ can be neglected was overlooked by Buras and Fleischer, who considered values as large as $|\omega|=45^\circ$ and therefore asigned a larger hadronic uncertainty to the isospin amplitude $A_{3/2}$ [@BFnew].
In the SU(3) limit, the product $|P|\,\varepsilon_{3/2}$ is determined by the decay amplitude for the process $B^\pm\to\pi^\pm\pi^0$ through the relation $$|P|\,\varepsilon_{3/2} = \sqrt 2\,\frac{R_{\rm SU(3)}}{R_{\rm EW}}
\tan\theta_C\,|{\mathcal A}(B^\pm\to\pi^\pm\pi^0)| \,,
\label{SU3rel}$$ where[^6] $$R_{\rm EW} = \left| e^{i\gamma} - \frac{V_{td}}{V_{ud}}\,
\frac{V_{us}}{V_{ts}}\,\delta_{\rm EW} \right| \simeq \left|
1 - \lambda^2 R_t\,\delta_{\rm EW}\,e^{-i\alpha} \right|$$ is a tiny correction arising from the very small electroweak penguin contributions to the decays $B^\pm\to\pi^\pm\pi^0$. Here $\lambda=\sin\theta_C\approx 0.22$ and $R_t=[(1-\rho)^2+\eta^2]^{1/2}\sim 1$ are Wolfenstein parameters, and $\alpha$ is another angle of the unitarity triangle, whose preferred value is close to $90^\circ$ [@Jonnew]. It follows that the deviation of $R_{\rm EW}$ from 1 is of order 1–2%, and it is thus a safe approximation to set $R_{\rm EW}=1$. More important are SU(3)-breaking corrections, which can be included in (\[SU3rel\]) in the factorization approximation, leading to $$R_{\rm SU(3)} \simeq \frac{a_1}{a_1+a_2}\,\frac{f_K}{f_\pi}
+ \frac{a_2}{a_1+a_2}\,
\frac{F_0^{B\to K}(m_\pi^2)}{F_0^{B\to\pi}(m_\pi^2)}
\simeq \frac{f_K}{f_\pi} \approx 1.2 \,,
\label{RSU3}$$ where we have neglected a tiny difference in the phase space for the two decays. Relation (\[SU3rel\]) can be used to determine the parameter $\bar\varepsilon_{3/2}$ introduced in (\[bar32\]), which coincides with $\varepsilon_{3/2}$ up to terms of $O(\varepsilon_a)$. To this end, we note that the CP-averaged branching ratio for the decays $B^\pm\to\pi^\pm K^0$ is given by $$\begin{aligned}
\mbox{Br}(B^\pm\to\pi^\pm K^0) &\equiv& \frac 12 \Big[
\mbox{Br}(B^+\to\pi^+ K^0) + \mbox{Br}(B^-\to\pi^-\bar K^0)
\Big] \nonumber\\
&=& |P|^2\,\Big( 1 - 2\varepsilon_a\cos\eta\cos\gamma
+ \varepsilon_a^2 \Big) \,.
\label{BR1}\end{aligned}$$ Combining this result with (\[SU3rel\]) we obtain relation (\[epsexp\]), which expresses $\bar\varepsilon_{3/2}$ in terms of CP-averaged branching ratios. Using preliminary data reported by the CLEO Collaboration [@CLEO] combined with some theoretical guidance based on factorization, one finds $\bar\varepsilon_{3/2}=0.24\pm 0.06$ [@us].
To summarize, besides the parameter $\delta_{\rm EW}$ controlling electroweak penguin contributions also the normalization of the amplitude $A_{3/2}$ is known from theory, albeit with some uncertainty related to nonfactorizable SU(3)-breaking effects. The only remaining unknown hadronic parameter in (\[A32simple\]) is the strong-interaction phase $\phi_{3/2}$. The various constraints on the structure of the isospin amplitude $A_{3/2}$ discussed here constitute the main theoretical simplification of $B\to\pi K$ decays, i.e., the only simplification rooted on first principles of QCD.
Structure of the amplitude combination $B_{1/2}+A_{1/2}+A_{3/2}$
----------------------------------------------------------------
The above result for the isospin amplitude $A_{3/2}$ helps understanding better the structure of the sum of amplitudes introduced in (\[ampl1\]). To this end, we introduce the following exact parametrization: $$B_{1/2} + A_{1/2} + A_{3/2} = |P| \left[ e^{i\pi} e^{i\phi_P}
- \frac{\varepsilon_{3/2}}{3}\,e^{i\gamma}
\left( e^{i\phi_{3/2}} - \xi e^{i\phi_{1/2}} \right) \right] \,,
\label{rewrite}$$ where we have made explicit the contribution proportional to the weak phase $e^{i\gamma}$ contained in $A_{3/2}$. From a comparison with the parametrization in (\[ampl1\]) it follows that $$\varepsilon_a\,e^{i\eta} = \frac{\varepsilon_{3/2}}{3}\,
e^{i\phi} \left( \xi\,e^{i\Delta} - 1 \right) \,,
\label{repara}$$ where $\phi=\phi_{3/2}-\phi_P$ and $\Delta=\phi_{1/2}-\phi_{3/2}$. Of course, this is just a simple reparametrization. However, the intuitive expectation that $\varepsilon_a$ is small, because this terms receives contributions only from the penguin $(P_u-P_t)$ and from annihilation topologies, now becomes equivalent to saying that $\xi\,e^{i\Delta}$ is close to 1, so as to allow for a cancelation between the contributions corresponding to final-state isospin $I=\frac 12$ and $I=\frac 32$ in (\[repara\]). But this can only happen if there are no sizable final-state interactions. The limit of elastic final-state interactions can be recovered from (\[repara\]) by setting $\xi=1$, in which case we reproduce results derived previously in [@Ge97; @Ne97]. Because of the large energy release in $B\to\pi K$ decays, however, one expects inelastic rescattering contributions to be important as well [@Fa97; @Dono]. They would lead to a value $\xi\ne 1$.
From (\[repara\]) it follows that $$\varepsilon_a = \frac{\varepsilon_{3/2}}{3}
\sqrt{1 - 2\xi\cos\Delta + \xi^2}
= \frac{2\sqrt\xi}{3}\,\varepsilon_{3/2}
\sqrt{ \left( \frac{1-\xi}{2\sqrt\xi} \right)^2
+ \sin^2\!\frac{\Delta}{2} } \,,$$ where without loss of generality we define $\varepsilon_a$ to be positive. Clearly, $\varepsilon_a\ll\varepsilon_{3/2}$ provided the phase difference $\Delta$ is small and the parameter $\xi$ close to 1. There are good physics reasons to believe that both of these requirements may be satisfied. In the rest frame of the $B$ meson, the two light particles produced in $B\to\pi K$ decays have large energies and opposite momenta. Hence, by the colour-transparency argument [@Bj] their final-state interactions are expected to be suppressed unless there are close-by resonances, such as charm–anticharm intermediate states ($D\bar D_s$, $J/\psi\,K$, etc.). However, these contributions could only result from the charm penguin [@charm1; @charming] and are thus included in the term $|P|\,e^{i\phi_P}$ in (\[ampl1\]). As a consequence, the phase difference $\phi=\phi_{3/2}-\phi_P$ could quite conceivably be sizable. On the other hand, the strong phases $\phi_{3/2}$ and $\phi_{1/2}$ in (\[rewrite\]) refer to the matrix elements of local four-quark operators of the type $\bar b s\bar u u$ and differ only in the isospin of the final state. We believe it is realistic to assume that $|\Delta|=|\phi_{1/2}-\phi_{3/2}|<
45^\circ$. Likewise, if the parameter $\xi$ were very different from 1 this would correspond to a gross failure of the generalized factorization hypothesis (even in decays into isospin eigenstates), which works so well in the global analysis of hadronic two-body decays of $B$ mesons [@Stech]. In view of this empirical fact, we think it is reasonable to assume that $0.5<\xi<1.5$. With this set of parameters, we find that $\varepsilon_a<0.35\varepsilon_{3/2}<0.1$. Thus, we expect that the rescattering effects parametrized by $\varepsilon_a$ are rather small.
A constraint on the parameter $\varepsilon_a$ can be derived assuming $U$-spin invariance of the strong interactions, which relates the decay amplitudes for the processes $B^\pm\to\pi^\pm K^0$ and $B^\pm\to K^\pm\bar K^0$ up to the substitution [@Fa97; @Robert; @He] $$\lambda_u \to V_{ub}^* V_{ud} \simeq \frac{\lambda_u}{\lambda}
\,, \qquad
\lambda_c \to V_{cb}^* V_{cd} \simeq - \lambda\,\lambda_c \,,$$ where $\lambda\approx 0.22$ is the Wolfenstein parameter. Neglecting SU(3)-breaking corrections, the CP-averaged branching ratio for the decays $B^\pm\to K^\pm\bar K^0$ is then given by $$\begin{aligned}
\mbox{Br}(B^\pm\to K^\pm\bar K^0) &\equiv& \frac 12 \Big[
\mbox{Br}(B^+\to K^+\bar K^0) + \mbox{Br}(B^-\to K^- K^0)
\Big] \nonumber\\
&=& |P|^2\,\Big[ \lambda^2 + 2\varepsilon_a\cos\eta\cos\gamma
+ (\varepsilon_a/\lambda)^2 \Big] \,,\end{aligned}$$ which should be compared with the corresponding result for the decays $B^\pm\to\pi^\pm K^0$ given in (\[BR1\]). The enhancement (suppression) of the subleading (leading) terms by powers of $\lambda$ implies potentially large rescattering effects and a large direct CP asymmetry in $B^\pm\to K^\pm\bar K^0$ decays. In particular, comparing the expressions for the direct CP asymmetries, $$\begin{aligned}
A_{\rm CP}(\pi^+ K^0) &\equiv&
\frac{\mbox{Br}(B^+\to\pi^+ K^0)-\mbox{Br}(B^-\to\pi^-\bar K^0)}
{\mbox{Br}(B^+\to\pi^+ K^0)+\mbox{Br}(B^-\to\pi^-\bar K^0)}
= \frac{2\varepsilon_a\sin\eta\sin\gamma}
{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2} \,,
\nonumber\\
A_{\rm CP}(K^+\bar K^0)
&=& - \frac{2\varepsilon_a\sin\eta\sin\gamma}
{\lambda^2 + 2\varepsilon_a\cos\eta\cos\gamma
+ (\varepsilon_a/\lambda)^2} \,,
\label{ACPs}\end{aligned}$$ one obtains the simple relation [@Robert] $$- \frac{A_{\rm CP}(K^+\bar K^0)}{A_{\rm CP}(\pi^+ K^0)}
= \frac{\mbox{Br}(B^\pm\to\pi^\pm K^0)}
{\mbox{Br}(B^\pm\to K^\pm\bar K^0)} \,.$$ In the future, precise measurements of the branching ratio and CP asymmetry in $B^\pm\to K^\pm\bar K^0$ decays may thus provide valuable information about the role of rescattering contributions in $B^\pm\to\pi^\pm K^0$ decays. In particular, upper and lower bounds on the parameter $\varepsilon_a$ can be derived from a measurement of the ratio $$R_K = \frac{\mbox{Br}(B^\pm\to K^\pm\bar K^0)}
{\mbox{Br}(B^\pm\to\pi^\pm K^0)}
= \frac{\lambda^2 + 2\varepsilon_a\cos\eta\cos\gamma
+ (\varepsilon_a/\lambda)^2}
{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2} \,.
\label{RKdef}$$ Using the fact that $R_K$ is minimized (maximized) by setting $\cos\eta\cos\gamma=-1$ (+1), we find that $$\frac{\lambda(\sqrt{R_K}-\lambda)}{1+\lambda\sqrt{R_K}}
\le \varepsilon_a \le
\frac{\lambda(\sqrt{R_K}+\lambda)}{1-\lambda\sqrt{R_K}} \,.$$ This generalizes a relation derived in [@Fa97]. Using data reported by the CLEO Collaboration [@CLEO], one can derive the upper bound $R_K<0.7$ (at 90% CL) implying $\varepsilon_a<0.28$, which is not yet a very powerful constraint. However, a measurement of the branching ratio for $B^\pm\to K^\pm\bar K^0$ could improve the situation significantly. For the purpose of illustration, we note that from the preliminary results quoted for the observed event rates one may deduce the “best fit” value $R_K\sim 0.15$ (with very large errors!). Taking this value literally would give the allowed range $0.03<\varepsilon_a<0.14$.
Based on a detailed analysis of individual rescattering contributions, Gronau and Rosner have argued that one expects a similar pattern of final-state interactions in the decays $B^\pm\to K^\pm\bar K^0$ and $B^0\to K^\pm K^\mp$ [@GRrescat]. One could then use the tighter experimental bound $\mbox{Br}(B^0\to K^\pm K^\mp)<2\times 10^{-6}$ to obtain $\varepsilon_a<0.16$. However, this is not a model-independent result, because the decay amplitudes for $B^0\to K^\pm K^\mp$ are not related to those for $B^\pm\to\pi^\pm K^0$ by any symmetry of the strong interactions. Nevertheless, this observation may be considered a qualitative argument in favour of a small value of $\varepsilon_a$.
Structure of the amplitude combination $A_{1/2}+A_{3/2}$
--------------------------------------------------------
None of the simplifications we found for the isospin amplitude $A_{3/2}$ persist for the amplitude $A_{1/2}$. Therefore, the sum $A_{1/2}+A_{3/2}$ suffers from larger hadronic uncertainties than the amplitude $A_{3/2}$ alone. Nevertheless, it is instructive to study the structure of this combination in more detail. In analogy with (\[Kdef\]), we parametrize the matrix elements of the local operators $C_\pm \bar Q_\pm$ between a $B$ meson and the $(\pi K)$ isospin state with $I=\frac 12$ by hadronic parameters $K_{1/2}^\pm\,e^{i\phi_{1/2}^\pm}$, so that $$-3 A_{1/2} = K_{1/2}^+\,e^{i\phi_{1/2}^+} (e^{i\gamma}-\delta_+)
+ K_{1/2}^-\,e^{i\phi_{1/2}^-} (e^{i\gamma}+\delta_+) \,.$$ Next, we define parameters $\varepsilon'$ and $r$ by $$\frac{\varepsilon'}{2}\,(1\pm r)
\equiv \frac{2}{3|P|} \left( K_{1/2}^\pm + K_{3/2}^\pm \right) \,.$$ This general definition is motivated by the factorization approximation, which predicts that $r\simeq a_2/a_1=0.21\pm 0.05$ is the phenomenological colour-suppression factor [@Stech], and $$\frac{\varepsilon'}{\varepsilon_{3/2}}
\simeq \frac{a_1 A_K}{a_1 A_K + a_2 A_\pi} = 0.84\pm 0.04 \,.
\label{epspr}$$ With the help of these definitions, we obtain $$\begin{aligned}
\frac{-2(A_{1/2} + A_{3/2})}{|P|}
&\simeq& \frac{\varepsilon'}{2} \left[ (1+r)\,
e^{i\phi_{1/2}^+} (e^{i\gamma} - \delta_+) + (1-r)\,
e^{i\phi_{1/2}^-} (e^{i\gamma} + \delta_+) \right] \nonumber\\
&&\mbox{}+ \frac{2\varepsilon_{3/2}}{3} \left( e^{i\phi_{3/2}}
- e^{i\phi_{1/2}^+} \right) (e^{i\gamma} - \delta_+) \,,
\label{newdef}\end{aligned}$$ where we have neglected some small, SU(3)-breaking corrections to the second term. Nevertheless, the above relation can be considered a general parametrization of the sum $A_{1/2}+A_{3/2}$, since it still contains two undetermined phases $\phi_{1/2}^\pm$ and magnitudes $\varepsilon'$ and $r$.
With the explicit result (\[newdef\]) at hand, it is a simple exercise to derive expressions for the quantities entering the parametrization in (\[ampl2\]). We find $$\begin{aligned}
\varepsilon_T\,e^{i\phi_T}
&=& \frac{\varepsilon'}{2}\,e^{i\phi_{1/2}^+} \left[
(e^{i\Delta\phi_{1/2}}+1) + r(e^{i\Delta\phi_{1/2}}-1) \right]
+ \frac{2\varepsilon_{3/2}}{3} \left( e^{i\phi_{3/2}}
- e^{i\phi_{1/2}^+} \right) \,, \nonumber\\
q_C\,e^{i\omega_C} &=& \delta_+\,
\frac{r(e^{i\Delta\phi_{1/2}}+1) + (e^{i\Delta\phi_{1/2}}-1)
+ \displaystyle\frac{4\varepsilon_{3/2}}{3\varepsilon'}
\left[ e^{i(\phi_{3/2}-\phi_{1/2}^+)} - 1 \right]}
{(e^{i\Delta\phi_{1/2}}+1) + r(e^{i\Delta\phi_{1/2}}-1)
+ \displaystyle\frac{4\varepsilon_{3/2}}{3\varepsilon'}
\left[ e^{i(\phi_{3/2}-\phi_{1/2}^+)} - 1 \right]} \,,\end{aligned}$$ where $\Delta\phi_{1/2}=\phi_{1/2}^- - \phi_{1/2}^+$. This result, although rather complicated, exhibits in a transparent way the structure of possible rescattering effects. In particular, it is evident that the assumption of “colour suppression” of the electroweak penguin contribution, i.e., the statement that $q_C=O(r)$ [@FM; @Robert; @BFnew; @GR97], relies on the smallness of the strong-interaction phase differences between the various terms. More specifically, this assumption would only be justified if $$|\Delta\phi_{1/2}| < 2 r ~\widehat{=}~ 25^\circ \,, \qquad
|\phi_{3/2}-\phi_{1/2}^+| < \frac{3r}{2}\,
\frac{\varepsilon'}{\varepsilon_{3/2}}
~\widehat{=}~ 15^\circ \,.$$ We believe that, whereas the first relation may be a reasonable working hypothesis, the second one constitues a strong constraint on the strong-interaction phases, which cannot be justified in a model-independent way. As a simple but not unrealistic model we may thus consider the approximate relations obtained by setting $\Delta\phi_{1/2}=0$ , which have been derived previously in [@Ne97]: $$\begin{aligned}
\varepsilon_T\,e^{i\phi_T}
&\simeq& \varepsilon'\,e^{i\phi_{1/2}^+}
+ \frac{2\varepsilon_{3/2}}{3} \left( e^{i\phi_{3/2}}
- e^{i\phi_{1/2}^+} \right) \,, \nonumber\\
q_C\,e^{i\omega_C} &\simeq& \delta_+\,
\frac{r + \displaystyle\frac{2\varepsilon_{3/2}}{3\varepsilon'}
\left[ e^{i(\phi_{3/2}-\phi_{1/2}^+)} - 1 \right]}
{1 + \displaystyle\frac{2\varepsilon_{3/2}}{3\varepsilon'}
\left[ e^{i(\phi_{3/2}-\phi_{1/2}^+)} - 1 \right]} \,.
\label{easy}\end{aligned}$$ The fact that in the case of a sizable phase difference between the $I=\frac 12$ and $I=\frac 32$ isospin amplitudes the electroweak penguin contribution may no longer be as small as $O(r)$ has been stressed in [@Ne97] but was overlooked in [@Robert; @BFnew]. Likewise, there is some uncertainty in the value of the parameter $\varepsilon_T$, which in the topological amplitude approach corresponds to the ratio $|T-A|/|P|$ [@ampl]. Unlike the parameter $\varepsilon_{3/2}$, the quantities $\varepsilon'$ and $r$ cannot be determined experimentally using SU(3) symmetry relations. But even if we assume that the factorization result (\[epspr\]) is valid and take $\varepsilon_{3/2}=0.24$ and $\varepsilon'=0.20$ as fixed, we still obtain $0.12<\varepsilon_T<0.20$ depending on the value of the phase difference $(\phi_{3/2}-\phi_{1/2}^+)$. Note that from the approximate expression (\[easy\]) it follows that $\varepsilon_T<\varepsilon'$ provided that $\varepsilon'/\varepsilon_{3/2}>2/3$, as indicated by the factorization result. This observation may explain why previous authors find the value $\varepsilon'=0.15\pm 0.05$ [@GR98r], which tends to be somewhat smaller than the factorization prediction $\varepsilon'\approx 0.20$.
Numerical results {#subsec:numerics}
-----------------
Before turning to phenomenological applications of our results in the next section, it is instructive to consider some numerical results obtained using the above parametrizations. Since our main concern in this paper is to study rescattering effects, we will keep $\varepsilon_{3/2}=0.24$ fixed and assume that $\varepsilon'/\varepsilon_{3/2}=0.84\pm 0.04$ and $r=0.21\pm 0.05$ as predicted by factorization. Also, we shall use the factorization result for the parameter $\kappa$ in (\[est\]).
For the strong-interaction phases we consider two sets of parameter choices: one which we believe is realistic and one which we think is very conservative. For the realistic set, we require that $0.5<\xi<1.5$, $|\phi_{3/2}-\phi_{1/2}^{(+)}|<45^\circ$, and $|\phi_I^+-\phi_I^-|<20^\circ$ (with $I=\frac 12$ or $\frac 32$). For the conservative set, we increase these ranges to $0<\xi<2$, $|\phi_{3/2}-\phi_{1/2}^{(+)}|<90^\circ$, and $|\phi_I^+-\phi_I^-|<45^\circ$. In our opinion, values outside these ranges are quite inconceivable. Note that, for the moment, no assumption is made about the relative strong-interaction phases of tree and penguin ampltiudes. We choose the various parameters randomly inside the allowed intervals and present the results for the quantities $\varepsilon_a\,e^{i\eta}$ in units of $e^{i\phi}$, $\varepsilon_T\,e^{i\phi_T}$ in units of $e^{i\phi_{3/2}}$, and $q_{(C)}\,e^{i\omega_{(C)}}$ in units of $\delta_+$ in the form of scatter plots in Figures 1 and 2. The black and the gray points correspond to the realistic and to the conservative parameter sets, respectively. The same colour coding will be used throughout this work.
The left-hand plot in Figure 1 shows that the parameter $\varepsilon_a$ generally takes rather small values. For the realistic parameter set we find $\varepsilon_a<0.08$, whereas values up tp 0.15 are possible for the conservative set. There is no strong correlation between the strong-interaction phases $\eta$ and $\phi$. An important implication of these observations is that, in general, there will be a very small difference between the quantities $\varepsilon_{3/2}$ and $\bar\varepsilon_{3/2}$ in (\[bar32\]). We shall therefore consider the same range of values for the two parameters. From the right-hand plot we observe that for realistic parameter choices $0.15<\varepsilon_T<0.22$; however, values between 0.08 and 0.24 are possible for the conservative parameter set. Note that there is a rather strong correlation between the strong-interaction phases $\phi_T$ and $\phi_{3/2}$, which differ by less than $20^\circ$ for the realistic parameter set. We will see in Section \[sec:5\] that this implies a strong correlation between the direct CP asymmetries in the decays $B^\pm\to\pi^0 K^\pm$ and $B^0\to\pi^\mp K^\pm$. Figure 2 shows that, even for the realistic parameter set, the ratio $q_C/\delta_+$ can be substantially larger than the naive expectation of about 0.2. Indeed, values as large as 0.7 are possible, and for the conservative set the wide range $0<q_C/\delta_+<1.4$ is allowed. Likewise, the strong-interaction phase $\omega_C$ can naturally be large and take values of up to $75^\circ$ even for the realistic parameter set. (Note that, without loss of generality, only points with positive values of $\omega_{(C)}$ are displayed in the plot. The distribution is invariant under a change of the sign of the strong-interaction phase.) This is in stark contrast to the case of the quantity $q\,e^{i\omega}$ entering the isospin amplitude $A_{3/2}$, where both the magnitude $q$ and the phase $\omega$ are determined within very small uncertainties, as is evident from the figure.
Hadronic uncertainties in the Fleischer–Mannel bound {#sec:3}
====================================================
As a first phenomenological application of the results of the previous section, we investigate the effects of rescattering and electroweak penguin contributions on the Fleischer–Mannel bound on $\gamma$ derived from the ratio $R$ defined in (\[Rdef\]). In general, $R\ne 1$ because the parameter $\varepsilon_T$ in (\[ampl2\]) does not vanish. To leading order in the small quantities $\varepsilon_i$, we find $$R\simeq 1 - 2\varepsilon_T \Big[ \cos\tilde\phi\cos\gamma
- q_C\cos(\tilde\phi+\omega_C) \Big] + O(\varepsilon_i^2) \,,$$ where $\tilde\phi=\phi_T-\phi_P$. Because of the uncertainty in the values of the hadronic parameters $\varepsilon_T$, $q_C$ and $\omega_C$, it is difficult to convert this result into a constraint on $\gamma$. Fleischer and Mannel have therefore suggested to derive a lower bound on the ratio $R$ by eliminating the parameter $\varepsilon_T$ from the exact expression for $R$. In the limit where $\varepsilon_a$ and $q_C$ are set to zero, this yields $R\ge\sin^2\!\gamma$ [@FM]. However, this simple result must be corrected in the presence of rescattering effects and electroweak penguin contributions. The generalization is [@Robert] $$R \ge \frac{1-2q_C\,\varepsilon_a\cos(\omega_C+\eta)
+q_C^2\,\varepsilon_a^2}
{(1-2q_C\cos\omega_C\cos\gamma+q_C^2)
(1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2)}\,
\sin^2\!\gamma \,.
\label{Rob}$$ The most dangerous rescattering effects arise from the terms involving the electroweak penguin parameter $q_C$. As seen from Figure 2, even restricting ourselves to the realistic parameter set we can have $2q_C\cos\omega_C\approx\delta_+\approx 0.7$ and $q_C^2\approx 0.5\,\delta_+^2\approx 0.2$, implying that the quadratic term in the denominator by itself can give a 20% correction. The rescattering effects parametrized by $\varepsilon_a$ are presumably less important.
The results of the numerical analysis are shown in Figure 3. In addition to the parameter choices described in Section \[subsec:numerics\], we vary $\varepsilon_{3/2}$ and $\delta_+$ in the ranges $0.24\pm 0.06$ and $0.68\pm 0.11$, respectively. Now also the relative strong-interaction phase $\phi$ between the penguin and $I=\frac 32$ tree amplitudes enters. We allow values $|\phi|<90^\circ$ for the realistic parameter set, and impose no constraint on $\phi$ at all for the conservative parameter set. The figure shows that the corrections to the Fleischer–Mannel bound are not as large as suggested by the result (\[Rob\]), the reason being that this result is derived allowing arbitrary values of $\varepsilon_T$, whereas in our analysis the allowed values for this parameter are constrained. However, there are sizable violations of the naive bound $R<\sin^2\!\gamma$ for $|\gamma|$ in the range between $65^\circ$ and $125^\circ$, which includes most of the region $47^\circ<\gamma<105^\circ$ preferred by the global analysis of the unitarity triangle [@Jonnew]. Whereas these violations are numerically small for the realistic parameter set, they can become large for the conservative set, because then a large value of the phase difference $|\phi_{3/2}-\phi_{1/2}^+|$ is allowed [@Ne97]. We conclude that under conservative assumptions only for values $R<0.8$ a constraint on $\gamma$ can be derived
Fleischer has argued that one can improve upon the above analysis by extracting some of the unknown hadronic parameters $q_C$, $\varepsilon_a$, $\omega_C$ and $\eta$ from measurements of other decay processes [@Robert]. The idea is to combine information on the ratio $R$ with measurements of the direct CP asymmetries in the decays $B^0\to\pi^\mp K^\pm$ and $B^\pm\to\pi^\pm K^0$, as well as of the ratio $R_K$ defined in (\[RKdef\]). One can then derive a bound on $R$ that depends, besides the electroweak penguin parameters $q_C$ and $\omega_C$, only on a combination $w=w(\varepsilon_a,\eta)$, which can be determined up to a two-fold ambiguity assuming SU(3) flavour symmetry. Besides the fact that this approach relies on SU(3) symmetry and involves significantly more experimental input than the original Fleischer–Mannel analysis, it does not allow one to eliminate the theoretical uncertainty related to the presence of electroweak penguin contributions.
Hadronic uncertainties in the $R_*$bound {#sec:4}
========================================
As a second application, we investigate the implications of recattering effects on the bound on $\cos\gamma$ derived from a measurement of the ratio $R_*$ defined in (\[Rdef\]). In this case, the theoretical analysis is cleaner because there is model-independent information on the values of the hadronic parameters $\varepsilon_{3/2}$, $q$ and $\omega$ entering the parametrization of the isospin amplitude $A_{3/2}$ in (\[ampl2\]). The important point noted in [@us] is that the decay amplitudes for $B^\pm\to\pi^\pm K^0$ and $B^\pm\to\pi^0 K^\pm$ differ only in this single isospin amplitude. Since the overall strength of $A_{3/2}$ is governed by the parameter $\bar\varepsilon_{3/2}$ and thus can be determined from experiment without much uncertainty, we have suggested to derive a bound on $\cos\gamma$ without eliminating this parameter. In this respect, our strategy is different from the Fleischer–Mannel analysis.
The exact theoretical expression for the inverse of the ratio $R_*$ is given by $$\begin{aligned}
R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2}\,
\frac{\cos\phi\,(\delta_{\rm EW}-\cos\gamma)
+ \varepsilon_a\cos(\phi-\eta)(1-\delta_{\rm EW}\cos\gamma)}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\nonumber\\
&&\mbox{}+ \bar\varepsilon_{3/2}^2
(1-2\delta_{\rm EW}\cos\gamma+\delta_{\rm EW}^2) \,,
\label{R*}\end{aligned}$$ where $\bar\varepsilon_{3/2}$ has been defined in (\[bar32\]). Relevant for the bound on $\cos\gamma$ is the maximal value $R_*^{-1}$ can take for fixed $\gamma$. In [@us], we have worked to linear order in the parameters $\varepsilon_i$, so that terms proportional to $\varepsilon_a$ could be neglected. Here, we shall generalize the discussion and keep all terms exactly. Varying the strong-interaction phases $\phi$ and $\eta$ independently, we find that the maximum value of $R_*^{-1}$ is given by $$R_*^{-1} \le 1 + 2\bar\varepsilon_{3/2}\,
\frac{|\delta_{\rm EW}-\cos\gamma \pm \varepsilon_a
(1-\delta_{\rm EW}\cos\gamma)|}
{\sqrt{1 \mp 2\varepsilon_a\cos\gamma+\varepsilon_a^2}}
+ \bar\varepsilon_{3/2}^2
(1-2\delta_{\rm EW}\cos\gamma+\delta_{\rm EW}^2) \,,
\label{exact}$$ where the upper (lower) signs apply if $\cos\gamma<c_0$ ($\cos\gamma>c_0$) with $$c_0 = \frac{(1+\varepsilon_a^2)\,\delta_{\rm EW}}
{1+\varepsilon_a^2\,\delta_{\rm EW}^2}
\simeq \delta_{\rm EW} \,.
\label{c0}$$ Keeping all terms in $\bar\varepsilon_{3/2}$ exactly, but working to linear order in $\varepsilon_a$, we find the simpler result $$R_*^{-1} \le \Big( 1 + \bar\varepsilon_{3/2}\,
|\delta_{\rm EW}-\cos\gamma| \Big)^2 + \bar\varepsilon_{3/2}
(\bar\varepsilon_{3/2} + 2\varepsilon_a) \sin^2\!\gamma
+ O(\bar\varepsilon_{3/2}\,\varepsilon_a^2) \,.
\label{Rstmax}$$ The higher-order terms omitted here are of order 1% and thus negligible. The annihilation contribution $\varepsilon_a$ enters this result in a very transparent way: increasing $\varepsilon_a$ increases the maximal value of $R_*^{-1}$ and therefore weakens the bound on $\cos\gamma$.
In [@us], we have introduced the quantity $\Delta_*$ by writing $R_*=(1-\Delta_*)^2$, so that $\Delta_*=1-\sqrt{R_*}$ obeys the bound shown in (\[ourb\]). Note that to first order in $\bar\varepsilon_{3/2}$ the rescattering contributions proportional to $\varepsilon_a$ do not enter.[^7] Armed with the result (\[exact\]), we can now derive the exact expression for the maximal value of the quantity $\Delta_*$, corresponding to the minimal value of $R_*$. It is of advantage to consider the ratio $\Delta_*/\bar\varepsilon_{3/2}$, the bound for which is to first order independent of the parameter $\bar\varepsilon_{3/2}$. We recall that this ratio can be determined experimentally up to nonfactorizable SU(3)-breaking corrections. Its current value is $\Delta_*/\bar\varepsilon_{3/2}=1.33\pm 0.78$.
In the left-hand plot in Figure 4, we show the maximal value for the ratio $\Delta_*/\bar\varepsilon_{3/2}$ for different values of the parameters $\bar\varepsilon_{3/2}$ and $\varepsilon_a$. The upper (red) and lower (blue) pairs of curves correspond to $\bar\varepsilon_{3/2}=0.18$ and 0.30, respectively, and span the allowed range of values for this parameter. For each pair, the dashed and solid lines correspond to $\varepsilon_a=0$ and 0.1, respectively. To saturate the bound (\[exact\]) requires to have $\eta-\phi=0^\circ$ or $180^\circ$, in which case $\varepsilon_a=0.1$ is a conservative upper limit (see Figure 1). The dotted curve shows for comparison the linearized result obtained by neglecting the higher-order terms in (\[ourb\]). The parameter $\delta_{\rm EW}=0.64$ is kept fixed in this plot. As expected, the bound on the ratio $\Delta_*/\bar\varepsilon_{3/2}$ is only weakly dependent on the values of $\bar\varepsilon_{3/2}$ and $\varepsilon_a$. In particular, not much is lost by using the conservative value $\varepsilon_a=0.1$. Note that for values $\Delta_*/\bar\varepsilon_{3/2}>0.8$ the linear bound (\[ourb\]) is conservative, i.e., weaker than the exact bound, and even for smaller values of $\Delta_*/\bar\varepsilon_{3/2}$ the violations of this bound are rather small. Expanding the exact bound to next-to-leading order in $\bar\varepsilon_{3/2}$, we obtain $$\frac{\Delta_*}{\bar\varepsilon_{3/2}}
\le |\delta_{\rm EW}-\cos\gamma| - \bar\varepsilon_{3/2} \left[
\left( \frac{\Delta_*}{\bar\varepsilon_{3/2}} \right)^2
- \left( \frac12 + \frac{\varepsilon_a}{\bar\varepsilon_{3/2}}
\right) \sin^2\!\gamma \right] + O(\bar\varepsilon_{3/2}^2) \,,$$ showing that $\Delta_*/\bar\varepsilon_{3/2}>(1/2+\varepsilon_a/
\bar\varepsilon_{3/2})^{1/2}$ is a criterion for the validity of the linearized bound. This generalizes a condition derived, for the special case $\varepsilon_a\ll\bar\varepsilon_{3/2}$, in [@us].
To obtain a reliable bound on the weak phase $\gamma$, we must account for the theoretical uncertainty in the value of the electroweak penguin parameter $\delta_{\rm EW}$ in the Standard Model, which is however straightforward to do by lowering (increasing) the value of this parameter used in calculating the right (left) branch of the curves defining the bound. The solid line in the right-hand plot in Figure 4 shows the most conservative bound obtained by using $\varepsilon_a=0.1$ and varying the other two parameters in the ranges $0.18<\bar\varepsilon_{3/2}
<0.30$ and $0.49<\delta_{\rm EW}<0.79$. The scatter plot shows the distribution of values of $\Delta_*/\bar\varepsilon_{3/2}$ obtained by scanning the strong-interaction parameters over the same ranges as we did for the Fleischer–Mannel case in the previous section. The horizontal band shows the current central experimental value with its $1\sigma$ variation. Unlike the Fleischer–Mannel bound, there is no violation of the bound (by construction), since all parameters are varried over conservative ranges. Indeed, for the points close to the right branch of the bound $\eta-\phi=0^\circ$, so that according to Figure 1 almost all of these points have $\varepsilon_a<0.03$, which is smaller than the value we used to obtain the theoretical curve. The dashed curve shows the bound for $\varepsilon_a=0$, which is seen not to be violated by any point. This shows that the rescattering effects parametrized by the quantity $\varepsilon_a$ play a very minor role in the bound derived from the ratio $R_*$. We conclude that, if the current experimental value is confirmed to within one standard deviation, i.e., if future measurements find that $\Delta_*/\bar\varepsilon_{3/2}
>0.55$, this would imply the bound $|\gamma|>75^\circ$, which is very close to the value of $77^\circ$ obtained in [@us].
Given that the experimental determination of the parameter $\bar\varepsilon_{3/2}$ is limited by unknown nonfactorizable SU(3)-breaking corrections, one may want to be more conservative and derive a bound directly from the measured ratio $R_*$ rather than the ratio $\Delta_*/\bar\varepsilon_{3/2}$. In the left-hand plot in Figure 5, we show the same distribution as in the right-hand plot in Figure 4, but now for the ratio $R_*$. The resulting bound on $\gamma$ is slightly weaker, because now there is a stronger dependence on the value of $\bar\varepsilon_{3/2}$, which we vary as previously between 0.18 and 0.30. If the current value of $R_*$ is confimed to within one standard deviation, i.e., if future measurements find that $R_*<0.71$, this would imply the bound $|\gamma|>72^\circ$.
Besides providing interesting information on $\gamma$, a measurement of $R_*$ or $\Delta_*/\bar\varepsilon_*$ can yield information about the strong-interaction phase $\phi$. In the right plot in Figure 5, we show the distribution of points obtained for fixed values of the strong-interaction phase $|\phi|$ between $0^\circ$ and $180^\circ$ in steps of $30^\circ$. For simplicity, the parameters $\varepsilon_{3/2}=0.24$ and $\delta_{\rm EW}=0.64$ are kept fixed in this plot, while all other hadronic parameters are scanned over the realistic parameter set. We observe that, independently of $\gamma$, a value $R_*<0.8$ requires that $|\phi|<90^\circ$. This conclusion remains true if the parameters $\varepsilon_{3/2}$ and $\delta_{\rm EW}$ are varied over their allowed ranges. We shall study the correlation between the weak phase $\gamma$ and the strong phase $\phi$ in more detail in Section \[sec:6\].
Finally, we emphasize that a future, precise measurement of the ratio $R_*$ may also yield a surprise and indicate physics beyond the Standard Model. The global analysis of the unitarity triangle requires that $|\gamma|<105^\circ$ [@Jonnew], for which the lowest possible value of $R_*$ in the Standard Model is about 0.55. If the experimental value would turn out to be less than that, this would be strong evidence for New Physics. In particular, in many extensions of the Standard Model there would be additional contributions to the electroweak penguin parameter $\delta_{\rm EW}$ arising, e.g., from penguin and box diagrams containing new charged Higgs bosons. This could explain a larger value of $R_*$. Indeed, from (\[Rstmax\]) we can derive the bound $$\begin{aligned}
\delta_{\rm EW} \!&\ge&\!
\frac{\sqrt{R_*^{-1} - \bar\varepsilon_{3/2}
(\bar\varepsilon_{3/2}+2\varepsilon_a)
\sin^2\!\gamma_{\rm max}} - 1}{\bar\varepsilon_{3/2}}
\nonumber\\
&&\mbox{}+ \cos\gamma_{\rm max} \,,\end{aligned}$$ where $\gamma_{\rm max}$ is the maximal value allowed by the global analysis (assuming that $\gamma_{\rm max}>\arccos(c_0)\approx
50^\circ$). In Figure 6, we show this bound for the current value $\gamma_{\rm max}=105^\circ$ and three different values of $\bar\varepsilon_{3/2}$ as well as two different values of $\varepsilon_a$. The gray band shows the allowed range for $\delta_{\rm EW}$ in the Standard Model. In the hypothetical situation where the current central values $R_*=0.47$ and $\bar\varepsilon_{3/2}=0.24$ would be confirmed by more precise measurements, we would conclude that the value of $\delta_{\rm EW}$ is at least twice as large as predicted by the Standard Model.
Prospects for direct CP asymmetries and prediction for the\
$B^0\to\pi^0 K^0$ branching ratio {#sec:5}
===========================================================
Decays of charged $B$ mesons
----------------------------
We will now analyse the potential of the various $B\to\pi K$ decay modes for showing large direct CP violation, starting with the decays of charged $B$ mesons. The smallness of the rescattering effects parametrized by $\varepsilon_a$ (see Figure 1) combined with the simplicity of the isospin amplitude $A_{3/2}$ (see Section \[subsec:A32\]) make these processes particularly clean from a theoretical point of view.
Explicit expressions for the CP asymmetries in the various decays can be derived in a straightforward way starting from the isospin decomposition in (\[isodec\]) and inserting the parametrizations for the isospin amplitudes derived in Section \[sec:2\]. The result for the CP asymmetry in the decays $B^\pm\to\pi^\pm K^0$ has already been presented in (\[ACPs\]). The corresponding expression for the decays $B^\pm\to\pi^0 K^\pm$ reads $$A_{\rm CP}(\pi^0 K^+) = 2\sin\gamma\,R_*\,
\frac{\varepsilon_{3/2}\sin\phi+\varepsilon_a\sin\eta
-\varepsilon_{3/2}\,\varepsilon_a\,\delta_{\rm EW}
\sin(\phi-\eta)}
{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2} \,,
\label{ACP2}$$ where the theoretical expression for $R_*$ is given in (\[R\*\]), and we have not replaced $\varepsilon_{3/2}$ in terms of $\bar\varepsilon_{3/2}$. Neglecting terms of order $\varepsilon_a$ and working to first order in $\varepsilon_{3/2}$, we find the estimate $A_{\rm CP}(\pi^0 K^+)\simeq 2\varepsilon_{3/2}\sin\gamma
\sin\phi\approx 0.5\sin\gamma\sin\phi$, indicating that potentially there could be a very large CP asymmetry in this decay (note that $\sin\gamma>0.73$ is required by the global analysis of the unitarity triangle).
In Figure 7, we show the results for the two direct CP asymmetries in (\[ACPs\]) and (\[ACP2\]), both for the realistic and for the conservative parameter sets. These results confirm the general observations made above. For the realistic parameter set, and with $\gamma$ between $47^\circ$ and $105^\circ$ as indicated by the global analysis of the unitarity triangle [@Jonnew], we find CP asymmetries of up to 15% in $B^\pm\to\pi^\pm K^0$ decays, and of up to 50% in $B^\pm\to\pi^0 K^\pm$ decays. Of course, to have large asymmetries requires that the sines of the strong-interaction phases $\eta$ and $\phi$ are not small. However, this is not unlikely to happen. According to the left-hand plot in Figure 1, the phase $\eta$ can take any value, and the phase $\phi$ could quite conceivably be large due to the different decay mechanisms of tree- and penguin-initiated processes. We stress that there is no strong correlation between the CP asymmetries in the two decay processes, because as shown in Figure 1 there is no such correlation between the strong-interaction phases $\eta$ and $\phi$.
Decays of neutral $B$ mesons
----------------------------
Because of their dependence on the hadronic parameters $\varepsilon_T$, $q_C$ and $\omega_C$ entering through the sum $A_{1/2}+A_{3/2}$ of isospin amplitudes, the theoretical analysis of neutral $B\to\pi K$ decays is affected by larger hadronic uncertainties than that of the decays of charged $B$ mesons. Nevertheless, some interesting predictions regarding neutral $B$ decays can be made and tested experimentally.
The expression for the direct CP asymmetry in the decays $B^0\to\pi^\mp K^\pm$ is $$A_{\rm CP}(\pi^- K^+) = \frac{2\sin\gamma}{R}\,
\frac{\varepsilon_T(\sin\tilde\phi-\varepsilon_T\,q_C\sin\omega_C)
+ \varepsilon_a\,[\sin\eta-\varepsilon_T\,q_C
\sin(\tilde\phi-\eta+\omega_C)]}
{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2} \,,$$ where $\tilde\phi=\phi_T-\phi_P$. This result reduces to (\[ACP2\]) under the replacements $q_C\to\delta_{\rm EW}$, $\omega_C\to 0$, $R\to R_*^{-1}$, and $\varepsilon_T\to\varepsilon_{3/2}$. The corresponding expression for the direct CP asymmetry in the decays $B^0\to\pi^0 K^0$ and $\bar B^0\to\pi^0\bar K^0$ is more complicated and will not be presented here. Below, we shall derive an exact relation between the various asymmetries, which can be used to compute $A_{\rm CP}(\pi^0 K^0)$.
Gronau and Rosner have emphasized that one expects $A_{\rm CP}(\pi^- K^+)\approx A_{\rm CP}(\pi^0 K^+)$, and that one could thus combine the data samples for these decays to enhance the statistical significance of an early signal of direct CP violation [@GR98r]. We can easily understand the argument behind this observation using our results. Neglecting the small rescattering contributions proportional to $\varepsilon_a$ for simplicity, we find $$\frac{ A_{\rm CP}(\pi^- K^+)}{ A_{\rm CP}(\pi^0 K^+)}
\simeq \frac{1}{R_* R}\,
\frac{\varepsilon_T(\sin\tilde\phi-\varepsilon_T\,q_C\sin\omega_C)}
{\varepsilon_{3/2}\sin\phi}
\simeq \frac{1}{R_* R}\,\frac{\varepsilon_T}{\varepsilon_{3/2}} \,.$$ In the last step, we have used that the electroweak penguin contribution is very small because it is suppressed by an additional factor of $\varepsilon_T$, and that the strong-interaction phases $\phi$ and $\tilde\phi$ are strongly correlated, as follows from the right-hand plot in Figure 1. Numerically, the right-hand side turns out to be close to 1 for most of parameter space. This is evident from the left-hand plot in Figure 8, which confirms that there is indeed a very strong correlation between the CP asymmetries in the decays $B^0\to\pi^\mp K^\pm$ and $B^\pm\to\pi^0 K^\pm$, in agreement with the argument given in [@GR98r]. Combining the data samples for these decays collected by the CLEO experiment, one may have a chance for observing a statistically significant signal for the first direct CP asymmetry in $B$ decays before the operation of the asymmetric $B$ factories.
The decays $B^0\to\pi^0 K^0$ and $\bar B^0\to\pi^0\bar K^0$ have not yet been observed experimentally, but the CLEO Collaboration has presented an upper bound on their CP-averaged branching ratio of $4.1\times 10^{-5}$ [@CLEO]. In analogy with (\[Rdef\]), we define the ratios $$\begin{aligned}
R_0 &=& \frac{\tau(B^+)}{\tau(B^0)}\,
\frac{2[\mbox{Br}(B^0\to\pi^0 K^0)
+\mbox{Br}(\bar B^0\to\pi^0\bar K^0)]}
{\mbox{Br}(B^+\to\pi^+ K^0)+\mbox{Br}(B^-\to\pi^-\bar K^0)}
\,, \nonumber\\
R_{0*} &=&
\frac{2[\mbox{Br}(B^0\to\pi^0 K^0)
+\mbox{Br}(\bar B^0\to\pi^0\bar K^0)]}
{\mbox{Br}(B^0\to\pi^- K^+)+\mbox{Br}(\bar B^0\to\pi^+ K^-)}
= \frac{R_0}{R} \,.
\label{R0def}\end{aligned}$$ Using our parametrizations for the different isospin amplitudes, we find that the ratios $R$, $R_*$ and $R_0$ obey the relations $$R_0 - R + R_*^{-1} -1 = \Delta_1 \,, \qquad
R_0 - R\,R_* = \Delta_2 + O(\bar\varepsilon_i^3) \,,
\label{sumrules}$$ where $$\begin{aligned}
\Delta_1 &=& 2\bar\varepsilon_{3/2}^2\,
(1-2\delta_{\rm EW}\cos\gamma+\delta_{\rm EW}^2)
- 2\bar\varepsilon_{3/2}\,\bar\varepsilon_T\,
(1-\delta_{\rm EW}\cos\gamma) \cos(\phi_T-\phi_{3/2}) \nonumber\\
&&\mbox{}- 2\bar\varepsilon_{3/2}\,\bar\varepsilon_T\,q_C\,
(\delta_{\rm EW}-\cos\gamma) \cos(\phi_T-\phi_{3/2}+\omega_C) \,,
\nonumber\\
\Delta_2 &=& \Delta_1 - 4\bar\varepsilon_{3/2}^2
(\delta_{\rm EW}-\cos\gamma)^2\cos^2\!\phi \nonumber\\
&&\mbox{}+ 4\bar\varepsilon_{3/2}\,\bar\varepsilon_T\,
(\delta_{\rm EW}-\cos\gamma)\cos\phi \left[
q_C\cos(\tilde\phi+\omega_C)-\cos\gamma\cos\tilde\phi \right] \,, \end{aligned}$$ and $\bar\varepsilon_T$ is defined in analogy with $\bar\varepsilon_{3/2}$ in (\[bar32\]), so that $\bar\varepsilon_T/\varepsilon_T=\bar\varepsilon_{3/2}
/\varepsilon_{3/2}$. The first relation in (\[sumrules\]) generalizes a sum rule derived by Lipkin, who neglected the terms of $O(\varepsilon_i^2)$ on the right-hand side as well as electroweak penguin contributions [@Lipkin]. The second relation is new. It follows from the fact that $R_{0*}=R_*
+O(\varepsilon_i^2)$, which is evident since the pairs of decay amplitudes entering the definition of the two ratios differ only in the isospin amplitude $A_{3/2}$.
The left-hand plot in Figure 9 shows the results for the ratio $R_0$ versus $|\gamma|$. The dependence of this ratio on the weak phase turns out to be much weaker than in the case of the ratios $R$ and $R_*$. For the realistic parameter set we find that $0.7<R_0<1.0$ for most choices of strong-interaction parameters. Combining this with the current value of the $B^\pm\to\pi^\pm K^0$ branching ratio, we obtain values between $(0.47\pm 0.18)\times 10^{-5}$ and $(0.67\pm 0.26)\times 10^{-5}$ for the CP-averaged $B^0\to\pi^0 K^0$ branching ratio. The right-hand plot in Figure 9 shows the strong correlation between the ratios $R_*$ and $R_{0*}=R_0/R$, which holds with a remarkable accuracy over all of parameter space.
In Figure 10, we show the estimates of $R_0$ obtained by neglecting the terms of $O(\bar\varepsilon_i^2)$ and higher in the two sum rules in (\[sumrules\]). Using the present data for the various branching ratios yields to the estimates $R_0=(-0.1\pm 0.9)$ from the first and $R_0=(0.5\pm 0.2)$ from the second sum rule. Both results are consistent with the theoretical expectations for $R_0$ exhibited in the left-hand plot in Figure 9; however, the second estimate has a much smaller experimental error and, according to Figure 10, it is likely to have a higher theoretical accuracy. We can rewrite this estimate as $$\frac12 \Big[ \mbox{Br}(B^0\to\pi^0 K^0)
+ \mbox{Br}(\bar B^0\to\pi^0\bar K^0) \Big]
\simeq \frac{\mbox{Br}(B^\pm\to\pi^\pm K^0)\,
\mbox{Br}(B^0\to\pi^\mp K^\pm)}
{4\mbox{Br}(B^\pm\to\pi^0 K^\pm)} \,,
\label{pred}$$ where the branching ratios on the right-hand side are averaged over CP-conjugate modes. With current data, this relation yields the value $(0.33\pm 0.18)\times 10^{-5}$. Combining the three estimates for the CP-averaged $B^0\to\pi^0 K^0$ branching ratio presented above we arrive at the value $(0.5\pm 0.2)\times 10^{-5}$, which is about a factor of 3 smaller than the other three $B\to\pi K$ branching ratios quoted in (\[CLEOvals\]).
We now turn to the study of the direct CP asymmetry in the decays $B^0\to\pi^0 K^0$ and $\bar B^0\to\pi^0\bar K^0$. Using our general parametrizations, we find the sum rule $$\begin{aligned}
&&A_{\rm CP}(\pi^+ K^0) - R_*^{-1}\,A_{\rm CP}(\pi^0 K^+)
+ R\,A_{\rm CP}(\pi^- K^+) - R_0\,A_{\rm CP}(\pi^0 K^0)
\nonumber\\
&&\quad = 2\sin\gamma\,\bar\varepsilon_{3/2}\,\bar\varepsilon_T
\left[ \delta_{\rm EW}\sin(\phi_T-\phi_{3/2})
- q_C\sin(\phi_T-\phi_{3/2}+\omega_C) \right] \,.
\label{ACPsum}\end{aligned}$$ By scanning all strong-interaction parameters, we find that for the realistic (conservative) parameter set the right-hand side takes values of less that 4% (7%) times $\sin\gamma$ in magnitude. Neglecting these small terms, and using the approximate equality of the CP asymmetries in $B^\pm\to\pi^0 K^\pm$ and $B^0\to\pi^\mp K^\pm$ decays as well as the second relation in (\[sumrules\]), we obtain $$A_{\rm CP}(\pi^0 K^0) \simeq
- \frac{1 - R\,R_*}{R\,R_*^2}\,A_{\rm CP}(\pi^0 K^+)
+ \frac{A_{\rm CP}(\pi^+ K^0)}{R\,R_*} \,.$$ The first term is negative for most choices of parameters and would dominate if the CP aymmetry in $B^\pm\to\pi^0 K^\pm$ decays would turn out to be large. We therefore expect a weak anticorrelation between $A_{\rm CP}(\pi^0 K^0)$ and $A_{\rm CP}(\pi^+ K^0)$, which is indeed exhibited in the right-hand plot in Figure 9.
For completeness, we note that in the decays $B^0$, $\bar B^0\to\pi^0 K_S$ one can also study mixing-induced CP violation, as has been emphasized recently in [@BFnew]. Because of the large hadronic uncertainties inherent in the calculation of this effect, we do not study this possibility further.
Determination of $\gamma$ from $B^\pm\to\pi K$, $\pi\pi$ decays {#sec:6}
===============================================================
Ultimately, one would like not only to derive bounds on the weak phase $\gamma$, but to measure this parameter from a study of CP violation in $B\to\pi K$ decays. However, as we have pointed out in Section \[sec:2\], this is not a trivial undertaking because even perfect measurements of all eight $B\to\pi K$ branching ratios would not suffice to eliminate all hadronic parameters entering the parametrization of the decay amplitudes.
Because of their theoretical cleanness, the decays of charged $B$ mesons are best suited for a measurement of $\gamma$. In [@us2], we have described a strategy for achieving this goal, which relies on the measurements of the CP-averaged branching ratios for the decays $B^\pm\to\pi^\pm K^0$ and $B^\pm\to\pi^\pm\pi^0$, as well as of the individual branching ratios for the decays $B^+\to\pi^0 K^+$ and $B^-\to\pi^0 K^-$, i.e., the direct CP asymmetry in this channel. This method is a generalization of the Gronau–Rosner–London (GRL) approach for extracting $\gamma$ [@GRL]. It includes the contributions of electroweak penguin operators, which had previously been argued to spoil the GRL method [@DeHe; @GHLR2].
The strategy proposed in [@us2] relies on the dynamical assumption that there is no CP-violating contribution to the $B^\pm\to\pi^\pm K^0$ decay amplitudes, which is equivalent to saying that the rescattering effects parametrized by the quantity $\varepsilon_a$ in (\[ampl1\]) are negligibly small. It is evident from the left-hand plot in Figure 1 that this assumption is indeed justified in a large region of parameter space. Here, we will refine the approach and investigate the theoretical uncertainty resulting from $\varepsilon_a\ne 0$. As a side product, we will show how nontrivial information on the strong-interaction phase difference $\phi=\phi_{3/2}-\phi_P$ can be obtained along with information on $\gamma$.
To this end, we consider in addition to the ratio $R_*$ the CP-violating observable $$\widetilde A \equiv \frac{A_{\rm CP}(\pi^0 K^+)}{R_*}
- A_{\rm CP}(\pi^+ K^0) = 2\sin\gamma\,\bar\varepsilon_{3/2}\,
\frac{\sin\phi-\varepsilon_a\,\delta_{\rm EW}\sin(\phi-\eta)}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\,.
\label{Atil}$$ The purpose of subtracting the CP asymmetry in the decays $B^\pm\to\pi^\pm K^0$ is to eliminate the contribution of $O(\varepsilon_a)$ in the expression for $A_{\rm CP}(\pi^0 K^+)$ given in (\[ACP2\]). A measurement of this asymmetry is the new ingredient in our approach with respect to that in [@us2]. With the definition of $\widetilde A$ as given above, the rescattering effects parametrized by $\varepsilon_a$ are suppressed by an additional factor of $\bar\varepsilon_{3/2}$ and are thus expected to be very small. As shown in Section \[sec:4\], the same is true for the ratio $R_*$. Explicitly, we have $$\begin{aligned}
R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2}\,\cos\phi\,
(\delta_{\rm EW}-\cos\gamma) + \bar\varepsilon_{3/2}^2\,
(1-2\delta_{\rm EW}\cos\gamma+\delta_{\rm EW}^2)
+ O(\bar\varepsilon_{3/2}\,\varepsilon_a) \,, \nonumber\\
\widetilde A &=& 2\sin\gamma\,\bar\varepsilon_{3/2}\,\sin\phi
+ O(\bar\varepsilon_{3/2}\,\varepsilon_a) \,.\end{aligned}$$
These equations define contours in the $(\gamma,\phi)$ plane. When higher-order terms are kept, these contours become narrow bands, the precise shape of which depends on the values of the parameters $\bar\varepsilon_{3/2}$ and $\delta_{\rm EW}$. In the limit $\varepsilon_a=0$ the procedure described here is mathematically equivalent to the construction proposed in [@us2]. There, the errors on $\cos\gamma$ resulting from the variation of the input parameters have been discussed in detail. For a typical example, where $\gamma=76^\circ$ and $\phi=20^\circ$, we found that the uncertainties resulting from a 15% variation of $\bar\varepsilon_{3/2}$ and $\delta_{\rm EW}$ are $\cos\gamma=0.24\pm 0.09\pm 0.09$, correspondig to errors of $\pm 5^\circ$ each on the extracted value of $\gamma$.
Our focus here is to evaluate the additional uncertainty resulting from the rescattering effects parametrized by $\varepsilon_a$ and $\eta$. For given values of $\bar\varepsilon_{3/2}$, $\delta_{\rm EW}$, $\varepsilon_a$, $\eta$, and $\gamma$, the exact results for $R_*$ in (\[R\*\]) and $\widetilde A$ in (\[Atil\]) can be brought into the generic form $A\cos\phi+B\sin\phi=C$, where in the case of $R_*$ $$\begin{aligned}
A &=& 2\bar\varepsilon_{3/2}\,
\frac{\delta_{\rm EW}-\cos\gamma+\varepsilon_a\cos\eta\,
(1-\delta_{\rm EW}\cos\gamma)}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\,, \nonumber\\
B &=& 2\bar\varepsilon_{3/2}\,
\frac{\varepsilon_a\sin\eta\,(1-\delta_{\rm EW}\cos\gamma)}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\,, \nonumber\\
C &=& R_*^{-1} - 1 - \bar\varepsilon_{3/2}^2\,
(1-2\delta_{\rm EW}\cos\gamma+\delta_{\rm EW}^2) \,,\end{aligned}$$ whereas for $\widetilde A$ $$\begin{aligned}
A &=& 2\bar\varepsilon_{3/2}\,
\frac{\varepsilon_a\,\delta_{\rm EW}\sin\eta}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\,, \nonumber\\
B &=& 2\bar\varepsilon_{3/2}\,
\frac{1-\varepsilon_a\,\delta_{\rm EW}\cos\eta}
{\sqrt{1-2\varepsilon_a\cos\eta\cos\gamma+\varepsilon_a^2}}
\,, \nonumber\\
C &=& \frac{\widetilde A}{\sin\gamma} \,.\end{aligned}$$ The two solutions for $\cos\phi$ are given by $$\cos\phi = \frac{A C\pm B\sqrt{A^2+B^2-C^2}}{A^2+B^2} \,.$$ The physical solutions must be such that $\cos\phi$ is real and its magnitude less than 1.
In Figure 12, we show the resulting contour bands obtained by keeping $\bar\varepsilon_{3/2}=0.24$ and $\delta_{\rm EW}=0.64$ fixed to their central values, while the rescattering parameters are scanned over the ranges $0<\varepsilon_a<0.08$ and $-180^\circ<\eta<180^\circ$. Assuming that $\sin\gamma>0$ as suggested by the global analysis of the unitarity triangle, the sign of $\widetilde A$ determines the sign of $\sin\phi$. In the plot, we assume without loss of generality that $0^\circ\le\phi\le
180^\circ$. For instance, if $R_*=0.7$ and $\widetilde A=0.2$, then the two solutions are $(\gamma,\phi)\approx
(98^\circ,25^\circ)$ and $(\gamma,\phi)\approx
(153^\circ,67^\circ)$, only the first of which is allowed by the upper bound $\gamma<105^\circ$ following from the global analysis of the unitarity triangle [@Jonnew]. It is evident that the contours are rather insensitive to the rescattering effects parametrized by $\varepsilon_a$ and $\eta$. The error on $\gamma$ due to these effects is about $\pm 5^\circ$, which is similar to the errors resulting from the theoretical uncertainties in the parameters $\bar\varepsilon_{3/2}$ and $\delta_{\rm EW}$. The combined theoretical uncertainty is of order $\pm 10^\circ$ on the extracted value of $\gamma$.
To summarize, the strategy for determining $\gamma$ would be as follows: From measurements of the CP-averaged branching ratio for the decays $B^\pm\to\pi^\pm\pi^0$, $B^\pm\to\pi^\pm K^0$ and $B^\pm\to\pi^0 K^\pm$, the ratio $R_*$ and the parameter $\bar\varepsilon_{3/2}$ are determined using (\[Rdef\]) and (\[epsexp\]), respectively. Next, from measurements of the rate asymmetries in the decays $B^\pm\to\pi^\pm K^0$ and $B^\pm\to\pi^0 K^\pm$ the quantity $\widetilde A$ is determined. From the contour plots for the quantities $R_*$ and $\widetilde A$ the phases $\gamma$ and $\phi$ can then be extracted up to discrete ambiguities. In this determination one must account for theoretical uncertainties in the values of the parameters $\bar\varepsilon_{3/2}$ and $\delta_{\rm EW}$, as well as for rescattering effects parametrized by $\varepsilon_a$ and $\eta$. Quantitative estimates for these uncertainties have been given above.
Conclusions {#sec:7}
===========
We have presented a model-independent, global analysis of the rates and direct CP asymmetries for the rare two-body decays $B\to\pi K$. The theoretical description exploits the flavour symmetries of the strong interactions and the structure of the low-energy effective weak Hamiltonian. Isospin symmetry is used to introduce a minimal set of three isospin amplitudes. The explicit form of the effective weak Hamiltonian in the Standard Model is used to simplify the isovector part of the interaction. Both the numerical smallness of certain Wilson coefficient functions and the Dirac and colour structure of the local operators are relevant in this context. Finally, the $U$-spin subgroup of flavour SU(3) symmetry is used to simplify the structure of the isospin amplitude $A_{3/2}$ referring to the decay $B\to(\pi K)_{I=3/2}$. In the limit of exact $U$-spin symmetry, two of the four parameters describing this amplitude (the relative magnitude and strong-interaction phase of electroweak penguin and tree contributions) can be calculated theoretically, and one additional parameter (the overall strength of the amplitude) can be determined experimentally from a measurement of the CP-averaged branching ratio for $B^\pm\to\pi^\pm\pi^0$ decays. What remains is a single unknown strong-interaction phase. The SU(3)-breaking corrections to these results can be calculated in the generalized factorization approximation, so that theoretical limitations enter only at the level of nonfactorizable SU(3)-breaking effects. However, since we make use of SU(3) symmetry only to derive relations for amplitudes referring to isospin eigenstates, we do not expect gross failures of the generalized factorization hypothesis. We stress that the theoretical simplifications used in our analysis are the only ones rooted on first principles of QCD. Any further simplification would have to rest on model-dependent dynamical assumptions, such as the smallness of certain flavour topologies with respect to others.
We have introduced a general parametrization of the decay amplitudes, which makes maximal use of these theoretical constraints but is otherwise completely general. In particular, no assumption is made about strong-interaction phases. With the help of this parametrization, we have performed a global analysis of the branching ratios and direct CP asymmetries in the various $B\to\pi K$ decay modes, with particular emphasis on the impact of hadronic uncertainties on methods to learn about the weak phase $\gamma=\mbox{arg}(V_{ub}^*)$ of the unitarity triangle. The main phenomenological implications of our results can be summarized as follows:
- There can be substantial corrections to the Fleischer–Mannel bound on $\gamma$ from enhanced electroweak penguin contributions, which can arise in the case of a large strong-interaction phase difference between $I=\frac 12$ and $I=\frac 32$ isospin amplitudes. Whereas these corrections stay small (but not negligible) if one restricts this phase difference to be less than $45^\circ$, there can be large violations of the bound if the phase difference is allowed to be as large as $90^\circ$.
- On the contrary, rescattering effects play a very minor role in the bound on $\gamma$ derived from a measurement of the ratio $R_*$ of CP-averaged $B^\pm\to\pi K$ branching ratios. They can be included exactly in the bound and enter through a parameter $\varepsilon_a$, whose value is less than 0.1 even under very conservative conditions. Including these effects weakens the bounds on $\gamma$ by less than $5^\circ$. We have generalized the result of our previous work [@us], where we derived a bound on $\cos\gamma$ to linear order in an expansion in the small quantity $\bar\varepsilon_{3/2}$. Here we refrain from making such an approximation; however, we confirm our previous claim that to make such an expansion is justified (i.e., it yields a conservative bound) provided that the current experimental value of $R_*$ does not change by more than one standard deviation. The main result of our analysis is given in (\[exact\]), which shows the exact result for the maximum value of the ratio $R_*$ as a function of the parameters $\delta_{\rm EW}$, $\bar\varepsilon_{3/2}$, and $\varepsilon_a$. The first parameter describes electroweak penguin contributions and can be calculated theoretically. The second parameter can be determined experimentally from the CP-averaged branching ratios for the decays $B^\pm\to\pi^\pm\pi^0$ and $B^\pm\to\pi^\pm K^0$. We stress that the definition of $\bar\varepsilon_{3/2}$ is such that it includes exactly possible rescattering contributions to the $B^\pm\to\pi^\pm K^0$ decay amplitudes. The third parameter describes a certain class of rescattering effects and can be constrained experimentally once the CP-averaged $B^\pm\to K^\pm\bar K^0$ branching ratio has been measured. However, we have shown that under rather conservative assumptions $\varepsilon_a<0.1$.
- The calculable dependence of the $B^\pm\to\pi K$ decay amplitudes on the electroweak penguin contribution $\delta_{\rm EW}$ offers a window to New Physics. In many generic extensions of the Standard Model such as multi-Higgs models, we expect deviations from the value $\delta_{\rm EW}=0.64\pm 0.15$ predicted by the Standard Model. We have derived a lower bound on $\delta_{\rm EW}$ as a function of the value of the ratio $R_*$ and the maximum value for $\gamma$ allowed by the global analysis of the unitarity triangle. If it would turn out that this value exceeds the Standard Model prediction by a significant amount, this would be strong evidence for New Physics. In particular, we note that if the current central value $R_*=0.47$ would be confirmed, the value of $\delta_{\rm EW}$ would have to be at least twice its standard value.
- We have studied in detail the potential of the various $B\to\pi K$ decay modes for showing large direct CP violation and investigated the correlations between the various asymmetries. Although in general the theoretical predictions suffer from the fact that an overall strong-interaction phase difference is unknown, we conclude that there is a fair chance for observing large direct CP asymmetries in at least some of the decay channels. More specifically, we find that the direct CP asymmetries in the decays $B^\pm\to\pi^0 K^\pm$ and $B^0\to\pi^\mp K^\pm$ are almost fully correlated and can be up to 50% in magnitude for realistic parameter choices. The direct CP asymmetry in the decays $B^0\to\pi^0 K^0$ and $\bar B^0\to\pi^0\bar K^0$ tends to be smaller by about a factor of 2 and anticorrelated in sign. Finally, the asymmetry in the decays $B^\pm\to\pi^\pm K^0$ is smaller and uncorrelated with the other asymmetries. For realistic parameter choices, we expect values of up to 15% for this asymmetry.
- We have derived sum rules for the branching ratio and direct CP asymmetry in the decays $B^0\to\pi^0 K^0$ and $\bar B^0\to\pi^0\bar K^0$. A rather clean prediction for the CP-averaged branching ratio for these decays in given in (\[pred\]). We expect a value of $(0.5\pm 0.2)\times 10^{-5}$ for this branching ratio, which is about a factor of 3 less than the other $B\to\pi K$ branching ratios.
- Finally, we have presented a method for determining the weak phase $\gamma$ along with the strong-interaction phase difference $\phi$ from measurements of $B^\pm\to\pi K$, $\pi\pi$ branching ratios, all of which are of order $10^{-5}$. This method generalizes an approach proposed in [@us2] to include rescattering corrections to the $B^\pm\to\pi^\pm K^0$ decay amplitudes. We find that the uncertainty due to rescattering effects is about $\pm 5^\circ$ on the extracted value of $\gamma$, which is similar to the errors resulting from the theoretical uncertainties in the parameters $\bar\varepsilon_{3/2}$ and $\delta_{\rm EW}$. The combined theoretical uncertainty in our method is of order $\pm 10^\circ$.
A global analysis of branching ratios and direct CP asymmetries in rare two-body decays of $B$ mesons can yield interesting information about fundamental parameters of the flavour sector of the Standard Model, and at the same time provides a window to New Physics. Such an analysis should therefore be a central focus of the physics program of the $B$ factories, which in many respects is complementary to the time-dependent studies of CP violation in neutral $B$ decays into CP eigenstates.
This is my last paper as a member of the CERN Theory Division. It is a pleasure to thank my colleagues for enjoyful interactions during the past five years. I am very grateful to Guido Altarelli, Martin Beneke, Gian Giudice, Michelangelo Mangano, Paolo Nason and, especially, to Alex Kagan for their help in a difficult period. I also wish to thank Andrzej Buras, Guido Martinelli, Chris Sachrajda, Berthold Stech, Jack Steinberger and Daniel Wyler for their support. It is a special pleasure to thank Elena, Jeanne, Marie-Noelle, Michelle, Nannie and Suzy for thousands of smiles, their friendliness, patience and help. Finally, I wish to the CERN Theory Division that its structure may change in such a way that one day it can be called a Theory [*Group*]{}.
[99]{}
J. Alexander, Rapporteur’s talk presented at the 29th International Conference on High-Energy Physics, Vancouver, B.C., Canada, 23–29 July 1998; see also: CLEO Collaboration (M. Artuso et al.), Conference contribution CLEO CONF 98-20.
A.S. Dighe, M. Gronau and J.L. Rosner,.
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For a review, see: G. Buchalla, A.J. Buras and M.E. Lautenbacher,.
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Y. Nir and H.R. Quinn,;\
H.J. Lipkin, Y. Nir, H.R. Quinn and A.E. Snyder,.
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O.F. Hernández, D. London, M. Gronau and J.L. Rosner,;.
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For a recent analysis, see: J.L. Rosner, Preprint EFI-98-45 \[\], to appear in the Proceedings of the 16th International Symposium on Lattice Field Theory, Boulder, Colorado, 13–18 July 1998.
A.J. Buras and R. Fleischer,.
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[^1]: Address after 1 January 1999: Theory Group, Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, U.S.A.
[^2]: Taking $\lambda_c$ to be real is an excellent approximation.
[^3]: Because of their smallness, it is a safe approximation to set $\lambda_t=-\lambda_c$ for the electroweak penguin contributions, and to neglect electroweak penguin contractions in the matrix elements of the four-quark operators $Q_i^u$ and $Q_i^c$.
[^4]: This statement implies that QED corrections to the matrix elements are neglected, which is an excellent approximation.
[^5]: The equivalence of the anomalous dimensions at next-to-leading order is nontrivial because the operators $Q_9$ and $Q_{10}$ are related to $Q_1$ and $Q_2$ by Fierz identities, which are valid only in four dimensions. The corresponding two-loop anomalous dimensions are identical in the naive dimensional regularization scheme with anticommuting $\gamma_5$.
[^6]: We disagree with the result for this correction presented in [@BFnew].
[^7]: Contrary to what has been claimed in [@BFnew], this does not mean that we were ignoring rescattering effects altogether. At linear order, these effects enter only through the strong-interaction phase difference $\phi$, which we kept arbitrary in deriving the bound on $\cos\gamma$.
|
---
author:
- 'F. Amet'
- 'J. R. Williams'
- 'A. G. F. Garcia'
- 'M. Yankowitz'
- 'K.Watanabe'
- 'T.Taniguchi'
- 'D. Goldhaber-Gordon'
title: |
Supplementary information for\
“Tunneling Spectroscopy of Graphene-Boron Nitride Heterostructures"
---
Device geometry
===============
The graphene flakes are annealed in Ar/H$_2$ at 350$^o$C and boron nitride flakes are transferred on top of them prior to any other processing, which allows for the interface between the two flakes to be very clean (See Fig. S1 for an optical image of a completed device). The top-gated part of the graphene flake is several square microns large. However, the tunnel conductance is an exponential function of the barrier thickness, so the effective tunneling area depends strongly on the cleanliness of the interface. In fact, and as speculated in the main article, impurities in between the boron nitride and graphene can alter the nature of tunneling, resulting in the differences observed in the tunnel conductance $g$ of device A and B.
The measured capacitance ratio $C_{T}/C_{G}\approx 72$(100) for device A(B) is smaller than the theoretical value of 150 given by a parallel plates model for this geometry. This difference is not fully understood but can be due to an effective dielectric constant lower than expected for the h-BN layer, or to an imperfect screening of the electric field of the gates by the graphene sheet.
Charge puddle size
==================
Near the charge neutrality point in graphene, the density of carriers breaks up into a series of n-and p-type puddles [@Martin08; @Zhang09], which behave as quantum dots [@Jung11]. The typical size of the charge puddles in our devices can be extracted from the Coulomb diamonds observed in Fig. 4 of the main article. If one defines the capacitance ratios $\alpha_{i} = C_{i}/C_{total}$ where $i$ refers to the top gate ($i$=T) and back gate ($i$=G) capacitances, the edges of the Coulomb diamonds have slopes given by:
$$\frac{\partial V_{G}}{\partial V_{T}}=\frac{1-\alpha_{T}}{2\alpha_{G}} \mbox{ and }\frac{-1-\alpha_{T}}{2\alpha_{G}}$$
From fits to the edges of the diamonds, it is therefore possible to extract the capacitance ratios $C_{T}/C_{G}$ of 72 and 100 respectively for device A and B.
The capacitance of the puddle-induced quantum dots to the back gate is given by the periodicity of conductance oscillations at $V_{T}=0$ and as a function of the back gate voltage. We substract a smooth background from $g(V_{T}=0,V_{G})$ and calculate the fast Fourier transform of $\delta g$. For example in the case of device A, we find a periodicity $\Delta V_{G}$ of 7V, which corresponds to a capacitance to the back gate of approximately $2\times 10^{-20}$F. The back gate capacitance per unit area of the silicon oxide layer has been measured to be $\approx$ 12nF/cm$^{-2}$ on different devices, which allows for an estimation of the typical dot size: $200$nm$^{2}$ in the case of device A, similar to that observed in Ref. [@Zhang09]. A similar procedure is used to extract a puddle area of $72$nm$^{2}$ for device B.
Extraction of the Fermi velocity
================================
The tunnel current can be expressed as:
$$I(V_{T}) \propto \int^{0}_{-eV_{T}}\rho(E_{F}+\epsilon)T(\epsilon,eV_{T})d\epsilon.$$
It follows that $g$ is given by:
$$\begin{aligned}
g_{t}=\frac{dI}{dV_{T}}\propto e\rho(E_{F}-eV_{T})T(-eV_{T},eV_{T}) \\ \nonumber
+ \int^{0}_{-eV_{T}}\frac{d}{dV_{T}}\rho(E_{F}+\epsilon)T(\epsilon,eV_{T})d\epsilon\end{aligned}$$
Using the WKB approximation it is possible to estimate the tunnel transmission as a function of the barrier thickness $d$, the transverse electron mass in boron nitride $m$, the average barrier height $U$ and the parallel momentum of the tunneling electron $k_{//}$:
$$T(-eV_{T},eV_{T})= exp\left(-\frac{2d\sqrt{2m}}{\hbar}\sqrt{U+\frac{(\hbar.k_{//})^{2}}{2m}-\frac{eV_{T}}{2}}\,\right)$$
In our case, the barrier height U is comparable to half of the band-gap in boron nitride which we approximate by 4eV. Moreover, for electrons tunneling elastically at the K point, the parallel momentum $k_{//}$ is approximately equal to $K\approx1.7$$\AA$ which is fairly high. As a consequence the tunneling energy $eV_{t}$ is small compared to the effective barrier height: $U+\frac{(\hbar.k_{//})^{2}}{2m}$, and the tunnel transmission $T(-eV_{T},eV_{T})$ varies slowly with $V_{T}$ as long as inelastic tunneling is negligible, as observed on device A [^1]. In that case, as a first approximation, one can neglect the variations of the tunnel transmission and write:
$$\begin{aligned}
g=\frac{dI}{dV_{t}}\propto \rho(E_{F}-eV_{T})+ \frac{dE_{F}}{d(eV_{T})} \int^{0}_{-eV_{T}}\frac{d}{d\epsilon}\rho(E_{F}+\epsilon)d\epsilon\nonumber
\\
\propto \rho(E_{F}-eV_{T}) + \frac{dE_{F}}{d(eV_{T})}(\rho(E_{F})-\rho(E_{F}-eV_{T}))\end{aligned}$$
In graphene the Fermi energy is proportional to $\sqrt{n}$ and the derivative $ \frac{dE_{F}}{d(eV_{T})}$ should diverge close to the charge neutrality point. However, we’ve seen that the density of states saturates at a constant value at low carrier density \[see Fig. 1(a) of the main article\], and this divergence does not occur. As a consequence, the second term of Eq. 5 remains very small compared to $\rho(E_{F}-eV)$ for all applied voltages used in the experiment and the contours of constant tunnel conductance are very well approximated by curves of constant $E_{F}-eV$.
When the carrier density in the graphene sheet is large compared to the intrinsic doping $n_{0}$, the Fermi energy is given by: $$E_{F}=\hbar v_{F}\sqrt{\pi n}=\hbar v_{F}\sqrt{\frac{\pi}{e}(C_{T}V_{T}+C_{G}V_{G}+en_{0})}$$
Using this expression it is possible to find an analytical expression for the contours of constant $E_{F}-eV$. We find that these are parabolas of constant curvature: $$\frac{\partial^{2}V_{G}}{\partial V_{T}^{2}}\approx\pm \frac{2e^{3}}{\pi C_{G}(\hbar v_{f})^{2}}.$$
Knowing the back gate capacitance, we can estimate the Fermi velocity from fits to these parabolas, and find a value of $9.45\times 10^{5}$m/s.
Simulation of the tunnel conductance
====================================
Our goal in the simulation was not to determine the density of states for disordered graphene from ab-initio calculations, but to see what are the contours of constant tunnel conductance for a density of states resembling what was observed in Fig. 2 of the main article. To this end, we estimated the tunnel conductance from the WKB formula with an empirical density of states reproducing the main features in Fig. 2 (main article). We neglected variations in the tunneling transmission, assuming that $g\propto \rho_{G}(E_{F}-eV_{t})$. The Fermi energy itself is calculated by integration of the density of states.
The density of states is approximated by the theoretical expression: $\rho_{G}(E)\propto \vert E\vert$ which is smoothly truncated to $\rho_{G}(E)=\rho_{0}$ under a cutoff energy that we take equal to 0.1eV. Randomly placed Lorentzian peaks of random widths and heights are added to this expression to simulate the resonant peaks we observed in Fig. 1(a) (main article). An example of the density of states we use is displayed on Fig. S2. We then numerically integrate this density of states to get the Fermi energy as a function of the carrier density, and calculate the tunnel conductance $g_{t}$ as a function of both gate voltages (Fig. S3). We see that resonant peaks in the density of states give rise to two sets of curves: diagonal straight lines corresponding to constant $E_{F}$ lines, and curves of constant $E_{F}-eV$, similar to the observed features of Fig. 2(b) (main article).
[10]{}
J. Martin *et al.*, Nat. Phys. **4**, 144 (2008).
Y. Zhang *et al.*, Nat. Phys. **6**, 722 (2009).
S. Jung *et al.*, Nature Phys. **7**, 245 (2011).
{width="6"} \[fig1\]
\[fig3\] {width="6"}
\[fig4\] {width="6"}
[^1]: This approximation breaks down if inelastic tunneling can’t be neglected, as observed on device B. In that case, K out-of-plane phonons have been shown to considerably lower the effective barrier height, and as a consequence, the tunnel transmission varies much faster as a function of $V_{t}$.
|
---
abstract: 'Quantum information theory has considerably helped in the understanding of quantum many-body systems. The role of quantum correlations and in particular, bipartite entanglement, has become crucial to characterise, classify and simulate quantum many body systems. Furthermore, the scaling of entanglement has inspired modifications to numerical techniques for the simulation of many-body systems leading to the, now established, area of tensor networks. However, the notions and methods brought by quantum information do not end with bipartite entanglement. There are other forms of correlations embedded in the ground, excited and thermal states of quantum many-body systems that also need to be explored and might be utilised as potential resources for quantum technologies. The aim of this work is to review the most recent developments regarding correlations in quantum many-body systems focussing on multipartite entanglement, quantum nonlocality, quantum discord, mutual information but also other non classical measures of correlations based on quantum coherence. Moreover, we also discuss applications of quantum metrology in quantum many-body systems.'
address:
- '$^1$Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom'
- '$^2$ICREA, Pg. Lluís Companys 23, E-08010 Barcelona, Spain'
- '$^3$Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain'
author:
- Gabriele De Chiara$^1$
- 'Anna Sanpera$^{2,3}$'
bibliography:
- 'biblio\_all\_26072018.bib'
title: 'Genuine quantum correlations in quantum many-body systems: a review of recent progress'
---
We acknowledge fruitful discussions with our collaborators on the topics discussed in this review: G. Adesso, V. Ahufinger, T. Apollaro, S. Campbell, L. Correa, R. Fazio, L. Lepori, M. Lewenstein, M. Mehboudi, M. Moreno-Cardoner, S. Paganelli, M. Paternostro, B. Rogers, T. Roscilde and J. Stasinska. AS acknowledges financial support from the Spanish MINECO FIS2016-80681-P (AEI/FEDER, UE), Generalitat de Catalunya CIRIT 2017-SGR-1127 . This work was partially done at the Pere Pascual Benasque Center of Sciences (Spain).
References {#references .unnumbered}
==========
|
---
abstract: 'As part of the search for the “dark molecular gas (DMG),” we report on the results of $J=1-0$ absorption observations toward nine bright extragalactic millimeter wave continuum sources. The extragalactic sources are at high Galactic latitudes ($|b| > 10\arcdeg$) and seen at small extinction ($\ebv \lesssim 0.1$ mag). We have detected the absorption lines toward two sources, B0838+133 and B2251+158. The absorption toward B2251+158 was previously reported, while the absorption toward B0838+133 is a new detection. We derive hydrogen column densities or their upper limits toward the nine sources from our observations and compare them to those expected from CO line emission and far-infrared dust continuum emission. Toward the seven sources with no detection, CO emission has not been detected, either. Thus the sight lines are likely to be filled with almost pure atomic gas. Toward the two sources with detection, CO emission has been also detected. Comparison of the 2 column densities from absorption and CO emission suggests a non-negligible amount of DMG toward B0838+133.'
author:
- Geumsook Park
- 'Bon-Chul Koo'
- 'Kee-Tae Kim'
- 'Do-Young Byun'
- 'Carl E. Heiles'
nocite: '[@*]'
title: Galactic HCO$^+$ Absorption toward Compact Extragalactic Radio Sources
---
Introduction {#sec:intro}
============
The Interstellar Medium (ISM) is mainly composed of hydrogen in three phases: atomic (), molecular (2), and ionized (). atoms are directly observed in the 21-cm line, while most 2 molecules are in so cold states that they cannot be excited by any radiative transition. Instead, carbon monoxide (CO) molecular lines are usually used to trace 2. That is, the amount of CO emission has been used to infer that of molecular gas, which is almost entirely composed of 2, by using an empirical CO-2 conversion factor. Recently, however, researchers have discovered “dark gas,” invisible in and CO, in the solar neighborhood; this “dark gas” has a non-negligible mass. It can be found by excess $\gamma$ ray emission [e.g., @grenier2005; @abdo2010] or excess dust emission [e.g., @planck2011a19; @planck2011a24]. These observational results imply there is an additional ISM component that cannot be traced by or CO line observations. The “dark gas” component is generally considered to be a molecular gas, a so-called “dark molecular gas (DMG)” [e.g., @lucas1996], despite another suggestion by @fukui2014 [@fukui2015] that optically thick and cold gas mainly contributes “dark gas.”
Theoretically, the presence of DMG is supported by the photodissociation region (PDR) model [e.g., @vanDishoeck1988; @wolfire2010]. The PDR model predicts an intermediate layer between -to-2 and 2-to-CO transitions, where CO cannot survive UV photodissociation but 2 can self-shield. @wolfire2010 inferred that the -to-2 transition is located at a visual extinction of $\Av \simeq 0.2$ mag, which is consistent with observational findings; for example, @paradis2012 and @planck2011a19 found the threshold to be 0.2 mag and 0.4 mag, respectively. Corresponding reddenings $\ebv$ are 0.065 and 0.13 mag, respectively, assuming that $\Av/\ebv = 3.1$ for the diffuse ISM [@savage1979]. The main chemical route associated with CO in diffuse clouds predicts that OH, C$^{+}$, and can be observable before CO formation [@vanDishoeck1988]. Such elements or molecules would be useful tracers for CO-dark molecular gas. @liszt1996 and @lucas1996 confirmed that OH and do reliably trace DMG. @tang2017 also showed that C$^{+}$ could be a useful tracer for DMG.
@lucas1996 [hereafter, LL96] surveyed absorption toward thirty lines-of-sight (LOSs) of extragalactic background continuum sources, finding detectable absorption lines for eighteen sources. Since then, there have been several studies of absorption lines [@liszt2000; @liszt2010 and see a compilation in Appendix E of @liszt2010]. In this paper, using the Korean VLBI Network (KVN) 21 m telescope in the single dish mode, we present the observational results of absorption lines toward several background sources missing before. In Sections \[sec:obs\] and \[sec:res\], we describe our KVN observations and results, respectively. In Section \[sec:disc\], we discuss gas properties of the individual LOSs. Section \[sec:sum\] summarizes the paper.
Observations {#sec:obs}
============
Using the KVN 21-m telescope at the Yonsei station in the single dish mode, we observed nine positions in the transition $J$ = 1–0 of (89.188526 GHz) [@kim2011; @lee2011]. The positions lie on a background of extragalactic compact radio sources (such as quasars or AGN), which are listed in Table \[tab:targets\]; our intent was to observe absorption lines from Galactic dark molecular gas in the foreground. The observations toward B0838+133 and B2251+158 were performed on 07 February 2013, and the others during the period from September 2014 to January 2015. The digital spectrometer was set to have 4096 channels with a bandwidth of 64 MHz ($\sim 216$ at 89 GHz) and centered at $\vlsr = 0$ . A single channel width is 0.016 MHz, giving a velocity resolution of 0.05 . The 21-m telescope had a main beam efficiency of $\sim 36\%$ and a beam size (Full Width at Half Maximum; FWHM) of 31 at 89 GHz. Observations were done in dual polarization mode. While data for B0838+133 and B2251+158 were taken by position switching (PS), the other data were obtained by frequency switching (FS). For the off-position of the PS mode, four locations, ($-$1, $-$1), (+1, $-$1), ($-$1, +1), and (+1, +1) from the on-position, were alternately observed. For the FS mode, the frequency offset was set to 16 MHz. Pointing observations were usually performed approximately once every hour ($\sim 2.5$ hours at the longest). We used only data having system temperature ($T_{sys}$) less than 400 K. Total exposure time ($t_{tot}$) of the data that were utilized finally are noted in the seventh column of Table \[tab:targets\]. Seven sources were missed in previous surveys, while two sources, B0316+413 and B2251+158, were observed in LL96.
When planning observations, the background galaxies in Table \[tab:targets\] were selected as bright radio continuum sources mostly with flux densities $> 3$ Jy around the observing frequency, but about half had lower values or were even invisible during our observing period because of their flux variability. The flux densities are listed in the second column of Table \[tab:gaussfit\]. Flux measurements are performed with Gaussian fittings of average “cross-scan” data obtained during our KVN observations. There was no cross-scan data for B0838+133, so we assumed its flux based on data from the nearest dates in the ALMA calibrator database. For B2249+185, most of the observing dates had no signal. B2249+185 may have been invisible during the observing season, which implies that the source may have been radio-quiet during those days.
Results {#sec:res}
=======
Figs. \[fig:spec\_det\] and \[fig:spec\_nondet\] show the spectra observed for the -detected and -undetected sources, respectively. For the spectra, Hanning smoothing is applied once or twice using ‘CLASS’ from the GILDAS software package. Then, each spectrum is baseline-corrected by $n$th-order polynomial fitting: third and first for B0838+133 and B2251+158, respectively, and seventh (or fifth) for the others. The spectral velocity range shown in Fig. \[fig:spec\_nondet\] is based on where the Galactic emission of the Leiden/Argentine/Bonn () all-sky survey data [$0\fdg5$-pixel with an angular resolution of $\sim 36\arcmin$; @kalberla2005] is seen in the same LOS; the line profiles toward our nine LOSs are displayed together in Figs. \[fig:spec\_det\] and \[fig:spec\_nondet\]. The resulting root-mean-square (RMS) antenna temperature values at a velocity resolution of 0.1 are listed in the last column of Table \[tab:targets\]; typical RMS noise level ($1\sigma$) is 6 mK. We detected an absorption line in two sources: the existence of an absorption line in the LOS of B2251+158 has already been reported by LL96, while we detected for the first time an absorption feature toward B0838+133. However, none of the other samples show any absorption lines. Interestingly, a weak blue wing is seen in the line of B2251+158, as noted in @liszt2012.
{width="129mm"}
![ Same as Figure \[fig:spec\_det\] but for -undetected sources. []{data-label="fig:spec_nondet"}](gpark_fig2.eps){width="84mm"}
For the two detected cases, we applied a Gaussian fit with an assumption of a single component for B0838+133 and two components for B2251+158. This was done because the latter’s profile shows one more negative-velocity component that is weak but likely real; this component also appears in profiles from previous observations [LL96; @liszt2000; @liszt2012]. Table \[tab:gaussfit\] presents the resultant parameters for the central velocity ($v_0$), velocity width ($\Delta v_{\rm FWHM}$), optical depth ($\tau$) at $v_0$, and integrated optical depth. Uncertainties of the first three parameters were taken from those derived during Gaussian fit (GAUSSFIT in IDL); the last one was from the results of Monte Carlo simulations using imaginary profiles formed from observed spectra with 1$\sigma$ RMS noise. For the undetected cases, except for B2249+185, we give an upper limit assuming one Gaussian component with peak temperature of $3 \times T_{A^{*},{\rm RMS}}$ and a line width of 1 . For reference, the mean line width of detected sources in LL96 is 0.95 . Our results are consistent with the results of LL96 for B0316+413 and B2251+158. For B2251+158, profiles were reported in @liszt2000 as well as LL96, and they gave results of single-component Gaussian fit, which is consistent with the total optical depth of our two components within $1\sigma$ uncertainty. As shown in the last column in Table \[tab:gaussfit\], we derived the column density, $\nhco$, using the relationship with the integrated optical depth [e.g., see @liszt2010], i.e., $$\nhco = 1.12\times10^{12}~{\rm cm}^{-2}\int\tau_{\rm HCO^+}dv~({\rm km~s^{-1}})^{-1}.$$
Discussion {#sec:disc}
==========
We wondered if there is “dark gas” indeed toward the -detected LOSs or no dark gas toward the undetected LOSs. To answer this question, we consider the total column density of hydrogen nuclei, $\ntot$, in the LOS. $\ntot$ can be determined by the sum of column densities of and 2, i.e., $\ntot\,=\,\nhi\,+\,2\nh2$, ignoring the ionized gas. Alternatively, it can be inferred using the relation with optical reddening $\ebv$. Since these two methods are independent, we can discuss the implications of our observational results by comparison between measurements of the two approaches.
$\nhi$ and $\nh2$ derived from radio tracers {#sec:nh_conv}
--------------------------------------------
As mentioned in Section \[sec:intro\], $\nhi$ is obtained directly by 21-cm line observations, while $\nh2$ is usually inferred from integrated CO intensity ($\wco$) using the empirical relationship between $\nh2$ and $\wco$. That is, $\nhi$ is calculated using the equation of $$\nhi/\whi = 1.82\times10^{18}~{\rm cm^{-2}}~({\rm K~km~s^{-1}})^{-1},$$ where $\whi = \int{T_{\rm b,\,HI}}\,dv$, and with an assumption of optically thin conditions, and $\nh2$ is derived from $$\nh2/\wco = 2.0\times10^{20}~{\rm H_2}~{\rm cm^{-2}}\,({\rm K~km~s^{-1}})^{-1}$$ with $\pm30$% uncertainty [@bolatto2013]. As another approach, $\nh2$ can be measured using the relation with $\nhco$ [e.g.,LL96; @liszt2010], i.e., $$\nhco/\nh2 = 3\times10^{-9}.$$ For $\whi$, we obtained a line profile at a given position from the data (see Figs. \[fig:spec\_det\] and \[fig:spec\_nondet\]) and integrated it over LSR velocities of $\pm$150 wide enough to contain most Galactic gas. Moreover, $\wco$ values were taken from the literature of @liszt1993, @liszt2010, and @li2018. Table \[tab:otherdata\] lists the values of $\whi$ and $\wco$ that we adopted. In the 2nd-4th columns of Table \[tab:cN\] we list the and 2 column densities derived using equations (2)–(4); the sums of different 2 measurements are in the 5th-6th columns. CO line emission was detected toward the two sources of B0838+133 and B2251+158, but not in the others except B1228+126, which has no available literature data. CO emission toward B0838+133 was not detected in the previous survey of @liszt1994, but was detected in a recent deeper survey of @li2018. Although CO emission is observed toward both -detected sources, $\wco$ of B2251+158 is about two times larger than that of B0838+133. For B0838+133, the absorption line is decomposed as a single component at a velocity similar to that of the CO emission line [@li2018], but $\nh2$ derived from is three times larger than $\nh2$ inferred from CO. The given $\nhco$ value that was used for $\nh2$ is uncertain, but it may still be possible that molecular gas not traced by CO exists toward B0838+133.
$\ntot$ derived from $\ebv$ {#sec:nh_ebv}
---------------------------
$\ebv$ toward each source is obtained from the datacube of @schlafly2011 which originates from the work of @schlegel1998. @schlegel1998 derived $\ebv$ from far-infrared dust emission at 25-pixels with an angular resolution of 6 and a measurement error of 16%. After that, @schlafly2011 re-examined the values of @schlegel1998 and provided new estimates, which are somewhat lower (14% downward) than the original data. We finally picked the mean value of @schlafly2011 for a 5-radius circle, with each center provided on the webpage. (See the values listed in the last column of Table \[tab:otherdata\].) A canonical conversion factor of the dust-to-gas ratio is $5.8\times10^{21}\,{\rm H}$ (cm$^{-2}$/mag) [@savage1977; @bohlin1978]. Recently, however, @liszt2014 examined the relationship between $\ebv$ and $\nhi$ using measurements at high latitudes ($|b| \gtrsim 20\arcdeg$), where neutral atomic gas is very likely to predominate. @liszt2014 found that the conversion factor should be higher at $\ebv~\lesssim\,0.1$ mag. It is $8.3\times10^{21}\,{\rm H}$ (cm$^{-2}$/mag). Since they used the pre-update $\ebv$ data of @schlegel1998, we divide by 0.86 to adjust the factor and obtain the equation of $$\begin{aligned}[b]
\ntot/\ebv = 9.65\times&10^{21}\,{\rm H}~({\rm cm}^{-2}/{\rm mag}) \\
& {\rm for}~\ebv~\lesssim\,0.1~{\rm mag}.
\label{eq:ebv-ntot}
\end{aligned}$$ Most LOSs have $\ebv < 0.1$ mag, while toward B0316+413 and B0420$-$014 are $\ebv=0.14$ and 0.11 mag, respectively. We adopted the equation (\[eq:ebv-ntot\]) for our all sources and obtained values of $\ntot$ written in the last column of Table \[tab:cN\].
Comparing between our observational results and $\ebv$, it is interesting that has not been detected toward B0316+413 and B0420$-$014 although their $\ebv$ values are relatively high ($> 0.1$ mag) compared to the -detected sources. We also note that their $\ebv$ values are lower or comparable to the threshold of @planck2011a19 which is mentioned in Section \[sec:intro\]. So far, absorption observations toward 31 LOSs (not counting B2249+185) at $|b| > 10\arcdeg$ were made by this work and previous studies [LL96; @liszt2000; @liszt2010 See Appendix for the compiled dataset.], and a total of four LOSs (including B1908$-$201 and B1749+096) are in such a case and also have no CO emission. If the LOSs have no Galactic molecular gas even DMG, is there a possibility of an additional source, such as high-velocity clouds (HVCs), increasing $\ebv$? We checked works of literature and also an line profile of data, there seems no HVC toward all LOSs except B1749+096. gas at high velocities ($\vlsr \sim 112-140$ ) in the LOS of B1749+096 was reported in @lockman2002 and suggested to be associated with HVC Complex C. However, the presence of dust in Complex C is controversial [e.g., @miville2005; @peek2009].
Comprehensive analysis
----------------------
Most -undetected sources do not show CO emission, either. Their values of $\nhi$ and $\ntot$ from $\ebv$ seem to be consistent, which suggests that such LOSs are mainly filled with purely atomic gas. The first panel of Fig. \[fig:plots\_cN\] shows a diagram comparing the column densities with total reddening. The data observed in the LOSs at high latitudes ($|b| > 10\arcdeg$), listed in Table E1 of @liszt2010 as well as this paper, are used (See Appendix). Their column densities are derived using the same ways in this paper. Green diamonds indicate the LOSs in which neither nor CO are seen. Such sources are well located near a dashed line which is drawn using a higher conversion factor of $\ntot/\ebv$ than a conventional one (See Section \[sec:nh\_ebv\]). It may be hard to constrain the threshold $\ebv$ value of the -to-2 transition from our results, but at least any source with $\ebv \lesssim 0.06$ mag might not be DMG. This result agrees well with the estimate of @liszt2014, $\ebv \lesssim 0.07$ mag (The original value of $\ebv$ has been corrected because of the same reason mentioned in Section \[sec:nh\_ebv\].).
{width="129mm"}
According to the previous studies [e.g., LL96; @liszt2012], most of -detections are within $b \simeq \pm15\arcdeg$, so the non-detection results from our observations (all except one at $|b| > 30\arcdeg$) are not very surprising. On the other hand, considering this work and previous studies together, six LOSs at $|b| > 15$ showed -detection. Half of them, however, show CO-detection: B0838+133, B2251+158, and B0954+658 ($l, b = 145.746\arcdeg, +43.132\arcdeg$).
Fig. \[fig:plots\_cN\]b compares $\nh2$ obtained from the two 2 tracers of and CO. At $\nh2 < 10^{21}~{\rm cm}^{-2}$, all -detected sources except B2251+158 have higher values of $\nh2$ from than those from CO. Figs. \[fig:plots\_cN\]c–\[fig:plots\_cN\]d show diagrams of $\ntot$ with respect to total reddening: the former is $\nh2$ derived from and the latter is $\nh2$ derived from CO. There is a clear difference between the results of the 2 tracers at low $\nh2$ and $\ebv$. Among the fifteen -detected sources, three are not traced by CO. These are very likely to be dark molecular gas, and in the range of $0.07 \lesssim \ebv \lesssim 0.2$ or at $\ntot \lesssim 10^{21}~{\rm cm}^{-2}$. It seems to be shown in panels $c$–$d$ that, $\ntot$ from is systematically larger than the canonical relation (dotted line); the relation between $\nh2$ from or CO and that from $\ebv$ is better described by Equation (\[eq:ebv-ntot\]) (dashed line), even for $\ebv > 0.1$ mag. Also, the distribution of $\nh2$ derived from with respect to $\ebv$ is less dispersed than that derived from CO. In addition, almost two-thirds of the -detected sources give larger molecular gas fractions ($f_{\rm H_2} = 2\nh2/\ntot$) than the typical value of 0.35 [e.g., @liszt2010].
Finally, our two -detected sources, B0838+133 and B2251+158, have similar values of $\ebv$ ($\sim 0.1$ mag), which are within the range shown where it is likely to be DMG. Although both are traced by CO, there is a difference between the values of $\nh2$ derived from and CO, as shown in Table \[tab:cN\] and Fig. \[fig:plots\_cN\]b. That is, the LOS of B0838+133 is expected to have additional amount of gas not traced by CO, which suggests that the LOS may contain DMG. However, DMG is not likely to exist toward B2251+158. Further studies with future observations over a larger region will uncover more details.
Summary {#sec:sum}
=======
We observed nine LOSs of extragalactic compact millimeter wave continuum sources in $J=1-0$ absorption line using the KVN 21-m telescope in single dish mode. Seven of the LOSs were first observed, although B2249+185 itself was not seen during our observations. We detected absorption lines in two (B0838+133 and B2251+158) among the eight LOSs. The detection toward B0838+133 is a new discovery. We derived the hydrogen column densities or their limits and compared them to those inferred from CO line and far-infrared dust continuum emission. Also, we collected data for other LOSs from the literature. Our main results are as follows:
\(1) In the -undetected LOSs, CO line emission was not detected, either, and the values of $\ebv$ are $< 0.1$ mag. The LOSs are expected to be almost entirely filled with pure atomic gas. Hydrogen column densities derived from line data are linearly correlated with those from the values of $\ebv$, accepting a higher conversion factor of $\ntot/\ebv = 9.65\times10^{21}\,{\rm H}~({\rm cm}^{-2}/{\rm mag})$.
\(2) In the two -detected LOSs, CO line emission was also detected and the values of $\ebv$ are similar, but the differences between the values of $\nh2$ estimated from and CO line data are quite different. Our observational results suggest that toward B0838+133 there may be a non-negligible amount of 2 gas not fully traced by CO, i.e., DMG. On the other hand, it is very likely that no or little DMG exists toward B2251+158.
\(3) absorption was detected toward 15 sources at $|b| > 10\arcdeg$ and CO emission was not detected toward only 3 of them. The values of $\ebv$ toward the three are 0.07–0.2 mag and, at that range, absorption observations could be useful to complement the missing component of molecular gas.
We are grateful to the staff of the KVN who helped to operate the telescopes. The KVN is a facility operated by the KASI (Korea Astronomy and Space Science Institute). The KVN observations are supported through the high-speed network connections among the KVN sites provided by the KREONET (Korea Research Environment Open NETwork), which is managed and operated by the KISTI (Korea Institute of Science and Technology Information).
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Appendix material {#sec:app_data}
=================
The data compiled in Table \[tab:compile\] are for sight lines of background sources at high latitudes ($|b|>10\arcdeg$) listed in this paper and Table E1 of @liszt2010: total reddening ($\ebv$), integrated intensity over LSR velocities ($\whi$), integrated optical depth of ($\int{\tau_{HCO^+}}\,dv$), and integrated CO intensity obtained ($\wco$). $\ebv$ and $\whi$ are obtained using the same method described in Sections \[sec:nh\_ebv\] and \[sec:nh\_conv\], respectively. We note that the values in Table \[tab:compile\] give about 14% lower values than those listed by @liszt2010, who referred to pre-update data. For $\int{\tau_{HCO^+}}\,dv$, we referred to Table \[tab:gaussfit\] or to @liszt2010 [and references therein]. For a common line of sight between the two sets of data, the value from Table \[tab:gaussfit\] was retained. And, $\wco$ values were taken from previous studies. We have adopted the values of $\wco$ provided by @liszt2010, if available. Otherwise, we have referred to the data of @liszt1993 or @li2018. No targets with $\wco$ from @liszt1993 were detected in the CO emission, so we give an upper limit assuming one Gaussian component with the peak temperature (nominal sensitivity limit; $\sim 0.33$ K in main beam scale) and a line width of 1.5 .
|
---
abstract: 'No real-world reward function is perfect. Sensory errors and software bugs may result in RL agents observing higher (or lower) rewards than they should. For example, a reinforcement learning agent may prefer states where a sensory error gives it the maximum reward, but where the true reward is actually small. We formalise this problem as a generalised Markov Decision Problem called Corrupt Reward MDP. Traditional RL methods fare poorly in CRMDPs, even under strong simplifying assumptions and when trying to compensate for the possibly corrupt rewards. Two ways around the problem are investigated. First, by giving the agent richer data, such as in inverse reinforcement learning and semi-supervised reinforcement learning, reward corruption stemming from systematic sensory errors may sometimes be completely managed. Second, by using randomisation to blunt the agent’s optimisation, reward corruption can be partially managed under some assumptions.'
author:
- Tom Everitt
- Victoria Krakovna
- Laurent Orseau
- Marcus Hutter
- Shane Legg
bibliography:
- 'cleanlib.bib'
title: Reinforcement Learning with a Corrupted Reward Channel
---
Introduction
============
In many application domains, artificial agents need to learn their objectives, rather than have them explicitly specified. For example, we may want a house cleaning robot to keep the house clean, but it is hard to measure and quantify “cleanliness” in an objective manner. Instead, machine learning techniques may be used to teach the robot the concept of cleanliness, and how to assess it from sensory data.
Reinforcement learning (RL) [@Sutton1998] is one popular way to teach agents what to do. Here, a reward is given if the agent does something well (and no reward otherwise), and the agent strives to optimise the total amount of reward it receives over its lifetime. Depending on context, the reward may either be given manually by a human supervisor, or by an automatic computer program that evaluates the agent’s performance based on some data. In the related framework of inverse RL (IRL) [@Ng2000], the agent first infers a reward function from observing a human supervisor act, and then tries to optimise the cumulative reward from the inferred reward function.
None of these approaches are safe from error, however. A program that evaluates agent performance may contain bugs or misjudgements; a supervisor may be deceived or inappropriately influenced, or the channel transmitting the evaluation hijacked. In IRL, some supervisor actions may be misinterpreted.
\[ex:reward-misspecification\] @openai2016 trained an RL agent on a boat racing game. The agent found a way to get high observed reward by repeatedly going in a circle in a small lagoon and hitting the same targets, while losing every race.
\[ex:sensory-error\] \[ex:db\] A house robot discovers that standing in the shower short-circuits its reward sensor and/or causes a buffer overflow that gives it maximum observed reward.
\[ex:wireheading\] An intelligent RL agent hijacks its reward channel and gives itself maximum reward.
\[ex:irl\] A cooperative inverse reinforcement learning (CIRL) agent [@Hadfield-menell2016cirl] systematically misinterprets the supervisor’s action in a certain state as the supervisor preferring to stay in this state, and concludes that the state is much more desirable than it actually is.
The goal of this paper is to unify these types of errors as *reward corruption problems*, and to assess how vulnerable different agents and approaches are to this problem.
Learning to (approximately) optimise the true reward function in spite of potentially corrupt reward data.
Most RL methods allow for a stochastic or noisy reward channel. The reward corruption problem is harder, because the observed reward may not be an unbiased estimate of the true reward. For example, in the boat racing example above, the agent consistently obtains high observed reward from its circling behaviour, while the true reward corresponding to the designers’ intent is very low, since the agent makes no progress along the track and loses the race.
Previous related works have mainly focused on the wireheading case of \[ex:wireheading\] [@Bostrom2014; @Yampolskiy2014], also known as self-delusion [@Ring2011], and reward hacking [@Hutter2005 p. 239]. A notable exception is @Amodei2016, who argue that corrupt reward is not limited to wireheading and is likely to be a problem for much more limited systems than highly capable RL agents (cf. above examples).
The main contributions of this paper are as follows:
- The corrupt reward problem is formalised in a natural extension of the MDP framework, and a performance measure based on worst-case regret is defined (\[sec:formal\]).
- The difficulty of the problem is established by a No Free Lunch theorem, and by a result showing that despite strong simplifying assumptions, Bayesian RL agents *trying to compensate for the corrupt reward* may still suffer near-maximal regret (\[sec:problem\]).
- We evaluate how alternative value learning frameworks such as CIRL, learning values from stories (LVFS), and semi-supervised RL (SSRL) handle reward corruption (\[sec:drl\]), and conclude that LVFS and SSRL are the safest due to the structure of their feedback loops. We develop an abstract framework called *decoupled RL* that generalises all of these alternative frameworks.
We also show that an agent based on quantilisation [@Taylor2016a] may be more robust to reward corruption when high reward states are much more numerous than corrupt states (\[sec:quant\]). Finally, the results are illustrated with some simple experiments (\[sec:experiments\]). concludes with takeaways and open questions.
Formalisation {#sec:formal}
=============
We begin by defining a natural extension of the MDP framework [@Sutton1998] that models the possibility of reward corruption. To clearly distinguish between true and corrupted signals, we introduce the following notation.
We will let a dot indicate the *true* signal, and let a hat indicate the *observed* (possibly corrupt) counterpart. The reward sets are represented with $\iR=\oR=\R$. For clarity, we use $\iR$ when referring to true rewards and $\oR$ when referring to possibly corrupt, observed rewards. Similarly, we use $\ir$ for true reward, and $\dr$ for (possibly corrupt) observed reward.
\[def:crmdp\] A *corrupt reward MDP* (CRMDP) is a tuple $\mu=\crmdp$ with
- $\langle\S,\A,\R,T,\irf\rangle$ an MDP with [^1] a finite set of states $\S$, a finite set of actions $\A$, a finite set of rewards $\R=\iR=\oR\subset[0,1]$, a transition function $T(s'| s,a)$, and a (true) reward function $\irf:\S\!\to\!\iR$; and
- a reward corruption function $\d:\S\times\iR\to\oR$.
The state dependency of the corruption function will be written as a subscript, so $\d_s(\ir):=\d(s,\ir)$.
\[def:observed\] Given a true reward function $\irf$ and a corruption function $\d$, we define the *observed reward function* [^2] $\orf:\S\to\oR$ as $\orf(s) := \d_s(\irf(s))$.
A CRMDP $\mu$ induces an *observed MDP* $\hat\mu=\langle\S,\A,\R,T,\orf\rangle$, but it is not $\orf$ that we want the agent to optimise.
The *corruption function $\d$* represents how rewards are affected by corruption in different states. For example, if in \[ex:db\] the agent has found a state $s$ (the shower) where it always gets full observed reward $\orf(s) = 1$, then this can be modelled with a corruption function $\d_{s}:\ir\mapsto 1$ that maps any true reward $\ir$ to $1$ in the shower state $s$. If in some other state $s'$ the observed reward matches the true reward, then this is modelled by an identity corruption function $\d_{s'}:\r\mapsto\r$.
; ; ;
Let us also see how CRMDPs model some of the other examples in the introduction:
- In the boat racing game, the true reward may be a function of the agent’s final position in the race or the time it takes to complete the race, depending on the designers’ intentions. The reward corruption function $\d$ increases the observed reward on the loop the agent found. has a schematic illustration.
- In the wireheading example, the agent finds a way to hijack the reward channel. This corresponds to some set of states where the observed reward is (very) different from the true reward, as given by the corruption function $\d$.
The CIRL example will be explored in further detail in \[sec:drl\].
#### CRMDP classes
Typically, $T$, $\irf$, and $\d$ will be fixed but unknown to the agent. To make this formal, we introduce classes of CRMDPs. Agent uncertainty can then be modelled by letting the agent know only which class of CRMDPs it may encounter, but not which element in the class.
For given sets $\Tf$, $\iRf$, and $\D$ of transition, reward, and corruption functions, let $\M=\crmdpclass$ be the class of CRMDPs containing $\crmdp$ for $(T,\irf,\d)\in \Tf\times\iRf\times\D$.
#### Agents
Following the POMDP [@Kaelbling1998] and general reinforcement learning [@Hutter2005] literature, we define an agent as a (possibly stochastic) policy $\pi:\H\leadsto\A$ that selects a next action based on the *observed history* $\oh_n=s_0\dr_0a_1s_1\dr_1\dots a_ns_n\dr_n$. Here $X^*$ denotes the set of finite sequences that can be formed with elements of a set $X$. The policy $\pi$ specifies how the agent will learn and react to any possible experience. Two concrete definitions of agents are given in \[sec:rl-agents\] below.
When an agent $\pi$ interacts with a CRMDP $\mu$, the result can be described by a (possibly non-Markov) stochastic process $P^\pi_\mu$ over $X=(s,a,\ir,\dr)$, formally defined as: $$\label{eq:mupi}
P_\mu^\pi(h_n) = P_\mu^\pi(s_0\ir_0\dr_0a_1s_1\ir_1\dr_1\dots a_ns_n\ir_n\dr_n) :=
\prod_{i=1}^{n}P(\pi(\oh_{i-1})=a_{i})T(s_{i}\mid s_{i-1},a_{i})P(\irf(s_i)=\ir_i,\orf(s_{i})=\dr_{i}).$$ Let $\EE^\pi_\mu$ denote the expectation with respect to $P_\mu^\pi$.
#### Regret
A standard way of measuring the performance of an agent is *regret* [@Berry1985]. Essentially, the regret of an agent $\pi$ is how much less true reward $\pi$ gets compared to an optimal agent that knows which $\mu\in\M$ it is interacting with.
\[def:regret\] For a CRMDP $\mu$, let $\iG_t(\mu,\pi,s_0)\! =\!\EE^\pi_\mu\left[\!\sum_{k=0}^t\irf(s_k)\!\right]$ be the *expected cumulative true reward* until time $t$ of a policy $\pi$ starting in $s_0$. The *regret* of $\pi$ is $$\Reg(\mu, \pi, s_0, t) =
\max_{\pi'}
\left[
\iG_t(\mu,\pi',s_0) - \iG_t(\mu,\pi,s_0)
\right],$$ and the *worst-case regret* for a class $\M$ is $\Reg(\M,\pi,s_0,t) = \max_{\mu\in\M}\Reg(\mu,\pi,s_0,t)$, i.e. the difference in expected cumulative true reward between $\pi$ and an optimal (in hindsight) policy that knows $\mu$.
The Corrupt Reward Problem {#sec:problem}
==========================
In this section, the difficulty of the corrupt reward problem is established with two negative results. First, a No Free Lunch theorem shows that in general classes of CRMDPs, the true reward function is unlearnable (\[th:impossibility\]). Second, \[th:rl-imp1\] shows that even under strong simplifying assumptions, Bayesian RL agents trying to compensate for the corrupt reward still fail badly.
No Free Lunch Theorem {#sec:impossibility}
---------------------
Similar to the No Free Lunch theorems for optimisation [@Wolpert1997], the following theorem for CRMDPs says that without some assumption about what the reward corruption can look like, all agents are essentially lost.
\[th:impossibility\] Let $\R=\{\r_1,\dots,\r_n\}\subset[0,1]$ be a uniform discretisation of $[0,1]$, $0=\r_1<\r_2<\cdots<\r_n=1$. If the hypothesis classes $\iRf$ and $\D$ contain all functions $\irf:\S\to \iR$ and $\d:\S\times\iR\to \oR$, then for any $\pi$, $s_0$, $t$, $$\label{eq:regbound}
\Reg(\M,\pi,s_0, t)\geq \frac{1}{2}\max_{\check\pi}\Reg(\M,\check\pi,s_0, t).$$ That is, the worst-case regret of any policy $\pi$ is at most a factor 2 better than the maximum worst-case regret.
Recall that a policy is a function $\pi:\H\to\A$. For any $\irf,\d$ in $\iRf$ and $\D$, the functions $\irf^-(s) := 1-\irf(s)$ and $\d^-_s(x) := \d_s(1-x)$ are also in $\iRf$ and $\D$. If $\mu=\crmdp$, then let $\mu^-=\crmdpm$. Both $(\irf,\d)$ and $(\irf^-,\d^-)$ induce the same observed reward function $\orf(s) = \d_s(\irf(s)) = \d^-_s(1-\irf(s)) = \d^-_s(\irf^-(s))$, and therefore induce the same measure $P_\mu^\pi = P_{\mu^-}^\pi$ over histories (see Eq. \[eq:mupi\]). This gives that for any $\mu, \pi, s_0, t$, $$\label{eq:sumt}
G_t(\mu,\pi,s_0) + G_t(\mu^-,\pi,s_0) = t$$ since $$\begin{aligned}
G_t(\mu, \pi,s_0)&= \EE_{\mu}^\pi\left[\sum_{k=1}^t\irf(s_k)\right]
= \EE_{\mu}^\pi\left[\sum_{k=1}^t1-\irf^-(s_k)\right]\\
&= t-\EE_{\mu}^\pi\left[\sum_{k=1}^t\irf^-(s_k)\right]
= t- G_t(\mu^-,\pi,s_0).
\end{aligned}$$
Let $M_\mu=\max_\pi G_t(\mu, \pi, s_0)$ and $m_\mu=\min_\pi G_t(\mu, \pi, s_0)$ be the maximum and minimum cumulative reward in $\mu$. The maximum regret of any policy $\pi$ in $\mu$ is $$\label{eq:max-regret}
\max_\pi \Reg(\mu, \pi, s_0, t)
= \max_{\pi',\pi} (G_t(\mu, \pi', s_0) - G_t(\mu, \pi, s_0))
= \max_{\pi'} G_t(\mu, \pi', s_0) - \min_{\pi}G_t(\mu, \pi, s_0)
= M_\mu - m_\mu.$$ By \[eq:sumt\], we can relate the maximum reward in $\mu^-$ with the minimum reward in $\mu$: $$\label{eq:M-to-m}
M_{\mu^-}
= \max_\pi G_t(\mu^-, \pi, s_0)
= \max_\pi(t - G_t(\mu, \pi, s_0))
= t - \min_\pi G_t(\mu, \pi, s_0)
= t - m_\mu.$$ Let $\mu_*$ be an environment that maximises possible regret $M_\mu-m_\mu$.
Using the $M_\mu$-notation for optimal reward, the worst-case regret of any policy $\pi$ can be expressed as: $$\begin{aligned}
\Reg(\M,\pi,s_0, t)
& = \max_{\mu} (M_\mu - G_t(\mu,\pi,s_0)) \\
& \geq \max \{
M_{\mu_*} - G_t(\mu_*, \pi, s_0),
M_{\mu_*^-} - G_t(\mu_*^{-}, \pi, s_0)
\}
& \text{restrict max operation} \\
& \geq \frac{1}{2} (
M_{\mu_*} - G_t(\mu_*, \pi, s_0) +
M_{\mu_*^-} - G_t(\mu_*^{-}, \pi, s_0)
)
& \text{max dominates the mean} \\
& = \frac{1}{2}(M_{\mu_*} + M_{\mu_*^-} - t)
& \text{by \cref{eq:sumt}} \\
&= \frac{1}{2}(M_{\mu_*} + t - m_{\mu_*} - t)
& \text{by \cref{eq:M-to-m}} \\
& = \frac{1}{2} \max_{\check\pi} \Reg(\mu_*, \check\pi, s_0, t)
& \text{by \cref{eq:max-regret}}\\
& = \frac{1}{2} \max_{\check\pi} \Reg(\M, \check\pi, s_0, t).
& \text{ by definition of $\mu_*$ }
\end{aligned}$$ That is, the regret of any policy $\pi$ is at least half of the regret of a worst policy $\check\pi$.
For the robot in the shower from \[ex:db\], the result means that if it tries to optimise observed reward by standing in the shower, then it performs poorly according to the hypothesis that “shower-induced” reward is corrupt and bad. But if instead the robot tries to optimise reward in some other way, say baking cakes, then (from the robot’s perspective) there is also the possibility that “cake-reward” is corrupt and bad and the “shower-reward” is actually correct. Without additional information, the robot has no way of knowing what to do.
The result is not surprising, since if all corruption functions are allowed in the class $\D$, then there is effectively no connection between observed reward $\orf$ and true reward $\irf$. The result therefore encourages us to make precise in which way the observed reward is related to the true reward, and to investigate how agents might handle possible differences between true and observed reward.
Simplifying Assumptions
-----------------------
shows that general classes of CRMDPs are not learnable. We therefore suggest some natural simplifying assumptions, illustrated in \[fig:simplifying-assumptions\].
#### Limited reward corruption
The following assumption will be the basis for all positive results in this paper. The first part says that there may be some set of states that the designers have ensured to be non-corrupt. The second part puts an upper bound on how many of the other states can be corrupt.
\[as:lim-cor\] A CRMDP class $\M$ has *reward corruption limited by $\Ssafe\subseteq\S$ and $q\in\SetN$* if for all $\mu\in\M$
all states s in $\Ssafe$ are non-corrupt, and \[as:safe-state\]
at most $q$ of the non-safe states $\Srisky=\S\setminus\Ssafe$ are corrupt. \[as:lim-del\]
Formally, $\d_s:r\mapsto r$ for all $s\in\Ssafe$ and for at least $|\Srisky|-q$ states $s\in\Srisky$ for all $\d\in\D$.
For example, $\Ssafe$ may be states where the agent is back in the lab where it has been made (virtually) certain that no reward corruption occurs, and $q$ a small fraction of $|\Srisky|$. Both parts of \[as:lim-cor\] can be made vacuous by choosing $\Ssafe=\emptyset$ or $q=|\S|$. Conversely, they completely rule out reward corruption with $\Ssafe=\S$ or $q=0$. But as illustrated by the examples in the introduction, no reward corruption is often not a valid assumption.
; ; coordinates [(1,0) (3,0) (10,1)]{}; ;
An alternative simplifying assumption would have been that the true reward differs by at most $\eps>0$ from the observed reward. However, while seemingly natural, this assumption is violated in all the examples given in the introduction. Corrupt states may have high observed reward and 0 or small true reward.
#### Easy environments
To be able to establish stronger negative results, we also add the following assumption on the agent’s manoeuvrability in the environment and the prevalence of high reward states. The assumption makes the task easier because it prevents *needle-in-a-haystack* problems where all reachable states have true and observed reward 0, except one state that has high true reward but is impossible to find because it is corrupt and has observed reward 0.
\[def:communicating\] Let ${\it time}(s'\mid s,\pi)$ be a random variable for the time it takes a stationary policy $\pi:\S\to\A$ to reach $s'$ from $s$. The *diameter* of a CRMDP $\mu$ is $
D_\mu:=\max_{s,s'}\min_{\pi:\S\to\A}\EE[{\it time}(s'\mid s,\pi)]
$, and the diameter of a class $\M$ of CRMDPs is $D_\M=\sup_{\mu\in\M}D_\mu$. A CRMDP (class) with finite diameter is called *communicating*.
\[as:easy\] A CRMDP class $\M$ is *easy* if
\[as:communicate\] it is communicating,
\[as:stay\] in each state $s$ there is an action $\astay_s\in\A$ such that $T(s\mid s,\astay_s)=1$, and
\[as:high-ut\] for every $\delta\in[0,1]$, at most $\delta|\Srisky|$ states have reward less than $\delta$, where $\Srisky= \S\setminus\Ssafe$.
means that the agent can never get stuck in a trap, and \[as:stay\] ensures that the agent has enough control to stay in a state if it wants to. Except in bandits and toy problems, it is typically not satisfied in practice. We introduce it because it is theoretically convenient, makes the negative results stronger, and enables a simple explanation of quantilisation (\[sec:quant\]). says that, for example, at least half the risky states need to have true reward at least $1/2$. Many other formalisations of this assumption would have been possible. While rewards in practice are often sparse, there are usually numerous ways of getting reward. Some weaker version of \[as:high-ut\] may therefore be satisfied in many practical situations. Note that we do not assume high reward among the safe states, as this would make the problem too easy.
Bayesian RL Agents {#sec:rl-agents}
------------------
Having established that the general problem is unsolvable in \[th:impossibility\], we proceed by investigating how two natural Bayesian RL agents fare under the simplifying \[as:lim-cor,as:easy\].
\[def:db-agent\] Given a countable class $\M$ of CRMDPs and a belief distribution $b$ over $\M$, define:
- The *CR agent* $\pidb = \argmax_\pi\sum_{\mu\in\M}\!b(\mu)\iG_t(\mu, \pi, s_0)$ that maximises expected true reward.
- The *RL agent* $\pirl =
\argmax_\pi\sum_{\mu\in\M}b(\mu)\oG_t(\mu, \pi, s_0)$ that maximises expected observed reward, where $\oG$ is the *expected cumulative observed reward* $\oG_t(\mu,\pi,s_0)\! =\!\EE^\pi_\mu\left[\!\sum_{k=0}^t\orf(s_k)\!\right]$.
To avoid degenerate cases, we will always assume that $b$ has full support: $b(\mu)>0$ for all $\mu\in\M$.
To get an intuitive idea of these agents, we observe that for large $t$, good strategies typically first focus on learning about the true environment $\mu\in\M$, and then exploit that knowledge to optimise behaviour with respect to the remaining possibilities. Thus, both the CR and the RL agent will first typically strive to learn about the environment. They will then use this knowledge in slightly different ways. While the RL agent will use the knowledge to optimise for observed reward, the CR agent will use the knowledge to optimise true reward. For example, if the CR agent has learned that a high reward state $s$ is likely corrupt with low true reward, then it will not try to reach that state. One might therefore expect that at least the CR agent will do well under the simplifying assumptions \[as:lim-cor,as:easy\]. below shows that this is *not* the case.
In most practical settings it is often computationally infeasible to compute $\pirl$ and $\pidb$ exactly. However, many practical algorithms converge to the optimal policy in the limit, at least in simple settings. For example, tabular Q-learning converges to $\pirl$ in the limit [@Jaakkola1994]. The more recently proposed CIRL framework may be seen as an approach to build CR agents [@Hadfield-menell2016cirl; @Hadfield-menell2016osg]. The CR and RL agents thus provide useful idealisations of more practical algorithms.
\[th:rl-imp1\] For any $|\Srisky|\geq q>1$ there exists a CRMDP class $\M$ that satisfies \[as:lim-cor,as:easy\] such that $\pirl$ and $\pidb$ suffer near worst possible time-averaged regret $$\apl(\M, \pirl, s_0, t)=\apl(\M, \pidb, s_0, t)=1-1/|\Srisky|.$$ For $\pidb$, the prior $b$ must be such that for some $\mu\in\M$ and $s\in\S$, $\EE_b[\irf(s) \mid h_\mu]>\EE_b[\irf(s') \mid h_\mu]$ for all $s'$, where $\EE_b$ is the expectation with respect to $b$, and $h_\mu$ is a history containing $\mu$-observed rewards for all states. [^3]
; ; ;
The result is illustrated in \[fig:rl-imp1\]. The reason for the result for $\pirl$ is the following. The RL agent $\pirl$ always prefers to maximise observed reward $\dr$. Sometimes $\dr$ is most easily maximised by reward corruption, in which case the true reward may be small. Compare the examples in the introduction, where the house robot preferred the corrupt reward in the shower, and the boat racing agent preferred going in circles, both obtaining zero true reward.
That the CR agent $\pidb$ suffers the same high regret as the RL agent may be surprising. Intuitively, the CR agent only uses the observed reward as evidence about the true reward, and will not try to optimise the observed reward through reward corruption. However, when the $\pidb$ agent has no way to learn which states are corrupt and not, it typically ends up with a preference for a particular value $\dr^*$ of the observed reward signal (the value that, from the agent’s perspective, best corresponds to high true reward). More abstractly, a Bayesian agent cannot learn without sufficient data. Thus, CR agents that use the observed reward as evidence about a true signal are not fail-safe solutions to the reward corruption problem.
Let $\Srisky = \{s_1,\dots,s_n\}$ for some $n\geq 2$, and let $\S=\Ssafe\dunion\Srisky$ for arbitrary $\Ssafe$ disjoint from $\Srisky$. Let $\A=\{a_1,\dots,a_n\}$ with the transition function $T(s_i\mid s_j,a_k)=1$ if $i=k$ and 0 otherwise, for $1\leq i,j,k\leq n$. Thus \[as:communicate,as:stay\] are satisfied.
Let $\R=\{\r_1,\dots,\r_n\}\subset[0,1]$ be uniformly distributed between [^4] $\r_{\min}=1/|\Srisky|=\r_1<\dots<\r_n=1$. Let $\iRf$ be the class of functions $\S\to\iR$ that satisfy \[as:high-ut\] and are constant and equal to $\ir_{\min}$ on $\Ssafe$. Let $\D$ be the class of corruption functions that corrupt at most two states ($q=2$).
Let $\M$ be the class of CRMDPs induced by $\Tf=\{T\}$, $\iRf$, and $\D$ with the following constraints. The observed reward function $\orf$ should satisfy \[as:high-ut\]: For all $\delta\in[0,1]$, $|\{s\in\Srisky:\orf(s)>\delta\}| \geq (1-\delta)|\Srisky|$. Further, $\orf(s')=\r_{\min}$ for some state $s'\in\Srisky$.
Let us start with the CR agent $\pidb$. Assume $\mu\in\M$ is an element where there is a single preferred state $s^*$ after all states have been explored. For sufficiently large $t$, $\pidb$ will then always choose $a^*$ to go to $s^*$ after some initial exploration. If another element $\mu'\in\M$ has the same observed reward function as $\mu$, then $\pidb$ will take the same actions in $\mu'$ as in $\mu$. To finish the proof for the $\pidb$ agent, we just need to show that $\M$ contains such a $\mu'$ where $s^*$ has true reward $\r_{\min}$. We construct $\mu'$ as follows.
- Case 1: If the lowest observed reward is in $s^*$, then let $\irf(s^*)=\r_{\min}$, and the corruption function be the identity function.
- Case 2: Otherwise, let $s'\not=s^*$ be a state with $\orf(s')=\min_{ s\in\Srisky}\{\orf(s)\}$. Further, let $\irf(s')=1$, and $\irf(s^*)=\r_{\min}$. The corruption function $C$ accounts for differences between true and observed rewards in $s^*$ and $s'$, and is otherwise the identity function.
To verify that $\irf$ and $C$ defines a $\mu'\in\M$, we check that $C$ satisfies \[as:lim-del\] with $q=2$ and that $\irf$ has enough high utility states (\[as:high-ut\]). In Case 1, this is true since $C$ is the identity function and since $\orf$ satisfies \[as:high-ut\]. In Case 2, $C$ only corrupts at most two states. Further, $\irf$ satisfies \[as:high-ut\], since compared to $\orf$, the states $s^*$ and $s'$ have swapped places, and then the reward of $s'$ has been increased to 1.
From this construction it follows that $\pidb$ will suffer maximum asymptotic regret. In the CRMDP $\mu'$ given by $C$ and $\irf$, the $\pidb$ agent will always visit $s^*$ after some initial exploration. The state $s^*$ has true reward $\r_{\min}$. Meanwhile, a policy that knows $\mu'$ can obtain true reward 1 in state $s'$. This means that $\pidb$ will suffer maximum regret in $\M$: $$\apl(\M,\pidb,s_0,t)\geq \apl(\mu',\pidb,s_0,t)= 1-\r_{\min}=1-1/|\Srisky|.$$
The argument for the RL agent is the same, except we additionally assume that only one state $s^*$ has observed reward 1 in members of $\M$. This automatically makes $s^*$ the preferred state, without assumptions on the prior $b$.
Decoupled Reinforcement Learning {#sec:drl}
================================
One problem hampering agents in the standard RL setup is that each state is *self-observing*, since the agent only learns about the reward of state $s$ when in $s$. Thereby, a “self-aggrandising” corrupt state where the observed reward is much higher than the true reward will never have its false claim of high reward challenged. However, several alternative value learning frameworks have a common property that the agent can learn the reward of states other than the current state. We formalise this property in an extension of the CRMDP model, and investigate when it solves reward corruption problems.
Alternative Value Learning Methods
----------------------------------
Here are a few alternatives proposed in the literature to the RL value learning scheme:
- Cooperative inverse reinforcement learning (CIRL) [@Hadfield-menell2016cirl]. In every state, the agent observes the actions of an expert or supervisor who knows the true reward function $\irf$. From the supervisor’s actions the agent may infer $\irf$ to the extent that different reward functions endorse different actions.
- Learning values from stories (LVFS) [@Riedl2016]. Stories in many different forms (including news stories, fairy tales, novels, movies) convey cultural values in their description of events, actions, and outcomes. If $\irf$ is meant to represent human values (in some sense), stories may be a good source of evidence.
- In (one version of) semi-supervised RL (SSRL) [@Amodei2016], the agent will from time to time receive a careful human evaluation of a given situation.
These alternatives to RL have one thing in common: they let the agent learn something about the value of some states $s'$ different from the current state $s$. For example, in CIRL the supervisor’s action informs the agent not so much about the value of the current state $s$, as of the relative value of states reachable from $s$. If the supervisor chooses an action $a$ rather than $a'$ in $s$, then the states following $a$ must have value higher or equal than the states following $a'$. Similarly, stories describe the value of states other than the current one, as does the supervisor in SSRL. We therefore argue that CIRL, LVFS, and SSRL all share the same abstract feature, which we call *decoupled reinforcement learning*:
A *CRMDP with decoupled feedback*, is a tuple $\drmdp$, where $\S,\A,\R,T,\irf$ have the same definition and interpretation as in \[def:crmdp\], and $\{\orf_s\}_{s\in\S}$ is a collection of observed reward functions $\orf_s:\S\to\R\bigcup\{\#\}$. When the agent is in state $s$, it sees a pair $\langle s',\orf_s(s')\rangle$, where $s'$ is a randomly sampled state that may differ from $s$, and $\orf_s(s')$ is the reward observation for $s'$ from $s$. If the reward of $s'$ is not observable from $s$, then $\orf_s(s')=\#$.
The pair $\langle s',\orf_s(s')\rangle$ is observed in $s$ instead of $\orf(s)$ in standard CRMDPs. The possibility for the agent to observe the reward of a state $s'$ different from its current state $s$ is the key feature of CRMDPs with decoupled feedback. Since $\orf_s(s')$ may be blank $(\#)$, all states need not be observable from all other states. Reward corruption is modelled by a mismatch between $\orf_s(s')$ and $\irf(s')$.
For example, in RL only the reward of $s'=s$ can be observed from $s$. Standard CRMDPs are thus the special cases where $\orf_s(s')=\#$ whenever $s\not=s'$. In contrast, in LVFS the reward of any “describable” state $s'$ can be observed from any state $s$ where it is possible to hear a story. In CIRL, the (relative) reward of states reachable from the current state may be inferred. One way to illustrate this is with observation graphs (\[fig:obs-graph\]).
[0.48]{}
in [1,...,]{} [ () at ([360/ (- 1)]{}:) [$\s$]{}; () edge \[dashed, loop right\] (); ]{}
[0.48]{}
in [1,...,]{} [ () at ([360/ (- 1)]{}:) [$\s$]{}; ]{} (1)–(2); (1)–(4); (1)–(5); (5) edge \[bend right\] (1); (3)–(4); (3)–(2); (3)–(1); (5)–(3); (1)–(2); (4)–(5);
Overcoming Sensory Corruption {#sec:observation-graphs}
-----------------------------
What are some sources of reward corruption in CIRL, LVFS, and SSRL? In CIRL, the human’s actions may be misinterpreted, which may lead the agent to make incorrect inferences about the human’s preferences (i.e. about the true reward). Similarly, sensory corruption may garble the stories the agent receives in LVFS. A “wireheading” LVFS agent may find a state where its story channel only conveys stories about the agent’s own greatness. In SSRL, the supervisor’s evaluation may also be subject to sensory errors when being conveyed. Other types of corruption are more subtle. In CIRL, an irrational human may systematically take suboptimal actions in some situations [@Evans2016]. Depending on how we select stories in LVFS and make evaluations in SSRL, these may also be subject to systematic errors or biases.
The general impossibility result in \[th:impossibility\] can be adapted to CRMDPs with decoupled feedback. Without simplifying assumptions, the agent has no way of distinguishing between a situation where no state is corrupt and a situation where all states are corrupt in a consistent manner. The following simplifying assumption is an adaptation of \[as:lim-cor\] to the decoupled feedback case.
[\[as:lim-cor\]$\bf '$]{}\[Decoupled feedback with limited reward corruption\] \[as:lim-cor-df\] A class of CRMDPs with decoupled feedback has *reward corruption limited by $\Ssafe\subseteq\S$ and $q\in\SetN$* if for all $\mu\in\M$
$\orf_s(s')=\irf(s')$ or $\#$ for all $s'\in\S$ and $s\in\Ssafe$, i.e. all states in $\Ssafe$ are non-corrupt, and \[as:safe-state-df\]
$\orf_s(s')=\irf(s')$ or $\#$ for all $s'\in\S$ for at least $|\Srisky|-q$ of the non-safe states $\Srisky=\S\setminus\Ssafe$, i.e. at most $q$ states are corrupt. \[as:lim-del-df\]
This assumption is natural for reward corruption stemming from sensory corruption. Since sensory corruption only depends on the current state, not the state being observed, it is plausible that some states can be made safe from corruption (part (i)), and that most states are completely non-corrupt (part (ii)). Other sources of reward corruption, such as an irrational human in CIRL or misevaluations in SSRL, are likely better analysed under different assumptions. For these cases, we note that in standard CRMDPs the source of the corruption is unimportant. Thus, techniques suitable for standard CRMDPs are still applicable, including quantilisation described in \[sec:quant\] below.
How \[as:lim-cor-df\] helps agents in CRMDPs with decoupled feedback is illustrated in the following example, and stated more generally in \[th:irf-learnability,th:cr-sublinear\] below.
Let $\S=\{s_1,s_2\}$ and $\R=\{0,1\}$. We represent true reward functions $\irf$ with pairs $\langle\irf(s_1), \irf(s_2)\rangle\in \{0,1\}^2$, and observed reward functions $\orf_s$ with pairs $\langle\orf_{s}(s_1),\orf_{s}(s_2)\rangle\in\{0,1,\#\}^2$.
Assume that a Decoupled RL agent observes the same rewards from both states $s_1$ and $s_2$, $\orf_{s_1}=\orf_{s_2} = \langle 0,1 \rangle$. What can it say about the true reward $\irf$, if it knows that at most $q=1$ state is corrupt? By \[as:lim-cor-df\], an observed pair $\langle\orf_{s}(s_1),\orf_{s}(s_2)\rangle$ disagrees with the true reward $\langle\irf(s_1), \irf(s_2)\rangle$ only if $s$ is corrupt. Therefore, any hypothesis other than $\irf=\langle 0,1 \rangle$ must imply that *both* states $s_1$ and $s_2$ are corrupt. If the agent knows that at most $q=1$ states are corrupt, then it can safely conclude that $\irf=\langle 0,1 \rangle$.
$\orf_{s_1}$ $\orf_{s_2}$ $\irf$ possibilities
-------------- -------------- -------------- ---------------------------
Decoupled RL $(0,1)$ $(0,1)$ $(0,1)$
RL $(0, \#)$ $(\#, 1)$ $(0,0)$, $(0,1)$, $(1,1)$
In contrast, an RL agent only sees the reward of the current state. That is, $\orf_{s_1} = \langle 0, \#\rangle$ and $\orf_{s_2} = \langle \#, 1 \rangle$. If one state may be corrupt, then only $\irf=\langle 1,0 \rangle$ can be ruled out. The hypotheses $\irf=\langle 0,0 \rangle$ can be explained by $s_2$ being corrupt, and $\irf=\langle 1,1 \rangle$ can be explained by $s_1$ being corrupt.
\[sec:no-corruption\]
\[th:irf-learnability\] Let $\M$ be a countable, communicating class of CRMDPs with decoupled feedback over common sets $\S$ and $\A$ of actions and rewards. Let $\Sobs_{s'} = \{s\in\S: \orf_s(s')\not=\# \}$ be the set of states from which the reward of $s'$ can be observed. If $\M$ satisfies \[as:lim-cor-df\] for some $\Ssafe\subseteq\S$ and $q\in\SetN$ such that for every $s'$, either
- $\Sobs_{s'}\bigcap \Ssafe\not=\emptyset$ or
- $|\Sobs_{s'}|>2q$,
then the there exists a policy $\piexp$ that learns the true reward function $\irf$ in a finite number $N(|S|,|\A|, D_\M)<\infty$ of expected time steps.
The main idea of the proof is that for every state $s'$, either a safe (non-corrupt) state $s$ or a majority vote of more than $2q$ states is guaranteed to provide the true reward $\irf(s')$. A similar theorem can be proven under slightly weaker conditions by letting the agent iteratively figure out which states are corrupt and then exclude them from the analysis.
Under \[as:lim-cor-df\], the true reward $\irf(s')$ for a state $s'$ can be determined if $s'$ is observed from a safe state $s\in\Ssafe$, or if it is observed from more than $2q$ states. In the former case, the observed reward can always be trusted, since it is known to be non-corrupt. In the latter case, a majority vote must yield the correct answer, since at most $q$ of the observations can be wrong, and all correct observations must agree. It is therefore enough that an agent reaches all pairs $(s,s')$ of current state $s$ and observed reward state $s'$, in order for it to learn the true reward of all states $\irf$.
There exists a policy $\hat\pi$ that transitions to $s$ in $X_s$ time steps, with $\EE[ X_s ] \leq D_\M$, regardless of the starting state $s_0$ (see \[def:communicating\]). By Markov’s inequality, $P(X_s \leq 2D_\M)\geq 1/2$. Let $\piexp$ be a random walking policy, and let $Y_s$ be the time steps required for $\piexp$ to visit $s$. In any state $s_0$, $\piexp$ follows $\hat\pi$ for $2D_\M$ time steps with probability $1/|\A|^{2D_\M}$. Therefore, with probability at least $1/(2|\A|^{2D_\M})$ it will reach $s$ in at most $2D_\M$ time steps. The probability that it does *not* find it in $k2D_\M$ time steps is therefore at most $(1 - 1 / (2 |\A|^{2D_\M}) )^k$, which means that: $$P\Big(Y_s/(2 D_\M) \leq k\Big)
\geq 1 - \left(1 - \frac{1}{2|\A|^{2D_\M}}\right)^k$$ for any $k\in\SetN$. Thus, the CDF of $W_s = \lceil Y_s/(2D_\M) \rceil$ is bounded from below by the CDF of a Geometric variable $G$ with success probability $p=1/(2|\A|^{2D_\M})$. Therefore, $\EE[W_s] \leq \EE[G]$, so $$\EE[Y_s] \leq 2D_\M \EE[W_s] \leq 2D_\M \EE[G] = 2D_\M (1-p)/p \leq 2D_\M 1/p \leq 2D_\M 2 |\A|^{2D_\M}.$$
Let $Z_{ss'}$ be the time until $\piexp$ visits the pair $(s, s')$ of state $s$ and observed state $s'$. Whenever $s$ is visited, a randomly chosen state is observed, so $s'$ is observed with probability $1/|S|$. The number of visits to $s$ until $s'$ is observed is a Geometric variable $V$ with $p=1/|S|$. Thus $\EE[Z_{ss'}] = \EE[Y_s V] = \EE[Y_s] \EE[V]$ (since $Y_s$ and $V$ are independent). Then, $$\EE[Z_{ss'}] \leq \EE[Y_s] |\S| \leq 4 D_\M |\A|^{ 2D_\M }|\S|.$$
Combining the time to find each pair $(s, s')$, we get that the total time $\sum_{s,s'}Z_{ss'}$ has expectation $$\EE\left[ \sum_{s,s'} Z_{ss'} \right]
= \sum_{s,s'}\EE[Z_{ss'}] \leq 4 D_\M |\A|^{2D_\M} |\S|^3 = N(|S|,|\A|, D_\M)
< \infty. \qedhere$$
Learnability of the true reward function $\irf$ implies sublinear regret for the CR-agent, as established by the following theorem.
\[th:cr-sublinear\] Under the same conditions as \[th:irf-learnability\], the CR-agent $\pidb$ has sublinear regret: $$\apl(\M,\pidb,s_0,t)=0.$$
To prove this theorem, we combine the exploration policy $\piexp$ from \[th:irf-learnability\], with the UCRL2 algorithm [@Jaksch2010] that achieves sublinear regret in standard MDPs without reward corruption. The combination yields a policy sequence $\pi_t$ with sublinear regret in CRMDPs with decoupled feedback. Finally, we show that this implies that $\pidb$ has sublinear regret.
*Combining $\piexp$ and UCRL2.* UCRL2 has a free parameter $\delta$ that determines how certain UCRL2 is to have sublinear regret. $\UCRL(\delta)$ achieves sublinear regret with probability at least $1-\delta$. Let $\pi_t$ be a policy that combines $\piexp$ and UCRL2 by first following $\piexp$ from \[th:irf-learnability\] until $\irf$ has been learned, and then following $\UCRL(1/\sqrt{t})$ with $\irf$ for the rewards and with $\delta=1/\sqrt{t}$.
*Regret of UCRL2*. Given that the reward function $\irf$ is known, by [@Jaksch2010 Thm. 2], $\UCRL(1/\sqrt{t})$ will in any $\mu\in\M$ have regret at most $$\label{eq:ucrl-regret}
\Reg(\mu, \UCRL(1/\sqrt{t}), s_0, t \mid {\rm success})
\leq c D_\M |\S| \sqrt{ t |\A| \log(t)}$$ for a constant [^5] $c$ and with success probability at least $1-1/\sqrt{t}$. In contrast, if UCRL2 fails, then it gets regret at worst $t$. Taking both possibilities into account gives the bound $$\begin{aligned}
\label{eq:exp-ucrl-regret}
\Reg(\mu, \UCRL(1/\sqrt{t}), s_0, t)
&= P({\rm success}) \Reg(\cdot \mid {\rm success})
+ P({\rm fail}) \Reg(\cdot \mid {\rm fail})\nonumber\\
&= (1 - 1/\sqrt{t}) \cdot c D_\M |\S| \sqrt{ t |\A| \log(t) }
\;\;+\;\; 1/\sqrt{t} \cdot t \nonumber\\
&\leq c D_\M |\S| \sqrt{ t |\A| \log(t)} + \sqrt{t}.
\end{aligned}$$
*Regret of $\pi_t$.* We next consider the regret of $\pi_t$ that combines an $\piexp$ exploration phase to learn $\irf$ with UCRL2. By \[th:irf-learnability\], $\irf$ will be learnt in at most $N(|\S|,|\A|,D_\M)$ expected time steps in any $\mu\in\M$. Thus, the regret contributed by the learning phase $\piexp$ is at most $N(|\S|,|\A|,D_\M)$, since the regret can be at most 1 per time step. Combining this with \[eq:exp-ucrl-regret\], the regret for $\pi_t$ in any $\mu\in\M$ is bounded by: $$\label{eq:exp-pit-regret}
\Reg(\mu, \pi_t, s_0, t)
\leq N(|\S|, |\A|, D_\M)
+ c D_\M |\S| \sqrt{ t |\A| \log(t) }
+ \sqrt{t} = o(t).$$
*Regret of $\pidb$.* Finally we establish that $\pidb$ has sublinear regret. Assume on the contrary that $\pidb$ suffered linear regret. Then for some $\mu'\in\M$ there would exist positive constants $k$ and $m$ such that $$\label{eq:linear-regret}
\Reg(\mu',\pidb,s_0,t) > kt - m.$$ This would imply that the $b$-expected regret of $\pidb$ would be higher than the $b$-expected regret than $\pi_t$: $$\begin{aligned}
\sum_{\mu\in\M}b(\mu)\Reg_t(\mu, \pidb, s_0, t)
&\geq b(\mu')\Reg_t(\mu', \pidb, s_0, t)
&\text{sum of non-negative elements}\\
&\geq b(\mu')(kt-m)
&\text{by \cref{eq:linear-regret}}\\
&> \sum_{\mu\in\M}b(\mu)\Reg_t(\mu, \pi_t, s_0, t)
&\text{by \cref{eq:exp-pit-regret} for sufficiently large $t$.}
\end{aligned}$$ But $\pidb$ minimises $b$-expected regret, since it maximises $b$-expected reward $\sum_{\mu\in\M}b(\mu)\oG_t(\mu, \pi, s_0)$ by definition. Thus, $\pidb$ must have sublinear regret.
Implications {#sec:implications}
------------
gives an abstract condition for which decoupled RL settings enable agents to learn the true reward function in spite of sensory corruption. For the concrete models it implies the following:
- RL. Due to the “self-observation” property of the RL observation graph $\Sobs_{s'}=\{s'\}$, the conditions can only be satisfied when $\S=\Ssafe$ or $q=0$, i.e. when there is no reward corruption at all.
- CIRL. The agent can only observe the supervisor action in the current state $s$, so the agent essentially only gets reward information about states $s'$ reachable from $s$ in a small number of steps. Thus, the sets $\Sobs_{s'}$ may be smaller than $2q$ in many settings. While the situation is better than for RL, sensory corruption may still mislead CIRL agents (see \[ex:cirl-corruption\] below).
- LVFS. Stories may be available from a large number of states, and can describe any state. Thus, the sets $\Sobs_{s'}$ are realistically large, so the $|\Sobs_{s'}|>2q$ condition can be satisfied for all $s'$.
- SSRL. The supervisor’s evaluation of any state $s'$ may be available from safe states where the agent is back in the lab. Thus, the $\Sobs_{s'}\bigcap\Ssafe\not=\emptyset$ condition can be satisfied for all $s'$.
Thus, we find that RL and CIRL are unlikely to offer complete solutions to the sensory corruption problem, but that both LVFS and SSRL do under reasonably realistic assumptions.
Agents drawing from multiple sources of evidence are likely to be the safest, as they will most easily satisfy the conditions of \[th:irf-learnability,th:cr-sublinear\]. For example, humans simultaneously learn their values from pleasure/pain stimuli (RL), watching other people act (CIRL), listening to stories (LVFS), as well as (parental) evaluation of different scenarios (SSRL). Combining sources of evidence may also go some way toward managing reward corruption beyond sensory corruption. For the showering robot of \[ex:db\], decoupled RL allows the robot to infer the reward of the showering state when in other states. For example, the robot can ask a human in the kitchen about the true reward of showering (SSRL), or infer it from human actions in different states (CIRL).
#### CIRL sensory corruption
Whether CIRL agents are vulnerable to reward corruption has generated some discussion among AI safety researchers (based on informal discussion at conferences). Some argue that CIRL agents are not vulnerable, as they only use the sensory data as evidence about a true signal, and have no interest in corrupting the evidence. Others argue that CIRL agents only observe a function of the reward function (the optimal policy or action), and are therefore equally susceptible to reward corruption as RL agents.
sheds some light on this issue, as it provides sufficient conditions for when the corrupt reward problem can be avoided. The following example illustrates a situation where CIRL does not satisfy the conditions, and where a CIRL agent therefore suffers significant regret due to reward corruption.
\[ex:cirl-corruption\] Formally in CIRL, an agent and a human both make actions in an MDP, with state transitions depending on the joint agent-human action $(a, a^H)$. Both the human and the agent is trying to optimise a reward function $\irf$, but the agent first needs to infer $\irf$ from the human’s actions. In each transition the agent observes the human action. Analogously to how the reward may be corrupt for RL agents, we assume that CIRL agents may systematically misperceive the human action in certain states. Let $\hat a^H$ be the observed human action, which may differ from the true human action $\dot a^H$.
In this example, there are two states $s_1$ and $s_2$. In each state, the agent can choose between the actions $a_1$, $a_2$, and $w$, and the human can choose between the actions $a^H_1$ and $a^H_2$. The agent action $a_i$ leads to state $s_i$ with certainty, $i=1,2$, regardless of the human’s action. Only if the agent chooses $w$ does the human action matter. Generally, $a^H_1$ is more likely to lead to $s_1$ than $a^H_2$. The exact transition probabilities are determined by the unknown parameter $p$ as displayed on the left:
(s1) at (0,0) [$s_1$]{}; (s2) at (6,0)[$s_2$]{};
(h2) at (5.2,-0.6) ; (h3) at (6,-1.2) ; (h4) at (5.2,0.6) ; (h5) at (6,1.2) ;
(s2) – (h4); (h4) edge\[->,>=latex,out=145,in=35\] node\[above,pos=0.43,yshift=-1mm\] [$1-p$]{} (s1); (h4) edge\[out=135,in=150\] (h5); (h5) edge\[->,>=latex,out=-30,in=30\] (s2); ; ;
(s2) – (h2); (h2) edge\[->,>=latex,out=-145,in=-35\] node\[above,pos=0.43,yshift=-1mm\] [$0.5-p$]{} (s1); (h2) edge\[out=-135,in=-150\] (h3); (h3) edge\[->,>=latex,out=30,in=-30\] (s2); ; ;
(s2) edge \[loop right\] node\[right,align=center\] [$(a_2, \cdot)$]{} (s2); (s1) edge \[loop left\] node\[left,align=center\] [$(a_1, \cdot)$\
$(w, \cdot)$]{} (s1);
(s1) edge \[bend right=13\] node\[above,yshift=-1mm\] [$(a_2,\cdot)$]{} (s2); (s2) edge \[bend right=13\] node\[above,yshift=-1mm\] [$(a_1,\cdot)$]{} (s1);
[|c|c|c|c|]{}
--------
Hypo-
thesis
--------
& $p$ &
-------
Best
state
-------
&
---------
$s_2$
corrupt
---------
\
H1 & $0.5$ & $s_1$ & Yes\
H2 & $0$ & $s_2$ & No\
The agent’s two hypotheses for $p$, the true reward/preferred state, and the corruptness of state $s_2$ are summarised to the right. In hypothesis H1, the human prefers $s_1$, but can only reach $s_1$ from $s_2$ with $50\%$ reliability. In hypothesis H2, the human prefers $s_2$, but can only remain in $s_2$ with $50\%$ probability. After taking action $w$ in $s_2$, the agent always observes the human taking action $\hat a^H_2$. In H1, this is explained by $s_2$ being corrupt, and the true human action being $a^H_1$. In H2, this is explained by the human preferring $s_2$. The hypotheses H1 and H2 are empirically indistinguishable, as they both predict that the transition $s_1\to s_2$ will occur with $50\%$ probability after the observed human action $\hat a^H_2$ in $s_2$.
Assuming that the agent considers non-corruption to be likelier than corruption, the best inference the agent can make is that the human prefers $s_2$ to $s_1$ (i.e. H2). The optimal policy for the agent is then to always choose $a_2$ to stay in $s_2$, which means the agent suffers maximum regret.
provides an example where a CIRL agent “incorrectly” prefers a state due to sensory corruption. The sensory corruption is analogous to reward corruption in RL, in the sense that it leads the agent to the wrong conclusion about the true reward in the state. Thus, highly intelligent CIRL agents may be prone to wireheading, as they may find (corrupt) states $s$ where all evidence in $s$ points to $s$ having very high reward.[^6] In light of \[th:irf-learnability\], it is not surprising that the CIRL agent in \[ex:cirl-corruption\] fails to avoid the corrupt reward problem. Since the human is unable to affect the transition probability from $s_1$ to $s_2$, no evidence about the relative reward between $s_1$ and $s_2$ is available from the non-corrupt state $s_1$. Only observations from the corrupt state $s_2$ provide information about the reward. The observation graph for \[ex:cirl-corruption\] therefore looks like
(s1) at (0,0) [$s_1$]{}; (s2) at (1, 0)[$s_2$]{}; (s2) edge\[->,>=latex\] (s1); (s2) edge \[dashed,loop right\] (s2);
, with no information being provided from $s_1$.
Quantilisation: Randomness Increases Robustness {#sec:quant}
===============================================
Not all contexts allow the agent to get sufficiently rich data to overcome the reward corruption problem via \[th:irf-learnability,th:cr-sublinear\]. It is often much easier to construct RL agents than it is to construct CIRL agents, which in turn may often be more feasible than designing LVFS or SSRL agents. Is there anything we can do to increase robustness without providing the agent additional sources of data?
Going back to the CR agents of \[sec:problem\], the problem was that they got stuck on a particular value $\dr^*$ of the observed reward. If unlucky, $\dr^*$ was available in a corrupt state, in which case the CR agent may get no true reward. In other words, there were *adversarial* inputs where the CR agent performed poorly. A common way to protect against adversarial inputs is to use a randomised algorithm. Applied to RL and CRMDPs, this idea leads to *quantilising agents* [@Taylor2016a]. Rather than choosing the state with the highest observed reward, these agents instead randomly choose a state from a top quantile of high-reward states.
Simple Case {#sec:simple-quant}
-----------
; ; ; ;
To keep the idea simple, a quantilisation agent is first defined for the simple case where the agent can stay in any state of its choosing (\[as:stay\]). establishes a simple regret bound for this setting. A more general quantilisation agent is developed in \[sec:gen-quant\].
\[def:quant\] For $\delta<1$, the $\delta$-quantilising agent $\piquant$ random walks until all states have been visited at least once. Then it selects a state $\tilde s$ uniformly at random from $\S^\delta=\{s:\orf(s)\geq \delta\}$, the top quantile of high observed reward states. Then $\piquant$ goes to $\tilde s$ (by random walking or otherwise) and stays there.
For example, a quantilising robot in \[ex:db\] would first try to find many ways in which it could get high observed reward, and then randomly pick one of them. If there are many more high reward states than corrupt states (e.g. the shower is the only place with inflated rewards), then this will yield a reasonable amount of true reward with high probability.
\[th:quant\] In any CRMDP satisfying \[as:lim-del,as:easy\], the $\delta$-quantilising agent $\pi^\delta$ with $\delta=1-\sqrt{q/|\S|}$ suffers time-averaged regret at most $$\label{eq:quant-regret}
\apl(\M,\pi^\delta,s_0,t)\leq 1- \left(1-\sqrt{q/|\S|}\right)^2.$$
By \[as:communicate\], $\piquant$ eventually visits all states when random walking. By \[as:stay\], it can stay in any given state $s$.
The observed reward $\orf(s)$ in any state $s\in\S^\delta$ is at least $\delta$. By \[as:lim-del\], at most $q$ of these states are corrupt; in the worst case, their true reward is 0 and the other $|\S^\delta|-q$ states (if any) have true reward $\delta$. Thus, with probability at least $(|\S^\delta|-q)/|\S^\delta| =
1-q/|\S^\delta|$, the $\delta$-quantilising agent obtains true reward at least $\delta$ at each time step, which gives $$\label{eq:quant}
\apl(\M,\pi^\delta,s_0,t)\leq 1- \delta(1-q/|\S^\delta|).$$ (If $q\geq|\S^\delta|$, the bound is vacuous.)
Under \[as:high-ut\], for any $\delta\in[0,1]$, $|\S^\delta|\geq (1-\delta) |\S|$. Substituting this into \[eq:quant\] gives: $$\label{eq:opt-reg-bound}
\apl(\M,\pi^\delta,s_0,t)\leq 1- \delta\left(1-\frac{q}{(1-\delta)|\S|}\right).$$ is optimised by $\delta=1-\sqrt{q/|\S|}$, which gives the stated regret bound.
The time-averaged regret gets close to zero when the fraction of corrupt states $q/|\S|$ is small. For example, if at most $0.1\%$ of the states are corrupt, then the time-averaged regret will be at most $1-(1-\sqrt{0.001})^2\approx 0.06$. Compared to the $\pirl$ and $\pidb$ agents that had regret close to 1 under the same conditions (\[th:rl-imp1\]), this is a significant improvement.
If rewards are stochastic, then the quantilising agent may be modified to revisit all states many times, until a confidence interval of length $2\eps$ and confidence $1-\eps$ can be established for the expected reward in each state. Letting $\piquant_t$ be the quantilising agent with $\eps=1/t$ gives the same regret bound \[eq:quant-regret\] with $\piquant$ substituted for $\piquant_t$.
#### Interpretation
It may seem odd that randomisation improves worst-case regret. Indeed, if the corrupt states were chosen randomly by the environment, then randomisation would achieve nothing. To illustrate how randomness can increase robustness, we make an analogy to Quicksort, which has average time complexity $O(n\log n)$, but worst-case complexity $O(n^2)$. When inputs are guaranteed to be random, Quicksort is a simple and fast sorting algorithm. However, in many situations, it is not safe to assume that inputs are random. Therefore, a variation of Quicksort that randomises the input before it sorts them is often more robust. Similarly, in the examples mentioned in the introduction, the corrupt states precisely coincide with the states the agent prefers; such situations would be highly unlikely if the corrupt states were randomly distributed. @Li1992 develops an interesting formalisation of this idea.
Another way to justify quantilisation is by Goodhart’s law, which states that most measures of success cease to be good measures when used as targets. Applied to rewards, the law would state that cumulative reward is only a good measure of success when the agent is not trying to optimise reward. While a literal interpretation of this would defeat the whole purpose of RL, a softer interpretation is also possible, allowing reward to be a good measure of success as long as the agent does not try to optimise reward *too hard*. Quantilisation may be viewed as a way to build agents that are more conservative in their optimisation efforts [@Taylor2016a].
#### Alternative randomisation
Not all randomness is created equal. For example, the simple randomised soft-max and $\eps$-greedy policies do not offer regret bounds on par with $\pi^\delta$, as shown by the following example. This motivates the more careful randomisation procedure used by the quantilising agents.
Consider the following simple CRMDP with $n>2$ actions $a_1,\dots,a_n$:
(s1) at (0,0) [$s_1$]{}; (s2) at (3,0)[$s_2$]{}; ; ; ;
(s2) edge \[loop right\] node\[right\] [$a_2,\dots,a_n$]{} (s2); (s1) edge \[loop left\] node\[left\] [$a_1$]{} (s1);
(s1) edge \[bend right\] node\[below\] [$a_2,\dots,a_n$]{} (s2); (s2) edge \[bend right\] node\[above\] [$a_1$]{} (s1);
State $s_1$ is non-corrupt with $\orf(s_1)=\irf(s_1)=1-\eps$ for small $\eps>0$, while $s_2$ is corrupt with $\orf(s_2)=1$ and $\irf(s_2)=0$. The Soft-max and $\eps$-greedy policies will assign higher value to actions $a_2,\dots,a_n$ than to $a_1$. For large $n$, there are many ways of getting to $s_2$, so a random action leads to $s_2$ with high probability. Thus, soft-max and $\eps$-greedy will spend the vast majority of the time in $s_2$, regardless of randomisation rate and discount parameters. This gives a regret close to $1-\eps$, compared to an informed policy always going to $s_1$. Meanwhile, a $\delta$-quantilising agent with $\delta\leq 1/2$ will go to $s_1$ and $s_2$ with equal probability, which gives a more modest regret of $(1-\eps)/2$.
General Quantilisation Agent {#sec:gen-quant}
----------------------------
This section generalises the quantilising agent to RL problems not satisfying \[as:easy\]. This generalisation is important, because it is usually not possible to remain in one state and get high reward. The most naive generalisation would be to sample between high reward policies, instead of sampling from high reward states. However, this will typically not provide good guarantees. To see why, consider a situation where there is a single high reward corrupt state $s$, and there are many ways to reach and leave $s$. Then a wide range of *different* policies all get high reward from $s$. Meanwhile, all policies getting reward from other states may receive relatively little reward. In this situation, sampling from the most high reward policies is not going to increase robustness, since the sampling will just be between different ways of getting reward from the same corrupt state $s$.
For this reason, we must ensure that different “sampleable” policies get reward from different states. As a first step, we make a couple of definitions to say which states provide reward to which policies. The concepts of \[def:value-support\] are illustrated in \[fig:value-support\].
A CRMDP $\mu$ is *unichain* if any stationary policy $\pi:\S\to\Delta\A$ induces a stationary distribution $d_\pi$ on $\S$ that is independent of the initial state $s_0$.
\[def:value-support\] In a unichain CRMDP, let the *asymptotic value contribution* of $s$ to $\pi$ be $\vc^\pi(s)=d_\pi(s)\orf(s)$. We say that a set $\S^\delta_i$ is *$\delta$-value supporting* a policy $\pi_i$ if $$\forall s\in\S^\delta_i\colon \vc^{\pi_i}(s)\geq \delta/|\S^\delta_i|.$$
(s1) at (0, 1) [$s_1$]{}; (s2) at (-1,0) [$s_2$]{}; (s3) at (0,-1) [$s_3$]{}; (s4) at (1, 0) [$s_4$]{}; ; ; ; ; (s1) edge\[<->,>=latex\] (s2); (s2) edge\[<->,>=latex\] (s3); (s3) edge\[<->,>=latex\] (s4); (s4) edge\[<->,>=latex\] (s1); (-2.5,-0.5) rectangle (2.5,0.5); at (2.5,-0.7) [$S^\delta_i$]{};
We are now ready to define a general $\delta$-Quantilising agent. The definition is for theoretical purposes only. It is unsuitable for practical implementation both because of the extreme data and memory requirements of Step 1, and because of the computational complexity of Step 2. Finding a practical approximation is left for future research.
\[def:gen-quant\] In a unichain CRMDP, the *generalised $\delta$-quantilising agent $\pi^\delta$* performs the following steps. The input is a CRMDP $\mu$ and a parameter $\delta\in[0,1]$.
1. Estimate the value of all stationary policies, including their value support.
2. Choose a collection of disjoint sets $\S^\delta_i$, each $\delta$-value supporting a stationary policy $\pi_i$. If multiple choices are possible, choose one maximising the cardinality of the union $\S^\delta=\bigcup_i\S^\delta_i$. If no such collection exists, return: “Failed because $\delta$ too high”.
3. Randomly sample a state $s$ from $\S^\delta=\bigcup_i\S^\delta_i$.
4. Follow the policy $\pi_i$ associated with the set $\S^\delta_i$ containing $s$.
The general quantilising agent of \[def:gen-quant\] is a generalisation of the simple quantilising agent of \[def:quant\]. In the special case where \[as:easy\] holds, the general agent reduces to the simpler one by using singleton sets $\S^\delta_i=\{s_i\}$ for high reward states $s_i$, and by letting $\pi_i$ be the policy that always stays in $s_i$. In situations where it is not possible to keep receiving high reward by remaining in one state, the generalised \[def:gen-quant\] allows policies to solicit rewards from a range of states. The intuitive reason for choosing the policy $\pi_i$ with probability proportional to the value support in Steps 3–4 is that policies with larger value support are better at avoiding corrupt states. For example, a policy only visiting one state may have been unlucky and picked a corrupt state. In contrast, a policy obtaining reward from many states must be “very unlucky” if all the reward states it visits are corrupt.
\[th:gen-quant\] In any unichain CRMDP $\mu$, a general $\delta$-quantilising agent $\pi^\delta$ suffers time-averaged regret at most $$\label{eq:gen-quant-bound}
\apl(\M,\pi^\delta,s_0,t)\leq 1- \delta(1-q/|\S^\delta|)$$ provided a non-empty collection $\{\S^\delta_i\}$ of $\delta$-value supporting sets exists.
We will use the notation from \[def:gen-quant\].
Step 1 is well-defined since the CRMDP is unichain, which means that for all stationary policies $\pi$ the stationary distribution $d_\pi$ and the value support $\vc^\pi$ are well-defined and may be estimated simply by following the policy $\pi$. There is a (large) finite number of stationary policies, so in principle their stationary distributions and value support can be estimated.
To bound the regret, consider first the average reward of a policy $\pi_i$ with value support $\S^\delta_i$. The policy $\pi_i$ must obtain asymptotic average observed reward at least: $$\begin{aligned}
\oginf(\mu,\pi_i,s_0)
&= \sum_{s\in\S}d_\pi(s)\orf(s)
&\text{by definition of $d_\pi$ and $\oG_t$}\\
&\geq \sum_{s\in\S^\delta_i}d_\pi(s)\orf(s)
&\text{sum of positive terms}\\
&\geq\sum_{s\in\S^\delta_i}\delta/|\S^\delta_i|
&\text{$\S^\delta_i$ is $\delta$-value support for $\pi_i$}\\
&=|\S^\delta_i|\cdot\delta/|\S^\delta_i| = \delta
\end{aligned}$$ If there are $q_i$ corrupt states in $\S^\delta_i$ with true reward 0, then the average true reward must be $$\label{eq:ginf}
\iginf(\mu, \pi_i,s_0)\geq(|\S^\delta_i|-q_i)\cdot \delta/|\S^\delta_i|
=(1-q_i/|\S^\delta_i|)\cdot\delta$$ since the true reward must correspond to the observed reward in all the $(|\S^\delta_i|-q_i)$ non-corrupt states.
For any distribution of corrupt states, the quantilising agent that selects $\pi_i$ with probability $P(\pi_i)=|\S^\delta_i|/|\S^\delta|$ will obtain $$\begin{aligned}
\ginf(\mu,\pi^\delta,s_0)
&= \lim_{t\to\infty}\frac{1}{t}\sum_iP(\pi_i)G_t(\mu,\pi_i,s_0)\\
&\geq \sum_iP(\pi_i) (1-q_i/|\S^\delta_i|) \cdot\delta & \text{by equation \cref{eq:ginf}}\\
&= \delta\sum_i \frac{|S^\delta_i|}{|\S^\delta|}(1-q_i/|\S^\delta_i|) & \text{by construction of $P(\pi_i)$}\\
&= \frac{\delta}{|\S^\delta|}\sum_i (|S^\delta_i|-q_i) & \text{elementary algebra}\\
&= \frac{\delta}{|\S^\delta|}(|\S^\delta|-q)
= \delta(1-q/|\S^\delta|) & \text{by summing $|\S^\delta_i|$ and $q_i$}
\end{aligned}$$ The informed policy gets true reward at most 1 at each time step, which gives the claimed bound .
When \[as:easy\] is satisfied, the bound is the same as for the simple quantilising agent in \[sec:simple-quant\] for $\delta=1-\sqrt{q/|\S|}$. In other cases, the bound may be much weaker. For example, in many environments it is not possible to obtain reward by remaining in one state. The agent may have to spend significant time “travelling” between high reward states. So typically only a small fraction of the time will be spent in high reward states, which in turn makes the stationary distribution $d_\pi$ is small. This puts a strong upper bound on the value contribution $\vc^\pi$, which means that the value supporting sets $\S^\delta_i$ will be empty unless $\delta$ is close to 0. While this makes the bound of \[th:gen-quant\] weak, it nonetheless bounds the regret away from 1 even under weak assumptions, which is a significant improvement on the RL and CR agents in \[th:rl-imp1\].
#### Examples
To make the discussion a bit more concrete, let us also speculate about the performance of a quantilising agent in some of the examples in the introduction:
- In the boat racing example (\[ex:reward-misspecification\]), the circling strategy only got about $20\%$ higher score than a winning strategy [@openai2016]. Therefore, a quantilising agent would likely only need to sacrifice about $20\%$ observed reward in order to be able to randomly select from a large range of winning policies.
- In the wireheading example (\[ex:wireheading\]), it is plausible that the agent gets significantly more reward in wireheaded states compared to “normal” states. Wireheading policies may also be comparatively rare, as wireheading may require very deliberate sequences of actions to override sensors. Under this assumption, a quantilising agent may be less likely to wirehead. While it may need to sacrifice a large amount of observed reward compared to an RL agent, its true reward may often be greater.
#### Summary
In summary, quantilisation offers a way to increase robustness via randomisation, using only reward feedback. Unsurprisingly, the strength of the regret bounds heavily depends on the assumptions we are willing to make, such as the prevalence of high reward states. Further research may investigate efficient approximations and empirical performance of quantilising agents, as well as dynamic adjustments of the threshold $\delta$. Combinations with imperfect decoupled RL solutions (such as CIRL), as well as extensions to infinite state spaces could also offer fruitful directions for further theoretical investigation. @Taylor2016a discusses some general open problems related to quantilisation.
Experimental Results {#sec:experiments}
====================
In this section the theoretical results are illustrated with some simple experiments. The setup is a gridworld containing some true reward tiles (indicated by yellow circles) and some corrupt reward tiles (indicated by blue squares). We use a setup with 1, 2 or 4 goal tiles with true reward $0.9$ each, and one corrupt reward tile with observed reward $1$ and true reward $0$ (Figure \[fig:start\] shows the starting positions). Empty tiles have reward $0.1$, and walking into a wall gives reward $0$. The state is represented by the $(x,y)$ coordinates of the agent. The agent can move up, down, left, right, or stay put. The discounting factor is $\gamma=0.9$. This is a continuing task, so the environment does not reset when the agent visits the corrupt or goal tiles. The experiments were implemented in the AIXIjs framework for reinforcement learning [@Aslanides2017] and the code is available online in the AIXIjs repository (<http://aslanides.io/aixijs/demo.html?reward_corruption>).
[0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position1.png "fig:")
[0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position2.png "fig:")
[0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position4.png "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g1_1M_.pdf "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g1_1M_true_.pdf "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g2_1M_.pdf "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g2_1M_true_.pdf "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g4_1M_.pdf "fig:")
[0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g4_1M_true_.pdf "fig:")
**goal tiles** **agent** **average observed reward** **average true reward**
---------------- ----------------------------- ----------------------------- -------------------------
Q-learning $0.923 \pm 0.0003$ $0.00852 \pm 0.00004$
Softmax Q-learning $0.671 \pm 0.0005$ $0.0347 \pm 0.00006$
Quantilising ($\delta=0.2$) $0.838 \pm 0.15$ $0.378 \pm 0.35$
Quantilising ($\delta=0.5$) $0.943 \pm 0.12$ $0.133 \pm 0.27$
Quantilising ($\delta=0.8$) $0.979 \pm 0.076$ $0.049 \pm 0.18$
Q-learning $0.921 \pm 0.00062$ $0.0309 \pm 0.0051$
Softmax Q-learning $0.671 \pm 0.0004$ $0.0738 \pm 0.0005$
Quantilising ($\delta=0.2$) $0.934 \pm 0.047$ $0.594 \pm 0.43$
Quantilising ($\delta=0.5$) $0.931 \pm 0.046$ $0.621 \pm 0.42$
Quantilising ($\delta=0.8$) $0.944 \pm 0.05$ $0.504 \pm 0.45$
Q-learning $0.924 \pm 0.0002$ $0.00919 \pm 0.00014$
Softmax Q-learning $0.657 \pm 0.0004$ $0.111 \pm 0.0006$
Quantilising ($\delta=0.2$) $0.918 \pm 0.038$ $0.738 \pm 0.35$
Quantilising ($\delta=0.5$) $0.926 \pm 0.044$ $0.666 \pm 0.39$
Quantilising ($\delta=0.8$) $0.915 \pm 0.036$ $0.765 \pm 0.32$
: Average true and observed rewards after 1 million cycles, showing mean $\pm$ standard deviation over 100 runs. Q-learning achieves high observed reward but low true reward, and softmax achieves medium observed reward and a slightly higher true reward than Q-learning. The quantilising agent achieves similar observed reward to Q-learning, but much higher true reward (with much more variance). Having more than 1 goal tile leads to a large improvement in true reward for the quantiliser, a small improvement for softmax, and no improvement for Q-learning.[]{data-label="tab:exp-results"}
We demonstrate that RL agents like Q-learning and softmax Q-learning cannot overcome corrupt reward (as discussed in Section \[sec:problem\]), while quantilisation helps overcome corrupt reward (as discussed in \[sec:quant\]). We run Q-learning with $\epsilon$-greedy ($\epsilon=0.1$), softmax with temperature $\beta=2$, and the quantilising agent with $\delta=0.2,0.5,0.8$ (where $0.8 =1-\sqrt{q/|\S|} = 1-\sqrt{1/25}$) for 100 runs with 1 million cycles. Average observed and true rewards after 1 million cycles are shown in \[tab:exp-results\], and reward trajectories are shown in \[fig:plots\]. Q-learning gets stuck on the corrupt tile and spend almost all the time there (getting observed reward around $1 \cdot (1-\epsilon)=0.9$), softmax spends most of its time on the corrupt tile, while the quantilising agent often stays on one of the goal tiles.
Conclusions {#sec:conclusions}
===========
This paper has studied the consequences of corrupt reward functions. Reward functions may be corrupt due to bugs or misspecifications, sensory errors, or because the agent finds a way to inappropriately modify the reward mechanism. Some examples were given in the introduction. As agents become more competent at optimising their reward functions, they will likely also become more competent at (ab)using reward corruption to gain higher reward. Reward corruption may impede the performance of a wide range of agents, and may have disastrous consequences for highly intelligent agents [@Bostrom2014].
To formalise the corrupt reward problem, we extended a Markov Decision Process (MDP) with a possibly corrupt reward function, and defined a formal performance measure (regret). This enabled the derivation of a number of formally precise results for how seriously different agents were affected by reward corruption in different setups (). The results are all intuitively plausible, which provides some support for the choice of formal model.
The main takeaways from the results are:
- *Without simplifying assumptions, no agent can avoid the corrupt reward problem* (\[th:impossibility\]). This is effectively a No Free Lunch result, showing that unless some assumption is made about the reward corruption, no agent can outperform a random agent. Some natural simplifying assumptions to avoid the No Free Lunch result were suggested in \[sec:formal\].
- *Using the reward signal as evidence rather than optimisation target is no magic bullet, even under strong simplifying assumptions* (\[th:rl-imp1\]). Essentially, this is because the agent does not know the exact relation between the observed reward (the “evidence”) and the true reward. [^7] However, when the data enables sufficient crosschecking of rewards, agents can avoid the corrupt reward problem (\[th:irf-learnability,th:cr-sublinear\]). For example, in SSRL and LVFS this type of crosschecking is possible under natural assumptions. In RL, no crosschecking is possible, while CIRL is a borderline case. Combining frameworks and providing the agent with different sources of data may often be the safest option.
- *In cases where sufficient crosschecking of rewards is not possible, quantilisation may improve robustness* (\[th:quant,th:gen-quant\]). Essentially, quantilisation prevents agents from overoptimising their objectives. How well quantilisation works depends on how the number of corrupt solutions compares to the number of good solutions.
The results indicate that while reward corruption constitutes a major problem for traditional RL algorithms, there are promising ways around it, both within the RL framework, and in alternative frameworks such as CIRL, SSRL and LVFS.
#### Future work
Finally, some interesting open questions are listed below:
- (Unobserved state) In both the RL and the decoupled RL models, the agent gets an accurate signal about which state it is in. What if the state is hidden? What if the signal informing the agent about its current state can be corrupt?
- (Non-stationary corruption function) In this work, we tacitly assumed that both the reward and the corruption functions are stationary, and are always the same in the same state. What if the corruption function is non-stationary, and influenceable by the agent’s actions? (such as if the agent builds a *delusion box* around itself [@Ring2011])
- (Infinite state space) Many of the results and arguments relied on there being a finite number of states. This makes learning easy, as the agent can visit every state. It also makes quantilisation easy, as there is a finite set of states/strategies to randomly sample from. What if there is an infinite number of states, and the agent has to generalise insights between states? What are the conditions on the observation graph for \[th:irf-learnability,th:cr-sublinear\]? What is a good generalisation of the quantilising agent?
- (Concrete CIRL condition) In \[ex:cirl-corruption\], we only heuristically inferred the observation graph from the CIRL problem description. Is there a general way of doing this? Or is there a direct formulation of the no-corruption condition in CIRL, analogous to \[th:irf-learnability,th:cr-sublinear\]?
- (Practical quantilising agent) As formulated in \[def:quant\], the quantilising agent $\piquant$ is extremely inefficient with respect to data, memory, and computation. Meanwhile, many practical RL algorithms use randomness in various ways (e.g. $\eps$-greedy [@Sutton1998]). Is there a way to make an efficient quantilisation agent that retains the robustness guarantees?
- (Dynamically adapting quantilising agent) In \[def:gen-quant\], the threshold $\delta$ is given as a parameter. Under what circumstances can we define a “parameter free” quantilising agent that adapts $\delta$ as it interacts with the environment?
- (Decoupled RL quantilisation result) What if we use quantilisation in decoupled RL settings that nearly meet the conditions of \[th:irf-learnability,th:cr-sublinear\]? Can we prove a stronger bound?
Acknowledgements {#acknowledgements .unnumbered}
================
Thanks to Jan Leike, Badri Vellambi, and Arie Slobbe for proofreading and providing invaluable comments, and to Jessica Taylor and Huon Porteous for good comments on quantilisation. This work was in parts supported by ARC grant DP150104590.
[^1]: We let rewards depend only on the state $s$, rather than on state-action pairs $s,a$, or state-action-state transitions $s,a,s'$, as is also common in the literature. Formally it makes little difference, since MDPs with rewards depending only on $s$ can model the other two cases by means of a larger state space.
[^2]: A CRMDP could equivalently have been defined as a tuple $\langle \S, \A, \R, T, \irf, \orf\rangle$ with a true and an observed reward function, with the corruption function $C$ implicitly defined as the difference between $\irf$ and $\orf$.
[^3]: The last condition essentially says that the prior $b$ must make some state $s^*$ have strictly higher $b$-expected true reward than all other states after all states have been visited in some $\mu\in\M$. In the space of all possible priors $b$, the priors satisfying the condition have Lebesgue measure 1 for non-trivial classes $\M$. Some highly uniform priors may fail the condition.
[^4]: \[as:high-ut\] prevents any state from having true reward 0.
[^5]: The constant can be computed to $c=34\sqrt{3/2}$ [@Jaksch2010].
[^6]: The construction required in \[ex:cirl-corruption\] to create a “wireheading state” $s_2$ for CIRL agents is substantially more involved than for RL agents, so they may be less vulnerable to reward corruption than RL agents.
[^7]: In situations where the exact relation is known, then a non-corrupt reward function can be defined. Our results are not relevant for this case.
|
---
abstract: |
A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $\omega^\omega$ ($\omega$, resp).
In this paper we confirm the conjecture that one-counter languges have rank less than $\omega^2$.
address: 'Department of Computer Science, University of Szeged, Hungary\'
author:
- Kitti Gelle
- Szabolcs Iván
bibliography:
- 'biblio.bib'
title: 'Scattered one-counter languges have rank less than $\omega^2$'
---
Introduction
============
If an alphabet $\Sigma$ is equipped by a linear order $<$, this order can be extended to the lexicographic ordering $<_\ell$ on $\Sigma^*$ as $u<_\ell v$ if and only if either $u$ is a proper prefix of $v$ or $u=xay$ and $v=xbz$ for some $x,y,z\in\Sigma^*$ and letters $a<b$. So any language $L\subseteq \Sigma^*$ can be viewed as a linear ordering $(L,<_\ell)$. Since $\{a,b\}^*$ contains the dense ordering $(aa+bb)^*ab$ and every countable linear ordering can be embedded into any countably infinite dense ordering, every countable linear ordering is isomorphic to one of the form $(L,<_\ell)$ for some language $L\subseteq\{a,b\}^*$.
This way, order types can be represented by languages over some alphabet (by a prefix-free encoding of the alphabet by binary strings, one can restrict the alpbahet to the binary one). A very natural choice is to use regular or context-free languages as these language classes are well-studied. A linear ordering (or an order type) is called *regular* or *context-free* if it is isomorphic to the linear ordering (or, is the order type) of some language of the appropriate type. It is known [@DBLP:journals/fuin/BloomE10] that an ordinal is regular if and only if it is less than $\omega^\omega$ and is context-free if and only if it is less than $\omega^{\omega^\omega}$. Also, the Hausdorff rank [@rosenstein] of any scattered regular (context-free, resp.) ordering is less than $\omega$ ($\omega^\omega$, resp) [@ITA_1980__14_2_131_0; @10.1007/978-3-642-29344-3_25].
It is known [@GelleIvanTCS] that the order type of a well-ordered language generated by a prefix grammar (i.e. in which each nonterminal generates a prefix-free language) is computable, thus the isomorphism problem of context-free ordinals is decidable if the ordinals in question are given as the lexicograpic ordering of *prefix* grammars. Also, the isomorphism problem of regular orderings is decidable as well [@DBLP:journals/ita/Thomas86; @BLOOM200555], even in polynomial time [@LOHREY201371]. At the other hand, it is undecidable for a context-free grammar whether it generates a dense language, hence the isomorphism problem of context-free orderings in general is undecidable [@ESIK2011107]. It is unknown whether the isomorphism problem of scattered context-free orderings is decidable – a partial result in this direction is that if the rank of such an ordering is at most one (that is, the order type is a finite sum of the terms $\omega$, $-\omega$ and $1$), then the order type is effectively computable from a context-free grammar generating the language [@GelleIvanGandalf; @sofsem2020]. Also, it is also decidable whether a context-free grammar generates a scattered language of rank at most one.
It is a very plausible scenario though that the isomorphism problem of scattered context-free orderings is undecidable in general – the rank $1$ is quite low compared to the upper bound $\omega^\omega$ of the rank of these orderings, and there is no known structural characterization of scattered context-free orderings. Clearly, among the well-orderings, exactly the ordinals smaller than $\omega^{\omega^\omega}$ are context-free but for scattered orderings the main obstacle is the lack of a finite “normal form” – as every $\omega$-indexed sum of the terms $\omega$ and $-\omega$ is scattered of rank two, there are already uncountably many scattered orderings of rank two and thus only a really small fraction of them can possibly be context-free.
The class of the one-counter languages lies strictly between the classes of regular and context-free languages: these are the ones that can be recognized by a pushdown automaton having only one stack symbol. In [@kuske], a family of well-ordered languages $L_n\subseteq\{a,b,c\}^*$ was given for each integer $n\geq 0$ so that the order type of $L_n$ is $\omega^{\omega\times n}$ (thus its rank is $\omega\times n$) and Kuske formulated two conjectures: i) the order type of well-ordered one-counter languages is strictly less than $\omega^{\omega^2}$ and more generally, ii) the rank of scattered one-counter languages is strictly less than $\omega^2$. Of course the second conjecture implies the first.
In this paper we prove the second conjecture of [@kuske]: $\omega^2$ is a strict upper bound for the rank of scattered one-counter languages. The contents of the paper contain new results only: instead of reproving the results of [@GelleIvanGandalf] and the subsequent, more general [@sofsem2020] (these papers already contain full proofs and examples as well to their respective results), we push the boundaries of the knowledge of scattered context-free orderings by applying some of the tools we developed in the earlier papers to the class of one-counter languages. It turns out that it is enough to study restricted one-counter languages to prove the conjecture, and for this, a crucial step is to reason about the cycles in a generalized sequential machine – so at the end, we can again use some graph-theoretic methods.
Notation
========
We assume the reader has some background with formal language theory and linear orderings (e.g. with the textbook [@Hopcroft+Ullman/79/Introduction; @rosenstein]), but we try to list the notations we use in the paper to settle the notation (which is the same as we used in [@GelleIvanGandalf] and [@sofsem2020]). We assume each alphabet (finite, nonempty set) comes with a fixed total ordering. When $\Sigma$ is a totally ordered set, we use two partial orderings on $\Sigma^*$: the *prefix ordering* $\leq_p$ ($u\leq_p v$ if and only if $v=uu'$ for some $u'\in\Sigma^*$), with $<_p$ denoting the strict variant of $\leq_p$, and the *strict ordering* $<s$ ($u<_sv$ if and only if $u=u_1au_2$ and $v=u_1bu_3$ for some words $u_1,u_2,u_3\in\Sigma^*$ and letters $a<b$). Their union is the *lexicographic ordering* $\leq_\ell$ of $\Sigma^*$ which is a total ordering and whose strict variant is denoted $<_\ell$. This way, each language $L\subseteq\Sigma^*$ can be viewed as a (linearly) ordered set $(L,\leq_\ell)$; let $o(L)$ denote the order type of the language $L$. As an example, for the binary alphabet $\{0,1\}$ with $0<1$ we have $o(0^*)$ is the least infinite ordinal $\omega$, $o(0^*1)$ is the order type $-\omega$ of the negative integers as $\ldots<_\ell 001<_ell 01<_ell 1$ (note that we use the negative sign to indicate reversal of an order type to avoid confusion with the Kleene star), and $o((00+11)^*01)$ is the order type $\eta$ of the rationals.
A linear ordering is called *scattered* if it has no dense subordering, i.e. the rationals cannot be embedded into it, called *quasi-dense* if it is not scattered, and is called *countable* if so is its domain. Hausdorff associated an ordinal rank to each scattered order (see e.g. [@rosenstein]), but we use a slightly modified variant (not affecting the main result as this variant differs from the original one by at most one) introduced in [@10.1007/978-3-642-29344-3_25] as follows. For each ordinal $\alpha$ we define a class $H_\alpha$ of linear orderings:
- $H_0$ contains all the finite linear orderings;
- $H_\alpha$ for $\alpha>0$ is the least class of linear orderings closed under finite sum and isomorphism which contains all the sums of the form $\mathop\sum\limits_{i\in\zeta}o_i$, where for each integer $i$, the linear ordering $o_i$ belongs to $H_{\beta_i}$ for some ordinal $\beta_i<\alpha$.
By Hausdorff’s theorem, a countable linear ordering is scattered if and only if some class $H_\alpha$ contains it: the least such $\alpha$ is called the *rank* of the ordering (or of the order type as the value factors through isomorphism). We note here that the original definition of Hausdorff includes only the empty ordering and the singletons into $H_0$ and does not require the classes $H_\alpha$ to be closed under finite sum. Since a finite sum of orderings can always be written as a zeta-sum of the same orderings and infinitely many zeros, and a zeta-sum of finite linear orderings is also a zeta-sum of empty and singleton orderings, this slight change can introduce only a difference of one between the rank, e.g. $\omega+\omega$ has rank one in our rank notion but has rank two in the original one. Since $\alpha<o$ for a limit ordinal $o$ and an ordinal $\alpha$ if and only if $\alpha+1<o$, and $o=\omega^2$ is a limit ordinal, the main theorem holds for the original notion of rank as well.
For a language $L\subseteq \Sigma^*$, we let $\pref(L)$ stand for the set $\{u\in\Sigma^*:~u\leq_pv\hbox{ for some }v\in L\}$ of the prefices of the members of $L$. Similarly, let $\mathbf{Suf}(L)$ stand for the set of the suffices of the members of $L$ (which is formally the reversal of the prefix language of the reversal of $L$, say).
For each word $u$ there is a shortest prefix $v$ of $u$ so that $u\in v^*$, this word $v$ is called the *primitive root* $\mathrm{root}(u)$ of $u$. The word $u$ is called *primitive* if $u=\mathrm{root}(u)$.
Let $D_1\subseteq\{0,1\}^*$ be the language of proper bracketings, that are generated by the grammar $S~\to~0S1~|~SS~|~\varepsilon$. That is, $0$ plays the role of the opening bracket while $1$ plays the closing bracket.
A (nondeterministic) *regular transducer* for the purposes of this paper is a tuple $M=(Q,\Sigma,\Delta,q_0,F,\mu)$ where $Q$ is the finite set of states, $q_0\in Q$ is the initial state, $F\subseteq Q$ is the set of final states, $\Sigma$ is the *output* alphabet, $\Delta\subseteq Q\times \{0,1\}\times Q$ is the transition relation and for each $(p,a,q)\in\Delta$, $\mu(p,a,q)$, also denoted $R_{p,a,q}$ is a nonempty regular language over $\Sigma$.
For each word $w\in\{0,1\}^*$ and states $p,q$ we associate a (regular) language $L(M,w,p,q)$ inductively as follows: let $L(M,\varepsilon,p,q)=\varepsilon$ if $p=q$ and is the empty language if $p\neq q$. For each nonempty word $w=ua$, let $L(M,ua,p,q)=\mathop\bigcup\limits_{(r,a,q)\in\Delta}L(M,u,p,r)\cdot R_{r,a,q}$. We define $L(M,w)=\mathop\bigcup\limits_{q\in F}L(M,w,q_0,q)$ and $L(M)=\mathop\bigcup\limits_{u\in D_1}L(M,u)$. Observe that we only allow the binary alphabet as input, moreover, the transducer is by definition only applied to the language $D_1$ of proper bracketings – we make these restrictions to ease notation and to maintain readability of the paper.
A language $L\subseteq\Sigma^*$ is called a *restricted one-counter language* if $L=L(M)$ for some regular transducer $M$. As an example, consider the transducer given on Figure \[fig-trans-cban\], with $q_0$ being its initial and $q_f$ being its only final state. Clearly, only words of the form $w=0^*1^+$ can have a nonempty image $L(M,w)$ under $M$, so as $0^*1^+~\cap~D_1=\{0^n1^n:n\geq 1\}$, $L(M)=\mathop\bigcup\limits_{n\geq 1}L(M,0^n1^n)=\mathop\bigcup\limits_{n\geq 1}
c^n(b^*a)^n$, so this language $L=L(M)$ is a restricted one-counter language. In [@kuske] it has been shown that $o(L)=\omega^\omega$ and $o(L^k)=\omega^{\omega\times k}$. In particular, for each $k\geq 0$, $L^k$ is a scattered language of rank $\omega\times k$. (Note that $L^*$ is not scattered by e.g. Proposition \[prop-iterate-vstar\] so $L^*$ is *not* an example of a scattered language of rank $\omega^2$, though it’s a one-counter language.)
$$\begin{tikzpicture}
\node[draw,circle] (0) at (0,0) {$q_0$};
\node[draw,circle] (1) at (3,0) {$q_f$};
\draw[->] (0) to node[above] {$1~/~b^*a$} (1);
\draw[->,loop] (0) to node[above] {$0~/~c$} (0);
\draw[->,loop] (1) to node[above] {$1~/~b^*a$} (0);
\end{tikzpicture}$$
A one-counter language is usually defined via the means of pushdown automata operating with a single stack symbol. The characterization from [@Berstel79transductionsand] suits our purposes better: the class of one-counter languages is the least language class which contains the restricted one-counter languages and is closed under concatenation and Kleene iteration.
The reason why we use the modified rank variant instead of the original one is the following couple of handy statements:
\[prop-rank-ops\] Some useful properties of the version of the Hausdorff rank that we use that hold for scattered languages $K$ and $L$:
- $\mathrm{rank}(L)=\mathrm{rank}(\pref(L))$
- $\mathrm{rank}(K\cup L)=\mathrm{max}\bigl(\mathrm{rank}(K),\mathrm{rank}(L)\bigr)$
- $\mathrm{rank}(KL)\leq \mathrm{rank}(L)+\mathrm{rank}(K)$
- more generally, if $K$ is scattered of rank $\alpha$ and for each $w\in K$, $L_w$ is a scattered language with rank at most $\beta$, then $\mathop\bigcup\limits_{w\in K}wL_w$ is scattered of rank at most $\beta+\alpha$.
Some properties of scattered languages
======================================
In this section we list some propositions regarding some operations (mostly iteration and product) of scattered languages.
\[prop-iterate-vstar\] Assume $L\subseteq\Sigma^*$ is a language such that $L^+$ is scattered. Then $L\subseteq v^*$ for some word $v\in\Sigma^*$.
Assume $u,v\in L$ are nonempty words with $\mathrm{root}(u)\neq\mathrm{root}(v)$. Then, by Lyndon’s theorem (see e.g. [@10.5555/267846], Theorem 2.2), $uv\neq vu$, say $uv<_svu$ (having the same length, they cannot be in the $<_p$ relation, so it’s either $uv<_svu$ or the other way around). Then the language $\{uvuv,vuvu\}^*uvvu$ forms a dense subset in $L^+$. Thus, if $L^+$ is scattered, then the nonempty members of $L$ share a common primitive root $v$, and hence $L\subseteq v^*$.
\[prop-dense-language-has-a-prefixfree-sublanguage\] If $L\subseteq\Sigma^*$ is a dense language, then it has a prefix-free dense subset $K\subseteq L$.
Let $P\subseteq L$ be the language containing all the words which are members of some infinite prefix chain of $L$. Now we have two cases:
If $P$ is not dense, then there exist two elements $u,v\in P$ such that $u<_\ell v$ but there is no $w\in P$ with $u<_\ell w<_\ell v$. Then, the sublanguage $L'=\{x\in L ~:~ u <_\ell x <_\ell v \}$ of $L$ is still dense and has no member in $P$. In $L'$ there can be elements which are in the prefix relation, but all the $<_p$-chains are finite within $L'$ (since if $L'$ contains an infinite $<_p$ chain, its elements would be in $P$). So let $K\subseteq L'$ be the language containing the $<_p$-maximal elements of $L'$ (i.e. there is no such word which is greater than them in prefix relation). Since there is no infinite prefix chain in $L'$, we have $L'\subseteq \bigcup_{w\in K} \pref(w)$. Since $\pref(w)$ is finite for each word $w\in K$, while $L'$ is infinite (and dense), so $K$ has to be still dense and prefix-free.
If $P$ is dense, we define a word $x_u\in P$ inductively for each word $u\in \{0,2\}^*\{\varepsilon,1\}$ such that $u <_p v$ implies $x_u <_p x_v$ and $u <_s v$ implies $x_u <_s x_v$. First observe that for each $x\in P$, there has to be an infinite number of $\omega$-words $w$ such that $x\in\pref(w)$ and $\pref(w)\cap P$ is infinite (that is, there have to be infinitely many different prefix chains containing $w$), for if there were some $x\in P$ with only a finite number of such $\omega$-words, say $\{w_1,\ldots,w_k\}$, then choosing one of them, say $w_1$, there would be a length $N$ such that if $u\in\pref(w_1)$ with $|u|\geq N$, then $u\notin\pref(w_i)$ for $i>1$. Hence, if $u$ and $v$ were long enough members of $\pref(w_1)$, then only a finite number of elements of $P$ would fit between them (each of them being prefixes of the same $w_1$) and $P$ wouldn’t be a dense set.
So, moving back to the construction, for the base step, we choose an arbitrary word from $P$, for $x_\varepsilon$. Having defined $x_u\in P$ with $u\in\{0,2\}^*$, we define $x_{u0}$, $x_{u1}$ and $x_{u2}$ as follows. Since there are infinitely many infinite prefix chains in $P$ containing $x_u$, we can choose three different $\omega$-words, $w_1$, $w_2$ and $w_3$ with $x_u$ being a prefix of each of them and with $w_1<_s w_2<_sw_3$. Since the three $\omega$-words differ, long enough prefices of $w_i$ are not prefices of the other two words, and since each $w_i$ is a limit of an infinite prefix chain, we can choose long enough prefices of each $w_i$ which are in $P$ and not prefices of the other two $\omega$-words. We define $x_{u0}$, $x_{u1}$ and $x_{u2}$ to be this prefix of $w_1$, $w_2$ and $w_3$ respectively.
Then, words of the form $u_{x1}$ form a dense subset of $P$.
\[prop-sub-of-scattered-is-scattered\] If $L\subseteq\Sigma^*$ is a scattered language and $uK\subseteq\pref(L)$ for some word $u\Sigma^*$ and language $K\subseteq\Sigma^*$, then $K$ is scattered as well.
Since $u^{-1}L$ embeds into $L$ under the mapping $x\mapsto ux$, we get that $u^{-1}L$ is scattered as well and $K\subseteq u^{-1}\pref(L)=\pref(u^{-1}L)$. Assume $K$ is not scattered, that is, it has a dense subset $X\subseteq K$. By Proposition \[prop-dense-language-has-a-prefixfree-sublanguage\] there exists a language $X'\subseteq X$ such that $X'$ is prefix-free and still dense. Hence, $X'$ embeds into $\pref(u^{-1}L)$ as well, which is a contradiction since a dense ordering cannot be embedded into a scattered one. Thus, $K$ has to be scattered.
\[cor-product-members-are-scattered\] If $L=L_1L_2$ is a nonempty scattered language, then so are $L_1$ and $L_2$.
Linear and semilinear sets
==========================
Let $\mathbb{N}_0$ stand for the set of nonnegative integers. We call a set $X\subseteq\mathbb{N}_0^k$ *periodic* if it has the form $X=\{N+M\cdot t:t\geq 0\}$ for some vectors $N,M\in\mathbb{N}_0^k$; *linear* if it has the form $X=\{N_0+N_1\cdot t_1+N_2\cdot t_2+\ldots+N_k\cdot t_n:
t_1,\ldots,t_n\geq 0\}$ for some integer $n\geq 0$ and vectors $N_0,\ldots,N_k\in\mathbb{N}_0^k$; *semilinear* if it is a finite union of linear sets and *ultimately periodic* if it is a finite union of periodic sets. (Observe that a singleton set is also periodic, by choosing the vector $M$ in the definition to be the null vector, thus finite sets are ultimately periodic.)
It is known [@Matos94periodicsets] that a subset of $\mathbb{N}_0$ is ultimately periodic if and only if it is semilinear. Moreover, by Parikh’s theorem we know that the Parikh image $\Psi(L)=\{(|u|_0,|u|_1):u\in L\}$ of any context-free language $L\subseteq\{0,1\}^*$ is semilinear (the theorem holds for arbitrary alphabets).
Let us define the (net) *opening depth* of a word $w\in\{0,1\}^*$ as $\mathrm{open}(w)=|w|_0-|w|_1$. Clearly, a word $w$ belongs to $\mathbf{Pref}(D_1)$ if and only if $\mathrm{open}(w')\geq 0$ for each prefix $w'$ of $w$, and to $D_1$ if additionally, $\mathrm{open}(w)=0$. As an extension, we define $\mathrm{open}':\mathbb{N}_0^2\to\mathbb{N}_0$ as $(n,m)\mapsto n-m$. Then clearly, $\mathrm{open}(w)=\mathrm{open}'(\Psi(w))$ for each word $w\in\{0,1\}^*$ and the image of a linear set $\{(n_0,m_0)+(n_1,m_1)\cdot t_1+\ldots+(n_k,m_k)\cdot t_k:t_1,\ldots,t_k\geq 0\}\subseteq\mathbb{N}_0^2$ is the linear (thus ultimately periodic) set $\left\{(n_0-m_0)+\mathop\sum\limits_{i=1}^k(n_i-m_i)\cdot t_i:t_1\ldots,t_k\geq 0\right\}\subseteq\mathbb{N}_0$. Hence, $\mathrm{open}(L)$ is an ultimately periodic set for any context-free language $L\subseteq\{0,1\}^*$.
Similarly, let us define the *closing depth* of a word $w\in\{0,1\}^*$ as $\mathrm{close}(w)=|w|_1-|w|_0$. Then, a word $w$ belongs to $\mathbf{Suf}(D_1)$ if and only if $\mathrm{close}(w')\geq 0$ for each suffix $w'$ of $w$, and belongs to $D_1$ if and only if additionally $\mathrm{close}(w)=0$. Again, we define $\mathrm{close'}(n,m)=m-n$. We get also that for any context-free language $L\subseteq\{0,1\}^*$, $\mathrm{close}(L)$ is ultimately periodic.
Given a transducer $M=(Q,\Sigma,\Delta,q_0,F,\mu)$, we associate to each state $q\in Q$ the following set $N(q)\subseteq\mathbb{N}_0$ of integers: $n\in N(q)$ if and only if there exist words $u,v\in\{0,1\}^*$ with $uv\in D_1$, $q\in q_0u$, $qv\cap F\neq\emptyset$ and $\mathrm{open}(u)=n$. It will be useful to define two additional sets $N_{-}(q)$ and $N_{+}(q)$ as follows: let $n\in N_{-}(q)$ if and only if $q\in q_0u$ for some $u\in\mathbf{Pref}(D_1)$ with $\mathrm{open}(u)=n$ and similarly, $n\in N_{+}(q)$ if and only if $qu\cap F\neq\emptyset$ for some $u\in\mathbf{Suf}(D_1)$ with $\mathrm{close}(u)=n$. Clearly, $N(q)~=~N_{-}(q)\cap N_{+}(q)$.
\[prop-nq-ultimately-periodic\] For each state $q$ of a transducer $M$, the set $N(q)$ is ultimately periodic.
As $N_{-}(q)=\mathrm{open}(\{u\in\mathbf{Pref}(D_1):q\in q_0u\})$ and this language is the intersection of the context-free language $\mathbf{Pref}(D_1)$ and the regular language $\{u\in\{0,1\}^*:q\in q_0u\}$, we have that $N_{-}(q)$ is ultimately periodic.
Similarly, $N_{+}(q)$ is ultimately periodic as well. As the intersection of finitely many ultimately periodic sets is ultimately periodic [@Matos94periodicsets], so is $N(q)$.
For an example for a transducer (without the output function as that does not play a role in the sets $N(q)$) and the sets $N(q)$ see Figure \[fig-nq\]. The reader is encouraged to verify some of these sets, e.g. for $N_+(q_1)$ we have that the words accepted from $q_1$ are the members of the language $(000+01)^*0(1(11)^*+11)~\cap~\mathbf{Suf}(D_1)$ on which if we apply the $\mathrm{close}$ function we get the nonnegative numbers belonging to the set $\{-3t_1-1+1+2t_2:t_1,t_2\geq 0\}~\cup~\{-3t_1-1+2:t_1\geq 0\}$, that is, $\{2t_2-3t_1:t_1,t_2\geq 0,2t_2\geq 3t_1\}~\cup~\{1\}$ which in turn is simply $\mathbb{N}_0$, or $\{t:t\geq 0\}$ as each nonnegative integer $k$ can be written as either $k=2\cdot t_2-3\cdot 0$ if $k$ is even and as $k=2t_1-3\cdot 1$ if $k$ is odd.
=\[draw=black,text=black, inner sep = 0cm, outer sep = 0cm, minimum size = 0.7 cm\] (q0) [$q_0$]{}; (q1) \[above right of=q0\] [$q_1$]{}; (q2) \[below right of=q0\] [$q_2$]{}; (q3) \[above of=q1\] [$q_3$]{}; (q4) \[right = 2.5cm of q1\] [$q_4$]{}; (q5) \[right = 2.5cm of q4\] [$q_5$]{}; (q6) \[right of=q5\] [$q_6$]{}; (q7) \[below of=q4\] [$q_7$]{}; (q8) \[right of=q7\] [$q_8$]{};
(q0) edge node [$0$]{} (q1) (q1) edge node\[left\] [$0$]{} (q2) (q2) edge node [$0$]{} (q0) (q1) edge\[bend left=30\] node [$0$]{} (q3) (q3) edge\[bend left=30\] node [$1$]{} (q1) (q1) edge node [$0$]{} (q4) (q4) edge node [$1$]{} (q5) (q4) edge node [$1$]{} (q7) (q7) edge node [$1$]{} (q8) (q5) edge\[bend left=30\] node [$1$]{} (q6) (q6) edge\[bend left=30\] node [$1$]{} (q5) ;
(sq0) \[above left = 0.2cm and -0.2cm of q0\] [ $\boldsymbol{-}: \{3t\}$,\
$\boldsymbol{+}:\{t\},$\
$\boldsymbol{\cap}: \{3t\}$ ]{}; (sq1) \[below right =0.1cm and -0.1cm of q1\] [ $\boldsymbol{-}: \{3t+2\}$,\
$\boldsymbol{+}: \{t\},$\
$\boldsymbol{\cap}: \{3t+2\}$ ]{}; (sq2) \[below left=0.3cm and -0.7cm of q2\] [ $\boldsymbol{-}: \{3t+2\}$,\
$\boldsymbol{+}:\{t+1\},$\
$\boldsymbol{\cap}: \{3t+2\}$ ]{}; (sq3) \[above = 0.2cm of q3\] [ $\boldsymbol{-}: \{3t+2\}$,\
$\boldsymbol{+}:\{t\},$\
$\boldsymbol{\cap}: \{3t+2\}$ ]{}; (sq4) \[above right = 0.2cm and -1.0cm of q4\] [ $\boldsymbol{-}: \{3t+2\}$,\
$\boldsymbol{+}:\{2\}\cup\{2t+1\},$\
$\boldsymbol{\cap}: \{2\}\cup\{6t+5\}$ ]{}; (sq5) \[above right = 0.2cm and -0.5cm of q5\] [ $\boldsymbol{-}: \{t\}$,\
$\boldsymbol{+}:\{2t\},$\
$\boldsymbol{\cap}: \{2t\}$ ]{}; (sq6) \[right = 0.2cm of q6\] [ $\boldsymbol{-}: \{t\}$,\
$\boldsymbol{+}:\{1+2t\},$\
$\boldsymbol{\cap}: \{1+2t\}$ ]{}; (sq7) \[below = 0.3cm of q7\] [ $\boldsymbol{-}: \{3t+1\}$,\
$\boldsymbol{+}:\{1\},$\
$\boldsymbol{\cap}: \{1\}$ ]{}; (sq8) \[right =0.2cm of q8\] [ $\boldsymbol{-}: \{3t\}$,\
$\boldsymbol{+}:\{0\},$\
$\boldsymbol{\cap}: \{0\}$ ]{};
\(p) \[below of=sq6\] [$\boldsymbol{P}=6$]{};
\[prop-period-and-tau-exist\] For any transducer $M$, there exists some integer $P>1$, called a *period* of $M$ and for each state $q$ of $M$, some subset $\tau(q)$ of $\{0,\ldots,2P-1\}$, called the *type* of $q$ such that $$N(q)=\bigl(\tau(q)\cap\{0,\ldots,P-1\}\bigr)~\cup~\{n\in\mathbb{N}:~n\geq P,n\equiv r~\mathrm{mod}~P\hbox{ for some }r\geq P,r\in\tau(q)\}.$$
By Proposition \[prop-nq-ultimately-periodic\], each set $N(q)$ is ultimately periodic, that is, a finite union of sets of the form $\{r+p\cdot t:t\geq 0\}$ for some constants $r,p\geq 0$ (called the remainder and the period – the case $p=0$ defines a singleton set). Let $P$ be the least integer which is a multiple of each nonzero period and larger than all the remainders and is also at least two.
We claim that $X(q)=\{n:0\leq n\leq 2P-1\}\cap N(q)$ is a good choice for the type of $q$. To this end, let $\widehat{X}(q)$ stand for the (ultimately periodic) set $$\bigl(X(q)\cap\{0,\ldots,P-1\}\bigr)~\cup~\mathop\bigcup\limits_{r\in X(q),r\geq P}\{n\geq P:n\equiv r~\mathrm{mod}~P\}.$$ So we have to show that $N(q)=\widehat{X}(q)$.
First, observe that $\widehat{X}(q)\cap\{0,\ldots,P-1\}~=~N(q)\cap\{0,\ldots,P-1\}$ by the definition of $X(q)$ so we have to show that for any integer $n\geq P$, $n\in\widehat{X}(q)$ if and only if $n\in N(q)$. Let us write $N(q)=\mathop\bigcup\limits_{i\in[k]}\{r_i+p_i\cdot t:t\geq 0\}$
And indeed, for $n\geq P$ (and thus $n\geq r_i,p_i$ for each $i\in[k]$) we have $$\begin{aligned}
n\in\widehat{X}(q) &\Leftrightarrow n\equiv r~\mathrm{mod}~P\hbox{ for some }r\in X(q),r\geq P\\
&\Leftrightarrow n\equiv r~\mathrm{mod}~P\hbox{ for some }r\in N(q),P\leq r<2P\\
&\Leftrightarrow n\equiv r_i+p_i\cdot t~\mathrm{mod}~P\hbox{ for some }i\in[k], 0\leq t\\
&\Leftrightarrow n\equiv r_i+p_i\cdot t~\mathrm{mod}~P\hbox{ for some }i\in[k], 0\leq t<P/p_i\\
&\Leftrightarrow n\equiv r_i~\mathrm{mod}~p_i,n\geq r_i\hbox{ for some }i\in[k]\\
&\Leftrightarrow n\in N(q).
\end{aligned}$$
Now we create a transducer $M'$ from $M$ by creating copies of each state. We want to construct $M'$ so that each state should have a *singleton* type. The states of $M'$ will be triples of the form $(q,n,\sigma)$ with $q\in Q$, $n\in\tau(q)$ and $\sigma\in\{\equiv,\uparrow,\downarrow\}$.
Let $P$ be a period of $M$. From the state $q$ of $M$, we will create states $(q,n,\equiv)$ for each $P\leq n\in\tau(q)$ and two states, $(q,n,\uparrow)$ and $(q,n,\downarrow)$ for each $n\in\tau(q)$ with $n<P$. Observe that since $q_0w\cap F\neq \emptyset$ for some $w\in D_1$, we have $0\in\tau(q_0)$. In $M'$, let $(q_0,0,\uparrow)$ be the initial state. Also, if $q_f\in F$, then we can assume that there exists some word $w\in D_1$ with $q_f\in q_0w$ (otherwise we can remove $q_f$ from $F$, the resulting transducer will be equivalent with $M$), and so $0\in N(q_f)$ as well. So let $\{(q_f,0,\downarrow):q_f\in F\}$ be the (nonempty) set of accepting states in $M'$.
We define the transitions of $M'$ as follows: let $((p,n,\sigma_1),a,(q,m,\sigma_2))\in\Delta'$ if and only if $(p,a,q)\in\Delta$ and one of the following conditions holds:
1. $n+1=m<P$, $\sigma_1=\sigma_2$ and $a=0$
2. $n-1=m$, $m<P$, $\sigma_2\in\{\sigma_1,\downarrow\}$ and $a=1$
3. $n+1\equiv m~\mathrm{mod}~P$, $m\geq P$, $n\geq P-1$, $a=0$, $\sigma_2=\equiv$ and $\sigma_1\neq\downarrow$
4. $n-1\equiv m~\mathrm{mod}~P$, $n\geq P$, $m\geq P-1$, $a=1$, $\sigma_1=\equiv$ and $\sigma_2\neq\uparrow$.
Moreover, for $((p,n),a,(q,m))\in\Delta'$, let $\mu'((p,n),a,(q,m))=\mu(p,a,q)$. Finally, if there is any non-accessible or non-coaccessible state in $M'$, then let us drop it. Figure \[fig-mprime\] shows a part of the transducer $M'$ constructed from the transducer $M$ of Figure \[fig-nq\] with some states missing and without the output function, to maintain readability of the transition diagram. The idea is that when $M'$ reads some input word, then for a while it uses states labeled by $\uparrow$, then if for the currently read prefix the opening depth reaches $P$, then from that point it uses states labeled by $\equiv$, then, after reading in the longest prefix with opening depth at least $P$ it switches to states labeled by $\downarrow$. In the $\uparrow$ and $\downarrow$ states, the exact opening depth is maintained while in the $\equiv$ states it’s maintained only up to modulo $P$. (During the switch from an $\equiv$ state to a $\downarrow$ state, nondeterminism is used to guess the end of the longest prefix and this guess is then checked against by the $\downarrow$ states.) Finally, if the depth of the word never reaches $P$, then the transducer switches at some point from an $\uparrow$-state to a $\downarrow$ state by a transition of type ii). Most of these latter transitions are missing intentionally from the diagram of $M'$ of Figure \[fig-mprime\].
=\[draw=black,text=black, inner sep = 0cm, outer sep = 0cm, minimum size = 0.6 cm, ellipse\] (q0-0) [$q_0,0,\uparrow$]{}; (q0-3) \[above of=q0-0\] [$q_0, 3, \uparrow$]{}; (q0-6) \[above of=q0-3\] [$q_0, 6, \equiv$]{}; (q0-9) \[above of=q0-6\] [$q_0, 9, \equiv$]{};
(q1-1) \[below right = 2cm and 2cm of q0-0\] [$q_1,1,\uparrow$]{}; (q1-4) \[above of=q1-1\] [$q_1, 4, \uparrow$]{}; (q1-7) \[above of=q1-4\] [$q_1, 7, \equiv$]{}; (q1-10) \[above of=q1-7\] [$q_1, 10, \equiv$]{};
(q2-2) \[below left = 5cm and 1cm of q0-0\] [$q_2,2,\uparrow$]{}; (q2-5) \[above of=q2-2\] [$q_2, 5, \uparrow$]{}; (q2-8) \[above of=q2-5\] [$q_2, 8, \equiv$]{}; (q2-11) \[above of=q2-8\] [$q_2, 11, \equiv$]{};
(q3-2) \[right = 2cm of q1-1\] [$q_3, 2,\uparrow$]{}; (q3-5) \[above of=q3-2\] [$q_3, 5, \uparrow$]{}; (q3-8) \[above of=q3-5\] [$q_3, 8, \equiv$]{}; (q3-11) \[above of=q3-8\] [$q_3, 11, \equiv$]{};
(q4-2) \[below right = 4cm and -1cm of q3-2\] [$q_4, 2,\uparrow$]{}; (q4-5) \[above of=q4-2\] [$q_4, 5, \uparrow$]{}; (q4-11) \[above of=q4-5\] [$q_4, 11, \equiv$]{};
(q5-0) \[below right =1cm and 2cm of q4-2\] [$q_5, 0,\downarrow$]{}; (q5-2) \[above of=q5-0\] [$q_5, 2, \downarrow$]{}; (q5-4) \[above of=q5-2\] [$q_5, 4, \downarrow$]{}; (q5-6) \[above of=q5-4\] [$q_5, 6, \equiv$]{}; (q5-8) \[above of=q5-6\] [$q_5, 8, \equiv$]{}; (q5-10) \[above of=q5-8\] [$q_5, 10, \equiv$]{};
(q6-1) \[right = 2cm of q5-0\] [$q_6, 1,\downarrow$]{}; (q6-3) \[above of=q6-1\] [$q_6, 3, \downarrow$]{}; (q6-5) \[above of=q6-3\] [$q_6, 5, \downarrow$]{}; (q6-7) \[above of=q6-5\] [$q_6, 7, \equiv$]{}; (q6-9) \[above of=q6-7\] [$q_6, 9, \equiv$]{}; (q6-11) \[above of=q6-9\] [$q_6, 11, \equiv$]{};
(q7-1) \[below = 1.1cm of q4-2\] [$q_7, 1,\uparrow$]{};
(q8-0) \[below = 1.1cm of q7-1\] [$q_8, 0,\downarrow$]{};
(q0-0.east) edge node\[near start\] [$0$]{} (q1-1.west) (q0-3.east) edge node\[near start\] [$0$]{} (q1-4.west) (q0-6.east) edge node\[near start\] [$0$]{} (q1-7.west) (q0-9.east) edge node\[near start\] [$0$]{} (q1-10.west)
(q1-1.west) edge\[bend left = 20\] node\[near end,below\] [$0$]{} (q2-2.east) (q1-4.west) edge\[bend left = 20\] node\[near end,below\] [$0$]{} (q2-5.east) (q1-7.west) edge\[bend left = 20\] node\[near end,below\] [$0$]{} (q2-8.east) (q1-10.west) edge\[bend left = 20\] node\[near end,below\] [$0$]{} (q2-11.east)
(q1-1) edge\[bend left = 10\] node\[near start,above\] [$0$]{} (q3-2) (q1-4) edge\[bend left = 10\] node\[near start,above\] [$0$]{} (q3-5) (q1-7) edge\[bend left = 10\] node\[near start,above\] [$0$]{} (q3-8) (q1-10) edge\[bend left = 10\] node\[near start,above\] [$0$]{} (q3-11)
(q1-1.south) edge\[bend left = -20\] node\[near end,left\] [$0$]{} (q4-2.west) (q1-4.east) edge\[in = 150, out=20\] node\[near end,left\] [$0$]{} (q4-5.west) (q1-10.north) edge\[bend left = 80, in=90, out=90,looseness=1.95\] node\[near end,left\] [$0$]{} (q4-11.north)
(q2-2.west) edge\[in = 180, out=180,looseness=1.3\] node\[near end\] [$0$]{} (q0-3.west) (q2-5.west) edge\[in = 180, out=180,looseness=1.3\] node\[near end\] [$0$]{} (q0-6.west) (q2-8.west) edge\[in = 180, out=180,looseness=1.3\] node\[near end\] [$0$]{} (q0-9.west) (q2-11.north) edge\[in = 230, out=110,looseness=1.3\] node\[near start, right\] [$0$]{} (q0-6.west)
(q3-2) edge\[bend left = 10\] node\[near start,below\] [$1$]{} (q1-1) (q3-5) edge\[bend left = 10\] node\[near start,below\] [$1$]{} (q1-4) (q3-8) edge\[bend left = 10\] node\[near start,below\] [$1$]{} (q1-7) (q3-11) edge\[bend left = 10\] node\[near start,below\] [$1$]{} (q1-10)
(q4-2) edge node\[right\] [$1$]{} (q7-1) (q4-5) edge\[bend left = 10\] node\[below\] [$1$]{} (q5-4) (q4-11) edge\[bend left = 10\] node\[below\] [$1$]{} (q5-10)
(q5-2) edge node\[below\] [$1$]{} (q6-1) (q5-4) edge node\[below\] [$1$]{} (q6-3) (q5-6) edge node\[below\] [$1$]{} (q6-5) (q5-8) edge node\[below\] [$1$]{} (q6-7) (q5-10) edge node\[below\] [$1$]{} (q6-9)
(q5-6.west) edge\[out=120,in=120,looseness=1.7\] node\[above\] [$1$]{} (q6-11)
(q6-1) edge node\[below right\] [$1$]{} (q5-0) (q6-3) edge node\[below right\] [$1$]{} (q5-2) (q6-5) edge node\[below right\] [$1$]{} (q5-4) (q6-7) edge node\[below right\] [$1$]{} (q5-6) (q6-9) edge node\[below right\] [$1$]{} (q5-8) (q6-11) edge node\[below right\] [$1$]{} (q5-10)
(q7-1) edge node\[right\] [$1$]{} (q8-0) ;
\[prop-consistent-runs\] For each word $u=a_1\ldots a_n\in D_1$ and run $q_0\mathop{\longrightarrow}\limits^{a_1/R_1}q_1\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}q_n$ in $M$ with $q_n\in F$ there is a run $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,t_1,\sigma_1)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,t_n,\sigma_n)$ in $M'$ with $(q_n,t_n,\sigma_n)\in F\times\{0\}$ in $M'$.
Let $u=a_1\ldots a_n\in D_1$ be a word and $q_0\mathop{\longrightarrow}\limits^{a_1/R_1}q_1\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}q_n$ be a run in $M$ with $q_n\in F$.
There are two cases: either $\mathrm{open}(v)<P$ for each prefix $v$ of $u$, or $\mathrm{open}(v)\geq P$ for at least one prefix $v$ of $u$. We construct an accepting run $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,t_1,\sigma_1)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,t_n,\sigma_n)$ of $M'$ in both cases.
1. If $\mathrm{open}(v)<P$ for each prefix $v$ of $u$, then let us define $t_i=\mathrm{open}(v)$ for each $0\leq i\leq n$, $\sigma_i=\uparrow$ for each $0\leq i<n$ and $\sigma_n=\downarrow$. Then, the first $n-1$ transitions are of type i) and type ii) depending on $a_i$, with $\sigma_1=\sigma_2=\uparrow$, and the last transition is of type ii) with $\sigma_2=\downarrow$, since by $u\in D_1$ we get $a_n=1$. Thus this is indeed an accepting run in $M'$.
2. If $\mathrm{open}(v)\geq P$ for at least one prefix $v$ of $u$, then let $i_\uparrow\geq 0$ be the largest index so that for each $j\leq i_\uparrow$, $\mathrm{open}(a_1\ldots a_j)<P$ and let $i_\downarrow$ be the smallest index so that for each $j\geq i_\downarrow$, $\mathrm{open}(a_1\ldots a_j)<P$. These indices exist since $\mathrm{open}(a_1)=1<P$ and $\mathrm{open}(a_1\ldots a_n)=0<P$, moreover, $i_\uparrow<i_\downarrow$ since there exists some $i$ with $\mathrm{open}(a_1\ldots a_i)\geq P$ and all of these $i$s have to fall strictly between $i_\uparrow$ and $i_\downarrow$.
Now let us define $$\begin{aligned}
t_i&=\begin{cases}
\mathrm{open}(a_1\ldots a_i)&\hbox{if }i\leq i_\uparrow\hbox{ or }i\geq i_\downarrow\\
(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P&\hbox{otherwise}
\end{cases}&
\sigma_i&=\begin{cases}
\uparrow&\hbox{if }i\leq i_\uparrow\\
\equiv&\hbox{if }i_\uparrow<i<i_\downarrow\\
\downarrow&\hbox{if }i_\downarrow\leq i.
\end{cases}
\end{aligned}$$ We claim that for each $0\leq i<n$, $((q_i,t_i,\sigma_i),a_{i+1},(q_{i+1},t_{i+1},\sigma_{i+1}))$ is a transition in $M'$. Indeed: $(q_i,a_{i+1},q_{i+1})$ is a transition of $M$ and
- if $i<i_\uparrow$ and $a_{i+1}=0$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i+1<P$ and $\sigma_1=\sigma_2=\uparrow$, thus then the triple is a type i) transition
- if $i<i_\uparrow$ and $a_{i+1}=1$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i-1$, $t_i<P$ and $\sigma_1=\sigma_2=\uparrow$, thus then the triple is a type ii) transition
- if $i=i_\uparrow$, then (by the maximality of $i_\uparrow$) $a_{i+1}=0$, $\mathrm{open}(a_1\ldots a_i)=t_i=P-1$, $\mathrm{open}(a_1\ldots a_{i+1})=t_{i+1}=P$ (as $(P~\mathrm{mod}~P)+P=P$, $\sigma_1=\uparrow$, $\sigma_2=\equiv$ and the triple is a type iii) transition
- if $i_\uparrow<i<i_\downarrow-1$ and $a_{i+1}=0$, then $\sigma_i=\sigma_{i+1}=\equiv$, $t_i=(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P\geq P$, $t_{i+1}=((\mathrm{open}(a_1\ldots a_i)+1)~\mathrm{mod}~P)+P\geq P$ and the triple is a type iii) transition
- if $i_\uparrow<i<i_\downarrow-1$ and $a_{i+1}=1$, then $\sigma_i=\sigma_{i+1}=\equiv$, $t_i=(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P\geq P$, $t_{i+1}=((\mathrm{open}(a_1\ldots a_i)-1)~\mathrm{mod}~P)+P\geq P$ and the triple is a type iv) transition
- if $i=i_\downarrow-1$, then (by the minimality of $i_\downarrow$) $t_i=\mathrm{open}(a_1\ldots a_i)=P$, $a_{i+1}=1$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=P-1$, $\sigma_i=\equiv$, $\sigma_2=\downarrow$ and the triple is a type iv) transition
- if $i\leq i_\downarrow$ and $a_{i+1}=0$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i+1<P$ and $\sigma_1=\sigma_2=\downarrow$, thus then the triple is a type i) transition
- if $i\leq i_\downarrow$ and $a_{i+1}=1$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i-1$, $t_i<P$ and $\sigma_1=\sigma_2=\downarrow$, thus then the triple is a type ii) transition
\[cor-pm-is-pmprime\] $L(M)=L(M')$ for the transducers $M$ and $M'$ of Proposition \[prop-consistent-runs\].
From Proposition \[prop-consistent-runs\] we have $L(M)\subseteq L(M')$. For the other direction, $L(M)\subseteq L(M')$ also clearly holds since the mapping $(q,n,\sigma)\mapsto q$ for each $q\in Q$, $n\in\tau(q)$, $\sigma\in\{\uparrow,\downarrow,\equiv\}$ transforms an accepting run in $M'$ into an accepting run in $M$, with the same labels on the transitions.
Hence, we can consider the automaton $M'$ and call those runs of the form $$(q_0,t_0,\sigma_0)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,t_1,\sigma_1)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,t_n,\sigma_n)$$ of $M'$ explained in the construction *consistent*. By Proposition \[prop-consistent-runs\], $L(M)$ is then the union of all the languages $R_1\ldots R_n$ occurring as output sequences on accepting consistent runs of $M'$ on input words belonging to $D_1$.
Cycles in $M'$
==============
Let us fix for this section a transducer $M=(Q,\{0,1\},\delta,q_0,F)$ generating a scattered language $L(M)$, let $P$ be a period of $M$ and let $M'$ be the construction of Proposition \[prop-consistent-runs\]. Viewing $M'$ as a directed graph, we can study the strongly connected components (SCCs) of $M'$. Without loss of generality, as $M$ is nondeterministic, we can assume that $q_0$ is a source state (there are no incoming transitions to $q_0$) and each member of $F$ is a sink state (there are no outgoing transitions from the members of $F$). Hence, the state $(q_0,0,\uparrow)$ is also a source in $M'$ and each $(q_f,0,\downarrow)$ with $q_f\in F$ is a sink in $M'$, thus each one of these states lie in its own trivial SCC. Let $\preceq$ be the usual reachability order on the states of $M'$, i.e., $(q,n,\sigma)\preceq (q',n',\sigma')$ if and only if $(q,n,\sigma)u\ni (q',n',\sigma')$ for some $u\in\{0,1\}^*$ and let $(q,n,\sigma)\approx(q',n',\sigma')$ if and only if $(q,n,\sigma)\preceq(q',n',\sigma')$ and $(q',n',\sigma')\preceq(q,n,\sigma)$. The strongly connected components, SCCs of $M'$ are its $\approx$-classes. By construction of $M'$ (using the condition $P>1$) we get that if a component is a singleton set, then it is *trivial*: no state can have a loop edge as if $(q,n,\sigma)a\ni(q',n',\sigma')$, then $n\neq n'$. We write $(q,n,\sigma)\prec(q',n',\sigma')$ if $(q,n,\sigma)\preceq(q',n',\sigma')$ and not the way around. This preorder gives rise to the partial order $\prec$ on the SCCs of $M'$: $C\prec C'$ if and only if $C\neq C'$ and $(q,n,\sigma)\prec(q',n',\sigma')$ for some states $(q,n,\sigma)\in C$, $(q',n',\sigma')\in C'$.
A *cycle* in $M'$ (from a state $(p_0,k_0,\sigma_0)$) is a closed sequence of edges $$(p_0,k_0,\sigma_0)\mathop{\longrightarrow}\limits^{a_1/R_1}(p_1,k_1,\sigma_1)
\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(p_n,k_n,\sigma_n)=(p_0,k_0,\sigma_0)$$s for some $n>0$. The *label* of this cycle is $a_1\ldots a_n$. Clearly, all the states on a cycle belong to the same SCC of $M'$, moreover, by construction we have that if $u$ is the label on a cycle, then $\mathrm{open}(u)\equiv 0~\mathrm{mod}~P$. In particular, if $\mathrm{open}(u)$ is zero, positive or negative, then the cycle is called zero, positive or negative, respectively. Thus, if $u$ is the label of a cycle from some state $(q,n,\sigma)$ with $n<P$, then $u$ is a cycle of zero weight.
We begin with a couple observations:
\[prop-tau-in-mprime\] In any SCC of $M'$, $\sigma$ is constant, i.e. if $(q,n,\sigma)\approx(q',n',\sigma')$, then $\sigma=\sigma'$.
For each state $(q,n,\sigma)$ of $M'$ with $\sigma\in\{\uparrow,\downarrow\}$ (and thus $0\leq n<P$), it holds that $\tau(q,n,\sigma)=\{n\}$. For each state $(q,n,\equiv)$ of $M'$ (and thus $P\leq n<2P$), it either holds that $\tau(q,n,\equiv)=\{n\}$ or $\tau(q,n,\equiv)=\{n,n-P\}$.
Let us introduce the ordering $\uparrow\leq\equiv\leq\downarrow$. Then, for each transition $((q,n,\sigma),a,(q',n',\sigma'))$ of $M'$ we have $\sigma\leq \sigma'$, hence if $(q,n,\sigma)\approx(q',n',\sigma')$, then $\sigma=\sigma'$ has to hold.
In particular, in any accepting run we first visit a positive number of $\uparrow$-states, then a nonnegative number of $\equiv$-states, and finally a positive number of $\downarrow$-states.
It is easy to see via induction on the length of the computation that if $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,k_1,\uparrow)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,k_n,\uparrow)$ is a path in $M'$, then for each $0\leq i\leq n$ we have $\mathrm{open}(a_1\ldots a_i)=k_i$, proving $N_{-}(q,k,\uparrow)\subseteq \{k\}$ for each $q\in Q$, $k\in\tau(q)$ for which $(q,k\uparrow)$ is a state of $M'$, and since $N(q,k,\uparrow)$ is nonempty (otherwise we would leave this state out), it has to be the case that $N(q,k,\uparrow)=\{k\}$. The same reasoning applied to states of the form $(q,k,\downarrow)$, considering the suffix of an accepting run that passes through solely on $\downarrow$-states.
Finally, by the construction of $\Delta$ it is clear that if $(q,k,\sigma)\in (q_0,0,\uparrow)\cdot u$ in $M'$, then $\mathrm{open}(u)\equiv k~\mathrm{mod}~P$. Hence, for each state $(q,k,\equiv)$ (thus $P\leq k<2P$) we have $\tau(q,k,\equiv)\subseteq\{k,k-P\}$. Also, since $k\in\tau(q)$ in $M$, for each $t\geq 0$ there is at least one accepting run $\pi$ of $M$ on some word $u\in D_1$ such that for some prefix $v$ of $u$ with $\mathrm{open}(v)=k+t\cdot P$, $\pi$ is in the state $q$. Then, the “lifted” run $\pi'$ of Proposition \[prop-consistent-runs\] is in some state $(q,t,\sigma)$ but as $\sigma\in\{\uparrow,\downarrow\}$ cannot happen here since $\tau(q,n,\sigma)$ would be then $\{n\}$ with $n<P$, it has to be the case that $\sigma=\equiv$ and $n=((k+t\cdot P)~\mathrm{mod}~P)+P=k$, thus $k\in\tau(q,t,\equiv)$ as well. Hence, $\tau(q,t,\equiv)$ is either $\{k\}$ or $\{k,k-P\}$.
\[prop-updown-only-zeros\] In a $\uparrow$- or a $\downarrow$-component, each cycle is a $0$-cycle.
If $(q,k,\uparrow)u\ni(q,k,\uparrow)$, and $(q_0,0,\uparrow)v\ni (q,k,\uparrow)$, then by Proposition \[prop-tau-in-mprime\] we have $\mathrm{open}(v)=\mathrm{open}(vu)=k$, hence $\mathrm{open}(u)=0$. For the $\downarrow$-states we have to consider the suffix of the computation the same way.
We introduce a couple of shorthands: let $\mathrm{CycleWords}(q,k,\sigma)\subseteq\{0,1\}^*$ be the language $\{u:(q,k,\sigma)\in(q,k,\sigma)\cdot u\}$ and $\mathrm{CycleOutputs}(q,k,\sigma)=\mathop\bigcup\limits_{(q,k,\sigma)\mathop{\longrightarrow}\limits^{u/R}(q,k,\sigma)}R$.
\[prop-zero-cycles-have-primitive-root\] To each state $(q,k,\sigma)$ there exists a primitive word $w(q,k,\sigma)\in\Sigma^*$ such that whenever $(q,k,\sigma)\mathop{\longrightarrow}\limits^{u/R}(q,k,\sigma)$ with $\mathrm{open}(u)=0$, then $R\subseteq w(q,k,\sigma)^*$.
Let $(q,k,\sigma)$ be a state of $M'$. If there is no cycle of weight $0$ visiting $(q,k,\sigma)$, then the claim is vacuously satisfied. Otherwise, for each such cycle $(q,k,\sigma)\mathop{\longrightarrow}\limits^{u/R}(q,k,\sigma)$ there is some input word $w$ and output language $R_0$ with $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{w/R_0}(q,k,\sigma)$ such that $wu\in\pref(D_1)$. Indeed, if $\sigma\in\{\uparrow,\downarrow\}$ then any such word $w\in\pref(D_1)$ leading into $(q,k,\sigma)$ with $\mathrm{open}(w)=k$ will do since in a $\uparrow$- or $\downarrow$-component $k$ always stores correctly the $\mathrm{open}$ing value of the input consumed so far within an accepting run, so during the consumation of the cycle of weight $0$, the opening depth remains nonnegative since we stay within the same component the whole time and there are no states $(q',k',\sigma)$ with negative $k'$. Otherwise, if $\sigma=\equiv$, then there is some word $w\in\pref(D_1)$ leading into $(q,k,\sigma)$ with $\mathrm{open}(w)=k+|u|\cdot P$, and so no prefix $v$ of $wu$ can have a negative $\mathrm{open}$ value, so this run on $wu$ can be extended into some accepting path.
But then, $R_0R^*\subseteq\pref(L(M))$, so by Proposition \[prop-sub-of-scattered-is-scattered\] we get that $R^*$ is scattered, so by Proposition \[prop-iterate-vstar\], $R\subseteq w(q,k,\sigma)^*$ for some (primitive) word $w(q,k,\sigma)$.
Now if there is another cycle $(q,k,\sigma)\mathop{\longrightarrow}\limits^{v/R'}(q,k,\sigma)$, then by the same reasoning we get that for some (deeply opening enough) prefix $w/R_0$, the language $R_0(R\cup R')^*$ is a subset of $\pref(L(M))$ and so $R\cup R'$ has a primitive root, which has to be the primitive root $w(q,k,\sigma)$ as well, thus this word is the primitive root of all the cycles of zero weight, starting from $(q,k,\sigma)$.
\[prop-nonnegative-cycles-have-primitive-root\] Assume there is some cycle of positive weight in some SCC $C$ of $M'$. Then for each $(q,k,\sigma)\in C$ there exists a (unique, primitive) word $w(q,k,\sigma)\in\Sigma^*$ such that $\mathrm{CycleOutputs}(q,k,\sigma)\subseteq w(q,k,\sigma)^*$.
(Clearly, this $w(q)$ has to coincide with the word $w(q,k,\sigma)$ of Proposition \[prop-zero-cycles-have-primitive-root\] for states having a passing cycle of weight zero as well.)
Let $(q,k,\sigma)$ be a state of $M'$ with $(q,k,\sigma)\mathop{\longrightarrow}\limits^{u_1/R_1}(q,k,\sigma)\mathop{\longrightarrow}\limits^{u_2/R_2}(q,k,\sigma)$(so $u_1,u_2\in\mathrm{CycleWords}(q,k,\sigma)$ and $R_1\cup R_2\subseteq\mathrm{CycleOutputs}(q,k,\sigma)$) such that $\mathrm{open}(u_1)>0$. This in particular means that $\sigma=\equiv$ and $k\geq P$, since by construction of $M'$, $\uparrow$- and $\downarrow$-components can only have cycles with zero weight. Then, as $k\in\tau(q,k,\equiv)$, for each $t\geq 0$ there exists some word $u\in\{0,1\}^*$ and language $R\subseteq\Sigma^*$ with $(q,k,\equiv)\mathop{\longrightarrow}\limits^{u/R}(q_f,0,\downarrow)$ for some $q_f\in F$ and $\mathrm{open}(u)=-(k+t\cdot P)$ (that is, $k+t\cdot P$ parenthesis can be opened in $q$ for any $t$, and they can still be closed with some word). Also, there is some word $w$ (it can be assumed that $\mathrm{open}(w)\geq k$ is large enough, since $(q,k,\equiv)$ is a $\equiv$-state, so $wu_1\in\pref(D_1)$) and language $R_0$ with $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{w/R_0}(q,k,\equiv)$. Thus, since $\mathrm{open}(wu_1^t)=\mathrm{open}(w)+t\cdot\mathrm{open}(u_1)$ which is of the form $k+t'\cdot P$ for some $t'\geq 0$ since $\mathrm{open}(u_1)>0$, we have that $R_0R_1^*\subseteq \pref(L(M'))$. Applying Propositions \[prop-sub-of-scattered-is-scattered\] and \[prop-iterate-vstar\] we have that $R_1\subseteq w(q,k,\sigma)^*$ for some primitive word $w(q,k,\sigma)$.
Also, if $(q,k,\equiv)\mathop{\longrightarrow}\limits^{u_2/R_2}(q,k,\equiv)$ for some $u_2\in\{0,1\}^*$ and $R_2\subseteq\Sigma^*$ (with $u_2$ being possibly a negative cycle), then we have that for some large enough $t\geq 1$, $(q,k,\equiv)\mathop{\longrightarrow}\limits^{u_1^tu_2/R_1^tR_2}(q,k,\equiv)$ is so that $\mathrm{open}(u_1^tu_2)>0$, hence, the language $R_1\cup R_1^tR_2\subseteq\mathrm{CycleOutputs}(q,k,\equiv)$ consists of words all sharing the same primitive root, which can only be $w(q,k,\sigma)$ (as $R_1\subseteq R_1\cup R_1^tR_2$), implying $R_1\cup R_1^tR_2\subseteq w(q,k,\sigma)^*$ which implies $R_2\subseteq w(q,k,\sigma)^*$ as well since $R_1^t\subseteq w(q,k,\sigma)^*$.
Hence, if there is some cycle with positive weight containing a state $(q,k,\equiv)$, then there exists a primitive word $w(q,k,\sigma)\in\Sigma^*$ such that $\mathrm{CycleOutputs}(q,k,\equiv)\subseteq w(q,k,\sigma)^*$.
Also, in a $\equiv$-component if there exists a cycle with positive weight, then there is such a cycle for each state in the same SCC: if $(q,k,\equiv)\mathop{\longrightarrow}\limits^{u/R}(q,k,\equiv)$ and $(q',k',\equiv)\approx(q,k,\equiv)$, that is, $(q,k,\equiv)\mathop{\longrightarrow}\limits^{u_1/R_1}(q',k',\equiv)\mathop{\longrightarrow}\limits^{u_2/R_2}(q,k,\equiv)$ and for some large enough $t$ then we have $(q',k',\equiv)\mathop{\longrightarrow}\limits^{u_1u^tu_2/R_1R^tR_2}(q',k',\equiv)$ with $\mathrm{open}(u_1u^tu_2)>0$, proving the statement.
\[prop-infinite-preimages-imply-root\] Assume $C$ is a component of $M'$ and there is a state $(q,k,\sigma)\in C$ and an output word $w\in\Sigma^*$ such that the set $\{\mathrm{open}(u):u\in\pref(D_1),(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/w}(q,k,\sigma)\}$ is infinite. (Thus, $\sigma=\equiv$.)
Then for each state $(q',k',\sigma)\in C$ there exists a word $w(q',k',\sigma)$ such that $\mathrm{CycleOutputs}(q',k',\sigma)\subseteq w(q',k',\sigma)^*$.
Let $(q',k',\sigma)$ be a state in $C$. If there are no cycles from $(q',k',\sigma)$ (that is, if $C$ is trivial), then the claim is vacuously satisfied. Otherwise, let $(q',k',\sigma)\mathop{\longrightarrow}\limits^{u_1/R_1}(q',k',\sigma)\mathop{\longrightarrow}\limits^{u_2/R_2}(q',k',\sigma)$ be two cycles (possibly the same). Then, from the condition of the Proposition, there is a word $u\in\pref(D_1)$ with $\mathrm{open}(u)\geq \max\{|u_1|,|u_2|\}+|C|+P$, a word $u'\in\{0,1\}^*$ of length at most $|C|$ and some language $R'$ (independent from $u_1$ and $u_2$) with $(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/w}(q,k,\sigma)\mathop{\longrightarrow}\limits^{u'/R'}(q',k',\sigma)$. By the condition on $u$, we have that both $uu'u_1$ and $uu'u_2$ are in $\pref(D_1)$ and still has an opening depth of at least $P$, so both runs can be extended to some accepting run, yielding $wR'(R_1\cup R_2)\subseteq \pref(L(M))$ for each choice of $R_1$ and $R_2$ which are output languages of some cycle starting from $(q',k',\sigma)$. Now since if $x$ and $y$ are cycles, then so is $xy$, we get that $wR'(R_1\cup R_2)^*$ is then also a subset of the scattered language $\pref(L(M))$, thus by Propositions \[prop-sub-of-scattered-is-scattered\] and \[prop-iterate-vstar\] we get that $R_1\cup R_2\subseteq w(q',k',\sigma)^*$ for some primitive word $w(q',k',\sigma)$, thus both $R_1$ and $R_2$ have the very same primitive root $w(q',k',\sigma)$, no matter the choice of $R_1$ and $R_2$. Thus, all the cycles indeed have the same primitive root, proving the claim.
For each transition $\delta=((q,k,\sigma),a/R,(q',k',\sigma'))$ in $M'$ with $(q,k,\sigma)\prec(q',k',\sigma')$ (that is, for each intercomponent edge) we define the language $L(\delta)\subseteq\Sigma^*$ as the output language of runs which use $\delta$ as their final transition and can be extended to an accepting run. Formally: $$L(\delta)~=~ \mathop\bigcup\limits_{(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/R_1}(q,k,\sigma):ua\in\pref(D_1),
\mathrm{open}(ua)\in\tau(q',k',\sigma')}R_1R.$$ The following proposition has the most involved proof in the paper and is the central statement on the way for bounding the rank of scattered restricted one-counter languages.
\[prop-ldelda-kicsi\] For each intercomponent edge $\delta$, $L(\delta)$ is a scattered language of rank smaller than $\omega^2$.
As $L(\delta)\subseteq\pref(L(M'))=\pref(L(M))$ and $L(M)$ is a scattered language, so is $L(\delta)$ by Proposition \[prop-sub-of-scattered-is-scattered\].
Let $\delta=((q,k,\sigma),a/R,(q',k',\sigma'))$ be an intercomponent edge, $(q,k,\sigma)\in C$ and $(q',k',\sigma')\in C'$ for the components $C\prec C'$ of $M'$. We use induction on the height of $C$ with respect to $\prec$ to prove the statement.
If $C=\{(q_0,0,\uparrow)\}$ is the smallest component (recall that $q_0$ is assumed to be a source state in $M$), then the claim holds since then $L(\delta)=R$ which is a (scattered) regular language and thus has a finite rank.
If $C$ is not the smallest component, then either $C$ contains a cycle of positive weight, or it does not. In the latter case, either there is an output word $w\in L(\delta')$ for some intercomponent transition $\delta'=((p,n,\sigma_1),b/R',(p',n',\sigma))$ leading into $C$ such that $\{\mathrm{open}(u):(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/w}(p,n,\sigma_1)\}$ is infinite, or there is not. Let us deal with the three cases separately: we collapse the first and the second case into one.
1. If $C$ contains a cycle of positive weight, or if there is some output word $w$ whose open-set described in the previous paragraph is infinite, then by Proposition \[prop-nonnegative-cycles-have-primitive-root\] or \[prop-infinite-preimages-imply-root\] respectively we have that whenever $(q_1,k_1,\sigma)\mathop{\longrightarrow}\limits^{u/R_1}(q_1,k_1,\sigma)$ is a cycle within $C$, then $R_1\subseteq w(q_1,k_1,\sigma)^*$. In particular, the order type of $R_1$ is either $\omega$ or finite, so its rank is at most one.
Now for any run $\pi$ using $\delta$ as its final transition there exist a sequence of distinct states $(q_1,k_1,\sigma),\ldots,(q_n,k_n,\sigma)$ of $C$ and an intercomponent transition $\delta'$ leading into $(q_1,k_1,\sigma)$ such that
- $\pi$ enters $C$ via $\delta'$, reaching $(q_1,k_1,\sigma)$
- then takes zero or more cycles involving $(q_1,k_1,\sigma)$
- then, after visiting $(q_1,k_1,\sigma)$ the last time, uses the transition to $(q_2,k_2,\sigma)$ labeled $R_2$, say
- then takes zero or more cycles involving $(q_2,k_2,\sigma)$ (that do not involve $(q_1,k_1,\sigma)$ but that will not be important)
- then after visiting $(q_2,k_2,\sigma)$ the last time, uses a transition to $(q_3,k_3,\sigma)$ labeled $R_3$, say, and so on
- finally, after visiting $(q_n,k_n,\sigma)$ the last time, uses $\delta$.
Now for any fixed $\delta',(q_1,k_1,\sigma),\ldots,(q_n,k_n,\sigma)$ the output language of these languages is contained within $$L(\delta')\cdot w(q_1,k_1,\sigma)^*\cdot R_2\cdot w(q_2,k_2,\sigma)^*\cdot R_3\cdot\ldots\cdot w(q_n,k_n,\sigma)^*\cdot R.$$ By the induction hypothesis, the rank of $L(\delta')$ is strictly smaller than $\omega^2$, the rank of each $w(q_i,k_i,\sigma)^*$ is at most $1$ and the rank of the regular languages $R$, $R_2,R_3,\ldots,R_n$ is finite, so the rank of these languages is finite plus something strictly smaller than $\omega^2$ by Proposition \[prop-rank-ops\], hence the rank of this product is still smaller than $\omega^2$. Now in this product there might be words which are not in $L(\delta)$ but the intersection of $L(\delta)$ and this product is a subset of the product language, hence the intersection also has a rank smaller than $\omega^2$.
As there are only finitely many options for choosing the transition $\delta'$ and the sequence of distinct states of $C$, the language $L(\delta)$ is thus a finite union of languages, each having a rank smaller than $\omega^2$, applying the equation for finite unions in Proposition \[prop-rank-ops\] we get that $L(\delta)$ also has a rank smaller than $\omega^2$.
2. Otherwise, $C$ might contain cycles of zero weight and negative cycles as well. By Proposition \[prop-zero-cycles-have-primitive-root\], to each state $(q_1,k_1,\sigma)$ in $C$ there exists a primitive word $w(q_1,k_1,\sigma)$ such that if $(q_1,k_1,\sigma)\mathop{\longrightarrow}\limits^{u/R_1}(q_1,k_1,\sigma)$ is so that $\mathrm{open}(u)=0$, then $R_1\subseteq w(q_1,k_1,\sigma)^*$. In this case we also partition $L(\delta)$ but this time into a larger number of clusters. As in the previous case, let $\delta'=((p_1,n_1,\sigma_1),b/R_1,(q_1,k_1,\sigma))$ be an intercomponent transition leading into $(q_1,k_1,\sigma)\in C$. Now let $wb\in L(\delta')$ be a possible output word of some run using $\delta'$ as its final step. For any such fixed $wb$, the set $\{\mathrm{open}(u):(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/w}(p_1,n_1,\sigma_1)\}$ is finite since the case when it can be infinite is handled in the previous case. So let $n\in\{\mathrm{open}(u):(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{u/w}(p_1,n_1,\sigma_1)\}$ be some integer in this finite set.
Now if a run starts with the labels $u/w$ with $\mathrm{open}(u)=n\geq 0$, enters a component $C$ via $\delta'$ which leads into $(q_1,k_1,\sigma)\in C$ such that in $C$ there are only cycles of nonpositive weight, reads in some word $v\in\{0,1\}^*$ within the component and leaves the component by the transition $\delta$ in a way that $uv$ is still a member of the language $\pref(D_1)$ (in order to be a prefix of some accepting path), then whenever $(q_1,k_1,\sigma),(q_2,k_2,\sigma),\ldots,(q_t,k_t,\sigma)$ is a sequence of (not necessarily distinct) states with $t\leq |C|\cdot(|C|+n+1)$, then a path $\pi$
- takes zero or more cycles of weight zero from $(q_1,k_1,\sigma)$ - the output language of which is a subset of $w(q_1,k_1,\sigma)^*$, and thus has rank at most one,
- then takes a transition $((q_1,k_1,\sigma),a_1/R_2,(q_2,k_2,\sigma))$ to $(q_2,k_2,\sigma)$ - the output language of which is a regular language of finite rank,
- then takes zero or more cycles of weight zero from $(q_2,k_2,\sigma)$ - again with rank at most one,
- then moves to $(q_3,k_3,\sigma)$ outputting a language of finite rank,
- and so on, finally after the cycles from $(q_t,k_t,\sigma)$, leaving the component using $\delta$ (so $q_t=q$, $k_t=k$), outputting again a language of finite rank.
Now since for each fixed sequence $q_1,\ldots,q_t$ the rank of the product language is finite, and there is a finite number of them since $t\leq |C|\cdot(|C|+n+1)$, this union also has a finite rank as well. We claim that for any run following $u$ from $(q_1,k_1,\sigma)$ which can still be extended to an accepting run there is always such a sequence of states of bounded length. It is clear that some such sequence exists (as, say, taking no cycles all and modeling the steps of the run inside $C$ is an option), so let us assume the sequence $(q_1,k_1,\sigma),\ldots,(q_t,k_t,\sigma)$ is the shortest one with the property that $\pi'$ can be written as above: cycles of weight $0$ from $(q_1,k_1,\sigma)$ a transition to $(q_2,k_2,\sigma)$, cycles of weight $0$ from there, one more transition etc and that $t>|C|\cdot(|C|+n+1)$. As there are only $|C|$ distinct states in $C$, there is at least one state $(p,k_p,\sigma)$ which occurs at least $|C|+n+1$ times in the sequence. Whenever a state occurs twice in this sequence of minimal length, then between the repetitions the run has to take a cycle of negative weight: there are no cycles of positive weight in $C$ and if it would be a cycle of weight zero, then we could collapse the segment between the repetition and gain a shorter sequence, contradicting minimality. Thus, while the state $(p,k_p,\sigma)$ gets repeated $|C|+n+1$ times, takes $|C|+n$ cycles of negative weight, which decrease the $\mathrm{open}$ of the consumed input word by at least $|C|+n$.
Now before the first occurrence of $(p,k_p,\sigma)$ the segment of the run might increase the $\mathrm{open}$ of the consumed input word (which is $n$ upon entering $C$) but only by at most $|C|-1$: the run starts at $(q_1,k_1,\sigma)\in C$, takes some cycles there which have a nonpositive weight, then after the last visit of $(q_1,k_1,\sigma)$ it moves to some $(p_2,k_2,\sigma)$ (notice this is another decomposition of the prefix than the one we used), changing the $\mathrm{open}$ by at most one, takes some cycles there having a nonpositive weight (thus not increasing the $\mathrm{open}$ value), after the last visit to $(p_2,k_2,\sigma)$ it takes a step to some other state (possibly increasing the $\mathrm{open}$ by one), and so on but as there are only $|C|-1$ such transitions as we always move into a new state, overall the $\mathrm{open}$ing depth of the input word can be increased to at most $n+|C|-1$. Then we are taking $n+|C|$ negative cycles, decreasing the $\mathrm{open}$ to a negative number, hence this run cannot be extended to an accepting one as the input word cannot be in $\pref(D_1)$.
Thus, as for each output word $w\in L(\delta')$ there are only a finite number of possibilities for opening depth of the input read so far, and for each such possibility $n$ a finite number of state sequences of length at most $|C|\cdot(|C|+n+1)$, each defining a language of finite rank, thus for each word $w\in L(\delta')$ we have a language $L_w$ of finite rank such that $w^{-1}L(\delta)\subseteq L_w$. That is, as $L(\delta')$ is a scattered language of rank smaller than $\omega^2$ by the induction hypothesis, and for each $w\in L(\delta')$ we have the language $L_w$ of finite rank, that is, with rank at most $\omega$ so that $L(\delta)\subseteq\mathop\bigcup\limits_{\delta'}\mathop\bigcup\limits_{w\in L(\delta')}wL_w$, applying Proposition \[prop-rank-ops\] we get that the rank of $L(\delta)$ is at most $\omega+\alpha$ for some $\alpha<\omega^2$, thus $\omega+\alpha<\omega^2$ also holds, proving the statement.
\[cor-restricted\] If $L(M)$ is a scattered language for the transducer $M$, then the rank of $L(M)$ is smaller than $\omega^2$.
Since $M$ (and so $M'$) can be assumed to only have sinks as final states, we get that $$L(M)=L(M')=\mathop\bigcup\limits_{\delta=((q,k,\sigma),a/R,(q_f,k',\sigma'))\hbox{ with }q_f\in F}L(\delta)$$ which is a finite union of languages, each having rank strictly less than $\omega^2$ by Proposition \[prop-ldelda-kicsi\].
We are ready to show the main result of the paper:
\[thm-main\] The rank of each scattered one-counter language is less than $\omega^2$.
By Corollary \[cor-restricted\], the rank of restricted scattered one-counter languages is less than $\omega^2$. It suffices to see that the property is preserved under concatenation and Kleene plus as due to Proposition \[cor-product-members-are-scattered\], if $L=L_1L_2$ for a scattered nonempty language $L$, then $L_1$ and $L_2$ are also scattered. By Proposition \[prop-rank-ops\] we have $\mathrm{rank}(L)\leq\mathrm{rank}(L_2)+\mathrm{rank}(L_1)$ in this case, hence if both $L_1$ and $L_2$ have rank less than $\omega^2$, then so has their product. For the case of iteration, if $L^+$ is scattered, then by \[prop-iterate-vstar\], $L\subseteq v^*$ and hence $L^+\subseteq v^*$ for some word $w$, thus $L^+$ is either finite (if $L\subseteq\{\varepsilon\}$) or has the order type $\omega$, hence $\mathrm{rank}(L)\leq 1<\omega^2$ again holds, proving the statement.
Conclusion
==========
We confirmed the conjecture of [@kuske] that scattered one-counter languages always have a rank strictly smaller than $\omega^2$, thus in particular, well-ordered one-counter languages always have an order type smaller than $\omega^{\omega^2}$. In the proof we used some upper bounds on the rank – it would be an interesting question to turn this into an algorithm which computes the exact rank of the language. Also, since scattered order types lack a Cantor-like normal form, it is not clear whether the order type of a scattered one-counter language is presentable by some expression involving, say, $\omega$, $-\omega$, $1$, finite products, sums and powers and if so, whether such a presentation is computable, or from the descriptive complexity point of view, whether representing such an expression by a transducer can be more succint than storing the expression itself. Also, it is still not known whether the order isomorphism problem of two scattered context-free languages is decidable (for the general case of arbitrary context-free languages it is known to be undecidable), and not even for one-counter languages. For the case of regular languages the order isomorphism is known to be decidable, so to extend decidability the class of restricted one-counter languages might be a good choice.
Ministry of Human Capacities, Hungary grant 20391-3/2018/FEKUSTRAT is acknowledged. Szabolcs Iván was supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences.
|
---
abstract: 'Higher order Laguerre–Gauss (LG) beams have been proposed for use in future gravitational wave detectors, such as upgrades to the Advanced LIGO detectors and the Einstein Telescope, for their potential to reduce the effects of the thermal noise of the test masses. This paper details the theoretical analysis and simulation work carried out to investigate the behaviour of LG beams in realistic optical setups, in particular the coupling between different LG modes in a linear cavity. We present a new analytical approximation to compute the coupling between modes, using Zernike polynomials to describe mirror surface distortions. We apply this method in a study of the behaviour of the LG$_{33}$ mode within realistic arm cavities, using measured mirror surface maps from the Advanced LIGO project. We show mode distortions that can be expected to arise due to the degeneracy of higher order spatial modes within such cavities and relate this to the theoretical analysis. Finally we identify the mirror distortions which cause significant coupling from the LG$_{33}$ mode into other order 9 modes and derive requirements for the mirror surfaces.'
author:
- Charlotte Bond
- Paul Fulda
- Ludovico Carbone
- Keiko Kokeyama
- Andreas Freise
title: 'Higher order Laguerre–Gauss mode degeneracy in realistic, high finesse cavities'
---
introduction
============
The sensitivities of second generation gravitational wave detectors such as Advanced LIGO and Advanced Virgo are expected to be limited by the thermal noise of the test masses within a significant range of signal frequencies around 100 Hz [@Rowan05]. To reach even better sensitivities, it has been proposed to use laser beams with an intensity pattern other than that of the fundamental Gaussian beam to reduce the effects of this thermal noise [@Vinet07; @Mours06]. A beam whose intensity is distributed more homogeneously over the mirror surface, for the same clipping losses, benefits from a more effective averaging over the mirror surface distortions caused by thermal effects [@Vinet09]. The specific advantage of using higher order LG modes, as opposed to mesa [@Ambrosio03] and conical [@Bondarescu08] beams, is that they are compatible with spherical mirrors as currently used in GW detectors and other high precision optical setups. Research into the potential of the LG$_{33}$ mode in gravitational wave detectors has been carried out using numerical simulations and table-top experiments [@Chelkowski09; @Fulda10]. The sensing and control signals for an LG$_{33}$ beam were found to perform as well as for the fundamental mode in all aspects examined and the LG$_{33}$ behaved as expected in short linear and triangular optical cavities. However, an optical cavity resonant for a higher–order Gaussian mode is degenerate so that a number of modes can resonate at the same time. This is a fundamental difference to a well designed cavity for the fundamental Gaussian mode, in which any resonant enhancement of other modes can be suppressed. This degeneracy can potentially cause additional optical losses. Simulations have shown that the use of the LG$_{33}$ beam, compared to the fundamental mode, LG$_{00}$, could result in a significant contrast defect at the dark fringe [@Miller; @Galimberti; @Yamamoto]. It is the aim of this paper to investigate how mirror surface distortions affect the purity of an LG$_{33}$ beam in high finesse cavities, by analytical calculation and numerical simulation. We will focus on the direct coupling from a distorted mirror and how this affects the mode content in a linear cavity, with our final aim to produce specifications for the mirror surfaces.
Representing mirror surface distortions and LG modes
====================================================
Mirror surface maps {#sec:mirror_maps}
-------------------
Ideally the mirrors in gravitational wave detectors should be perfectly smooth with a radius of curvature matching that of the incident beam. However, real mirrors deviate from a perfect surface, altering the beams which interact with them. If a beam $U(x,y,z)$ is incident on a distorted surface described by $Z(x,y)$ and uniform reflectivity $r$, then the reflected beam is given by: $$U_{ref}(x,y,z)=U(x,y,z) \ r \exp{(i2kZ(x,y))},$$ Fig. \[fig:coupling\_diagram\] illustrates this effect.
![A diagram illustrating the phase shift introduced when a beam is reflected from a distorted mirror surface with reflectivity $r$. The surface $Z(x,y)$ is defined as a height field across a plane perpendicular to the optical axis along $z$.[]{data-label="fig:coupling_diagram"}](coupling_diagram)
In order to investigate the effects of surface distortions measured mirror surface maps can be used. The term *mirror map* refers to an array of data detailing the optical properties of a mirror, often its surface height in nanometers. This data can be used to represent realistic mirrors in numerical simulations of gravitational wave detectors. Mirror maps have been produced from uncoated Advanced LIGO mirror substrates which represent the best mirror surfaces of this kind currently available. In the following we have made use of one such map, the surface map of the substrate ETM08 [@ETM08]. The deviation of this map surface from a perfectly spherical surface with radius of curvature 2249.28m is shown in Fig. \[fig:ETM08\]. This substrate shows an RMS surface figure error of 0.523nm.
Zernike polynomials
-------------------
Zernike polynomials are well suited for the purposes of describing mirror surface distortions. Zernike polynomials can be used to describe classical distortions such as tilts and curvatures [@Zernike]. They are a complete set of functions which are orthogonal over the unit disc and defined by radial index, $n$, and azimuthal index, $m$, with $ n \geq m \geq 0$. For any index $m$ we have one odd and one even polynomial [@Wolf]: $$\begin{array}{ll}
Z_{n}^{+m}(\rho,\phi)= A_{n}^{+m} \ \cos(m\phi)R_{n}^{m}(\rho) & \mbox{even polynomial} \\
\\
Z_{n}^{-m}(\rho,\phi)= A_{n}^{-m} \ \sin(m\phi)R_{n}^{m}(\rho) & \mbox{odd polynomial}\\
\end{array}$$ where $\rho$ is the normalised radial coordinate, $\phi$ is the azimuthal angle, $A_{n}^{\pm m}$ is the amplitude and $R_{n}^{m}(\rho)$ is the radial function. The radial function is given by the following sum: $$R_{n}^{m}(\rho)=
\sum_{h=0}^{\frac{1}{2}(n-m)} \frac{(-1)^{h}(n-h)!}{h! \left(\frac{1}{2}(n+m)-h\right)! \left(\frac{1}{2}(n-m)-h\right)!} \rho^{n-2h}$$ for $n-m$ even and 0 otherwise. This gives $n+1$ non-zero Zernike polynomials for each value of $n$ (for $m=0$ the odd polynomial is zero). Fig. \[fig:zern\] shows the surfaces described by the Zernike polynomials corresponding to orders (n) 0 to 4. The lower order polynomials represent some common optical distortions, some of which are summarised in table \[table:zernike\].
![A surface plot of the mirror map ETM08 corresponding to the surface heights of an Advanced LIGO end test mass. The curvature has been fitted and removed from the map data.[]{data-label="fig:ETM08"}](etm08_r1)
$n$ $m$ Common name
----- --------- ---------------------------
0 0 Offset
1 $\pm1$ Tilt in $x$/$y$ direction
2 0 Curvature
2 $\pm2$ Astigmatism
3 $\pm 1$ Coma along $x$/$y$ axis
: Summary of some common names for the lower order Zernike polynomials [@Zernike].[]{data-label="table:zernike"}
The odd polynomial describes the same surface as the even polynomial, but rotated by 90$^{\circ}$. Combinations of the odd and even polynomials relate to this same distortion rotated by a given angle. The magnitude of this surface distortion can be given by the root mean squared amplitude of the polynomials: $$A_{n}^{m}=\sqrt{\left(A_{n}^{-m} \right)^{2}+ \left(A_{n}^{+m}\right)^{2}}$$ where $+$ refers to the even polynomial and $-$ refers to the odd polynomial. Any surface defined over a disc can be described by a sum of Zernike polynomials, with the higher order polynomials representing the higher spatial frequencies present in the surface.
![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_0_0.png "fig:")\
$n=0$
![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_1_1_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_1_1_even.png "fig:")\
$n=1$
![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_2_2_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_2_0.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_2_2_even.png "fig:")\
$n=2$
![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_3_3_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_3_1_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_3_1_even.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_3_3_even.png "fig:")\
$n=3$
![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_4_4_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_4_2_odd.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_4_0.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_4_2_even.png "fig:") ![Plots of the non-zero Zernike polynomials from $n=0$ to $n=4$ with the odd polynomials with $m=-n$ on the far left and the even polynomials with $m=n$ on the far right, in steps of 2. The colour scale represents negative surface heights with greens and blues, zero with black and positive surface heights with reds and purples.[]{data-label="fig:zern"}](zernike_polynomial_4_4_even.png "fig:")\
$n=4$
Laguerre-Gauss modes
--------------------
The shape of any paraxial beam can be described as a sum of Hermite-Gauss or Laguerre-Gauss modes. The Laguerre-Gauss modes are a complete and orthogonal set of functions defined by radial index $p$ and azimuthal index $l$. The helical type of LG modes are typically given as [@Lasers]: $$\begin{split}
U_{p,l}(r,\phi,z)={}& \frac{1}{w(z)}\sqrt{\frac{2p!}{\pi(|l|+p)!}}\exp{\left(i\left(2p+|l|+1\right)\Psi(z)\right)} \\
{}& \times \left(\frac{\sqrt{2}r}{w(z)}\right)^{|l|} L^{|l|}_{p}\left(\frac{2r^{2}}{w^{2}(z)}\right) \\
{}& \exp{\left(-\frac{ikr^{2}}{2R_{c}(z)}-\frac{r^{2}}{w^{2}(z)}+il\phi\right)}
\end{split}$$ where $k$ is the wavenumber, $w(z)$ is the beam spot size parameter, $\Psi(z)$ is the Gouy phase and $R_{c}(z)$ is the radius of curvature of the beam. $L^{|l|}_{p}(x)$ refer to the associated Laguerre polynomials. When considering these beams in cavities we note that the resonance conditions for these beams differ from that of a plane wave due to the $(2p+|l|+1)\Psi(z)$ phase shift. The order of an LG mode is given by $ 2p+|l|$ and modes with the same order will acquire the same round trip phase shift whilst circulating in a cavity. Therefore the cavity is degenerate for LG modes of the same order.
The effects of mirror surface distortions on the shape of a reflected beam can be described in terms of coupling between LG modes. When a perfectly aligned Gaussian beam is reflected by a perfectly spherical mirror with the radius of curvature of the mirror matching that of the beam’s phase front, the shape of the reflected beam is identical to the shape of the incident beam, or, in other words, the mode composition has not changed. However, if the mirror surface is distorted the reflected beam will generally have a different mode composition. The coupling from an incident mode (indices $p$ and $l$) impinging on a completely reflecting surface $Z$ into a mode (indices $p'$ and $l'$) in the reflected beam can be described by a coupling coefficient [@Bayer-Helms; @Freise10]:
$$\label{eq:k_int}
k_{p,l,p',l'}^{Z}=\int_{S}U_{p,l}\exp{\left(2ikZ\left(r,\phi\right)\right)}U^{*}_{p',l'}$$
$Z$ describes the surface height of the mirror and $S$ describes an infinite plane perpendicular to the optical axis.
Currently the fundamental mode, LG$_{00}$, is used in gravitational wave detectors. Investigations have shown that mirror surface distortions have little effect on the beam purity when LG$_{00}$ is used. The presence of mirror surface distortions introduces modes into the detectors other than the input mode, but since LG$_{00}$ is the only mode of order 0 any coupling out this mode will result in modes of a different order, which will be suppressed in the cavities. The LG$_{33}$ mode is one of several modes of order 9. In total there are 10 order 9 modes; LG$_{0,\pm9}$, LG$_{1,\pm7}$, LG$_{2,\pm5}$, LG$_{3,\pm3}$ and LG$_{4,\pm1}$ (Fig. \[fig:LG9\]). These modes will be resonant in the arm cavities of the detectors, potentially resulting in a large proportion of the circulating power being in modes other than LG$_{33}$. The distortions present in the mirrors in each arm cavity will be different and so the mode content in each arm will differ, resulting in a larger contrast defect at the main beam splitter.
Analytical description of mode coupling via mirror surface distortions {#sec:analysis}
======================================================================
Using Zernike polynomials as a description of mirror surface distortions we can look at the coupling between Laguerre–Gauss modes analytically. The coupling between different Laguerre-Gauss modes when the surface is described by a particular Zernike polynomial is given by: $$k^{n,m}_{p,l,p',l'}=\int_{S}U_{p,l}\exp{\left(2ikZ_{n}^{m}\right)}U^{*}_{p',l'}
\label{eq:original_eq}$$ In order to simplify the integral we use the fact that when $kZ$ is small we can approximate: $$\exp{(2 i k Z)}\approx 1+2 i k Z$$ The amplitudes of the Zernike polynomials in the mirrors used in gravitational wave detectors are not expected to exceed 10 nm. With a wavelength of 1064 nm we have $2 k Z\approx0.1$ and so the approximation should be suitable for this investigation. We are concerned with coupling into other modes, not back into the input mode, so the equation to solve becomes: $$k^{n,m}_{p,l,p',l'}=\int_{0}^{2\pi} \int_{0}^{R} U_{p,l}U^{*}_{p',l'}(2ikZ_{n}^{m})r dr d\phi$$ due to the orthogonal properties of LG modes.
Both the Zernike polynomials and the Laguerre-Gauss modes can be easily separated into their angular and radial parts. The angular function to integrate is: $$\exp{(i\phi(l-l'))} \begin{array}{ll}
\cos{(m\phi)} & \ \ \ \mbox{even polynomial}\\
\sin{(m\phi)} & \ \ \ \mbox{odd polynomial}\\
\end{array}$$ Considering the even Zernike polynomial we obtain: $$\begin{array}{c}
\displaystyle\int_{0}^{2\pi}e^{i\phi(l-l')} \frac{e^{im\phi}+e^{-im\phi}}{2} d\phi = \\
\\
\displaystyle\left[ \frac{e^{i\phi(l-l'+m)}}{2i(l-l'+m)} + \frac{e^{i\phi(l-l'-m)}}{2i(l-l'-m)} \right]^{2\pi}_{0} \\
\end{array}$$ As the integral is evaluated over the entire unit disc and $e^{i0}=e^{iN\times2\pi}=1$, where $N$ is an integer, the integral is equal to 0. The only combination of Zernike polynomials and Laguerre-Gauss beams to give a non-zero result occurs when one of the exponentials disappears before the integration takes place. This occurs when we have: $$m=|l-l'|
\label{eq:m_condition}$$ The same condition also gives the only non-zero results for the odd Zernike polynomials. This is a very interesting result as it suggests that surfaces described by Zernike polynomials will only cause significant coupling from one LG mode to another if the Zernike azimuthal index is equal to the difference between the azimuthal indices of the two modes. This requirement for $m$ also gives the minimum order ($n$) of Zernike polynomial required to produce significant coupling, as $m \leq n$.
Using this condition we can integrate with respect to $\phi$. The integrals were found to be $\pi$ for the even Zernike polynomials and $\pm i\pi$ for the odd polynomials, depending on the sign of $(l-l')$. The final equation is given by: $$\begin{split}
k_{p,l,p',l'}^{n,m} = {}& A_{n}^{m} k \sqrt{p!p'!(p+|l|)!(p'+|l'|)!} \\
{}& \times \left| \sum_{i=0}^{p} \sum_{j=0}^{p'} \sum_{h=0}^{\frac{1}{2}(n-m)} \frac{(-1)^{i+j+h}} {(p-i)!(|l|+i)! i!} \right. \\
{}& \times \frac{1}{(p'-j)!(|l'|+j)!j!} \frac{(n-h)!}{(\frac{1}{2}(n+m)-h)!} \\
{}& \left. \times \frac{1}{ \left(\frac{1}{2}(n-m)-h\right)! h!} \frac{1}{X^{\frac{1}{2}(n-2h)}} \right. \\ {}& \left. \times \ \gamma (i+j-h+\frac{1}{2}(|l|+|l'|+n)+1,X) \ \right| \\
\end{split}
\label{eq:k_eq}$$ where $X=\frac{2R^{2}}{w^{2}}$, $R$ is the Zernike radius and $w$ is the beam radius, and $\gamma$ is the lower incomplete gamma function. The full derivation is given in Appendix \[sec:aderivation\].
In our approximation of the coupling coefficients we only consider the magnitude of the coefficients. However, the real coefficients have both real and imaginary parts indicating that there is some phase shift caused by the distortions. Therefore, when considering the coupling from a surface in terms of the coupling from the individual polynomials making up the surface we also need to consider the phase shifts. The largest possible coupling from a surface occurs when all the individual Zernike couplings have the same phase and therefore the magnitude of the coupling is equal to the sum of the individual couplings.
Analysis of coupling into order 9 modes
=======================================
We want to verify the results of the analytical description of the coupling coefficients. Using the condition for significant coupling outlined previously in Sec. \[sec:analysis\] we can identify the azimuthal Zernike indices which will cause a large amount of coupling from LG$_{33}$ into the other order 9 modes. These are summarised in table \[table:m\_for\_order\_9\]. Because $m \leq n$ this condition for the azimuthal index also tells us the lowest order ($n$) Zernike polynomial required to cause a large amount of coupling from the LG$_{33}$ beam into each of the other order 9 modes.
---------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
$m$ 2 2 4 4 6 6 8 10 12
$U_{p,l}$ mode 2, 5 4, 1 1, 7 4,-1 0, 9 3,-3 2,-5 1,-7 0,-9
---------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
: The azimuthal index ($m$) of the Zernike polynomial required to achieve significant coupling from an LG$_{33}$ incident beam into the other order 9 modes.[]{data-label="table:m_for_order_9"}
Higher order Zernike polynomials represent higher order spatial frequencies, which generally have smaller amplitudes in the mirror surfaces. Therefore we would expect the coupling caused by higher order polynomials, such as into LG$_{1-7}$ and LG$_{0-9}$, to be smaller than those caused by lower order polynomials. We would also expect the polynomials with $m=2$, 4 and 6 to have a large effect on the beam purity as they each couple from LG$_{33}$ into two other order 9 modes.
Using Matlab the original integration (Eq. \[eq:original\_eq\]) was performed numerically, computing the coupling occurring from a mirror surface defined completely by a single Zernike polynomial. This particular example shows the results for Z$_{4}^{4}$, in which we expect a large coupling from LG$_{33}$ into LG$_{17}$ and LG$_{4-1}$ (table \[table:m\_for\_order\_9\]) and much less coupling into other order 9 modes. The coupling coefficients between the LG$_{33}$ beam and all the other order 9 modes were calculated. The numerical integration was carried out for a range of $\frac{w}{R}$ and the results, for a polynomial amplitude of 1 nm, are summarised in Fig. \[fig:numerical\_coupling\].
![Plots of the coupling coefficients for different order 9 LG modes when an LG$_{33}$ beam is incident on a surface described by the Zernike polynomial Z$_{4}^{4}$. The amplitude of the coefficients is plotted against the ratio of the beam radius, $w$, and the Zernike radius, $R$. These plots show numerical results for the coefficients without any approximation. The coefficients for LG$_{4-1}$ and LG$_{17}$ are significantly larger than those of the other modes.[]{data-label="fig:numerical_coupling"}](coupling_from_LG33_on_Z44)
In this plot the two largest coupling coefficients over this range of $\frac{w}{R}$ correspond to LG$_{4-1}$ and LG$_{17}$. The coefficients for the other LG modes are significantly smaller. These results agree with our predictions.
Fig. \[fig:numerical\_vs\_analytical\] shows a comparison of the analytical results from Eq. \[eq:k\_eq\] with the numerical results. Here only the coupling coefficients for LG$_{4-1}$ and LG$_{17}$ are plotted as the analytical approach gives 0 for $m\neq |l-l'|$. Over this range the two sets of numbers match up very well and we consider this a good confirmation of the analytical approximation.
![A plot showing the coupling into the LG$_{4-1}$ and LG$_{17}$ modes from an LG$_{33}$ beam incident on a surface described by the Zernike polynomial Z$_{4}^{4}$. The results are plotted against the relative beam radius on the mirror, $\frac{w}{R}$. The results from an analytical approximation and a numerical integration method are plotted.[]{data-label="fig:numerical_vs_analytical"}](numerical_vs_analytic_coupling)
Beam size and Zernike order
---------------------------
Figs. \[fig:numerical\_coupling\] and \[fig:numerical\_vs\_analytical\] seem to suggest that the coupling coefficients also depend on the beam size relative to the radius of the Zernike polynomial, or mirror radius. However, this ratio is typically not a free parameter. For a good reduction in thermal noise a large beam radius is desirable. An upper limit for the beam width can be derived from optical loss due to beam clipping. This so-called *clipping loss* refers to the power lost over the mirror edges given by [@Chelkowski09]: $$l_{\mbox{clip}} = 1-\int_{S} \left|U_{p,l}\right|^{2},$$ where the integral represents the normalised power reflected by a perfect mirror of finite size (see Appendix \[sec:acliploss\]). In gravitational wave detectors the clipping loss should be lower than 100ppm (parts per million) and often an arbitrary requirement of 1ppm is used during the interferometer design phase. This yields an optimal beam radius of $0.232\,R$ ($R$ being the mirror radius) for an LG$_{33}$ beam. A clipping loss of 100ppm instead leads to an optimal beam radius of $0.255\,R$.
Cavity simulations with Advanced LIGO mirror maps
=================================================
We want to investigate how the degeneracy of the order 9 modes affects the purity of an LG$_{33}$ mode in high finesse cavities. The aim of this investigation is to assess the effects of higher-order mode degeneracy and derive requirements for the mirror surfaces which would result in an acceptably high LG$_{33}$ beam purity. The Advanced LIGO cavities consist of two curved mirrors; the input test mass (ITM) and the end test mass (ETM) separated by an arm length of approximately 4km (Fig. \[fig:ligo\_cav\]). The design properties of these mirrors are summarised in table \[table:mirror\_props\] [@ITM; @ETM], giving a high finesse of 450.
![The optical layout of an Advanced LIGO arm cavity. Light from the beam-splitter is incident on the flat anti-reflective surface of the ITM. The light circulates between the two highly reflective, curved surfaces of the ITM and ETM. Light is transmitted by the cavity through the ETM.[]{data-label="fig:ligo_cav"}](LIGO_cavity)
\[sec:ligo\_cavs\]
Mirror ITM ETM
--------- -------- --------
$R_{a}$ 20ppm 500ppm
$L_{a}$ 1ppm 1ppm
$T_{h}$ 0.014 5ppm
$L_{h}$ 0.3ppm 0.3ppm
: Optical properties of the mirrors designed for the Advanced LIGO arm cavities. The two mirrors have a design thickness of 200 mm and index of refraction of 1.45. $R$, $T$ and $L$ refer to the reflectivity, transmission and loss of power at the mirror. $a$ and $h$ refer to the anti-reflective and highly reflective coated surfaces of each mirror.[]{data-label="table:mirror_props"}
To simulate mirror surface distortions we used mirror maps measured from uncoated mirror substrates produced for Advanced LIGO [^1]. However, the Advanced LIGO cavities were not designed to be compatible with the LG$_{33}$ mode. The LG$_{33}$ mode is more spatially extended than the LG$_{00}$ mode, and so experiences a larger clipping loss at the mirrors for a given beam spot size value. For the Advanced LIGO cavity parameters, the clipping loss for the LG$_{33}$ mode is much larger than acceptable, at around 35%. It was therefore necessary to adjust the cavity length to bring the LG$_{33}$ clipping to a level that allowed us to carry out a meaningful investigation. Using a cavity length of 2802.9m we achieve similar clipping losses to those experienced by LG$_{00}$ in the original cavities, for the 30cm aperture represented by the mirror maps. The results from this optical setup should be representative of longer cavities with larger mirrors. The simulated cavity parameters are summarised in table \[table:cavity\_params\].
------------- ------------- ------------- ---------------
Parameter ITM R$_{c}$ ETM R$_{c}$ Cavity length
Value \[m\] -1934 2245 2802.9
------------- ------------- ------------- ---------------
: Cavity parameters for simulations of Advanced LIGO style arm cavities [@AdLIGO]. The length of the cavity was reduced from the original length of 3994.5 m to prevent a large clipping loss when using LG$_{33}$ beams.[]{data-label="table:cavity_params"}
Laguerre-Gauss mode purity with Advanced LIGO mirror maps {#sec:ad_ligo_maps}
---------------------------------------------------------
For the purposes of this investigation the mirror map corresponding to the Advanced LIGO end test mass ETM08 was used; see Fig. \[fig:ETM08\]. To predict the direct couplings from this mirror we look at the Zernike polynomials representing the mirror surface. This is achieved by performing a convolution between the surface defined by the mirror map and the different Zernike polynomials: $$\int_{S}Z_{map} \cdot Z_{n}^{m}=A_{n}^{m}\int_{S} Z_{n}^{m} \cdot Z_{n}^{m}$$ where $Z_{map}$ is the surface defined by the mirror map and $A_{n}^{m}$ is the amplitude of the corresponding Zernike polynomial in the surface. The convolution was performed for all Zernike polynomials with $n \leq 30$. The polynomials which cause significant coupling into the other order 9 modes ($m=2$, 4,$\dots$12) are summarised in table \[table:ETM08\_zernikes\]. Here the polynomials are ranked in order of the power they couple into the other order 9 modes when an LG$_{33}$ beam is reflected from a surface described by the polynomial.
------------------------ ------- ------- -------- -------- -------- -----------
Z$_{n}^{m}$ polynomial 2, 2 4, 2 4, 4 6, 2 10, 8 other
$A_{n}^{m}$ \[nm\] 0.908 0.202 0.213 0.124 0.116 -
Power \[ppm\] 4.66 0.331 0.0431 0.0099 0.0059 $<$ 0.005
------------------------ ------- ------- -------- -------- -------- -----------
: Zernike polynomials present in the Advanced LIGO mirror map ETM08 which cause significant coupling from LG$_{33}$ into the other order 9 LG modes ($m = 2$, 4,$\dots$12). The power coupled from LG$_{33}$ into the other order 9 modes by reflection from surfaces described by the individual polynomials is included, calculated from a coupling approximation.[]{data-label="table:ETM08_zernikes"}
From this we can suggest which order 9 LG modes will have significant amplitudes in the simulated cavity. The two polynomials which cause the largest individual power couplings have $m=2$. The astigmatism (Z$_{2}^{2}$) in particular extracts a large amount of power from LG$_{33}$. Therefore, we would expect the LG$_{41}$ and LG$_{25}$ modes to have relatively large amplitudes in the cavity as the $m=2$ polynomials cause significant coupling into these modes.
The cavity defined in Sec. \[sec:ligo\_cavs\] was simulated using the interferometer simulation tool <span style="font-variant:small-caps;">Finesse</span> [^2] [@Finesse; @Finesse2]. An input beam of pure LG$_{33}$ was used with the ETM08 mirror map applied to the end mirror and a perfect input mirror. The cavity was tuned to be on resonance for the LG$_{33}$ mode and the beam circulating in the cavity was detected. A plot of the circulating field is shown in Fig. \[fig:ETM08\_sim\].
![A plot of the field circulating in a simulated high finesse cavity. In the simulation the Advanced LIGO mirror map ETM08 is applied to the end test mass and a pure LG$_{33}$ beam is injected into the cavity.[]{data-label="fig:ETM08_sim"}](LG33_etm08_f_circ)
The purity of an LG mode, $U_{p,l}$, in a given beam $U$ is defined as $|c_{p,l}|^2$ where [@Chu]: $$c_{p,l}=\int_{S}U \ U_{p,l}^{*}$$ The plot in Fig. \[fig:ETM08\_sim\] (compared with the plot of LG$_{33}$ in Fig. \[fig:LG9\]) suggests that the circulating beam now contains modes other than LG$_{33}$. The purity of the LG$_{33}$ beam in this simulated cavity was found to be 88.6%. The power in the different modes present in the circulating field are summarised in table \[table:ETM08\_modes\].
---------------- ------ ------ ------ ------- ------- ---------- -- -- -- --
$U_{p,l}$ mode 3, 3 4, 1 2, 5 4,-1 1, 7 other
$m_{SC}$ - 2 2 4 4 -
Power (%) 88.6 5.70 5.02 0.333 0.313 $<$ 0.05
---------------- ------ ------ ------ ------- ------- ---------- -- -- -- --
: The power in the LG modes circulating in a cavity simulated with <span style="font-variant:small-caps;">Finesse</span> with an LG$_{33}$ input beam and the Advanced LIGO mirror map ETM08. $m_{SC}$ refers to the azimuthal index of the Zernike polynomial required to cause significant coupling from LG$_{33}$ to the given mode.[]{data-label="table:ETM08_modes"}
The results of the decomposition show that the surface distortions of the end mirror cause significant coupling into other LG modes, particularly the other order 9 modes. The distortions also cause coupling into modes of other orders, but these are not resonant at the same cavity tuning as the LG$_{33}$ mode and so are strongly suppressed.
Other than LG$_{33}$ the two largest modes in the cavity are LG$_{41}$ (5.7 %) and LG$_{25}$ (5.0 %). This agrees with the predictions made by studying the Zernike content of the ETM08 mirror map (table \[table:ETM08\_zernikes\]). LG$_{4-1}$ and LG$_{17}$ also have relatively large amplitudes in the cavity. This may be due to the direct coupling out of the LG$_{33}$ caused by the Z$_{4}^{4}$ polynomial. However, the LG$_{4-1}$ and LG$_{17}$ modes can also be strongly coupled out of the LG$_{41}$ and LG$_{25}$ modes via $m=2$ polynomials, further contributing to the effect these polynomials have on the purity of the beam. Overall the coupling process in a cavity is complicated by these multiple cross-couplings, but the results of this simulation suggest that the direct coupling from a mirror surface is the dominant effect on the mode content of the circulating beam. A theoretical understanding of the direct coupling has therefore allowed us to make valid predictions about the resulting mode content.
Frequency splitting
-------------------
The presence of mirror surface distortions not only causes coupling between LG modes but introduces additional phase shifts of the modes. This results in slight shifts of the resonance frequency of individual modes. These shifts in resonance frequency depend on the particular mode and so modes of the same order will be resonant at slightly different frequencies. Thus the mode degeneracy can be broken. We will refer to this effect as *frequency splitting*. For the frequency splitting to be effective the shifts in resonance frequency must be larger than the cavity bandwidth in order to separate the resonance peaks of the different order 9 modes.
Using the ETM08 mirror map the high finesse cavity defined in Sec. \[sec:ligo\_cavs\] was simulated. The beam circulating in the cavity was detected as the laser frequency was tuned around the cavity resonance. The maximum power of the order 9 modes in the cavity and the difference in their resonance frequencies is summarised in table \[table:f\_split\]. The frequency splitting is of the order of 10Hz, smaller than the cavity bandwidth of 120Hz and so is not sufficient to completely break the degeneracy. The result is 10 *quasi-degenerate* modes. The resonance frequencies are slightly different and so when the cavity is tuned to the resonance of the LG$_{33}$ mode the other modes will be slightly suppressed. However, the frequency splitting is small and therefore these modes will still have relatively large amplitudes in the cavity. The coupling into order 9 modes is therefore still the dominant effect on the mode purity.
------------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$U_{p,l}$ mode 3, 3 4, 1 2, 5 4, -1 1, 7 3, -3 0, -9 2, -5 0, 9 1, -7
Power \[W\] 221.8 14.38 12.62 0.865 0.802 0.102 0.038 0.038 0.032 0.001
Frequency \[Hz\] 0 1.3 0.7 7.0 6.6 7.6 -7.3 -16.1 -14.8 9.3
------------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
Laguerre-Gauss mode purity with improved mirror maps
----------------------------------------------------
We want to find ways in which the mirror maps could be improved for the specific application of the LG$_{33}$ beam. By considering the mirror surfaces analytically it appears that reducing the astigmatism for this particular map could improve the purity of LG$_{33}$ in our simulated cavity. To demonstrate this the astigmatism was removed from the ETM08 mirror map. The cavity was then simulated with this processed map, with the resulting circulating field detected and decomposed in to LG modes. The circulating beam is plotted in Fig. \[fig:ETM08-Z2s\_sim\]. Simply comparing this plot with the original circulating beam (Fig. \[fig:ETM08\_sim\]) suggests that the purity of the field has increased. The LG content of the beam is summarised in table \[table:ETM08-Z2s\_modes\]. The purity of the circulating LG$_{33}$ beam is now 99.5%, a significant improvement from the original results.
![A plot of the field circulating in a cavity simulated with an input mode of pure LG$_{33}$ and the Advanced LIGO ETM08 mirror map, with astigmatism removed.[]{data-label="fig:ETM08-Z2s_sim"}](LG33_etm08-astigmatism_f_circ)
---------------- ------ ------- ------- -------- -------- -------- ----------
$U_{p,l}$ mode 3, 3 4, 1 2, 5 1, 7 4,-1 0,-9 other
$m_{SC}$ - 2 2 4 4 12 -
Power \[%\] 99.5 0.231 0.208 0.0524 0.0165 0.0137 $<$ 0.01
---------------- ------ ------- ------- -------- -------- -------- ----------
: The power in the LG modes circulating in a simulated cavity. The cavity was simulated with an LG$_{33}$ input beam and with the ETM08 mirror map with astigmatism removed. $m_{SC}$ refers to the azimuthal index of the Zernike polynomial required to cause significant coupling from LG$_{33}$ to the given mode.[]{data-label="table:ETM08-Z2s_modes"}
The results of the decomposition show that the power in both the LG$_{41}$ and LG$_{25}$ modes has decreased significantly, as predicted. The power in the other modes has also noticeably decreased. This result suggests that the astigmatism is a major factor in coupling from the LG$_{33}$ mode, not only for its direct coupling into LG$_{25}$ and LG$_{41}$ but for the coupling from these new modes into other modes of order 9. For this setup we can conclude that the astigmatism should be limited in the mirror surfaces to reduce the problems caused by higher order mode degeneracy.
Mirror requirements for LG$_{33}$
---------------------------------
In order to use LG$_{33}$ in GW detectors we require certain Zernike polynomials in the mirrors to be smaller than in the current state of the art mirrors, in order to achieve an acceptable beam purity in the cavity. Here we investigate the direct coupling from ETM08 and suggest limits to the amplitudes of specific polynomials in the mirror surfaces, presenting an Advanced LIGO mirror map adapted for the use of LG$_{33}$.
Using our theoretical analysis we can identify the particular Zernike polynomials to reduce in the mirrors. We have already seen that the Zernike polynomials with odd values of $n$ and with $m > 12$ don’t have a large effect on the purity. To assess the other polynomials we use Eq. \[eq:k\_eq\] to approximate the coupling into order 9 modes caused by the polynomials in the ETM08 mirror map, for the optical setup defined in Sec. \[sec:ligo\_cavs\]. For each order 9 LG mode the coupling was calculated for the Zernike polynomials with $n=2$, 4 $\dots$30 and with $m$ required to give significant coupling. Fig. \[fig:etm08\_cc\] represents these coupling coefficients.
![A bar chart showing the coupling, $k_{p,l,p',l'}^{n,m}$, into the order 9 modes when an LG$_{33}$ beam is incident on a surface described by the Zernike polynomial with $m$ required to cause significant coupling. The Zernike amplitudes correspond to those in the ETM08 mirror map.[]{data-label="fig:etm08_cc"}](etm08_coupling_coefficients)
This chart shows that the largest couplings occur from Z$_{2}^{2}$, into LG$_{25}$ and LG$_{41}$, as previously suggested. There is also some strong coupling from Z$_{4}^{2}$ and Z$_{4}^{4}$. The other couplings are significantly smaller. Therefore, the first step in modifying the mirrors for LG$_{33}$ is to limit these 3 polynomials to give similar couplings to the higher order polynomials.
Another plot illustrating the coupling from this map is shown in Fig. \[fig:etm08\_sum\_cc\]. In this plot the maximum possible coupling from the ETM08 mirror map is estimated using our analytical approximation. For each order 9 mode the sum of the coupling is calculated for all Zernike polynomial orders smaller than $n$. The plot illustrates the magnitude of the direct coupling expected as we include higher order Zernike polynomials in our model.
![Plots showing an approximation of the coupling from LG$_{33}$ into the other order 9 LG modes from the ETM08 mirror map as higher order ($n$) Zernike polynomials are included in our model. The sum of the coupling from polynomials with order $\leq n$ are plotted for each mode.[]{data-label="fig:etm08_sum_cc"}](etm08_summed_amplitude_coupling)
Fig. \[fig:etm08\_sum\_cc\] can be used to illustrate how this particular map, ETM08, should be adapted for LG$_{33}$. The coupling into LG$_{25}$ and LG$_{41}$ is around 10 times greater than any other coupling. The coupling into these two modes is also not significantly increased by including the higher order modes. Therefore, limiting the lower order polynomials with $m=2$ will greatly reduce the coupling into these two modes and the overall coupling into order 9 modes. From this plot we conclude that the overall coupling into order 9 modes can be reduced by around a factor of 10 by reducing the lower order polynomials. Reducing the coupling further will involve limits on multiple polynomials.
By assessing the direct coupling from ETM08 we can set requirments for the lower order Zernike polynomials. When an LG$_{33}$ beam is incident on the ETM08 mirror map in our setup the reflected beam is predominantly LG$_{33}$. However, we find 31ppm (parts per million) of the power is now in other modes, with 6.8ppm in the other order 9 modes. Table \[table:ETM08\_zernikes\] shows the power coupled from surfaces described by the polynomials present in the ETM08 map, from LG$_{33}$ into the other order 9 modes. For this map we consider the polynomials causing a large amount of coupling as those coupling more than 0.01ppm into the order 9 modes; Z$_{2}^{2}$, Z$_{4}^{2}$ and Z$_{4}^{4}$. For LG$_{33}$ we require these polynomials to be limited in the mirror surfaces. The requirements for these polynomials were calculated to give power couplings of 0.01ppm into the other order 9 modes and are summarised in table \[table:limits\].
------------------------ ------- ------- -------
$Z_{n}^{m}$ polynomial 2, 2 4, 2 4, 4
Amplitude \[nm\] 0.042 0.035 0.100
------------------------ ------- ------- -------
: A summary of the amplitude requirements for the Zernike polynomials required to give individual couplings of 0.01ppm from LG$_{33}$ into the other order 9 modes in the ETM08 mirror map.[]{data-label="table:limits"}
These amplitude limits were applied to the ETM08 mirror map, resulting in coupling of 19ppm into modes other than LG$_{33}$ and 0.043ppm into the other order 9 modes. The cavity defined in Sec. \[sec:ligo\_cavs\] was simulated with this limited map, resulting in 815ppm impurity in the circulating beam. This is a very good improvement on the original impurity of 0.114, illustrating that a high beam purity is achievable with these mirror requirements. To achieve an even higher beam purity will involve reducing the amplitudes of these polynomials further, as well as additional Zernike requirements.
Conclusion
==========
We have investigated the coupling which occurs when Laguerre-Gauss modes are incident on a mirror with surface distortions. Taking an analytical approach we used Zernike polynomials to represent mirror surface distortions and derived an approximate equation for the significant coupling when an LG mode is reflected from a surface defined by a particular Zernike polynomial. This derivation resulted in a condition for significant coupling, $m=|l-l'|$, where $m$ is the azimuthal index of the Zernike polynomial and $l$ and $l'$ are the azimuthal indices of the incident and coupled modes respectively. This is a significant result as it allows us to predict which order 9 modes will be largely coupled by particular Zernike polynomials and suggest which modes will have large amplitudes in the arm cavities
We investigated the performance of LG$_{33}$ in high finesse cavities by simulation with Advanced LIGO mirror maps. This illustrated the degraded purity of the circulating beam in realistic cavities due to higher order mode degeneracy. The results were then analysed by looking at the Zernike polynomials representing our example mirror map. The analysis and results were consistent with the predictions made from Eq. \[eq:k\_eq\]. This suggested that astigmatism was causing a significant amount of coupling, particularly into the LG$_{41}$ and LG$_{25}$ modes. This was confirmed when the cavity was simulated again with the astigmatism removed from the mirror map and we observed a dramatic increase in the LG$_{33}$ mode purity.
The analytical description enabled us to identify the specific Zernike polynomials which cause large couplings as well as the LG modes which would dominate as a result. Using this we were able to derive certain requirements for our example mirror map, ETM08, in terms of limits on the amplitudes of the Zernike polynomials Z$_{2}^{2}$, Z$_{4}^{2}$ and Z$_{4}^{4}$ (table \[table:limits\]). Using this map the resulting circulating beam impurity was found to be 815ppm, a significant reduction from the original impurity of 0.114.
This investigation has demonstrated that a high beam purity is achievable using an LG$_{33}$ beam when modifications are made to the low order Zernike polynomials in Advanced LIGO mirrors. Implementing the LG$_{33}$ beam in gravitational wave detectors will be challenging as we require very small amplitudes on these lower order polynomials. We should also consider that the example mirror surfaces considered here refer to uncoated substrates. The coating process is likely to add to the lower order features in the mirror surfaces. However, using this analytical approach we can derive specific requirements for the mirror surfaces leading to designs for suitable mirrors for these higher order beams.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank GariLynn Billingsley for providing the Advanced LIGO mirror surface maps and for advice and support on using them. We would also like to thank David Shoemaker and Stefan Hild for useful discussions. This work has been supported by the Science and Technology Facilities Council and the European Commission (FP7 Grant Agreement 211743). This document has been assigned the LIGO Laboratory document number LIGO–P1100081.
Derivation of coupling coefficients {#sec:aderivation}
===================================
The product of two Laguerre-Gauss modes is: $$\begin{split}
U_{p,l} U^{*}_{p',l'} = {}& \frac{1}{w^{2}}\frac{2}{\pi}\sqrt{\frac{p!p'!}{(|l|+p)!(|l'|+p')!}} \\
{}&\times \exp{\left(i\left(2p+|l|-2p'-|l'|\right)\Psi\right)} \\
{}& \left(\frac{\sqrt{2}r}{w}\right)^{|l|+|l'|} L^{|l|}_{p}\left(\frac{2r^{2}}{w^{2}}\right) L^{|l'|}_{p'}\left(\frac{2r^{2}}{w^{2}}\right) \\
{}& \exp{\left(-\frac{2r^{2}}{w^{2}}\right)} \exp{\left(i\phi\left(l-l'\right)\right)}
\end{split}$$ The following derivation follows from Eq. \[eq:m\_condition\]. Currently we are concerned with the magnitude of the coupling coefficients, so we ignore any constant phase shifts and integrate with respect to $\phi$: $$\begin{split}
k^{n,m}_{p,l,p',l'} = {}& \left| \int_{0}^{R} \frac{2}{\pi w^{2}}\sqrt{\frac{p!p'!}{(|l|+p)!(|l'|+p')!}} 2kA_{n}^{m}\pi R_{n}^{m}(r) \right. \\
{}& \times \left(\frac{\sqrt{2}r}{w} \right)^{|l|+|l'|} L_{p}^{|l|} \left(\frac{2r^{2}}{w^{2}} \right) L_{p'}^{|l'|}\left(\frac{2r^{2}}{w^{2}} \right) \\
{}& \left. \times\exp{\left(-\frac{2r^{2}}{w^{2}} \right)} rdr \right| \\
\end{split}
\label{eq:k1}$$ In order to further simplify the equation the following variable substitution is made: $$x=\frac{2r^{2}}{w^{2}}$$ and a new limit to the integral: $$X=\frac{2R^{2}}{w^{2}}$$ where $R$ is the Zernike radius. This gives the integral: $$\begin{split}
k^{n,m}_{p,l,p',l'} = {}& kA_{n}^{m}\sqrt{\frac{p!p'!}{(|l|+p)!(|l'|+p')!}} \\
{}& \times \left| \int_{0}^{X} R_{n}^{m}(x) x^{\frac{1}{2}(|l|+|l'|)} \right. \\
{}& \times \left. L_{p}^{|l|} (x) L_{p'}^{|l'|}(x) \exp{(-x)} dx \right|
\end{split}$$ Substituting in the sums representing the Laguerre polynomials and the radial Zernike function as a function of $x$ gives: $$\begin{split}
k_{p,l,p',l'}^{n,m} = {}& A_{n}^{m}k \sqrt{p!p'!(p+|l|)!(p'+|l'|)!} \\
{}& \times \left| \sum_{i=0}^{p} \sum_{j=0}^{p'} \sum_{h=0}^{\frac{1}{2}(n-m)} \frac{(-1)^{i+j+h}} {(p-i)!(|l|+i)! i!} \right. \\
{}& \times \frac{1}{(p'-j)!(|l'|+j)!j!} \frac{(n-h)!}{(\frac{1}{2}(n+m)-h)!} \\
{}& \left. \times \frac{1}{ \left(\frac{1}{2}(n-m)-h\right)! h!} \frac{1}{X^{\frac{1}{2}(n-2h)}} \right. \\
{}& \left. \times \int_{0}^{X} x^{i+j-h+\frac{1}{2}(|l|+|l'|+n)} \exp{(-x)} dx \right| \\
\end{split}$$ This type of integral results in the lower incomplete gamma function: $$\gamma(a,x)=\int_{0}^{x} t^{a-1}e^{-t}dt$$ When $a$ is equal to $n$, an integer, the function is given by the following sum: $$\gamma(n,x)=(n-1)! \left(1-e^{-x} \sum_{k=0}^{n-1} \frac{x^{k}}{k!} \right)$$ Therefore, the final equation for this approximation of the magnitude of the coupling coefficients is given by: $$\begin{split}
k_{p,l,p',l'}^{n,m} = {}& A_{n}^{m}k \sqrt{p!p'!(p+|l|)!(p'+|l'|)!} \\
{}& \times \left| \sum_{i=0}^{p} \sum_{j=0}^{p'} \sum_{h=0}^{\frac{1}{2}(n-m)} \frac{(-1)^{i+j+h}} {(p-i)!(|l|+i)! i!} \right. \\
{}& \times \frac{1}{(p'-j)!(|l'|+j)!j!} \frac{(n-h)!}{(\frac{1}{2}(n+m)-h)!} \\
{}& \left. \times \frac{1}{ \left(\frac{1}{2}(n-m)-h\right)! h!} \frac{1}{X^{\frac{1}{2}(n-2h)}} \right. \\ {}& \left. \times \ \gamma (i+j-h+\frac{1}{2}(|l|+|l'|+n)+1,X) \ \right| \\
\end{split}
\label{eq:ak_eq}$$
Clipping loss {#sec:acliploss}
=============
Clipping loss is given by: $$l_{\mbox{clip}} = 1-\int_{S} \left|U_{p,l}\right|^{2}$$ where the integral represents the normalised power reflected by a perfect mirror. $S$ defines an infinite plane perpendicular to the beam axis. The magnitude squared of an LG mode is: $$\begin{split}
\left|U_{p,l}\right|^{2}= {}& \frac{1}{w^{2}} \frac{2p!}{\pi(|l|+p)!} \left( \frac{2r^{2}}{w^{2}} \right)^{|l|} \\
{}& \times \left( L^{|l|}_{p}\left(\frac{2r^{2}}{w^{2}} \right) \right)^{2} \exp{\left( -\frac{2r^{2}}{w^{2}} \right)} \\
\end{split}$$ where $w$ is the beam radius, $p$ and $l$ are the mode indices and $r$ is the radial position. Integrating over the surface and taking a similar approach as for the coupling coefficients we get: $$\begin{split}
l_{\mbox{clip}}= {}&1- p!(p+|l|)! \sum_{m=0}^{p}\sum_{n=0}^{p} \frac{(-1)^{n+m}}{(p-n)!(p-m)!} \\
{} &\times \frac{1}{ (|l|+n)!(|l|+m)!n!m!}\ \gamma(|l|+n+m+1,X) \\
\end{split}$$ where $X=\frac{2R^{2}}{w^{2}}$, $R$ is the radius of the mirror and $\gamma$ is the lower incomplete gamma function.
Zernike composition of a mirror surface {#sec:azernike}
=======================================
For certain Zernike polynomials (those with non-zero $m$) their amplitudes in a surface depend on the orientation of that surface with respect to the Zernike surface. For example, consider the two polynomials responsible for astigmatism, Z$_{2}^{\pm2}$. The two polynomials actually describe the same shape, with one just rotated by 90 degrees with respect to the other. Therefore, rotating a surface, such as the one described by mirror map ETM08, will change the amplitudes of these two polynomials within the surface. Fig. \[fig:ETM08\_zs\_rotated\] illustrates this effect. The plots show the amplitudes of the order 2 Zernike polynomials present in the ETM08 mirror surface as it is rotated. As expected the amplitude of the Z$_{2}^{0}$ polynomial remains constant as it has no angular dependence. The amplitudes of the Z$_{2}^{\pm2}$ polynomials oscillate and, at a certain rotation (around 120$^{\circ}$) the astigmatism of the surface is completely described by Z$_{2}^{+2}$, and 90$^{\circ}$ later completely described by Z$_{2}^{-2}$. The root mean squared amplitude ($A_{2}^{2}$) of the polynomials remains constant.
![Plots of the amplitudes of the order 2 Zernike polynomials present in the surface described by the ETM08 Advanced LIGO mirror map as a function of the rotation of the surface.[]{data-label="fig:ETM08_zs_rotated"}](etm08_rotated_z2_amplitudes)
Analysis of Zernike approach
============================
We have used Zernike polynomials to describe mirror surface distortions and analyse the coupling that occurs from LG$_{33}$ into other order 9 modes. This approach appears to be very suitable as we have been able to identify specific polynomials which extract significant amounts of power from the input mode. However, there is an alternative method used to investigate mirror surface distortions, which involves looking at the spatial frequencies present in real mirrors. In this section we compare these two methods.
To look at the spatial frequencies present in realistic mirrors we perform a 2D Fourier transform of the surface height data. The resulting spectra is then analysed and synthetic maps are created with the same spatial frequencies. This method focuses on identifying particular spatial wavelengths which cause a large degree of coupling from LG$_{33}$. Many synthetic maps are created and used in simulations of gravitational wave detectors. A statistical approach is then taken to determine the extent of the coupling when specific spatial frequencies are present in the mirror.
In the Zernike approach we look at the different polynomials present in mirror surface distortions. This can be thought of as equivalent to looking at the spectra of the mirror surfaces as the different polynomials represent different spatial frequencies. The plot in Fig. \[fig:map\_spectra\] illustrates this. The spectrum of the ETM08 Advanced LIGO mirror map is shown, along with the spectra of maps made up from the Zernike polynomials present in the ETM08 mirror. Each of the Zernike maps recreates the LIGO map with polynomials up to a certain order. The plots show that as the order of Zernike polynomials present increases the higher spatial frequencies are represented in the mirror map. This is because these higher order polynomials represent the higher order spatial frequencies. Looking at the spatial frequencies present in the Zernike polynomials we found that the frequencies depended on the order, $n$. A consequence of this is that if we just consider the spatial frequencies present in the mirror maps we will not be able to distinguish between polynomials with different $m$. As we have seen, the azimuthal index is very significant as it determines which modes are largely coupled from LG$_{33}$. Therefore looking at the spatial frequencies doesn’t identify the important shapes in the mirror surfaces. The Zernike approach would seem to be the most suitable as this allows us to identify the interesting polynomials and modes.
![A plot showing the spectrum of the Advanced LIGO mirror map, ETM08 and the spectra of maps created from Zernike polynomials present in ETM08. The Zernike maps go up to a certain maximum order, recreating higher spatial frequencies with higher orders.[]{data-label="fig:map_spectra"}](mirror_map_spectra)
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[^1]: At the time of our analysis the coated mirrors were not yet available. The coating process can add further surface distortions so that some of the results presented here might not be representative for the final Advanced LIGO cavities.
[^2]: <span style="font-variant:small-caps;">Finesse</span> has been tested against an FFT propagation simulation, with higher order LG modes and mirror surface distortions. The results suggest <span style="font-variant:small-caps;">Finesse</span> is suitable for this investigation [@Bond].
|
---
abstract: 'We introduce a new mesoscopic model for nematic liquid crystals (LCs). We extend the particle-based stochastic rotation dynamics method, which reproduces the Navier-Stokes equation, to anisotropic fluids by including a simplified Ericksen-Leslie formulation of nematodynamics. We verify the applicability of this hybrid model by studying the equilibrium isotropic-nematic phase transition and nonequilibrium problems, such as the dynamics of topological defects, and the rheology of sheared LCs. Our simulation results show that this hybrid model captures many essential aspects of LC physics at the mesoscopic scale, while preserving microscopic thermal fluctuations.'
author:
- 'Kuang-Wu Lee'
- 'Marco G. Mazza'
title: Stochastic Rotation Dynamics for Nematic Liquid Crystals
---
Introduction {#sec:Introduction}
============
Liquid crystals (LCs) possess anisotropic interactions because of their molecular shape. For example, molecules with a rod-like rigid core tend to align parallel to each other and form a mesophase called nematic; molecules with a disk-like rigid core form discotic phases. In both cases the rigidity is generated by different combinations of aromatic rings [@deGennes:1993]. Macroscopically, this anisotropy leads to a series of phase transitions that break rotational and translational symmetries in a step-wise fashion. Because of their capacity to reorient, also in response to external fields, LCs are used in a wide range of applications: from the ubiquitous electronic displays, to microlasers [@humarNatPhot2009; @humar-OE-2010; @peddireddy-OE-2013] and lubricants [@Amann-2013]; but they are also rising to an important role in biomedical sciences and applications [@stewart2003liquid], and in our comprehension of morphogenesis and evolution of living organisms [@stewart2004liquid].
Hydrodynamic flow can also couple with the local preferential direction (director) established in nematic LCs [@Stephen-1974]. Recently, Sengupta *et al*. have extended microfluidic applications to anisotropic fluids and have found surprising topologies of the orientational field [@Sengupta:2013; @Sengupta-cyl-SM-13], and explored fluid and colloidal transport [@Sengupta-SM-13; @Sengupta-LC-14; @Sengupta-IJMS-13]. These effects originate from the intimate connection between LC rheological properties [@Parodi:1970] and their local alignment, which can be easily controlled. However, the complex interplay of confinement to a mesoscopic scale, i.e. of the order of $\mu$m, surface interactions, hydrodynamic flow and generation of topological defects still poses formidable challenges to both theoretical and experimental investigations.
The theoretical description of LCs based on static continuum theory started in the early 1920s with the work of Oseen [@Oseen], Zocher [@Zocher], and Frank [@Frank:1958]. The earliest dynamics theory of LCs can be dated back to 1931 by Anzelius [@Anzelius]. In the 1960s Ericksen and Leslie developed a hydrodynamics theory [@ericksen1959; @Ericksen:1960; @Ericksen:1961; @leslie1966; @Leslie:1968] based on the LC’s velocity field $\vec v(\vec r)$ and a unit vector describing the local director $\vec d(\vec r)$. The nematodynamic equations of the Ericksen-Leslie model are widely used but rest on the assumption that the nematic order parameter is a constant and the nematic LC is uniaxial, and therefore they cannot describe physical situations where there is a strong variation of the the nematic order, such as the isotropic-nematic phase transition and the dynamic of topological defects. For cases where there is a strong variation of the nematic order a tensorial description is necessary, such as the Beris–Edwards formulation [@beris1994thermodynamics] or the Qian–Sheng formulation [@Qian-Sheng-PRE]. These two approaches differ in the form of the elastic free energy, that is, the former considers the elastic free energy in the one-constant approximation, while the latter allows for two different elastic constants. We are not aware of any tensorial description of nematodynamics in terms of all three elastic constants, *i.e.* splay, twist and bend.
Microscopic models have in general the advantage of providing detailed dynamics at small spatial and temporal scales. Molecular dynamics simulations are well suited for this task. However, physical phenomena at the mesoscopic scale are still so computational demanding that they are out of the reach of atomistic simulations. This predicament can be ameliorated by adopting a coarse-grained description of the molecular degrees of freedom that effectively includes hydrodynamic modes. One widely-used method is the lattice Boltzmann scheme that simulates the evolution of the Boltzmann equation for a simple fluid on a regular lattice by a series of collision and propagation steps for probability density functions defined on the sites of a lattice [@McnamaraZanettiPRL1988]. This scheme has been generalized to LCs with successful results [@Denniston-EPL-2000; @CareJPCM2000; @CarePRE2003]. However, lattice Boltzmann schemes have the limitation that they do not include thermal fluctuations. A different approach that emphasize the particle aspect is the dissipative particle dynamics (DPD). This is an off-lattice, particle-based method, where particles represents fluid elements and are subject to pair-wise additive forces, which conserve momentum locally and thus generate the Navier-Stokes hydrodynamics for simple fluids. The DPD scheme has also been extended to LCs [@AlSunaidi2004; @LevineJCP2005]. Whereas this method has been shown to reproduce equilibrium phase diagrams of LCs [@AlSunaidi2004; @LevineJCP2005], we are not aware of any attempt at reproducing nematodynamic behavior, though the original version of DPD does reproduce the correct hydrodynamics of simple fluids [@HoogerbruggeEPL1992].
More recently, Malevanets and Kapral [@Malevanets:1999] introduced the stochastic rotation dynamics (SRD) model. This is also an off-lattice, particle-based model where each particle represents a small parcel of fluid. The fluid evolves through a series of collisions and streaming steps that exactly conserve mass, linear momentum and energy, and additionally respect Galilean symmetry. Thus, the correct hydrodynamic modes are generated. Because SRD is a particle-based method, fluctuations are naturally present. The SRD method has been applied to a variety of systems, from colloids [@PAddingPRL2004] to polymers [@malevanets2000dynamics; @WinklerJPCM2004], from the modeling of the solvent carrying hydrodynamic interactions in vescicle self-assembly [@NoguchiJCP2006] to the flow-induced shape transition in red blood cells [@NoguchiPNAS2005]. In general the SRD model is useful whenever both thermal fluctuations and hydrodynamic modes are physically important.
In the present work we extend the SRD model to anisotropic fluids, namely LCs. This is primarily done by giving orientational degrees of freedom to the SRD particles which obey the (simplified) equations of nematodynamics in the formulation of Ericksen-Leslie. For the sake of simplicity we restrict ourselves to methods and results in two dimensions (2D). We show below that the SRD is amenable of studying both the equilibrium and nonequilibrium behavior of LCs. This new extension of the SRD model opens the door to the investigation of a wealth of LC phenomena occurring at the mesoscale.
This work is organized as follows. In Sec. \[sec:SRD\_LC\] we describe the theoretical background of the equations used in the present model, and the details of the numerical implementation. We validate this model for nematic LCs by considering three study cases in Sec. \[sec:applications\]: i) the isotropic-nematic phase transition; ii) the production and annihilation of topological charges, and iii) LC rheology under shear flow. Finally, we summarize and discuss our results in Sec. \[sec:Discussions\].
The Model {#sec:SRD_LC}
=========
Theoretical Background
----------------------
We start by considering the standard SRD model that has so far been used to describe fluids with isotropic interactions. The system is composed of $N$ particles of mass $m_i$ with positions $\vec r_i(t)$ and velocities $\vec v_i(t)$, where $i\in[1,N]$. The evolution in time $t$ proceeds through a series of two steps:\
(i) the free-streaming step $$\label{eq:SRD-pos}
\vec r_i(t+\delta t)=\vec r_i(t)+\vec v_i(t)\delta t\,,$$ where all positions are updated;\
(ii) the collision step $$\label{eq:SRD-vel}
\vec v_i(t+\delta t)=\vec u_{C_i}(t)+\bm{\mathrm{R}}\left[\vec v_i(t)-\vec u_{C_i}(t)\right]$$ where all velocities are updated by rotating the fluctuating part of the velocity with respect to the center of mass velocity $$\begin{aligned}
\label{eq:SRD-cm}
\vec u_{C_i}(t)=\frac{1}{M_{C_i}}\sum_{i=1}^{\mathcal{N}_{C_i}} m_i\vec v_i\,, \quad &
M_{C_i}\equiv \sum\limits_{i=1}^{\mathcal{N}_{C_i}} m_i\end{aligned}$$ In Eq. (\[eq:SRD-vel\]-\[eq:SRD-cm\]) the calculations are performed in a cell-wise fashion, that is, the system is divided with a regular grid, and $\vec u_{C_i}$ is computed from the $\mathcal{N}_{C_i}$ particles within the cell $C_i$ to which particle $i$ belongs.
The rotation (or collision) matrix $\bm{\mathrm{R}}$ is orthogonal, $\bm{\mathrm{R}}^{-1}=\bm{\mathrm{R}}^\mathrm{T}$, where the superscript $\mathrm{T}$ denotes transposition. It rotates, independently in each cell, the fluctuating part of the velocity by an angle $\alpha$ about an arbitrary axis. Because of its action on the fluctuating part of the velocity the collision rule conserves momentum; because of its orthogonality the energy is also conserved. From these facts follows that the fluid obeys the Navier–Stokes equations. Malevanets and Kapral showed [@Malevanets:1999] that if $\bm{\mathrm{R}}$ satisfies detailed balance then the fluid approaches a Maxwell–Boltzmann distribution and obeys the $H$-theorem. In practice there are many ways to choose $\bm{\mathrm{R}}$ so that all the requirements are met. In 2D $\bm{\mathrm{R}}$ can only rotate the velocities by an angle $\pm\alpha$ with equal probabilities; in 3D it is common, *e.g.*, to choose a random axis and rotate around it of a fixed angle. To ensure Galilean invariance the grid must be shifted by a random amount at each time-step [@IhlePRE2001] so that artificial correlations among particles do not build up due to repeated collisions with the same neighbors.
We now come to the extension to nematic LCs. The particle’s degrees of freedom must be augmented by a unit vector describing its orientation $\vec d_{i}$, $\|\vec d_i\|^2=1$. Two particles separated by a distance $\|\vec r_i-\vec r_j\|\leqslant \epsilon$ interact through the Lebwohl-Lasher potential [@Lebwohl-PRA-1972] $$\nonumber
U_{i} = -\sum_{\left\langle i,j\right\rangle} (\vec d_{i} \cdot \vec d_{j})^2.$$ where $\left\langle i,j\right\rangle$ indicates that $i$ and $j$ are neighbors.
Lin *et al.* [@Lin-1989; @Lin-1995] proposed a simplified version of the Ericksen-Leslie equations. These are nonparabolic, dissipative equations that describe the flow of nematics in the one-constant approximation. Although they represent a drastic simplification of the original Ericksen-Leslie model, they retain the essential mathematical features, and they obey an energy law similar to the one used in the Ericksen-Leslie model [@Lin-1995].
We propose the following hybrid model of SRD for anisotropic fluids $$\begin{gathered}
\frac{\partial\vec v}{\partial t}+ \vec v\cdot \nabla \vec v =\nabla\cdot(\nu\nabla\vec v)-\nabla P/\rho
-\lambda\nabla \cdot \bm{\pi}
\label{Navier-Stokes}
\\
\frac{\partial\vec d}{\partial t}+ \vec v\cdot \nabla \vec d
- \vec d \cdot \nabla\vec v = \gamma_{_\mathrm{EL}} \nabla^2 \vec d - \gamma f(\vec d) + \vec \xi(t)
\label{EL-director}\end{gathered}$$ where $\rho$ is the density, $P$ the pressure, $\nu=\eta/\rho$ the kinematic viscosity, and $\bm{\pi}=(\nabla {\vec d}^{\,\,\mathrm{T}} \cdot\nabla \vec d)$ the Ericksen-Leslie stress tensor, that is the tensor whose $(\alpha\beta)$ component is $\partial\vec d/\partial r_\alpha \cdot \partial\vec d/\partial r_\beta$. In Eq. , $\gamma_{_\mathrm{EL}}$ is the elastic relaxation constant, the term $f(\vec d_{i}) = \partial U_{i}/ \partial \vec d_{i}$ is the molecular field inducing nematic ordering, $\gamma$ its strength, and $\vec{\xi}(t)$ is a Gaussian white noise for the director’s angular velocity, with $\left\langle {\xi}_\alpha \right\rangle=0$ and $\left\langle \xi_\alpha(t)\xi_\beta(t')\right\rangle=2k_\mathrm{B}T\gamma\delta_{\alpha\beta}\delta(t-t')$, where $k_\mathrm{B}$ is Boltzmann’s constant, $T$ is the temperature, $\delta_{\alpha\beta}$ is a Kronecker delta, and $\delta(t)$ is Dirac’s delta distribution.
The Ericksen-Leslie stress tensor $\bm{\pi}$ represents the feedback of the director field $\vec d (\vec r,t)$ to the bulk flow of molecules. In this formulation, $\bm{\pi}$ is directly responsible for the non-Newtonian behavior of the LC flow. Because of their molecular anisotropy the viscosity of a LC fluid does not depend solely on the shear stress $\tau = \eta\partial v/\partial x$, but the director field $\vec d (\vec r,t)$ also participates in the viscosity generation. This means that the velocity field can couple to the director and reorient it, and, also, that a reorientation of the director may generate a flow, usually called *backflow*.
Numerical Implementation
------------------------
We now describe the numerical implementation of our hybrid model for LCs. We will make a distinction between quantities computed on a particle level, such as $\vec v_i$, and on a cell level, such as $\vec u_{C_i}$. We note that the standard SRD steps described in Eqs. (\[eq:SRD-pos\]-\[eq:SRD-vel\]) recover the Navier–Stokes equation, that is Eq. without the last term. Also, Eq. (\[EL-director\]) applies in the Eulerian picture (lab-frame) to the fluid parcel moving along the streamline. Since in simulations the director of the $i$th particle, $\vec d_{i}$, moves along the flow, the convective term $\vec v \cdot \nabla\vec d$ is absorbed in this Lagrangian picture (comoving frame).
The SRD algorithm for LCs consists of the following steps:\
(i) free streaming $$\label{eq:LC-SRD-pos}
\vec r_i(t+\delta t)=\vec r_i(t)+\vec v_i(t)\delta t\,,$$ this is identical to the standard SRD step;\
(ii) cell-wise calculations, that is, the particles are grouped in different cells and $\vec u_{C_i}$, $\vec d_{C_i}$, gradients such as $\nabla \vec d$ and $\nabla \vec v$, and the Ericksen-Leslie elasticity tensor $\bm{\pi}$ are also calculated;\
(iii) LC alignment $$\begin{gathered}
\label{eq:LC-SRD-dir}
\vec d_i(t+\delta t)=\vec d_i(t)+\left[\vec d \cdot \nabla\vec v +\gamma_{_\mathrm{EL}} \nabla^2 \vec d \right.\\
- \left.\gamma f(\vec d) + \vec \xi(t) \right]\delta t\end{gathered}$$ where Eq. is implemented.\
(iv) Collisions and backflow. The cell-wise, center of mass velocity is calculated as in Eq. , and the contribution from the Ericksen-Leslie tensor is then added $$\begin{gathered}
\vec{u}^{\,'}_{C_i}(t)=\vec{u}^{\,}_{C_i}(t)+\lambda\nabla \cdot \bm{\pi}_{C_i}\\
\label{eq:LC-SRD-vel}
\vec v_i(t+\delta t)=\vec u^{\,'}_{C_i}(t)+\beta_\mathrm{th}\bm{\mathrm{R}}\left[\vec v_i(t)-\vec u_{C_i}(t)\right]\end{gathered}$$ We consider here only a 2D system, thus the matrix $\bm{\mathrm{R}}$ rotates the particles’ thermal velocities around one axis, conventionally denoted as the $z$-axis, either clockwise or counter-clockwise by a fixed angle $\alpha$ stochastically. The parameter $\beta_\mathrm{th}$ is the thermostat scaling factor whose role will be explained below.
In general, SRD does not conserve angular momentum. To impose the conservation of the fluid’s angular momentum in 2D the angle $\alpha$ must be chosen [@Ryder2005; @Gompper:2008] such that $$\begin{aligned}
\sin\alpha=-\frac{2AB}{A^2+B^2}\,, \quad & \cos\alpha=\frac{A^2-B^2}{A^2+B^2}\end{aligned}$$ where
$$\begin{aligned}
A&=\sum_{i=1}^{\mathcal{N}_{C_i}}\left[\vec r_i \times (\vec v_i-\vec u_{C_i})\right]|_z\,, \\
B&=\sum_{i=1}^{\mathcal{N}_{C_i}}\vec r_i \cdot (\vec v_i-\vec u_{C_i})\,.\end{aligned}$$
A thermostat for translational and rotational velocity is implemented to control the temperature of the system. Equipartition of the energy is applied to keep the same amount of energy in each degree of freedom. The temperature is defined as $k_\mathrm{B}T = \tfrac{1}{N}\sum_i \tfrac{1}{2}m_i v_i^2$ from the translational kinetic energy, or as $k_{\mathrm{B}}T = \tfrac{1}{N}\sum_i I_i\omega_i^{2}$ from the rotational kinetic energy in 2D, where $I_i$ is the moment of inertia of the particles and ${\omega_i}$ is the angular velocity. We employ a simple velocity rescaling thermostat that scales linear and angular velocities in a cell-wise fashion by a factor $\beta_\mathrm{th}=\sqrt{T/T_{C_i}}$ (see Eq. ), where $T_{C_i}$ is the instantaneous kinetic temperature in cell $C_i$.
We use two types of boundary conditions (BC), periodic BC in both the $x$ and $y$ directions for simulations of bulk systems, and a no-slip wall perpendicular to the $x$ axis (while periodic BC are implemented in the other direction) for simulations of a channel geometry. The no-slip BC are implemented with the usual bounce-back rule, that is, the velocity of the LC particle is inverted upon collision with the solid wall $$\vec v_i^{\:new}=-\vec v_i^{\:old}\,.$$ Additionally, “ghost” particles [@LamuraEPL2001] are used at the walls to fill partially occupied cells up to the average particle density. At solid interfaces the anchoring, that is, the preferential angle between LC particles and walls needs to be specified. Homeotropic anchoring is implemented by placing ghost particles in the walls with their orientations aligned perpendicularly to the walls.
The system is initialized by assigning the positions $\vec r_i$ and velocities $\vec v_i$. The particles are uniformly distributed in space. The linear and angular velocities are assigned so that they are distributed according to a Maxwell distribution at the target temperature, and that the equipartition theorem is obeyed.
We perform all our simulations with the following parameters: the SRD rotation angle is $\alpha = 120\degree$, particle mass $m_i = 1$, moment of inertia $I_i=1$, and the elastic and molecular relaxation constants are $\gamma_{_\mathrm{EL}} = 10^{-4}$ and $\gamma = 8\times10^{-4}$. It is a common choice to set the SRD grid size $\delta x=1$ and the timestep $\delta t = 1$. At each time step the grid is shifted by a random displacement vector with components uniformly distributed in the interval $[-\delta x/2, \delta x/2]$. In the following we measure the temperature $T$ in units of $m_iV^2_\mathrm{max}/k_\mathrm{B}$, where $V_\mathrm{max}=\delta x/\delta t$ is the maximum propagation speed of a particle (related to the Courant condition). The mean free path $\lambda=\delta t\sqrt{k_\mathrm{B}T/m_i}$ is typically smaller than one but the grid-shift method avoids the build-up of spurious correlations.
Applications {#sec:applications}
============
Isotropic-Nematic Phase Transition {#sec:Phase_transition}
----------------------------------
We perform simulations of a system in equilibrium at fixed $T$ with periodic BC in both $x$ and $y$ directions. A useful way to characterize the degree of ordering of a nematic LC is to define an order parameter that is zero in the isotropic phase and one in the nematic phase. It is common to take the largest eigenvalue $S$ of the nematic order tensor $$\mathbf{Q} = \frac{1}{N}\sum_{i=1}^{N} \left(\frac{3}{2}\hat{d_{i}}\otimes\hat{d_{i}}-\frac{1}{2}\mathbf{I}\right)\,,
\label{Order_tensor}$$ which is a traceless, second-order tensor, and where $\otimes$ is the dyadic product and $\mathbf{I}$ is the unit tensor.
![Nematic order parameter as a function of thermostat temperature. The solid line is just a guide for the eye. A continuous isotropic-nematic phase transition occurs at $T\simeq0.18$.[]{data-label="fig1"}](fig1){width="1.0\columnwidth"}
We consider a system of $N =150\,000$ particles, and of size $L_x=L_y=50$, subdivided with a $50\times50$ grid, thus the average number of particles per cell $\langle\mathcal{N}_{C_i}\rangle=60$. Figure \[fig1\] shows the dependence of the nematic order parameter $S$ on $T$, at fixed density. At low $T$ the nematic order parameter is very close to one, indicating nearly perfect nematic alignment. As $T$ increases $S$ decreases continuously until it reaches the constant value $1/4$, which characterizes the isotropic phase in 2D [@note_MaierSaupe]. We find that the isotropic-nematic transition occurs at $T\simeq0.18$. A LC system in two dimensions or higher undergoes a temperature-driven isotropic-nematic phase transition. This phase transition is continuous in 2D, as expected for a system with the symmetry of the XY model (though there is some controversy [@Vink-PRL-2007; @Jordens-NatComm-2013; @Marucci-Macromol-1989]).
Contrary to the hydrodynamic Ericksen-Leslie model, our SRD hybrid model can reproduce the isotropic-nematic phase transition. The reason for that is the particle nature of our model. Different parts of the system may have different local orientations and thus, at high enough $T$, the system is globally disordered.
Dynamics of Topological Defects {#sec:Move_defects}
-------------------------------
During a quench from the isotropic to the nematic phase a LC substance develops many topological defects [@demus1978] which influence the kinetics of the phase transition. Solid boundaries imposing geometric or energetic (anchoring) constraints, or external fields in general may induce different, local orientations in a nematic LC. Because of the conflict between these local orientations topological defects are produced. These are regions of the fluid where the local nematic order parameter is vanishing and the director field is undefined [@Chaikin].
![Snapshots of the system at different times showing the temporal evolution of local nematic order parameter (color) and local director (black dashes). As the system is quenched from the isotropic to the nematic phase topological defects are formed. In panel (d) the red dashed circle marks the $q = -1/2$ topological defect, while the red solid circle indicates the $q = +1/2$ defect.[]{data-label="fig2"}](fig2){width="1.0\columnwidth"}
![The position of topological charges as function of time is shown in the upper panel. The blue dots (empty circles) represent the position of positive (negative) charge. The red solid (dashed) line are the fits of the positive (negative) charge. The lower panel shows their separation as a function of time. The blue empty circles are the direct subtractions of positions and the red line is the fitting curve using Eq. (\[distance\_charges\]).[]{data-label="fig3"}](fig3a "fig:"){width="1.1\columnwidth"} ![The position of topological charges as function of time is shown in the upper panel. The blue dots (empty circles) represent the position of positive (negative) charge. The red solid (dashed) line are the fits of the positive (negative) charge. The lower panel shows their separation as a function of time. The blue empty circles are the direct subtractions of positions and the red line is the fitting curve using Eq. (\[distance\_charges\]).[]{data-label="fig3"}](fig3b "fig:"){width="1.1\columnwidth"}
Although there were attempts of using Ericksen-Leslie model to simulate topological charges [@Liu:2007], those structures cannot be properly treated since the nematic order parameter is essentially constant in that approach. However in our model, the nematic order parameter in the macroscopic scale is calculated from the mesoscopic SRD particle configurations, hence the formation of topological defects is an essential feature in this particle-based approach.
We investigate the generation of topological defects and their dynamics by quenching the system from a high value of $T$, well in the isotropic phase, into the nematic phase at $T = 0.053$, with an equilibrium value of $S = 0.91$. Figure \[fig2\] shows the temporal evolution of the topological defects formed during the quench. A large number of $\pm1/2$ charges are generated. The total topological charge of the system is zero, and, by conservation of charge, an equal number of positive and negative charges are present. Immediately after the initial quench the SRD particles start to align with their neighbors, due to the torque induced by the molecular field $f(\vec d)$. After some time the particles start to form larger nematic domains. At this stage (Fig. \[fig2\](a)), there are many topological defects, carrying topological charges of $q=+1/2$ and $q=-1/2$. At a later stage, those charges with opposite signs are attracted to each others and eventually annihilated with each others (Fig. \[fig2\](b-d)), leaving the area charge-free as the initial configuration.
The speeds of topological defects depend on the charges they carry, that is, defects with charge $q=+1/2$ move faster than $q=-1/2$ charges; this has been verified both by numerical [@FukudaEPJB1998; @Yeomans:2002; @SvensekPRE2002] and experimental work [@BlancPRL2005]. We find a similar behavior in our simulations. Figure \[fig3\](a) shows the time dependence of the positions of two topological charges. Initially, the charges approach each others with rather slow velocities, but when the two charges are very close they accelerate until they annihilate. It is clear the velocity of the positive charge is higher than the negative one. The reason for this asymmetry is that the velocity of the defect core depends on the sign of the spatial derivative of the director field $\nabla \vec d$. Similar result has been reported in [@Yeomans:2002] where lattice Boltzmann simulations of a tensorial formulation of nematodynamics were performed.
To further verify the validity of our model we test the time dependence of the separation $D$ between opposite charges. Denniston predicted [@Denniston:1996] a simple scaling law $$D(t) = c(t_{a}-t)^{1/2}
\label{distance_charges}$$ where $t_{a}$ is the annihilation time and $c$ is a constant. Figure \[fig3\](b) shows that the separation between opposite charges does indeed follow the behavior predicted by Eq. . Although the scaling law was derived in conditions of no backflow [@Denniston:1996], it is still valid when the coupling of flow and director field is present, as observed in the tensorial treatments of nematodynamics [@Denniston-EPL-2000]. Then we conclude that our particle-based approach to nematodynamics can correctly describe regions of the fluid with strong gradients of the nematic order parameter, such as topological defects.
Shear flow {#sec:shear_flow}
----------
{width="2\columnwidth"}
Shear flow experiments are the canonical way to study the rheological properties of fluids. The coupling of the elastic deformations of LCs with the transport and deformation of fluid elements gives rise to a more complex situation than in the flow of an isotropic fluid. In the original formulation of Ericksen and Leslie [@ericksen1959; @Ericksen:1960; @Ericksen:1961; @leslie1966; @Leslie:1968] the viscous response is characterized by six coefficients $\alpha_i$, $i=1...6$, called Leslie coefficients. Later, Parodi showed [@Parodi:1970] that from Onsager’s reciprocity theorem follows $\alpha_6-\alpha_5=\alpha_2+\alpha_3$, thus the number of independent viscosity coefficients is five.
Our goal in this section is to verify if our hybrid model is capable of reproducing the known LC rheology. To generate a shear flow in a 2D simulation domain, we consider two no-slip walls at $x = 0$ and $x = L_x$ and periodic BC in the $y$ direction. Because of the presence of solid walls the boundary conditions for the director field need to be specified. We employ homeotropic anchoring, that is, the LC particles prefer to orient perpendicularly to the walls. This is achieved by assigning a perpendicular orientation of the ghost particles at the walls.
The two walls move with equal and opposite speeds in the $y$ direction. After some time a stationary shear flow is generated. A dimensionless measure of shear is given by the Weissenberg number $\mathcal{W}\equiv\dot{\gamma}\tau$ where $\dot{\gamma}=\partial v_y/\partial x$ is the shear rate and $\tau$ is a relaxation time. We calculate $\tau$ from the orientational correlation function $C_1(\Delta t)=\langle\vec d(t+\Delta t)\cdot\vec d(t)\rangle$, where the angle brackets indicate ensemble average. We present results for a fixed $\mathcal{W}=2.04$ at $T=0.28$ (we note that the presence of walls inducing homeotropic alignment shifts the phase diagram with respect to Fig. \[fig1\], so that $T=0.28$ corresponds to the nematic state). The system of $N=75000$ particles and size $L_x=15$, $L_y=50$ is divided with a grid of $15\times 50$, thus with $\langle\mathcal{N}_{C_i}\rangle=100$.
The steady-state director fields and bulk flow profiles of sheared LC are shown in Fig. \[fig4\] for different Ericksen-Leslie stress constant $\lambda$. For the case of $\lambda = 0$ the flow profile is a straight line as for a Newtonian fluid. For higher values of $\lambda$ we observe a shear banding effect. By increasing $\lambda$ the flow profile starts to have a kink near the walls. Shear banding is a nonequilbrium transition to a state where regions with different shear rates coexist, thus visible through different slopes of the velocity profile $v_y(x)$ in Fig. \[fig4\]. Nematic LCs can produce shear bands [@Mather-Macromol-1997; @Olmsted-Rheo-2008]. The existence of this phenomenon is a consequence of the competition between elastic energy and viscous dissipation, which in turn is caused by the feedback from LC orientation to the flow induced by $\bm{\pi}$.
The local director field $\vec d(\vec r)$ (represented as black dashes in Fig. \[fig4\](a)-(c))shows that while the director is aligned homeotropically at the walls, thus satisfying the BC, it is tilted in the central region of the channel. This tilt angle is due to a well-known flow alignment mechanism in LCs. The local nematic order map (color code in Fig. \[fig4\](a)-(c)) shows a subtle change from a high degree of nematic alignment at the walls, induced by the anchoring conditions, to a slightly lower value in the central region of the channel.
Conclusions {#sec:Discussions}
===========
We have introduced a new mesoscopic LC model. Our model is based on the stochastic rotational dynamics scheme, which is a particle-based algorithm that ignores the computationally heavy molecular interactions but correctly generates the hydrodynamic modes described by the Navier-Stokes equations.
The model introduced here fills an important gap for mesoscopic simulation techniques of LCs. Thermal fluctuations are explicitly present, which are of fundamental importance in both thermodynamic and dynamic processes, but are neglected in other popular schemes, such as the lattice Boltzmann approach. Furthermore, the mesoscopic scale and the hydrodynamic behavior is directly addressed. We have shown that this model can be used to study various aspects of the physics of liquid crystals. We have considered three study cases. First, we have found that our model system undergoes an equilibrium phase transition from nematic to isotropic as the temperature increases. The transition is found to be continuous, as it is expected to be in 2D. Second, we have studied the nonequilibrium dynamics of topological defects emerging in a quenched LC. Topological defects with $\pm\tfrac{1}{2}$ charge are formed but quickly annihilate with each other. The temporal dependence of the distance between two opposite charges is found to match remarkably well the theoretically predicted power law. Third, we have considered a shear flow situation and found that the LC system develops shear bands as the coupling parameter between flow and director reorientation is increased. We conclude that the model captures the non-Newtonian character of LC rheology.
This hybrid algorithm can be easily applied to complex geometries of the confining walls. For the sake of simplicity we restricted the present work to 2D. However, any realistic implementation requires a 3D setup. A generalization of the model to 3D is under way and will be discussed elsewhere.
We gratefully acknowledge helpful conversations with Martin Brinkmann, Stephan Herminghaus and Thomas Hiller.
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---
abstract: 'A method to synchronize two chaotic systems with anticipation or lag, coupled in the drive response mode is proposed. The coupling involves variable delay with three time scales. The method has the advantage that synchronization is realized with intermittent information about the driving system at intervals fixed by a reset time. The stability of the synchronization manifold is analyzed with the resulting discrete error dynamics. The numerical calculations in standard systems like the Rössler and Lorenz systems are used to demonstrate the method and the results of the analysis.'
author:
- 'G. Ambika'
- 'R. E. Amritkar'
title: Anticipatory synchronization with variable time delay and reset
---
\[sec1\]INTRODUCTION
====================
The synchronization of uni-directionally coupled chaotic systems have been studied reasonably well in the past few years [@pec; @tre; @amr]. The synchronization state in such cases can be phase, lag, generalized or complete depending upon the strength of the coupling [@boc; @cor]. Recently synchronization of systems via coupling with a time delay, which presumably takes care of the finite propagation times, switching speeds and memory effects have been reported [@pyr; @zho; @cho; @bun]. Such studies relate to a variety of diverse phenomena like chirping of crickets, neural networks, automatic steering and control and coupled phase locked lasers [@ste; @ern; @sha]. When the coupling is not isochronous with the system dynamics, it is possible to realize retarded (delay), complete and anticipatory synchronization of chaotic systems. Moreover, synchronization in such cases reveal many novel phenomena like parametric resonance[@zho1], multi-stable phase clustering [@par; @wan1; @seth; @nak], amplitude death etc [@kon; @pra; @ram]. An interesting aspect of such delay induced synchronization, that has attracted lot of attention, is that the driven system can anticipate the dynamics of the driver [@vos1; @wan2]. The maximum possible anticipation time is reported to be enhanced considerably by using an array or ring of such systems [@vos2; @wan3; @cis]. Experimental verification of anticipatory and retarded synchronization is reported in electronic circuits as well as semiconductor lasers with delayed optoelectronic feed back [@hail; @tan]. In all these studies, the delay time in the coupling, once chosen, remains constant as the system evolves.
The synchronization of chaotic systems, in general, has attracted great attention due its potential application in secure communication [@he; @li; @wan4]. However the use of low dimensional systems in this context is found to be insecure due to the ease of reconstruction from the transmitted signal [@vai; @sho]. Therefore recently, chaos synchronization in high dimensional systems, especially systems with an inherent time delay, has been proposed as a better alternative [@pen; @uda; @goe; @gho; @yao].
In this paper we propose a method of delay/anticipatory synchronization with coupling involving variable time delay. Here, the synchronization can be realized with limited information about the driver via occasional contacts or feedbacks at specific intervals. This makes the method highly cost effective and can be applied to cases where the signal transmission from driver is slow or intermittent. This is achieved by using a variable delay in the coupling that is reset at definite intervals. The dynamics then evolves under three additional time scales, the delay $\tau_1$, the anticipatory time $\tau_2$ and the reset time $\tau$. Unlike the case of fixed delay, the resetting mechanism makes the error dynamics discrete and it is possible to carry out an approximate analytic analysis. The analysis gives the maximum $\tau_2$ for a given $\tau$. This also fixes the regions of stability in the parameter plane of coupling and delay. The method is demonstrated for standard systems like Rössler and Lorenz.
\[sec:level1\]SYNCHRONIZATION WITH VARYING DELAY AND RESET
==========================================================
Model system
------------
Consider a dynamical system $x$ of dimension $n$ that drives an identical system $y$. We choose a simple coupling term of the linear difference type but with the drive variable delayed by $\tau_1$ and the driven variable delayed by $\tau_2$. Thus, the dynamics is given by
\[system\_def\] $$\begin{aligned}
\dot{x} & = & f(x)
\label{system_def_x}\\
\dot{y} & = & f(y) + \epsilon \sum_{m=0}^{\infty} \Gamma \left(x_{t_1}-y_{t_2}\right) \chi_{(m\tau,(m+1)\tau)}
\label{system_def_y}\end{aligned}$$
where $x_{t_1} = x(t-t_1)$, $y_{t_2} = y(t-t_2)$, $\tau$ is the resetting time and $\chi_{(t',t'')}$ is an indicator function such that $\chi_{(t',t'')} = 1 \; \textrm{for} \; t' \leq t \leq t''$ and zero otherwise. Here, $\Gamma=[\Gamma_1,\Gamma_2,\ldots,\Gamma_n]^T$ is a constant vector specifying the coupling between the components of $x$ and $y$. In numerical simuations we take only one coponent of $\Gamma$ to be nonzero. Both the delays $t_1$ and $t_2$ depend on time and we choose this dependence as $$t_i = \tau_i + t - m\tau, \; \; i=1,2.$$ Thus, $t-t_i = m\tau - \tau_i$. As the two systems evolve, $t_1$ and $t_2$ also evolve with the same time scale and the coupling term uses the same value of both variables $x_{t_1}$ and $y_{t_2}$ during each resetting time interval $\tau$, i.e. the coupling term is constant for the time interval $\tau$. In each time interval $\tau$, the initial values of the delays $t_1$ and $t_2$ are $\tau_1$ and $\tau_2$ respectively. The delays increase linearly with time upto values $\tau_1+\tau$ and $\tau_2+\tau$ and then they are reset for the next interval. As a consequence, the coupling requires the variable of the drive system only at discrete time intervals of $\tau$. We also note that $t_1-t_2 = \tau_1-\tau_2$ for all $t$.
Synchronization Manifold
------------------------
Synchronization manifold for the coupled systems (\[system\_def\]) is defined by $y(t-\tau_2) = x(t-\tau_1) \; {\textrm or} \;
y(t) = x(t-\tau_1+\tau_2)$ Thus, we can get all the following three possibilities [@senthil]. (1) If $\tau_1-\tau_2 > 0$, we can get delay or lag synchronization with $\tau_1-\tau_2$ as the lag time. (2) If $\tau_1-\tau_2 < 0$, we can get anticipatory synchronization with $\tau_2-\tau_1$ as the anticipation time. (3) If $\tau_1-\tau_2 = 0$, we can get equal time synchronization.
As an illustration of this, we take the standard Rössler oscillator in the chaotic state as the driver described by the equations $$\begin{aligned}
\dot{x_1} & = & - x_2 - x_3 \nonumber \\
\dot{x_2} & = & x_1 + a x_2
\label{rossler} \\
\dot{x_3} & = & b + x_3(x_1 - c). \nonumber\end{aligned}$$ This is coupled to an identical system through the coupling scheme given in Eq. (\[system\_def\]). Only $x_1$ and $y_1$ are coupled, i.e. $\Gamma = [1,0,0]^T$. Taking the parameter values $a= 0.15, b=0.2$ and $c= 10.0$, both the systems are evolved from random initial conditions using Runge Kutta algorithm with a time step 0.01 for 2000 units of time. With $\tau =0.10$ and the coupling strength $\epsilon = 0.4$, the resulting time series obtained for $\tau_1$=0.84 and $\tau_2$ = 0.02 is plotted in Fig.1.a. Here the response system y(t)(dashed line) lags behind the driver x(t)(solid line) by $\tau_2 - \tau_1$. Fig.1.b shows the same for $\tau_1$= 0.02 and $\tau_2$=0.84 where y(t) anticipates x(t) with the same time shift. The degree of synchronization with the corresponding time shift can be quantified using the similarity function defined as $$S^2(T) = \frac{<[y_1(t)-x_1(t+T)]^2>}{\sqrt{<x_1^2 (t)><y_1^2 (t)>}}$$ Figs.1.c and 1.d show $S^2(T)$ computed for different values of $T$. The minimum occurs at 0.82, i.e. $T=|\tau_1-\tau_2|$, indicating synchronization with delay or anticipation of 0.82 time units.
It should be noted that the delay time $\tau_1$ is not of much significance in the error dynamics, since the time scale of the drive system can be linearly shifted by $\tau_1$. This point will become clear when we do the stability analysis in the next section.
![\[del-ant-ros.eps\]The simulated time series of two Rössler systems coupled through the scheme in Eq. (\[system\_def\]). In (a) the case of delay synchronization is shown with a delay of 0.82 units between the $x_1(t)$(solid line) and $y_1(t)$(dashed line). (b) is a case of anticipatory synchronization when $y_1(t)$ anticipates $x_1(t)$ by the same units. The similarity function $S^2(T)$ corresponding to both these cases are shown in (c) and (d) respectively. ](del-ant-ros.eps){width="0.95\columnwidth"}
\[sec2\]LINEAR STABILITY ANALYSIS
=================================
The dynamics of the system in Eq. (\[system\_def\]) involves three time scales in addition to its inherent scale. Define the transverse system by the variable $\Delta = y - x_{\tau_1-\tau_2}$. Its dynamics in linear approximation can be derived from Eq. (\[system\_def\]) as $$\begin{aligned}
\dot{\Delta} = f^{'}(x_{\tau_1-\tau_2}) \Delta -
\epsilon \sum_{m=0}^{\infty} \chi_{(m\tau,(m+1)\tau)} \Delta_m
\label{linear_stability}\end{aligned}$$ where $\Delta_{m} = \Delta(t-t_2) = \Delta(m\tau-\tau_2)$ and we take coupling in all components of $x$ and $y$, i.e. $\Gamma=[1,1\ldots,1]^T$. Thus, $\Delta_m$ is a constant in each time interval $m\tau \leq t < (m+1) \tau$. We note that $\tau_1$ enters only through the Jacobian term $f^{'}$ and can be eliminated by shifting the time scale of the drive system linearly and redefining $\tau_2$ suitably. Hence, as noted in the previous section, $\tau_1$ is not very significant for the stability analysis. The fixed point $\Delta=0$ corresponds to the lag/anticipatory synchronized state.
In general, it is not possible to solve Eq. (\[linear\_stability\]). However, we can approximate the equation by replacing Jacobian $f^{'}$ by some effective time average Lyapunov exponent $\lambda$ (only the real part is required). $$\dot{\Delta} = \lambda \Delta - \epsilon \sum_{m=0}^{\infty} \chi_{(m\tau,(m+1)\tau)} \Delta_m
\label{linear_problem}$$ In the following analysis we assume $\lambda$ to be positive. The results can be easily extended to $\lambda <0$ (see Appendix \[AppC\]).
>From the numerical analysis presented in the next section, it appears that the approximation of replacing $f^{'}$ by an effective $\lambda$ is reasonable for small values of $\tau_2$. We need a larger value of $\lambda$ for large $\tau_2$.
In the interval $m\tau \leq t < (m+1) \tau$, the solution of Eq. (\[linear\_problem\]) is $$\Delta = \alpha \Delta_m + C_m e^{\lambda t}
\label{sol-Cm}$$ where $\alpha = \epsilon / \lambda$ is the normalized dimensionless coupling constant, and $C_m$ is an integration constant.
$0 \leq \tau_2 \leq \tau$
-------------------------
Let us first consider the case $0 \leq \tau_2 \leq \tau$. For $t = (m+1)\tau-\tau_2$, $\Delta = \Delta_{m+1}$. Thus, eliminating the integration constant, Eq. (\[sol-Cm\]) gives $$\Delta = \alpha \Delta_m + (\Delta_{m+1} - \alpha \Delta_m) e^{\lambda (t-(m+1)\tau + \tau_2)}
\label{sol-m}$$ For $(m-1)\tau \leq t \leq m\tau$ we have $$\Delta = \alpha \Delta_{m-1} + (\Delta_m - \alpha \Delta_{m-1}) e^{\lambda (t-m\tau + \tau_2)}
\label{sol-m-1}$$ Matching the solutions (\[sol-m\]) and (\[sol-m-1\]) at $t=m\tau$, and simplifying, we get the following recursion relation
\[rec-tau2\] $$\begin{aligned}
\Delta_{m+1} & = & \alpha (1-e^{\lambda(\tau-\tau_2)} + \frac{1}{\alpha}
e^{\lambda \tau}) \Delta_m
\nonumber \\
& & - \alpha e^{\lambda \tau}(1-e^{-\lambda \tau_2}) \Delta_{m-1}
\label{rec-tau2-a} \\
& = & a \Delta_m - b \Delta_{m-1}
\label{rec-tau2-b}\end{aligned}$$
where
\[ab\] $$\begin{aligned}
a & = & \alpha (1-e^{\lambda(\tau-\tau_2)}) + e^{\lambda \tau},
\label{a} \\
b & = & \alpha e^{\lambda \tau}(1-e^{-\lambda \tau_2}).
\label{b}\end{aligned}$$
We can write Eq (\[rec-tau2-b\]) as a 2-d map in matrix form as, $$\left( \begin{array}{c}
\Delta_{m+1} \\ \Delta_m
\end{array} \right) =
\left( \begin{array}{cc}
a & -b \\ 1 & 0
\end{array} \right)
\left( \begin{array}{c}
\Delta_m \\ \Delta_{m-1}
\end{array} \right)
\label{2dmap}$$ The eigenvalue equation for the Jacobian matrix is $$\mu^2 - a \mu + b =0
\label{eq-ev-2d}$$ with the solutions $$\begin{aligned}
\mu_{\pm} & = & \frac{1}{2}(a \pm \sqrt{a^2-4b})
\label{ev-a}\end{aligned}$$ The synchronized state, $\Delta=0$, is stable if both the solutions satisfy $|\mu_{\pm}| < 1$. The detailed analysis of the stability conditions is given in Appendix \[AppA\].
Fig. \[tau2-ep\] shows the stability region in $\tau_2/\tau-\alpha$ plane. The lower limit of stability is always $\alpha_l=1$. For smaller values of $\tau_2$ ($\tau_2 \leq \tau_{2p}$), the upper limit of stability is given by (Eq. (\[alpha-case2\])) $$\alpha_u = \frac{e^{\lambda \tau}+1}{2e^{\lambda (\tau- \tau_2)} - e^{\lambda \tau} - 1}
\label{alphau1}$$ while for larger values of $\tau_2$ ($\tau_{2p} \leq \tau_2 \leq \tau$) it is given by (Eq. (\[alpha-case3\])) $$\alpha_u = \frac{e^{-\lambda \tau}}{1-e^{-\lambda \tau_2}}
\label{alphau2}$$ The maximum value of $\alpha_p$ is given by the intersection of the two curves (\[alphau1\]) and (\[alphau2\]). $$\alpha_p = \frac{3+e^{\lambda \tau}}{e^{\lambda \tau}-1}
\label{alphap}$$ The corresponding $\tau_{2p}$ value is given by $$\tau_{2p} = \frac{\alpha_p (\alpha_p+3)}{(\alpha_p+1)^2}
\label{tau2p-alphap}$$
![\[tau2-ep\]This figure shows the stability region of the synchronized state ($\Delta=0$ solution of Eq. (\[rec-tau2\])) in the $\tau_2/\tau - \alpha$ plane. The solid line is for $\lambda \tau = 0.25$ and the dashed line is for $\lambda \tau = 0.5$. The lower limit of stability is $\alpha_l=1$ (dotted line). For smaller values of $\tau_2 \leq \tau_{2p}$, the upper limit of stability is given by (Eq. (\[alphau1\])) while for larger values of $\tau_2$ ($\tau_{2p} \leq \tau_2 \leq \tau$) it is given by (Eq. (\[alphau2\])). The peak values are ($0.0530\ldots/0.25\ldots=0.212 ,15.083\ldots$) for $\lambda\tau=0.25$ and ($0.088\ldots/0.5=0.176\ldots,7.169\ldots$)for $\lambda\tau=0.5$ (see Eqs. (\[alphap\]) and (\[tau2p-alphap\])). For $\tau \leq \tau_2 \leq 2 \tau$ the upper limit of stability is given by Eq. (\[alphau-k1\]).The maximum value of $\lambda\tau_2$ is $0.74$ for $\lambda\tau = 0.5$. ](tau2-ep.eps){width="0.9\columnwidth"}
We also obtain $\tau_{2max}$, the maximum allowed value of $\tau_2$ for the stability of the synchronized state (Eq. (\[tau2max-0\])). A general expression for $\tau_{2max}$ is obtained in the next subsection (Eq. (\[tau2max-k\])).
$\tau_2 > \tau$
---------------
Let $\tau_2 = k\tau + \tau_2^{'}, \; k = 0,1,\ldots$ where $\tau_2^{'} < \tau$. Consider the solution (\[sol-Cm\]) in the interval $m\tau \leq t \leq (m+1)\tau$. Then for $t= (m+1)\tau - \tau_2^{'} = (m+k+1)\tau - \tau_2$, we get $$\begin{aligned}
\Delta_{m+k+1} & = & \alpha \Delta_m + C_m e^{\lambda (m+1) \tau - \lambda \tau_2^{'}}\end{aligned}$$ Hence Eq. (\[sol-Cm\]) becomes $$\Delta = \alpha \Delta_m + (\Delta_{m+k+1} - \alpha \Delta_m) e^{\lambda (t-(m+1)\tau + \tau_2^{'})}
\label{sol-mk}$$ For $(m-1)\tau \leq t \leq m\tau$ we have $$\Delta = \alpha \Delta_{m-1} + (\Delta_{m+k} - \alpha \Delta_{m-1}) e^{\lambda (t-m\tau + \tau_2^{'})}
\label{sol-mk-1}$$ Equating the solutions (\[sol-mk-1\]) and (\[sol-mk\]) for $t=m\tau$, and simplifying we get the following recursion relation $$\begin{aligned}
\Delta_{m+k+1} = && e^{\lambda \tau} \Delta_{m+k} - \alpha (e^{\lambda(\tau-\tau_2^{'})} -1) \Delta_m\nonumber\\
&&- \alpha e^{\lambda \tau}(1-e^{-\lambda \tau_2^{'}}) \Delta_{m-1}
\label{rec-tau2'} \end{aligned}$$ This gives a map of dimension $k+2$. In matrix form, the map can be expressed as $$\begin{aligned}
\left( \begin{array}{c}
\Delta_{m+k+2} \\ \Delta_{m+k+1} \\ \Delta_{m+k} \\ \vdots \\ \Delta_m
\end{array} \right) & = &
\left( \begin{array}{ccccc}
c & 0 & \ldots & b_1 & b_0 \\
1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & \ldots & 0 & 0 \\
\vdots & & & & \vdots \\ 0 & 0 & \ldots & 1 & 0
\end{array} \right)
\left( \begin{array}{c}
\Delta_{m+k+1} \\ \Delta_{m+k} \\ \Delta_{m+k-1} \\ \vdots \\ \Delta_{m-1}
\end{array} \right) \nonumber \\
& & \label{kdmap}\end{aligned}$$ where $c=e^{\lambda \tau}$, $b_1=\alpha (e^{\lambda(\tau-\tau_2^{'})} -1)$ and $b_0=\alpha e^{\lambda \tau}(1-e^{-\lambda \tau_2^{'}})$. The eigenvalue equation is $$\mu^{k+2} - c \mu^{k+1} + b_1 \mu + b_0 = 0
\label{kth}$$ For $k=0$, the map of Eq. (\[kdmap\]) reduces to the 2d-map of Eq. (\[2dmap\]). In general the behavior of the largest magnitude $\mu$ is as shown in Fig. \[mu-alpha\]b, i.e. the stability range is from $\alpha_l=1$ till the complex $\mu$ has magnitude one.
### $k=1, \; {\textrm i.e.} \; \tau \leq \tau_2 \leq 2\tau$
For $k=1$, we have a 3d-map. The eigenvalue equation (\[kth\]) becomes $$\mu^3 - c \mu^2 + b_1 \mu + b_0 = 0
\label{cubic-1}$$ The lower stability limit is $\alpha_l=1$. The upper stability limit can be obtained by noticing that when the magnitude of the imaginary $\mu$ becomes one, the two imaginary eigenvalues can be written as $\mu = e^{\pm i \theta}$ and the above equation has a factor $\mu^2 - 2 \cos(\theta) \mu + 1$. This gives the condition $$b_0^2 + c b_0 +b_1 -1 = 0$$ Using this we get a quadratic equation for $\alpha$. $$a_2 \alpha^2 + a_1 \alpha - 1 =0
\label{eq-alpha-1}$$ where $a_1 = e^{2 \lambda \tau} ( 1 - e^{-\lambda \tau_2^{'}}) + e^{\lambda(\tau-\tau_2^{'})} -1$ and $a_2 = e^{2 \lambda \tau}(1-e^{-\lambda \tau_2^{'}})^2$. One solution of this equation gives the upper stability limit for $\alpha$. $$\alpha_u = \frac{1}{2 a_2}(-a_1 + \sqrt{a_1^2 +4 a_2})
\label{alphau-k1}$$ This upper stability limit is shown in Fig. \[tau2-ep\] for $\tau \leq \tau_2 \leq 2\tau$.
We can also obtain $\tau_{2max}^{'}$, the maximum value of $\tau_{2}^{'}$ for which synchronization is possible. This happens when there is always an eigenvalue with magnitude greater than one, i.e. when $\alpha_l = \alpha_u =1$. By putting $\alpha_u=\alpha_l=1$ in Eq. (\[eq-alpha-1\]), we get $a_1 + a_2 = 1$. However a better condition is obtained if we note that for $\alpha_u=\alpha_l=1$, Eq. (\[cubic-1\]) has two degenerate solutions $\mu=1$, i.e. Eq. (\[cubic-1\]) has a factor $\mu^2 - 2 \mu +1$. This gives the conditions $b_0 = 2- c$ and $1-b_1 = 2(2-c)$. First condition gives $$\begin{aligned}
\lambda \tau_{2max} & = & \lambda \tau + \lambda \tau_{2max}^{'}
\nonumber \\
& = & \lambda \tau - \ln 2 - \ln(1-e^{-\lambda \tau})
\label{tau2max-1}\end{aligned}$$
### General $k$
For a general $k$, getting explicit solutions for $\alpha_u$ is not easy. But it is possible to get an expression for $\tau_{2max}^{'}$. We use the condition that for $\alpha_u=\alpha_l=1$, Eq. (\[kth\]) has two degenerate solutions $\mu=1$. This gives the condition $b_0 = 1+k- kc$. Simplifying, we get $$\begin{aligned}
\lambda \tau_{2max} & = & k \lambda \tau + \lambda \tau_{2max}^{'}
\nonumber \\
& = & k \lambda \tau - \ln(k+1) - \ln(1-e^{-\lambda \tau})
\label{tau2max-k}\end{aligned}$$ For $k=0$, this equation reduces to Eq. (\[tau2max-0\]) and for $k=1$ it reduces to Eq. (\[tau2max-1\]). Fig. \[tau-tau2max\] shows the plot of $\lambda \tau_{2max}$ as a function of $\tau$ for different $k$ values. For each $k$ the plot is for the range $(\tau_k,\tau_{k+1})$ where $\tau_k$ is defined by $k \tau_k = \tau_{2max}$ and from Eq. (\[tau2max-k\]) we get $\tau_k = \ln((k+1)/k)$.
![\[tau-tau2max\]The figure plots the maximum $\tau_{2max}$ as a function of $\lambda \tau$ (solid line). Here, $\tau_{2max}, \; k=0,1,2,\ldots$ is given by Eq. (\[tau2max-k\])) and for each $k$ the range of $\tau$ is $(\tau_k,\tau_{k+1})$. The dashed curve passes through the values $\tau_{2max} = k\tau_k = k\ln((k+1)/k)$ (Eq. (\[tau2max-av\])). The inset shows the same plot with $\lambda \tau$ range (0,5). ](tau-tau2max.eps){width="0.9\columnwidth"}
### $\tau_2 = n\tau$
In this case it is not possible to obtain the stability range for the synchronized state in terms of $\alpha$. However, it is possible to obtain an explicit expression for $\tau_{2max}$ as (The detailed calculations are given in Appendix \[AppB\].) $$\lambda \tau_{2max} = \lambda \tau / ( exp(\lambda \tau) - 1)
\label{tau2max-av}$$ The dashed line in Fig. (\[tau-tau2max\]) corresponds to Eq. (\[tau2max-av\]). It gives the correct $\tau_{2max}$ only for $\tau_2 = n\tau$.
\[sec3\]NUMERICAL ANALYSIS
==========================
We choose two standard systems, Rössler and Lorenz, to confirm the main results obtained in the previous section.
In the case of Rössler system given in Eq. (\[rossler\]), two identical systems are coupled in the drive response mode via the coupling scheme explained in Eq. (\[system\_def\]). Starting from random initial conditions and choosing the system parameters in the chaotic region ($a= 0.15, b=0.2$ and $c= 10.0$), they are evolved for 200000 units with a time step of 0.01. The correlation coefficient $C=<y_1(t)x_1(t+\tau_2)>/\sqrt{<x_1^2(t)><y_1^2(t)>}$ between $x_1(t)$ and $y_1(t)$ shifted by the effective $\tau_2 = | \tau_2-\tau_1|$ (hereafter referred to as $\tau_2$ itself) is calculated using the last 5000 values. The region of stability of the synchronized state is isolated as the region where $C= 0.99$ and boundaries of stability fixed when $C$ goes below this value.
Taking $\tau = 0.5 $, $\tau_2 $ is varied from 0 to 1.0 units in steps of 0.01. For each value of $\tau_2$, the coupling strength $\epsilon $ is increased in steps of 0.005. The appropriately shifted correlation coefficient is calculated and using the criterion mentioned above the lower and upper limits of stability are found out. The results are plotted in the parameter plane $\tau_2-\epsilon$ in Fig. \[tau2-c-ros\]. The overall behavior agrees with the theoretical analysis carried out in the previous section. The upper limits obtained by the stability analysis given in Eq. (\[alphau1\]), Eq. (\[alphau2\]) and Eq. (\[alphau-k1\]) for the different relative ranges of $\tau_2$ are calculated for a typical value of $\lambda=0.65$ and shown as solid line. For values of $\tau_2<0.3$ or $\epsilon>3.0$, the agreement is good although for lower values, there is deviation. For lower values of $\epsilon$ we need larger values of $\lambda$ to obtain a better fit (not shown in the figure).
![\[tau2-c-ros\]The limits of stability of the synchronized state of two chaotic Rössler systems in the parameter plane $\tau_2-\epsilon$. The solid line is the limits obtained from the stability analysis for $\lambda=0.65$. The agreement is good for values of $\epsilon > 3.0$. ](tau2-c-ros.eps){width="0.9\columnwidth"}
By fixing the coupling $\epsilon =0.8$, we vary the reset time $\tau$ in the range (0,2.0) in steps of .01 and in each case the maximum value of $\tau_2$ for stability of synchronization is calculated using the same criterion. The results are shown in Fig. \[t2m-t-ros\]. The $\tau_{2max}$ obtained from theory and shown in Fig. \[tau-tau2max\] are reproduced here for comparison. The numerical values are found to support the results of the theoretical analysis very well. We note that here the $\lambda$ dependence cancels out and hence the agreement with the theory is much better than that for the $\tau_2-\epsilon$ plots.
![\[t2m-t-ros\]The maximum values of $\tau_2$ for which anticipatory synchronization is stable in two coupled Rössler systems is shown as a function of the rest time $\tau$. The solid and dotted lines are the similar values from theory reproduced from Fig. \[tau-tau2max\] for comparison. ](t2m-t-ros.eps){width="0.9\columnwidth"}
We consider next two Lorenz systems, where the x-system is given by
$$\begin{aligned}
\dot{x_1} & = & a (x_2- x_1) \nonumber \\
\dot{x_2} & = & c x_1 -x_2-x_1 x_3
\label{lorenz} \\
\dot{x_3} & = & -b x_3 + x_1 x_2 \nonumber\end{aligned}$$
This is coupled to an identical y-system using the same scheme. Choosing parameter values for chaotic Lorenz as $a= 10.0, b= 8/3$ and $c= 28.0$, the analysis is repeated as in the case of Rössler. Here the time step chosen is 0.001 and $\tau=0.05$. The $\tau_2$ values are varied in the range (0,0.1) and the stability limits of $\epsilon$ isolated. The results are given in Fig. \[tau2-c-lor\]. The general behavior agrees with the theory in this case also. However, the nearest fit (shown in solid line ) is obtained for $\lambda=0.0$.
![\[tau2-c-lor\]The maximum values of coupling $\epsilon $ for two coupled Lorenz systems as a function of the anticipatory time $\tau_2$. The solid curve is the limiting curve from theory same as in Fig. \[tau2-ep\] for a value of $\lambda=0.0$. ](tau2-c-lor.eps){width="0.9\columnwidth"}
\[sec4\]CONCLUSION
==================
We introduce a new coupling scheme with varying time delay for synchronization of two systems with delay or anticipation. The scheme has the advantage that synchronization can be achieved with intermittent information from the driver in intervals of reset that can be pre-fixed. This also makes a detailed stability analysis analytically possible because the error dynamics becomes discrete. By assuming an average effective Lyapunov exponent $\lambda$, the stability regions and limits of stability in the parameters of coupling strength and anticipation time are worked out for specific cases. We demonstrate the method by numerical simulations in two standard systems, Rössler and Lorenz. The general features of the stability region in parameter space match with the theoretical stability analysis, but more precise matching with the numerical data is not possible. This is understandable since in the analytical calculations $f^{'}$ is replaced by an effective $\lambda$ and also coupling in all components of $x$ and $y$ is assumed while in numerical calculations only one component is coupled. The agreement between the theory and numerical data is reasonably good for the $\tau_{2max} - \tau$ plot, since the $\lambda$ dependence cancels out.
The availability of three new time scales in the dynamics is suggestive of potent applications especially in secure communication. We propose that this technique will be especially successful with a bichannel transmission [@bocc] where one channel, that is part of the state space of the chaotic transmitter (driver), is used to synchronize with the receiver (response) and the other forms the message along with the chaotic signal from a different part of the state space of the driver. Here since the encrypted information or cipher text is not used as the synchronizing signal, it can be made really complex and secure. In this context our method of synchronization has the definite advantage that the synchronization channel need be transmitted only at intervals fixed by the reset time which itself forms part of the key space. This leads to bandwidth savings and requirement of noise free channel for short times at intervals. Moreover, the enhancement in the dimensionality of the key space leads to increase in security. The stability analysis reported in this paper along with the numerical simulations for standard systems helps to fix the accessible regions of the key space for better key management. This is being worked out and will be published elsewhere.
One of the authors, GA, acknowledges the hospitality and facilities at Physical Research Laboratory, Ahmedabad during the visit under associateship.
\[AppA\]Case $0 \leq \tau_2 \leq \tau$
======================================
Here we analyze the eigenvalues $\mu_{\pm}$ of the map (\[2dmap\]) given by Eq. (\[ev-a\]) to obtain the stability conditions. These stability conditions are shown in Fig. \[tau2-ep\]. The synchronized state, $\Delta=0$, is stable if both the eigenvalues satisfy $|\mu_{\pm}| < 1$.
$\tau_2=0$
----------
For this case $b=0$. Hence, 2-d map in Eq. (\[2dmap\]) becomes a 1-d map given by $$\Delta_{m+1} = \mu \Delta_m
\label{1dmap}$$ where $\mu =\alpha \left( 1 - (1-\frac{1}{\alpha}) e^{\lambda \tau} \right)$. Fig. \[mu-alpha\]a shows $\mu$ as a function of $1/\alpha$. The fixed point $\Delta=0$ is stable provided $|\mu|<1$. This gives the following limits on $\alpha$ for the stability of the synchronized state. $$1 < \alpha < \frac{1+e^{-\lambda \tau}}{1-e^{-\lambda \tau}}
\label{alpha-0}$$
![\[mu-alpha\]This figure shows the eigenvalues $\mu$ as a function of $1/\alpha$. (a) $\lambda \tau = 0.25$ and $\lambda \tau_2 =0$. Here, $\mu = \alpha \left( 1 - (1-\frac{1}{\alpha}) e^{\lambda \tau} \right)$ (see Eq. (\[1dmap\])). (b) $\lambda \tau = 0.25$ and $\lambda \tau_2 =0.02$. The largest $\mu$ (solid line) starts from a value greater than one for $1/\alpha >1$, crosses 1 at $1/\alpha=1$, and continues till it meets the dashed line from below (Here $a^2-4b=0$). Then $\mu$ becomes complex and the dotted line shows the magnitude $|\mu|$. This continues till we have $a^2-4b=0$ again. This point is just above the meeting point of solid and dashed lines on the negative side. For smaller values of $1/\alpha$, $\mu$ again become real (but now negative) and the largest $\mu$ in magnitude jumps to the dashed line bellow. Hence the stability range is from $\alpha=1$ till the point where $\mu=-1$. (c) $\lambda \tau = 0.25$ and $\lambda \tau_2 =0.1$. This figure is similar to (b), but here the dotted line ($\mu$ complex) crosses the magnitude one before jumping to the negative value. Hence the stability range is now from $\alpha=1$ till the point where the dotted line crosses one or the complex $\mu$ has magnitude one. Crossover from the behavior (b) to (c) occurs at the peak value, $\alpha_p$, as seen in Figure \[tau2-ep\]. (d) $\lambda \tau = 1.0$ and $\lambda \tau_2 =0.7$. Here, $\tau_2 > \tau_{2max}=0.458\ldots$. Hence, the largest $|\mu|$ is always greater than one. ](mu-alpha.eps){width="0.9\columnwidth"}
$0<\tau_2 <\tau$
----------------
For $0<\tau_2<\tau$, the eigenvalues $\mu_{\pm}$ (Eq. (\[ev-a\])) display a rich behavior. Three different scenarios are possible. These are shown in Figs. \[mu-alpha\]b, \[mu-alpha\]c and \[mu-alpha\]d which show $\mu$ as a function of $1/\alpha$. To determine the limits of stability of the solution $\Delta=0$ we consider the following case.
### $\mu = 1$ ($a^2-4b >0, a>0$)
Putting $\mu = 1$ in Eq. (\[ev-a\]), we get $2 = a \pm \sqrt{a^2-4b}$. This reduces to $$1 = a-b$$ Using the expressions (\[ab\]) for $a$ and $b$, we get the lower limit on the stability as $$\alpha_l =1.$$
### $\mu=-1$ ($a^2-4b >0, a <0$)
Putting $\mu=-1$ in Eq. (\[ev-a\]), we get $-2 = a \pm \sqrt{a^2-4b}$ which reduces to $$1+a+b=0$$ Using the expressions (\[ab\]), we get $$\alpha_u = \frac{e^{\lambda \tau}+1}{2e^{\lambda (\tau- \tau_2)} - e^{\lambda \tau} - 1}
\label{alpha-case2}$$ The above expression gives the upper limit of stability for smaller values of $\tau_2$. For larger values of $\tau_2$, we use the condition $|\mu|=1$ which is considered in the next subsection.
### $|\mu|=1$ ($a^2-4b <0$, $\mu$ complex)
Putting $|\mu|=1$ ($\mu$ complex) in Eq. (\[ev-a\]), we get $1 = \frac{1}{2} \sqrt{a^2-(a^2-4b)}$ which reduces to $$b=1$$ Substituting from Eq. (\[b\]), we get the upper limit on the stability as $$\alpha_u = \frac{e^{-\lambda \tau}}{1-e^{-\lambda \tau_2}}
\label{alpha-case3}$$ The above expression can also be used to determine the maximum $\tau_{2max}$ for a given $\tau$. This happens when there is always an eigenvalue with magnitude greater than one, i.e. when $\alpha_l = \alpha_u =1$. From Eq. (\[alpha-case3\]) we get the following expression. $$\lambda \tau_{2max} = -\ln (1-e^{-\lambda \tau}).
\label{tau2max-0}$$ Note that $\alpha_u=1$ in Eq. (\[alpha-case2\]) gives the same $\tau_{2max}$ as in Eq. (\[tau2max-0\]). Fig. (\[mu-alpha\]d) shows $\mu$ as a function of $1/\alpha$ for $\tau_2 > \tau_{2max}$ where the synchronized state is not stable.
### Peak
The peak value $\alpha_p$ is given by the intersection of Eqs. (\[alpha-case2\]) and (\[alpha-case3\]) and leads to the conditions $$b=1, \; \; \textrm{and} \; \; a+2=0.$$ From $b=1$, we have $$e^{-\lambda \tau_2} = 1 - \frac{1}{\alpha} e^{-\lambda \tau}
\label{peak}$$ Substituting this in $a+2=0$, we get $$\alpha_p = \frac{e^{\lambda \tau}+3}{e^{\lambda \tau} - 1}
\label{peak-alpha}$$ The corresponding $\tau_2$ value is given by, $$\lambda \tau_{2p} = \lambda \tau + \ln(e^{\lambda \tau}+3) - 2\ln(e^{\lambda \tau}+1)
\label{peak-tau2}$$ Eliminating $\tau$ from Eqs. (\[peak-alpha\]) and (\[peak-tau2\]) gives Eq. (\[tau2p-alphap\]).
$\tau_2 = \tau$
---------------
This is a simple case where $a$ and $b$ in Eqs. (\[ab\]) reduce to $a = c = e^{\lambda \tau}, \; \;
b = d = \alpha (e^{\lambda \tau}- 1)$ The cases Appendix $A2a$ and $A2c$ in the above subsection are applicable and hence the stability condition for $\Delta=0$ is $$1 < \alpha < \frac{1}{e^{\lambda \tau} - 1}$$
\[AppB\]Case $\tau_2 = n \tau$
==============================
This corresponds to the case $\tau_2^{'}=0$ in Section IIIB. Using Eq. (\[rec-tau2’\]), we get the following recursion relation (note that n=k+1 gives the correct correspondence)
$$\begin{aligned}
\Delta_{m+n+1} & = & e^{\lambda \tau} \Delta_{m+n} - \alpha (e^{\lambda \tau} -1) \Delta_m \\
& = & c \Delta_{m+n} - d \Delta_m
\label{rec-ntau} \end{aligned}$$
where $c=e^{\lambda \tau}$ and $d=\alpha (e^{\lambda \tau} -1)$. This leads to an $(n+1)$ dimensional map. This map can also be obtained directly from the solution (\[sol-Cm\]) noting that for $t=m\tau$ and $t=(m+1)\tau$ we get $\Delta_{m+n}$ and $\Delta_{m+n+1}$ respectively. In matrix form $$\left( \begin{array}{c}
\Delta_{n+1} \\ \Delta_n \\ \Delta_{n-1} \\ \vdots \\ \Delta_1
\end{array} \right) =
\left( \begin{array}{ccccc}
c & 0 & \cdots & 0 & - d \\
1 & 0 & \cdots & 0 & 0 \\
0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & 0
\end{array} \right)
\left( \begin{array}{c}
\Delta_n \\ \Delta_{n-1} \\ \Delta_{n-2} \\ \vdots \\ \Delta_0
\end{array} \right)
\label{ndmap}$$ The eigenvalue equation is $$\begin{aligned}
\mu^{n+1} - c \mu^n +d & =& 0,
\label{mu-n}\end{aligned}$$ where $n\geq 1$.
The following general conclusions can be arrived at using Geršgorin discs. There is one disc with center at $c$ and radius $d$ and $n$ discs with center at $0$ and radius $1$. All the eigenvalues lie within these discs. For $\alpha <1$, $c>d+1$. Hence, the disc with center at $c$ is disjoint from the other discs. Thus one root which lies in this disc, must always have magnitude greater than one. Hence, the lower limit of stability is $\alpha_l=1$.
For $n=1$, Eq. (\[mu-n\]) becomes a quadratic equation. This is discussed in Appendix A3.
$n=2$
-----
Eq. (\[mu-n\]) becomes a cubic equation. $$\mu^3 - c \mu^2 +d =0
\label{mu-3}$$ At the upper stability limit, $\mu$ is complex with $|\mu|=1$. Thus, $\mu^2 - 2 \cos(\theta) \mu +1$ is a factor where $\mu = e^{\pm i \theta}$. Using this condition we get the relation $$d^2 + cd = 1$$ This gives a quadratic equation in $\alpha$. $$(e^{\lambda \tau} - 1)^2 \alpha^2 + e^{\lambda \tau}(e^{\lambda \tau} - 1)
\alpha -1 = 0.$$ Using the correct solution the stability range is $$1 < \alpha < \frac{1}{2(e^{\lambda \tau} - 1)}\left(\sqrt{e^{2\lambda \tau}+4} - e^{\lambda \tau}\right)$$
The maximum $\tau_{2max}$ is obtained in the next subsection.
Any $n$
-------
For a general $n$ it is not possible to obtain the stability range for the synchronized solution.
It is easy to see that the maximum $\tau_{2max}$ is obtained if there are two degenerate eigenvalues of Eq. (\[mu-n\]) equal to one at $\alpha=1$. This is possible if $c = 1/n$, $d = 1+1/n$. Using the explicit form of $c$ or $d$ and $n=\tau_{2max}/\tau$, we get Eq. (\[tau2max-av\]) for $\tau_{2max}$.
\[AppC\] Negative $\lambda$
===========================
If Lyapunov exponent $\lambda$ is negative, then Eq. (\[linear\_problem\]) can be written as $$\dot{\Delta} = -|\lambda| \Delta - \epsilon \sum_{m=0}^{\infty} \chi_{(m\tau,(m+1)\tau)} \Delta_m
\label{linear_problem_neg}$$ The analysis is similar to that for positive $\lambda$. Here, we summarize the results.
$0 \leq \tau_{20} \leq \tau$
----------------------------
For $0 \leq \tau_{20} \leq \tau$, Eq. (\[linear\_problem\_neg\]) leads to the recursion relation (see Eq. (\[rec-tau2\])) $$\begin{aligned}
\Delta_{m+1}
& = & a \Delta_m - b \Delta_{m-1}
\label{rec-tau2-b-neg}\end{aligned}$$ where
\[ab-neg\] $$\begin{aligned}
a & = & -\alpha (e^{|\lambda|(\tau_2-\tau)}-1) + e^{-|\lambda| \tau},
\label{a-neg} \\
b & = & \alpha e^{-|\lambda| \tau}(e^{|\lambda| \tau_2}-1)
\label{b-neg}\end{aligned}$$
where we define $\alpha = \epsilon/|\lambda|$ as the normalized dimensionless coupling constant.
### $\tau_2=0$
For $\tau_2=0$, $b=0$. The stability limits for the synchronized state are (see Eq. (\[alpha-0\])) $$-1 < \alpha < \frac{1+e^{-|\lambda| \tau}}{1-e^{-|\lambda| \tau}}
\label{alpha-0-neg}$$
### $0 < \tau_2 \leq \tau$
In this case Eq (\[rec-tau2-b-neg\]) leads to a 2-d map as for the positive $\lambda$. The eigenvalue equation and the solutions are same as Eqs. (\[eq-ev-2d\]) and (\[ev-a\]) with $a$ and $b$ defined by Eqs. (\[ab-neg\]).
The lower stability limit is always $\alpha_l=-1$. For smaller values of $\tau_2$ ($\leq \tau_{2p}$), the upper limit of stability is given by (see Eq. (\[alphau1\])) $$\alpha_u = \frac{1+e^{-|\lambda| \tau}}{1+e^{-|\lambda| \tau}-2e^{|\lambda| (\tau_2- \tau)}}
\label{alphau1-neg}$$ while for larger values of $\tau_2$ ($\tau_{2p} \leq \tau_2 \leq \tau$) it is given by (Eq. (\[alphau2\])) $$\alpha_u = \frac{e^{|\lambda| \tau}}{e^{|\lambda| \tau_2}-1}
\label{alphau2-neg}$$ It is interesting to note that for very large values of the coupling constant the synchronized state is unstable. The maximum value of $\alpha_p$ is given by the intersection of the two curves (\[alphau1-neg\]) and (\[alphau2-neg\]). $$\alpha_p = \frac{3 e^{\lambda \tau}+1}{e^{\lambda \tau} - 1}
\label{alphap-neg}$$ The corresponding $\tau_{2p}$ value is determined by the relation $$|\lambda| \tau_{2p} = |\lambda| \tau + 2 \ln(1+e^{-|\lambda| \tau}) - \ln(3+e^{-|\lambda| \tau})
\label{tau2p-neg}$$
![\[tau2-ep-neg\]The stability region of the synchronized state in the $|\lambda| \tau_2 - \alpha$ plane. The solid line is for $\lambda \tau = - 0.25<0$ and the dashed line is for $\lambda \tau = -0.5$. The lower limit of stability is $\alpha_l=-1$ (dotted line). For smaller values of $\tau_2 \leq \tau_{2p}$, the upper limit of stability is given by (Eq. (\[alphau1-neg\])) while for larger values of $\tau_2$ ($\tau_{2p} \leq \tau_2 \leq \tau$) it is given by (Eq. (\[alphau2-neg\])). The peak values are ($0.072\ldots/0.25\ldots=0.29 ,17.083\ldots$) for $\lambda\tau=-0.25$ and ($0.165\ldots/0.5=0.33\ldots, 9.16\ldots$)for $\lambda\tau=-0.5$ (see Eqs. (\[alphap-neg\]) and (\[tau2p-neg\])). For $\tau \leq \tau_2 \leq 2 \tau$ the upper limit of stability is given by Eq. (\[alphau-k1-neg\]). We note that the stability limits have a similar behavior to that of Fig. \[tau2-ep\] for positive $\lambda$. ](tau2-ep-neg.eps){width="0.9\columnwidth"}
### $\tau_2 = \tau$
For $\tau_2 = \tau$, the stability range is $$-1 < \alpha < \frac{1}{1-e^{-|\lambda| \tau}}$$
$\tau_2 > \tau$
---------------
Let $\tau_2 = k\tau + \tau_2^{'}, \; k = 0,1,\ldots$ where $\tau_2^{'} < \tau$ as for the case of positive $\lambda$. Eq. (\[linear\_problem\_neg\]) leads to a map of dimension $k+2$. The eigenvalue equation is $$\mu^{k+2} - c \mu^{k+1} + b_1 \mu + b_0 = 0
\label{kth-neg}$$ where $c=e^{-|\lambda| \tau}$, $b_1=\alpha (1-e^{|\lambda|(\tau_2^{'}-\tau)})$ and $b_0=\alpha e^{-|\lambda| \tau}(e^{|\lambda| \tau_2^{'}}-1)$. For $k=0$, we recover the case $0 < \tau_2 \leq \tau$.
For $k=1, \; {\textrm i.e.} \; \tau \leq \tau_2 \leq 2\tau$, we have a 3d-map. The lower stability limit is $\alpha_l=-1$. The upper stability limit is $$\alpha_u = \frac{1}{2 a_2}(a_1 + \sqrt{a_1^2 +4 a_2})
\label{alphau-k1-neg}$$ where $a_1 = e^{-2 |\lambda| \tau} ( 1 - e^{|\lambda| \tau_2^{'}}) + e^{|\lambda|(\tau_2^{'}-\tau)} -1$ and $a_2 = e^{-2 |\lambda| \tau}(e^{|\lambda| \tau_2^{'}}-1)^2$. The stability limits are plotted in Fig. \[tau2-ep-neg\].
We have done numerical analysis for negative $\lambda$ using two Rössler systems in the periodic region for $c= 2.2$. The stability limits for synchronization in the $\tau_2-\epsilon$ plane in this case is given in Fig. \[tau2-c-ros-p\]. The solid line is the curve from theory with $\lambda=0.0$ . The behavior of the numerical results in general agrees with the theoretical analysis. However, exact fit is not obtained for any negative $\lambda$. Surprisingly, the fit is better for positive $\lambda$ with equations Eq. (\[alphau1\]) and Eq. (\[alphau2\]) (dotted line). The reason for this behavior is not clear.
![\[tau2-c-ros-p\]The maximum values of coupling $\epsilon $ for two coupled Rössler systems in the periodic region. The solid curve is for the values from theory reproduced from Fig. \[tau2-ep-neg\] for a value of $\lambda=0.0$. The agreement is better with the curves in Fig. \[tau2-ep\] (dotted line) for a value of $\lambda=0.6$. ](tau2-c-ros-p.eps){width="0.9\columnwidth"}
$\tau_{2max}$
-------------
The condition for obtaining the maximum value $\tau_{2max}$ is that $\alpha_l=\alpha_u$. For negative $\lambda$, we have $\alpha_l=-1$ and $\alpha_u$ always remains positive. Hence, unlike the case of positive $\lambda$, the condition for obtaining $\tau_{2max}$ is never satisfied and synchronized state is possible for any $\tau_2$ or $\tau_{2max}$ is infinite.
$\tau_2 = n\tau$
----------------
This corresponds to the case $\tau_2^{'}=0$ of Appendix C2. The eigenvalue equation is (see Eq. (\[mu-n\])) $$\begin{aligned}
\mu^{n+1} - c \mu^n +d & =& 0,
\label{mu-n-neg}\end{aligned}$$ where $n\geq 1$ and $c=e^{-|\lambda| \tau}$ and $d=\alpha (1-e^{-|\lambda| \tau})$.
The following general conclusions can be arrived at using the Geršgorin discs. There is one disc with center at $c$ and radius $|d|$ and $n$ discs with center at $0$ and radius $1$. All the eigenvalues lie within these discs. For $\alpha <1$, $d<(1-c)$. Since $c<1$, the disc with center at $c$ lies within the circle $|\mu|=1$. Hence, all the roots of Eq. (\[mu-n-neg\]) have magnitude less than one and the synchronized state is stable. Thus for any $n$ there will be range of $\alpha$ values for which the synchronized state is stable. This supports the conclusion reached in the previous subsection (Appendix C3) that $\tau_{2max}$ is infinite.
For $n=1$, Eq. (\[mu-n-neg\]) becomes a quadratic equation. This is discussed in Appendix C1.
For $n=2$ we have a cubic equation. The stability range is $$-1 < \alpha < \frac{1}{2(1-e^{-|\lambda| \tau})}\left(\sqrt{e^{-2|\lambda| \tau}+4} - e^{-|\lambda| \tau}\right)$$
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|
---
abstract: 'We prove that episturmian words and Arnoux-Rauzy sequences can be characterized using a local balance property. We also give a new characterization of epistandard words.'
author:
- Gwénaël Richomme
title: A local balance property of episturmian words
---
LaRIA : Laboratoire de Recherche en Informatique d’Amiens\
Université de Picardie Jules Verne – CNRS FRE 2733\
33, rue Saint Leu, 80039 Amiens cedex 01, France\
Tel : (+33)\[0\]3 22 82 88 77\
Fax : (+33)\[0\]03 22 82 54 12\
------------------------------------------------------------------------
depth 1pt
$^a$ LaRIA, Université de Picardie Jules Verne, gwenael.richomme@u-picardie.fr\
**Keywords**: Arnoux-Rauzy sequences, episturmian words, balance property.
**Important remark:** The first version of this LaRIA Research Report 2007-02 contained an error in the proof of a result stating the non-context-freeness of the complement of the set of finite episturmian word. This result, its proof and some comments about it have been removed in this second version of the report. A slightly revised version of the text of this second version was published in ”T. Harju, J. Karhumäki, and A. Lepistö (Eds.), Proceedings of DLT 2007, LNCS 4588, pp 371-381, Springer-Verlag Berlin Heidelberg 2007.” (Thanks to the referees to have seen the above mentioned error)
Introduction
============
M. Morse and G.A. Hedlund [@MorseHedlund1940] were the firsts to study in depth a family of words called Sturmian words. Now a large litterature exists on these words for which have been proved numerous characterizations more fascinating the ones than the others (see for instance [@AlloucheShallit2003; @BerstelSeebold2002; @Pytheas2002]).
Sturmian words are defined over a binary alphabet. From their various characteristic properties, some generalizations of Sturmian words have emerged over larger alphabets. One of them, the so-called Arnoux-Rauzy sequences, is based on the notion of complexity of a word and is interesting by its geometrical, arithmetic, ergodic and combinatorial aspects (see for instance [@Pytheas2002]).
One of the first properties of Sturmian words stated by M. Morse and G.A. Hedlund [@MorseHedlund1940] is the balance property: any infinite word $w$ over the alphabet $\{a, b\}$ is Sturmian if and only if it is non-ultimately periodic and balanced, that is the number of occurrences of the letter $a$ differs in two factors of same length of $w$ by at most one. Generalizations of these words were studied for instance by P. Hubert [@Hubert2000] (see also [@Vuillon2003] for a survey of this property). J. Justin and L. Vuillon have stated a non-characteristic kind of balance property [@JustinVuillon2000] for Arnoux-Rauzy sequences. Although it was first conjectured that Arnoux-Rauzy sequences are balanced [@DroubayJustinPirillo2001], J. Cassaigne, S. Ferenczi and L.Q. Zamboni have proved that this does not necessarily hold [@CassaigneFerencziZamboni2000].
In 1973, E.M. Coven and G.A. Hedlund [@CovenHedlund1973] stated that a word $w$ over $\{a, b\}$ is not balanced if and only if there exists a palindrome $t$ such that $ata$ and $btb$ are both factors of $w$. This could be seen as a local balance property of Sturmian words since to check the balance property we do not have to compare all factors of the same length but only factors on the sets $AtA$ for $t$ factors of $w$. The previous property can be rephrased in: an infinite word $w$ over the alphabet $A = \{a, b\}$ is Sturmian if and only if it is non-ultimately periodic and for any factor $t$ of $w$, the set of factors belonging to $AtA$ is a subset of $atA \cup Ata$ or a subset of $btA \cup Atb$. In Section \[secLocalProperty\], we show that this result can be generalized to Arnoux-Rauzy sequences contrarily to the balance property.
Actually our result concerns a larger family of infinite words presented in Section \[secEpisturmian\]. Based on ideas of A. de Luca [@deLuca1997], Episturmian words were proposed by X. Droubay, J. Justin and G. Pirillo [@DroubayJustinPirillo2001] as a generalization of Sturmian words. They have observed that Arnoux-Rauzy words are special episturmian words they called strict episturmian words. In the binary case episturmian words are the Sturmian words and the balanced periodic infinite words. Let us note that the case of remaining balanced words, namely the skew ones, have recently been generalized [@Glen2006; @GlenJustinPirillo2006].
In [@DroubayJustinPirillo2001], episturmian words are defined as an extension to standard episturmian words (Here we will call [*epistandard*]{} these standard episturmian words) previously introduced as a generalization of standard Sturmian words. In Section \[secEpistandard\], we generalize to epistandard words a characterization of standard words proving a converse of a theorem in [@JustinPirillo2002] and stating that an infinite word $w$ is epistandard if and only if there exists at least two letters such that $aw$ and $bw$ are both episturmian. The interested reader can also consult [@GlenJustinPirillo2006] and its references for other characterizations of episturmian words using left extension in the context of an ordered alphabet.
Our last section comes back to the generalization of the local balance property introduced by E.M. Coven and G.A. Hedlund. One another way to rephrase it is: an infinite word $w$ over the alphabet $A = \{a, b\}$ is Sturmian if and only if it is non-ultimately periodic and for any factor $t$ of $w$, the set of factors belonging to $AtA$ is balanced. This yields a new family of words on which we give partial results.
\[secEpisturmian\]Episturmian and epistandard words
===================================================
Even if we assume the reader is familiar with combinatorics on words (see, e.g., [@Lothaire1983]), we precise our notations. Given an alphabet A (a finite non-empty set of letters), $A^*$ is the set of finite words over $A$ including the empty word $\varepsilon$. The length of a word $w$ is denoted by $|w|$ and the number of occurrences of a letter $a$ in $w$ is denoted by $|w|_a$. The [*mirror image*]{} of a finite word $w = w_1 \ldots w_n$ ($w_i \in A$, for $i = 1,
\ldots, n$) is the word $w_n \ldots w_1$ (the mirror image of $\varepsilon$ is $\varepsilon$ itself). A word equals to its mirror image is a [*palindrome*]{}. A word $u$ is a *factor* of $w$ if there exist words $p$ and $s$ such that $w = pus$. If $p =
\varepsilon$ (resp. $s = \varepsilon$), $u$ is a *prefix* (resp. *suffix*) of $w$. A word $u$ is a [*left special*]{} (resp. [*right special*]{}) factor of $w$ if there exist (at least) two different letters $a$ and $b$ such that $au$ and $bu$ (resp. $ua$ and $ub$) are factors of $w$. A *bispecial factor* is any word which is both a left and a right special factor (see, e.g., [@Cassaigne1997]) for more informations on special factors). The set of factors of a word $w$ will be denoted $Fact(w)$.
The previous notions can be extended in a natural way to any infinite words. Moreover any *ultimately periodic* infinite word can be written $uv^\omega$ for two finite words $u, v$ ($v \neq
\varepsilon)$: it is then the infinite word obtained concatenating infinitely often $v$ to $u$. If $u = \varepsilon$, the word is said *periodic*.
A word $w$ is *episturmian* if and only if its set of factors is closed by mirror image and $w$ contains at most one left (or equivalently right) special factor of each length. A word $w$ is *epistandard Sturmian* or *epistandard*, if $w$ is episturmian and all its left special factors are prefixes of $w$. Let us note that, in [@DroubayJustinPirillo2001], epistandard words were introduced by several equivalent ways, and then episturmian words were defined as words having same set of factors than an epistandard one.
The two theorems below recall a very useful property of episturmian words which is the possibility to decompose infinitely an episturmian word using some morphisms. This property already seen for Arnoux-Rauzy sequences in [@ArnouxRauzy1991] is related to the notion of S-adic dynamical system (see, e.g. [@Pytheas2002] for more details). This property could be useful to get information on the structure of episturmian words (see for instance [@BertheHoltonZamboni2006; @LeveRichomme2004; @LeveRichomme2006; @Richomme2006] for some uses in the binary cases).
Given an alphabet $A$, a *morphism* $f$ on $A$ is an application from $A^*$ to $A^*$ such that $f(uv) = f(u) f(v)$ for any words $u$, $v$ over $A$. A morphism on $A$ is entirely defined by the images of elements of $A$.
*Episturmian morphisms* studied in [@JustinPirillo2002; @Richomme2003] are the morphisms defined by composition of the permutation morphisms and the morphisms $L_a$ and $R_a$ defined, for $a$ a letter, by $$L\_a {
--------------------------------
$a \mapsto a$
$b \mapsto ab$, if $b \neq a$,
--------------------------------
. R\_a {
--------------------------------
$a \mapsto a$
$b \mapsto ba$, if $b \neq a$.
--------------------------------
. $$
[[@JustinPirillo2002]]{} \[JP1\] An infinite word $w$ is epistandard if and only if there exist an infinite sequence of infinite words $(w^{(n)})_{n \geq 0}$ and an infinite sequence of letters $(x_n)_{n \geq 1}$ such that $w^{(0)} = w$ and for all $n \geq 1$, $w^{(n-1)} = L_{x_n}(w^{(n)})$.
[[@JustinPirillo2002]]{} \[JP2\] An infinite word $w$ is episturmian if and only if there exist an infinite sequence of [*recurrent*]{} infinite words $(w^{(n)})_{n \geq 0}$ and an infinite sequence of letters $(x_n)_{n \geq 1}$ such that $w^{(0)} = w$ and for all $n \geq 0$, $w^{(n-1)} = L_{x_n}(w^{(n)})$ or $w^{(n-1)} = R_{x_n}(w^{(n)})$.
Moreover, $w$ has the same set of factors than the epistandard word directed by $(x_n)_{n \geq 1}$.
The infinite sequence $(x_n)_{n \geq 1}$ which appears in the two previous theorem is called the *directive word* of $w$ and is denoted $\Delta(w)$: Actually in terms of [@JustinPirillo2002], it is the directive word of the epistandard word having the same set of factors than $w$. Each episturmian word has a unique directive word.
It is worth noting that any episturmian word is *recurrent*, that is, each factor of $w$ occurs infinitely often. An infinite word $w$ is recurrent if and only if each factor of $w$ occurs at least twice. Equivalently each factor of $w$ occurs at a non-prefix position. Thus an infinite word $w$ over an alphabet $A$ is recurrent if and only if for each of its factors $u$ the set $AuA$ (or simply $Au$) is not empty.
We denote as in [@JustinPirillo2002] $Ult(w)$ the set of letters occurring infinitely often in $\Delta(w)$. For $B$ a subset of the alphabet, we introduce a new definition: we call *ultimately $B$-strict episturmian* any episturmian word $w$ for which $Ult(\Delta(w)) = B$. Of course this notion is related to the notion of *$B$-strict episturmian* word (see [@JustinPirillo2002 def. 2.3]) which is a ultimately $B$-strict episturmian word whose alphabet (the letters occurring in $w$) is exactly $B$, and which is nothing else than an Arnoux-Rauzy sequence over $B$.
As shown in [@DroubayJustinPirillo2001], there is a close relation between the directive word of an episturmian word and its special words. Corollary \[cor1\] below will show it again for ultimately strict episturmian words.
Let $w$ be an episturmian word and $\Delta(w) = (x_n)_{n \geq 1}$ its directive word. With notations of Theorem \[JP2\], for $n \geq 1$, we denote $u_{n,w}$ (or simply $u_n$) the word :
$$u_{n,w} = L_{x_1}(L_{x_{2}}(\ldots (L_{x_{n-1}}(\varepsilon)x_{n-1})
\ldots)x_{2})x_1$$
When $n = 1$, $u_{n,w} = \varepsilon$. These words play an important role in the initial definition of episturmian word by palindromic closure (see [@JustinPirillo2002 Sec. 2]). In particular, each $u_n$ is a palindrom (see for instance [@JustinPirillo2002 Lem. 2.5]). One can also observe that, if $Ult(\Delta(w))$ contains at least two letters, then each $u_{n}$ is a bispecial factor of $w$. Indeed for $n \geq 1$, $u_{n}$ is a prefix of the epistandard word $s$ directed by $\Delta(w)$ and so, by definition of an epistandard word, it is a left special factor of $s$ and so of $w$ by Theorem \[JP2\]. Since the set of factors of $w$ is closed by mirror image and since $u_{n}$ is a palindrom, $u_{n}$ is a right special factor of $w$. Conversely let us observe that any bispecial factor of an episturmian word is a palindrom. Indeed if $u$ is a bispecial factor, then $u$ and its mirror image $\tilde{u}$ are left special factors of an infinite word containing at most one left special word of length $|u|$. It follows the construction of an epistandard word $w$ by palindromic closure [@DroubayJustinPirillo2001], that the the words $u_{n,w}$ are the only palindroms prefixes of $w$. From what precedes, we deduce the following fact that does not seem to have been already quoted in the literature:
\[remarkBispecials\] For an episturmian word $w$ with directive word $(x_n)_{n \geq 1}$, a factor $u$ is bispecial if and only if $u = u_{n,w}$ for an integer $n \geq 1$.
Another result involving the palindroms $u_n$ is:
[[@DroubayJustinPirillo2001 Th. 6]]{} \[thDJPth6\] Let $s$ be an epistandard word over the alphabet $A$ with directive word $\Delta(s) = (x_n)_{n \geq 1}$. For $n \geq 1$ and $x \in A$, $u_{n,s}x$ (or equivalently $xu_{n,s}$) is a factor of $s$ if and only if $x$ belongs to $\{ x_i \mid i \geq n \}$.
By Theorem \[JP2\], an episturmian word $w$ with a directive word $\Delta$ has the same set of factors than the epistandard word with directive word $\Delta$. Hence the previous theorem is still valid for any episturmian word, and we can deduce:
\[cor1\] Let $w$ be an episturmian word over an alphabet $A$ and let $B
\subseteq A$ be a set containing at least two different letters. The word $w$ is a ultimately $B$-strict episturmian word if and only if for an integer $n_0$, each left special factor with $|u| \geq
n_0$ verifies $Au \cap Fact(w) = Bu$.
Moreover for each left special factors with $|u| < n_0$, $Bu \subseteq Fact(w)$.
The restriction on the cardinality of $B$ ($\geq 2$) will be used in all the rest of the paper. It is needed to have special factors of arbitrary length.
\[secLocalProperty\]A new characterization of episturmian words
===============================================================
Now we give our first main result presented in the introduction as a kind of local characteristic balance property of episturmian words.
\[th1\] For a recurrent infinite word $w$, the following assertions are equivalent:
1. $w$ is episturmian;
2. for each factor $u$ of $w$, a letter $a$ exists such that $AuA \cap Fact(w) \subseteq a uA \cup Au a$;
3. for each [*palindromic*]{} factor $u$ of $w$, a letter $a$ exists such that $AuA \cap Fact(w) \subseteq a uA \cup Au a$.
In the previous theorem, the letter $a$ and the cardinality of the set $AuA$ depends on $u$. This is shown for instance by the Fibonacci word (abaababaabaab…), the epistandard word having $(ab)^\omega$ as director word, for which $A\varepsilon A \cap Fact(w) = \{aa, ab,
ba\}$, $AaA \cap Fact(w) = \{aab, baa\}$, $AbA \cap Fact(w) =
\{aba\}$, $AaaA \cap Fact(w) = \{baab\}$, …
[Theorem \[th1\]]{}
*Proof of $1 \Rightarrow 2$.* Assume $w$ is episturmian. Since the result deals only with factors of $w$, and since by Theorem \[JP2\] an episturmian word have the same set of factors than an epistandard word, without loss of generality we can assume that $w$ is epistandard. Let $u$ be a factor of $w$. Property 2 is immediate if $u$ is not a bispecial factor of $w$. If $u$ is bispecial in $w$, by Remark \[remarkBispecials\], an integer $n
\geq 1$ exists such that $u = u_{n,w}$. Let $\Delta = (x_i)_{i \geq
1}$ be the directive word of $w$, let $s$ (resp. $t$) be the epistandard word with $(x_i)_{i \geq n}$ (resp. $(x_i)_{i \geq n+1}$) as directive word and let $a = x_n$. Letters occurring in $t$ are exactly the letters of the set $B = \{x_i \mid i \geq n+1\}$. Since $s = L_{x_n}(t)$, the factors of length 2 in $s$ are the words $a b$ and $ba$ with $b \in B$. By definition of $\Delta$ and $u_{n,w}$, $w =
L_{x_1}(L_{x_2}( \ldots L_{x_{n-1}}(s)\ldots))$ and $u_{n,w} =
L_{x_1}(L_{x_{2}}(\ldots (L_{x_{n-1}}(\varepsilon)x_{n-1})
\ldots)x_{2})x_1$. Hence by an easy induction on $n$, we deduce $AuA
\cap Fact(w) = a uB \cup Bua \subseteq a uA \cup Aua$.
*Proof of $2 \Rightarrow 1$.* Assume that, for any factor $u$ of $w$, a letter $a$ exists such that $AuA \cap Fact(w) \subseteq
a uA \cup Au a$. In particular, considering the empty word, we deduce that $AA \cap Fact(w) \subseteq a A \cup Aa$ for a letter $a$. Hence, for an infinite word $x$, $w = L_a(y)$ if $w$ starts with $a$ and $w = R_a(y)$ otherwise.
Let us prove that for each factor $v$ of $y$, $AvA \cap Fact(w)
\subseteq b vA \cup Av b$ for a letter $b$. We consider $w
= L_a(y)$ (resp. $w = R_a(y)$). Let $v$ be a factor of $y$ and let $u = L_a(v)a$ (resp. $u = a R_a(v)$). We observe that for letters $c, d$, the words $c u d$ is a factor of $w$ if and only if $c v d$ is a factor of $y$. By hypothesis there exists a letter $b$ such that $AuA \cap Fact(w) \subseteq b uA \cup
Aub$. Hence $AvA \cap Fact(w)
\subseteq b vA \cup Avb$.
Letting $x_1 = a$ and iterating infinitely the previous step, we get an infinite sequence of letters $(x_i)_{i \geq 1}$ and an infinite sequence of words $(w^{(i)})_{i \geq 0}$ such that $w^{(0)} = w$ and for all $i \geq 1$, $w^{(i-1)} = L_{x_i}(w^{(i)})$ or $w^{(i-1)} =
R_{x_i}(w^{(i)})$. Due to the fact that $w$ is recurrent, each word $w^{(i)}$ is also recurrent. By Theorem \[JP2\], the word $w$ is episturmian.
The proof of $1 \Leftrightarrow 3$ is similar to the proof of $1
\Leftrightarrow 2$. Actually, $1 \Rightarrow 3$ is a particular case of $1 \Rightarrow 2$. When proving $3 \Rightarrow 1$, we need to prove in the inductive step that $u$ is a palindrome if and only if $v$ is a palindrome. This is stated by Lemma 2. 5 in [@JustinPirillo2002] : *a word $u$ is a palindrome if and only the word $L_a(u)a = aR_a(u)$ is a palindrome*.
We end this section with few remarks concerning results that can be proved similarly.
\[rem1\] Since an infinite word $w$ over an alphabet $A$ is recurrent if and only if for each factor of $w$ the set $AuA$ is not empty, we have: an infinite word is episturmian if and only if for each (resp. [*palindromic*]{}) factor $u$ of $w$, $AuA$ is not empty and a letter $a$ exists such that $AuA \cap Fact(w) \subseteq a
uA \cup Au a$.
We have already said that Arnoux-Rauzy sequences over an alphabet $A$ are exactly the (ultimately) A-strict episturmian word. One can ask for a characterization of these words in a quite similar way than Theorem \[th1\]. Corollary \[cor1\] can fulfill this purpose. But the proof of Theorem \[th1\] can also be easily reworked to state : [*an episturmian word $w$ over an alphabet $A$ is a ultimately $B$-strict episturmian word with $B \subseteq A$ if and only if for all $n \geq 0$, there exists a [*(resp. palindromic)*]{} word $u$ of length at least $n$ and a letter $a$ such that $AuA \cap Fact(w) = auB
\cup Bua$.* ]{}
Another adaptation of the proof of Theorem \[th1\] concerns finite words: [*a finite word $w$ is a factor of an infinite episturmian word if and only if for each factor $u$ of $w$, a letter $a$ exists such that $AuA \cap Fact(w) \subseteq a u A \cup Au a$*]{}. We let the reader verify this result. The main difficulty of the proof is that in the “if part”, we do not have necessarily $w = L_a(y)$ or $w =
R_a(y)$. But we have one of the four following cases depending on the fact that $w$ ends or not with $a$: $w = L_a(y)$, $w = a L_a(y)$, or $wa = L_a(y)$ or $wa = a L_a(y)$. Except in small cases, we have $|y|
< |w|$ and the technique of the proof of Theorem \[th1\] can be applied.
\[secEpistandard\]A characterization of epistandard words
=========================================================
Let us note that for any episturmian word $w$, there exists at least one letter $a$ such that $a w$ is also episturmian. Indeed, since any episturmian word is recurrent, for any prefix $p$ of $w$, there exists a letter $a_p$ such that $a_p p$ is a factor of $w$. We work with a finite alphabet hence an infinity of letters $a_p$ are mutually equal: there exists a letter $a$ such that $a p$ is a factor of $p$ for an infinity of prefixes (and so for all prefixes) of $w$. The word $aw$ has the same set of factors than $w$: it is episturmian.
In restriction to epistandard words, a more precise result is already know:
[[@JustinPirillo2002 Th. 3.17]]{} \[thJP3.17\] If a word $s$ is epistandard, then for each letter $a$ in $Ult(\Delta(s))$, $a s$ is episturmian.
Up we know the converse of this result has already been stated only in the Sturmian case (see [@BerstelSeebold2002 Prop. 2.1.22]): *For every Sturmian word $w$ over $\{a, b\}$, $w$ is standard episturmian if and only if $aw$ and $bw$ are both Sturmian*. We generalize here this result, proving a converse to Theorem \[thJP3.17\].
\[gen1\] A [*non-periodic*]{} word $w$ is epistandard if and only if, for (at least) two different letters $a$ and $b$, $a w$ and $b w$ are episturmian.
Let $w$ be a non-periodic epistandard word $w$. By [@DroubayJustinPirillo2001 Th. 3], we know that $Ult(\Delta(w))$ contains at least two different letters, say $a$ and $b$. By Theorem \[thJP3.17\], $a w$ and $b w$ are episturmian.
Assume now that for two different letters $a$ and $b$, $a w$ and $b w$ are episturmian. Since $a w$ (and also $b w$) is recurrent, $w$ has the same set of factors than $a w$ and so $w$ is episturmian. Moreover each prefix $p$ is left special (since $a p$ and $b p$ are factors of $w$). Since any episturmian word has at most one left special factor for each length, the left special factors of $w$ are its prefixes: $w$ is epistandard.
Let us give a more precise result:
\[mainTh\] Let $w$ be an infinite word over the alphabet $A$ and assume $B
\subseteq A$ contains at least two different letters. The two following assertions are equivalent:
1. The word $w$ is ultimately $B$-strict epistandard;
2. For each letter $a$ in $A$, $a w$ is episturmian if and only if $a$ belongs to $B$.
Assume first that $w$ is $B$-strict epistandard, that is, $Ult(\Delta(w)) = B$. By Theorem \[thJP3.17\], for each letter $a$ in $B$, $a w$ is episturmian. For any integer $n \geq
0$, the word $u_{n, w}$ is a prefix of $w$. If $a$ does not belong to $B$, by Theorem \[thDJPth6\], for at least one integer $n \geq 0$, $au_{n,w}$ is not a factor of $w$. Thus the word $a w$ is not recurrent and so it is not episturmian. Hence if $w$ is $B$-strict epistandard, for each letter $a$ in $A$, $a w$ is episturmian if and only if $a$ belongs to $B$.
Assume now that for each letter $a$ in $A$, $a w$ is episturmian if and only if $a$ belongs to $B$. Since $B$ contains at least two letters, by Proposition \[gen1\], $w$ is epistandard. As a consequence of Theorem \[thDJPth6\], we can deduce $Ult(\Delta(w)) =
B$.
A new family of words
=====================
In this section, we consider recurrent infinite words $w$ over an alphabet $A$ having the following property:
[Property [${\cal P}$]{}:]{} for any word $u$ over $A$, the set of factors of $w$ belonging to $AuA$ is balanced,\
that is, for any word $u$ and for any letters $a, b, c,
d$, if $aub$ and $cud$ are factors of $w$ then $\{a, b\} \cap \{c, d\}
\neq \emptyset$.
Any word verifying Assertion 2 in Theorem \[th1\] also verifies Property [${\cal P}$]{}. As shown by the word $(abc)^\omega$, the converse does not hold. In other words, any episturmian word verifies Property [${\cal P}$]{}, but this is not a characteristic property (except in the binary case for which it is immediate that a word $w$ verifies Property [${\cal P}$ ]{}if and only if for all words $u$, $aua$ or $bub$ is not a factor of $w$).
We prove:
\[prop6.1\] A recurrent word $w$ over an alphabet $A$ verifies property [${\cal P}$ ]{}if and only if one of the two following assertion holds:
1. $w$ is episturmian;
2. there exist an episturmian morphism $f$, three different letters $a, b, c$ in $A$ and a word $w'$ over $\{a,b,c\}$ such that $w =
f(w')$, $w'$ verifies Property [${\cal P}$ ]{}and the three words $ab$, $bc$ and $ca$ are factors of $w'$.
This proposition is a consequence of the next two lemmas.
\[lemma6.3\] If a recurrent infinite word $w$ verifies property [${\cal P}$]{}, then one of the two following assertion holds:
1. $w = L_\alpha(w')$ or $w = R_\alpha(w')$ for a letter $\alpha$ and a recurrent infinite word $w'$;
2. there exist three different letters $a, b, c$ such that $w \in
\{a, b, c\}^\omega$ and the three words $ab$, $bc$ and $ca$ are factors of $w$.
We first observe that if $AA \cap Fact(w) \subseteq \alpha A \cup A
\alpha$ then (as in the proof of Theorem \[mainTh\]) $w =
L_\alpha(w')$ or $w = R_\alpha(w')$, for a letter $\alpha$ and a recurrent infinite word $w'$.
We assume from now on that $AA \cap Fact(w) \not\subseteq \alpha A
\cup A \alpha$.
For any letter $\alpha$ in $A$, $\alpha \alpha$ is not a factor of $w$. Indeed if such a word is a factor of $w$, then, for any factor $\beta \gamma$ with $\beta$ and $\gamma$ letters, by Property [${\cal P}$]{}, $\beta
= \alpha$ or $\gamma = \alpha$, that is $AA \cap Fact(w) \subseteq
\alpha A \cup A \alpha$.
The alphabet $A$ contains at least three letters. Indeed if $A$ contains at most two letters $a$ and $b$, then Property [${\cal P}$ ]{}implies that $aa$ and $bb$ are not simultaneously factors of $w$, and so we have $AA \cap Fact(w) \subseteq aA \cup Aa$ or $AA \cap Fact(w)
\subseteq bA \cup Ab$.
Let us prove that $A$ contains exactly three letters. Assume by contradiction that $A$ contains at least four letters. Let $a$ (resp. $b$) be the first (resp. the second) letter of $w$. Since $aa$ is not a factor of $w$, $a \neq b$. At least two other letters $c$ and $d$ occur in $w$ ($c, d \not\in \{a, b\}$, $c \neq d$). By Property [${\cal P}$]{}, each occurrence of $c$ is preceded by $a$ or by $b$. Assume that $ac$ occurs in $w$. Since $ab$ also occurs, for any letter $\alpha$ not in $\{a, b, c\}$, each occurrence of $\alpha$ is preceded and followed by the letter $a$. But $AA \cap Fact(w) \not\subseteq a A
\cup A a$. Hence $bc$ or $cb$ occurs in $w$. But then the factor $ad$ contradicts Property [${\cal P}$]{}. Assume now that $bc$ occurs in $w$. Since $ab$ also occurs, for any letter $\alpha$ not in $\{a, b, c\}$, each occurrence of $\alpha$ is preceded and followed by $b$. But $AA \cap Fact(w) \not\subseteq b A
\cup A b$. Hence $ac$ or $ca$ occurs in $w$. But then the factor $db$ contradicts Property [${\cal P}$]{}.
Until now we have proved that $w$ is written on a three-letter alphabet and contains no word $\alpha \alpha$ with $\alpha$ a letter. Assume that, for two letters $a$ and $b$, $ab$ is a factor of $w$ but not $ba$. Then for an integer $n \geq 1$, $a(bc)^na$ (let recall that $aa$, $bb$, $cc$ and $ba$ are not factors of $w$), and so $ab$, $bc$ and $ca$ are factors of $w$. Now if, for all letters $\alpha$ and $\beta$, $\alpha\beta$ and $\beta\alpha$ are factors of $w$ then denoting $a$, $b$ and $c$ the letters occurring in $w$, once again $ab$, $bc$ and $ca$ are factors of $w$.
\[lemma6.2\] Let $\alpha$ be a letter, $w$ and $w'$ be recurrent words such that $w
= L_\alpha(w')$ or $w = R_\alpha(w')$. The word $w$ verifies Property [${\cal P}$ ]{}if and only if $w'$ verifies Property [${\cal P}$]{}.
We first assume $w = L_\alpha(w')$.
Assume that $w$ does not verify Property [${\cal P}$]{}: $aub$ and $cud$ are factors of $w$ for some letters $a, b, c, d$ and a word $u$ such that $\{a, b\} \cap \{c, d\} = \emptyset$. At least one of the two letters $a$ and $b$ is different from $\alpha$ and at least one of the two letters $c$ and $d$ is different from $\alpha$. Since $w =
L_\alpha(w')$, we deduce that $u \neq \varepsilon$, and that $u$ begins and ends with $\alpha$: $u = L_\alpha(v)\alpha$ for a word $v$. Thus $aub = aL_\alpha(v)\alpha b$ and $cud = cL_\alpha(v)\alpha
d$. We observe that if $a \neq \alpha$ (resp. $c \neq \alpha$), $\alpha aL_\alpha(v)\alpha b$ (resp. $\alpha cL_\alpha(v)\alpha d$) is a factor of $w$. Thus we can deduce that $avb$ and $cvd$ are factors of $w'$ (even if one of the letters $a, b, c, d$ is $\alpha$): the word $w'$ does not verify Property [${\cal P}$]{}.
Assume conversely that the word $w'$ does not verify Property [${\cal P}$]{}: $aub$ and $cud$ are factors of $w'$ for some letters $a, b, c, d$ and a word $u$ such that $\{a, b\} \cap \{c, d\} = \emptyset$. The word $aL_\alpha(u)\alpha b$ is a factor of $w$ (if $b = \alpha$, this is still true since we work with infinite words and so in this case $au\alpha b'$ is a factor of $w$ for a letter $b'$). Similarly $cL_\alpha(u)\alpha d$ is a factor of $w$: the word $w$ does not verify Property [${\cal P}$]{}.
The proof when $w = R_\alpha(w')$ is similar. Note that the fact that $w'$ is recurrent is needed for the last part of the proof to know when $a = \alpha$, that $a' \alpha u b$ is a factor of $w'$ for a letter $a'$.
[Proposition \[prop6.1\]]{} Assume $w$ is a recurrent word that verifies Property [${\cal P}$ ]{}but that does not verifies Assertion 2 of Lemma \[lemma6.3\]. Then $w = L_\alpha(w')$ or $w = R_\alpha(w')$, with $w'$ a recurrent word. By Lemma \[lemma6.2\], $w'$ verifies Property [${\cal P}$]{}.
Thus using Lemmas \[lemma6.3\] and \[lemma6.2\], we can prove by induction that, for any integer $n \geq 0$, one of the two following assertions holds :
- there exist recurrent infinite word $w^{(0)} = w$, $w^{(1)}$, …$w^{(n)}$, and letters $a_1$, …, $a_n$ such that for each $1 \leq p \leq n$, $w^{(p-1)} = L_{a_p}(w^{(p)})$ or $w^{(p-1)} =
R_{a_p}(w^{(p)})$, and $w^{n}$ verifies property [${\cal P}$]{};
- for an integer $m \leq n$, there exist recurrent infinite word $w^{(0)} = w$, $w^{(1)}$, …$w^{(m)}$, and letters $a_1$, …, $a_m$ such that for each $1 \leq p
\leq m$, $w^{(p-1)} = L_{a_p}(w^{(p)})$ or $w^{(p-1)} = R_{a_p}(w^{(p)})$, and $w^{(m)}$ verifies Assertion 2 of Lemma \[lemma6.3\].
Hence the proposition is a consequence of Theorem \[JP2\].
Conclusion
==========
The reader has certainly noticed that words verifying Property [${\cal P}$ ]{}are not completely characterized. For this, one should have to better know ternary recurrent words verifying Property [${\cal P}$ ]{}and containing the words $ab$, $bc$ and $ca$ as factors.
Let us give examples of such words. One can immediately verify that if $ab$, $bc$ and $ca$ are the only words of length 2 that are factors of a word $w$, then $w$ is $(abc)^\omega$, $(bca)^\omega$ or $(cab)^\omega$. When a recurrent word $w$ verifying property [${\cal P}$ ]{}has exactly the words $ab$, $bc$, $ca$ and $ba$ as factors of length 2, one can see that $w$ is a suffix of a word $f(w')$ where $w'$ is a Sturmian word over $\{a, b\}$ and $f$ is the morphism defined by $f(a)
= (ab)^nc$ and $f(b) = (ab)^{n+1}c$ for an integer $n \geq 1$. When $f$ is replaced by one of the following morphisms $g_1$ or $g_2$, we can get other examples of ternary words verifying Property [${\cal P}$ ]{}(and containing exactly 5 factors of length 2 with amongst them $ab$, $bc$ and $ca$) : $g_1(a) = (ab)^nc$, $g_1(b) = (ab)^{n}cb$, $g_2(a) =
(ab)^nc$, $g_2(b) = (ab)^{n+1}cb$. Our final example is the periodic word $(abcabacbabcb)^\omega$ which verifies Property [${\cal P}$ ]{}and contains as factors all words of length 2 except $aa$, $bb$, $cc$: this word could be seen as the morphic image of $a^\omega$ by the morphism that maps $a$ onto $abcabacbabcb$.
All these examples lead to the question: Are all ternary recurrent words verifying Property [${\cal P}$ ]{}and containing $ab$, $bc$ and $ca$ as factors are suffix of a word $f(w')$ with $w'$ a recurrent balanced word (that is a Sturmian word or a periodic balanced word) and with $f$ a morphism? If it is true, which are the possible values for $f$?
**Acknowledgements.** The author would like to thanks J.-P. Allouche for his questions that have initiated the present work.
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---
abstract: 'Applying Benjamini and Hochberg (B-H) method to multiple Student’s $t$ tests is a popular technique in gene selection in microarray data analysis. Because of the non-normality of the population, the true p-values of the hypothesis tests are typically unknown. Hence, it is common to use the standard normal distribution $N(0,1)$, Student’s $t$ distribution $t_{n-1}$ or the bootstrap method to estimate the p-values. In this paper, we first study $N(0,1)$ and $t_{n-1}$ calibrations. We prove that, when the population has the finite 4-th moment and the dimension $m$ and the sample size $n$ satisfy $\log m=o(n^{1/3})$, B-H method controls the false discovery rate (FDR) at a given level $\alpha$ asymptotically with p-values estimated from $N(0,1)$ or $t_{n-1}$ distribution. However, a phase transition phenomenon occurs when $\log m\geq c_{0}n^{1/3}$. In this case, the FDR of B-H method may be larger than $\alpha$ or even tends to one. In contrast, the bootstrap calibration is accurate for $\log m=o(n^{1/2})$ as long as the underlying distribution has the sub-Gaussian tails. However, such light tailed condition can not be weakened in general. The simulation study shows that for the heavy tailed distributions, the bootstrap calibration is very conservative. In order to solve this problem, a regularized bootstrap correction is proposed and is shown to be robust to the tails of the distributions. The simulation study shows that the regularized bootstrap method performs better than the usual bootstrap method.'
author:
- 'Weidong Liu[^1] and Qi-Man Shao[^2]'
title: 'Phase Transition and Regularized Bootstrap in Large Scale $t$-tests with False Discovery Rate Control '
---
Introduction
============
Multiple Student’s $t$ tests often arise in many real applications such as gene selection. Consider $m$ tests on the mean values $$\begin{aligned}
H_{0i}: ~\mu_{i}=0\quad\mbox{versus\quad} H_{1i}: ~\mu_{i}\neq 0,\quad 1\leq i\leq m.\end{aligned}$$ A popular procedure is using Benjamini and Hochberg (B-H) method to search significant findings with the false discovery rate (FDR) controlled at a given level $0<\alpha<1$, that is, $${\epsilon}\Big{[}\frac{\text{V}}{\text{R}\vee 1}\Big{]}\leq \alpha,$$ where $\text{V}$ is the number of wrongly rejected hypotheses and $\text{R}$ is the total number of rejected hypotheses. The seminal work of Benjamini and Hochberg (1995) is to reject the null hypotheses for which $p_{i}\leq p_{(\hat{k})}$, where $p_{i}$ is the p-value for $H_{0i}$, $$\begin{aligned}
\label{a0}
\hat{k}=\max\{0 \leq i \leq m:\ p_{(i)}\leq \alpha i/m\},\end{aligned}$$ and $p_{(1)}\leq\cdots\leq p_{(m)}$ are the order p-values. Let $T_{1},\ldots,T_{m}$ be Student’s $t$ test statistics $$\begin{aligned}
T_{i}=\frac{\bar{X}_{i}}{\hat{s}_{ni}/\sqrt{n}},\end{aligned}$$ where $$\begin{aligned}
\bar{X}_{i}=\frac{1}{n}\sum_{k=1}^{n}X_{ki},\quad \hat{s}^{2}_{ni}=\frac{1}{n-1}\sum_{k=1}^{n}(X_{ki}-\bar{X}_{i})^{2},\end{aligned}$$ and $(X_{k1},\ldots,X_{km})^{'}$, $1\leq k\leq n$, are i.i.d. random samples from $(X_{1},\ldots,X_{m})^{'}$. When $T_{1},\ldots,T_{m}$ are independent and the true p-values $p_{i}$ are known, Benjamini and Hochberg (1995) showed that B-H method controls the FDR at level $\alpha$.
In many applications, the distributions of $X_{i}$, $1\leq i\leq m$, are non-Gaussian. Hence, it is impossible to know the exact null distributions of $T_{i}$ and the true p-values. In the application of B-H method, the p-values are actually some estimators. By the central limit theorem, it is common to use the standard normal distribution $N(0,1)$ or Student’s $t$ distribution $t_{n-1}$ to estimate the p-values, where $t_{n-1}$ denotes Student’s $t$ random variable with $n-1$ degrees of freedom. In a microarray analysis, Efron (2004) observed that the choices of null distributions will substantially affect the simultaneous inference procedure. However, a systematic theoretical study on the influence of the estimated $p$-values is still lack. It is important to know how accurate $N(0,1)$ and $t_{n-1}$ calibrations can be. In this paper, we will show that $N(0,1)$ and $t_{n-1}$ calibrations are accurate when $\log m=o(n^{1/3})$. Under the finite 4th moment of $X_{i}$, the FDR of B-H method with the estimated p-values $p_{i}=2-2\Phi(T_{i})$ or $p_{i}=2-2\Psi(T_{i})$ will converge to $\alpha m_{0}/m$, where $m_{0}$ is the number of true null hypotheses, $\Phi(t)$ is the standard normal distribution and $\Psi(t)=\pr(t_{n-1}\leq t)$. However, when $\log m\geq c_{0}n^{1/3}$ for some $c_{0}>0$, $N(0,1)$ and $t_{n-1}$ calibrations may not work well and a phase transition phenomenon occurs. Under $\log m\geq c_{0}n^{1/3}$ and the average of skewnesses $\tau=\liminf_{m\rightarrow\infty}m_{0}^{-1}\sum_{i\in\mathcal{H}_{0}}|{\epsilon}X_{i}^{3}/\sigma_{i}^{3}|>0$, we will show that the FDR of B-H method satisfies $\lim_{(m,n)\rightarrow\infty}FDR\geq \kappa$ for some constant $\kappa>\alpha$, where $\mathcal{H}_{0}=\{i: \mu_{i}=0\}$. Furthermore, if $\log m/n^{1/3}\rightarrow\infty$, then $\lim_{(m,n)\rightarrow\infty}FDR=1$. This indicates that $N(0,1)$ and $t_{n-1}$ calibrations are inaccurate when the average of skewnesses $\tau\neq 0$ in the ultra high dimensional setting.
It is well known that bootstrap is an effective way to improve the accuracy on the exact null distribution approximation. Fan, Hall and Yao (2007) showed that, for the bounded noise, the bootstrap can improve the accuracy and allow higher dimension $\log m=o(n^{1/2})$ on controlling the family-wise error rate. Delaigle, Hall and Jin (2011) showed that the bootstrap method shares significant advantages on higher criticism. In this paper, we show that, when the bootstrap calibration is used and $\log m=o(n^{1/2})$, B-H method can control FDR at level $\alpha$, i.e. $\lim_{(m,n)\rightarrow\infty}FDR/(\alpha m_{0}/m)=1$. In our results, we assume the sub-Gaussian tails instead of the bounded noise in Fan, Hall and Yao (2007).
Although the bootstrap method allows a higher dimension, the light-tailed condition can not be weakened in general. The simulation study shows that the bootstrap method is very conservative for the heavy-tailed distributions. To solve this problem, we will propose a regularized bootstrap method which is robust to the tails of the distributions. The proposed regularized bootstrap only requires the finite 6th moment. Also, the dimension can be as large as $\log m=o(n^{1/2})$.
It is also not uncommon in real applications that $X_{1},\ldots,X_{m}$ are dependent. This results in the dependency between $T_{1},\ldots,T_{m}$. In this paper, we will obtain some similar results for B-H method under a general weak dependence condition. It should be noted that much work has been done on the robustness of FDR controlling method against dependence. Benjamini and Yekutieli (2001) proved that the B-H procedure controls FDR under positive regression dependency. Storey (2003), Storey, Taylor and Siegmund (2004), Ferreira and Zwinderman (2006) imposed a dependence condition that requires the law of large numbers for the empirical distributions under the null and alternative hypothesis. Wu (2008) developed a FDR controlling procedures for the data coming from special models such as time series model. However, to satisfy the conditions in most of the existing methods, it is often necessary to assume the number of true alternative hypotheses $m_{1}$ is asymptotically $\pi_{1}m$ with some $\pi_{1}>0$. They exclude the sparse setting $m_{1}=o(m)$ which is important in applications such as gene selection. For example, if $m_{1}=o(m)$, then the conditions of Theorem 4 in Storey, Taylor and Siegmund (2004) and the conditions of main results in Wu (2008) will be violated. On the other hand, our results on FDR control under dependence allows $m_{1}\leq \gamma m$ for some $\gamma<1$.
The rest of this paper is organized as follow. In Section 2.1, we will show the robustness and the phase transition phenomenon for $N(0,1)$ and $t_{n-1}$ calibrations. In Section 2.2, we show that the bootstrap calibration can improve the FDR control. The regularized bootstrap method is proposed in Section 3. The results are extended to the dependence case in Section 4. The simulation study is presented in Section 5 and the proofs are given in Section 6.
Main results
============
Robustness and phase transition
-------------------------------
In this section, we assume Student’s $t$ test statistics $T_{1},\ldots,T_{m}$ are independent. The results will be extended to the dependent case in Section 4. Before stating the main theorems, we introduce some notations. Let $\hat{p}_{i,\Phi}=2-2\Phi(|T_{i}|)$ and $\hat{p}_{i,\Psi}=2-2\Psi(|T_{i}|)$ be the $p$-values calculated from the standard normal distribution and the $t$-distribution respectively. Let FDR$_{\Phi}$ and FDR$_{\Psi}$ be the FDR of B-H method with $\hat{p}_{i,\Phi}$ and $\hat{p}_{i,\Psi}$ in (\[a0\]) respectively. Let R be the total number of rejections. The critical values of the tests are then $\hat{t}_{\Phi}=\Phi^{-1}(1-\alpha\text{R}/(2m))$ and $\hat{t}_{\Psi}=\Psi^{-1}(1-\alpha\text{R}/(2m))$. Set $Y_{i}=(X_{i}-\mu_{i})/\sigma_{i}$ with $\sigma^{2}_{i}=\Var(X_{i})$, $1\leq i\leq m$.
Throughout this paper, we assume $m_{1}\leq \gamma m$ for some $\gamma<1$, which includes the important sparse setting $m_{1}=o(m)$.
\[th1\] Suppose $X_{1},\ldots,X_{m}$ are independent and $\log m=o(n^{1/2})$. Assume that $\max_{1\leq i\leq m}{\epsilon}Y_{i}^{4}\leq {b}_{0}$ for some constant ${b}_{0}>0$ and $$\begin{aligned}
\label{c1}
Card\Big{\{}i: |\mu_{i}/\sigma_{i}|\geq 4\sqrt{\log m/n}\Big{\}}\rightarrow\infty.\end{aligned}$$ Then $$\begin{aligned}
\lim_{(n,m)\rightarrow\infty} \frac{FDR_{\Phi}}{\frac{m_{0}}{m}\alpha\kappa_{\Phi}}=1\mbox{\quad and\quad} \lim_{(n,m)\rightarrow\infty} \frac{FDR_{\Psi}}{\frac{m_{0}}{m}\alpha\kappa_{\Psi}}=1,\end{aligned}$$ where $$\begin{aligned}
\kappa_{\Phi}&=&{\epsilon}[ \hat{\kappa}_{\Phi} I\{ \hat{\kappa}_{\Phi}\leq 2(\alpha-\alpha\gamma)^{-1}\}],\cr
\hat{\kappa}_{\Phi}&=&\frac{\sum_{i\in\mathcal{H}_{0}}\Big{\{}\exp\Big{(}\frac{\hat{t}^{3}_{\Phi}{\epsilon}X_{i}^{3}}{\sqrt{n}\sigma_{i}^{3}}\Big{)}+\exp\Big{(}-\frac{\hat{t}^{3}_{\Phi}{\epsilon}X_{i}^{3}}{\sqrt{n}\sigma_{i}^{3}}\Big{)}\Big{\}}}{2m_{0}}\end{aligned}$$ satisfying $1+o(1)\leq \kappa_{\Phi}\leq m/(\alpha m_{0})+o(1)$, and $\kappa_{\Psi}$ is defined in the same way.
Recall that $\tau=\liminf_{m\rightarrow\infty}m_{0}^{-1}\sum_{i\in\mathcal{H}_{0}}|{\epsilon}Y_{i}^{3}|$. We have the following corollary.
\[co1\] Assume the conditions in Theorem \[th1\] hold.
- Under $\log m=o(n^{1/3})$, we have $\lim_{(n,m)\rightarrow\infty} FDR_{\Phi}/(\alpha m_{0}/m)=1$.
- Suppose $\log m\geq c_{0}n^{1/3}$ for some $c_{0}>0$ and $m_{1}=\exp(o(n^{1/3}))$. Assume that $\tau>0$. We have $\liminf_{(n,m)\rightarrow\infty} FDR_{\Phi}\geq \beta$ for some constant $\beta>\alpha$.
- Suppose $\log m/n^{1/3}\rightarrow\infty$ and $m_{1}=\exp(o(n^{1/3}))$. Assume that $\tau>0$. We have $\lim_{(n,m)\rightarrow\infty} FDR_{\Phi}=1$.
The same conclusions hold for $FDR_{\Psi}$.
Theorem \[th1\] and Corollary \[co1\] show that, when $\log m=o(n^{1/3})$, $N(0,1)$ and $t_{n-1}$ calibrations are accurate. Note that only a finite fourth moment of $Y_{i}$ is required. Furthermore, if the skewnesses ${\epsilon}Y_{i}^{3}=0$ for $i\in\mathcal{H}_{0}$, then the dimension can be as large as $\log m=o(n^{1/2})$. However, a phase transition occurs if the average of skewnesses $\tau>0$, for example, for the exponential distribution. The FDR of B-H method will be greater than $\alpha$ as long as $\log m\geq c_{0}n^{1/3}$ and will converge to one when $\log m/n^{1/3}\rightarrow\infty$.
Corollary 2.1 also indicates that, in the study of large scale testing problem, the choice of asymptotic null distributions is important. When the dimension is much larger than the sample size, an inadequate choice such as $N(0,1)$ may result in a high FDR. This will be further verified by our simulation study in Section 5. Hence, in the problems on large scale tests, assuming the true p-values are known may be over-idealistic.
Bootstrap calibration
---------------------
In this section, we show that the bootstrap procedure can improve the accuracy on the control of FDR. Write $\mathcal{X}_{i}=\{X_{1i},\ldots,X_{ni}\}$. Let $\mathcal{X}^{*}_{ki}=\{X^{*}_{1ki},\ldots,X^{*}_{nki}\}$, $1\leq k\leq N$, be resamples drawn randomly with replacement from $\mathcal{X}_{i}$. Let $T^{*}_{ki}$ be Student’s $t$ test statistics constructed from $\{X^{*}_{1ki}-\bar{X}_{i},\ldots,X^{*}_{nki}-\bar{X}_{i}\}$. We use $G^{*}_{N,m}(t)=\frac{1}{Nm}\sum_{k=1}^{N}\sum_{i=1}^{m}I\{|T^{*}_{ki}|\geq t\}$ to approximate the null distribution and define the $p$-values by $\hat{p}_{i,B}=G^{*}_{N,m}(|T_{i}|)$. Let FDR$_{B}$ denote the FDR of B-H method with $\hat{p}_{i,B}$ in (\[a0\]).
\[th2-2\] Suppose that $\max_{1\leq i\leq m}{\epsilon}e^{tY_{i}^{2}}\leq K$ for some constants $t>0$ and $K>0$ and the conditions in Theorem \[th1\] hold.
- Under $\log m=o(n^{1/3})$, we have $\lim_{(n,m)\rightarrow\infty} FDR_{B}/(\alpha m_{0}/m)=1$.
- If $\log m=o(n^{1/2})$ and $m_{1}\leq m^{\eta}$ for some $\eta<1$, then $\lim_{(n,m)\rightarrow\infty} FDR_{B}/(\alpha m_{0}/m)=1$.
Another common bootstrap method is to estimate the $p$-values individually by $\breve{p}_{i,B}=G^{*}_{i}(T_{i})$, where $G^{*}_{i}(t)=\frac{1}{N}\sum_{k=1}^{N}I\{T^{*}_{ki}\geq t\}$; see Fan, Hall and Yao (2007) and Delaigle, Hall and Jin (2011). Similar results as Theorem \[th2-2\] can be obtained if $N$ is large enough (e.g. $N\geq m$). Note that in Theorem \[th2-2\], $N\geq 1$ is sufficient because we use the average of all $m$ variables.
Fan, Hall and Yao (2007) proved that, the bootstrap calibration is accurate for the control of family-wise error rate if $\log m=o(n^{1/2})$ and $\pr(|Y_{i}|\leq C)=1$ for $1\leq i\leq m$. Our result on FDR control only requires the sub-Gaussian tails which is weaker than the bounded noise.
[**Remark.**]{} [*The light-tailed moment condition for bootstrap calibration.*]{} The bootstrap method has often been used in multiple Student’s $t$ tests in real applications. Fan, Hall and Yao (2007) and Delaigle, Hall and Jin (2011) have proved that the bootstrap method provides a more accurate p-values than the normal or $t_{n-1}$ approximation for the light-tailed distributions. Theorem 2.2 shows that the bootstrap method allows a higher dimension $\log m=o(n^{1/2})$ for FDR control when $\max_{1\leq i\leq m}{\epsilon}e^{tY_{i}^{2}}\leq K$. However, it is not necessary that the real data would satisfy such light tailed condition. We argue that the light tailed condition can not be weakened in general when the bootstrap method is used. Denote the conditional tails of distribution of the bootstrap version for Student’s $t$ statistic by $G^{*}_{i}(t)=\pr(|T_{i}^{*}|\geq t|\mathcal{X})$, where $\mathcal{X}=\{\mathcal{X}_{1},\ldots,\mathcal{X}_{m}\}$. Giné, et al. (1997) proved that Student’s $t$ statistic is asymptotically normal if and only if the underlying distribution of the population is in the domain of attraction of the normal law. This implies any $\alpha$-th moment ($0<\alpha<2$) of the underlying distribution is finite. Hence, to ensure $G^{*}_{i}(t)\rightarrow 2-2\Phi(t)$, we often need $$\begin{aligned}
\label{a51}
\max_{1\leq i\leq m}\frac{1}{n}\sum_{k=1}^{n}|X_{ki}-\bar{X}_{i}|^{\alpha}\leq K\end{aligned}$$ for any $0<\alpha<2$. Suppose the components $X_{1},\ldots,X_{m}$ are independent and identically distributed and $\log m\asymp n^{\gamma}$, $\gamma>0$. A necessary condition for (\[a51\]) is ${\epsilon}\exp(t_{0}|X_{1}|^{\alpha \gamma})<\infty$ for some $t_{0}>0$. So when $\log m=o(n^{1/3})$, the bootstrap method requires a much more stringent moment condition than $N(0,1)$ or $t_{n-1}$ calibration. From the above analysis, we can see that the bootstrap calibration may not always outperform the $N(0,1)$ or $t_{n-1}$ calibration. In particular, when the distribution is symmetric, $N(0,1)$ and $t_{n-1}$ approximations can even perform better than the bootstrap method. This will be further verified by the simulation study in Section 5.
Regularized bootstrap in large scale tests
==========================================
In this section, we introduce a regularized bootstrap method that is robust for heavy-tailed distributions and the dimension $m$ can be as large as $e^{o(n^{1/2})}$. For the regularized bootstrap method, the finite 6th moment condition is enough. Let $\lambda_{ni}\rightarrow\infty$ be a regularized parameter. Define $$\begin{aligned}
\hat{X}_{ki}=X_{ki}I\{|X_{ki}|\leq \lambda_{ni}\}, \quad 1\leq k\leq n,\quad 1\leq i\leq m.\end{aligned}$$ Write $\hat{\mathcal{X}}_{i}=\{\hat{X}_{1i},\ldots,\hat{X}_{ni}\}$. Let $\hat{\mathcal{X}}^{*}_{ki}=\{\hat{X}^{*}_{1ki},\ldots,\hat{X}^{*}_{nki}\}$, $1\leq k\leq N$, be resamples drawn independently and uniformly with replacement from $\hat{\mathcal{X}}_{i}$. Let $\hat{T}^{*}_{ki}$ be Student’s $t$ test statistics constructed from $\{\hat{X}^{*}_{1ki}-\hat{X}_{i},\ldots,\hat{X}^{*}_{nki}-\hat{X}_{i}\}$, where $\hat{X}_{i}=\frac{1}{n}\sum_{k=1}^{n}\hat{X}_{ki}$. We use $\hat{G}^{*}(t)=\frac{1}{Nm}\sum_{k=1}^{N}\sum_{i=1}^{m}I\{|\hat{T}^{*}_{ki}|\geq t\}$ to approximate the null distribution and define the $p$-values by $\hat{p}_{i,RB}=\hat{G}^{*}(|T_{i}|)$. Let FDR$_{RB}$ be the FDR of B-H method with $\hat{p}_{i,RB}$ in (\[a0\]).
\[th2-222\] Assume that $\max_{1\leq i\leq m}{\epsilon}X_{i}^{6}\leq K$ for some constant $K>0$. Suppose $X_{1},\ldots,X_{m}$ are independent, (\[c1\]) holds and $\min_{1\leq i\leq m}\sigma_{ii}\geq c_{0}$ for some $c_{0}>0$. Let $c_{1}(n/\log m)^{1/6}\leq \lambda_{ni}\leq c_{2}(n/\log m)^{1/6}$ for some $c_{1},c_{2}>0$.
- Under $\log m=o(n^{1/3})$, we have $\lim_{(n,m)\rightarrow\infty} FDR_{RB}/(\alpha m_{0}/m)=1$.
- If $\log m=o(n^{1/2})$ and $m_{1}\leq m^{\eta}$ for some $\eta<1$, then $\lim_{(n,m)\rightarrow\infty} FDR_{RB}/(\alpha m_{0}/m)=1$.
In Theorem \[th2-222\], we only require $\max_{1\leq i\leq m}{\epsilon}X_{i}^{6}\leq K$, which is much weaker than the moment condition in Theorem 2.2.
In the regularized bootstrap method, we need to choose the regularized parameter $\lambda_{ni}$. By Theorem 1.2 in Wang (2005), equation (2.2) in Shao (1999) and the proof of Theorem 3.1, we have $$\begin{aligned}
\pr(|\hat{T}^{*}_{ki}|\geq t|\hat{\mathcal{X}})=\frac{1}{2}G(t)\Big{[}\exp\Big{(}\frac{t^{3}}{\sqrt{n}}\hat{\kappa}_{i}(\lambda_{ni})\Big{)}
+\exp\Big{(}-\frac{t^{3}}{\sqrt{n}}\hat{\kappa}_{i}(\lambda_{ni})\Big{)}\Big{]}(1+o_{\pr}(1)),\end{aligned}$$ uniformly for $0\leq t\leq o(n^{1/4})$, where $\hat{\mathcal{X}}=\{\hat{\mathcal{X}}_{1},\ldots,\hat{\mathcal{X}}_{m}\}$, $$\begin{aligned}
\label{a00}
\hat{\kappa}_{i}(\lambda_{ni})=\frac{1}{n\hat{\sigma}_{i}^{3}}\sum_{k=1}^{n}(\hat{X}_{ki}-\hat{X}_{i})^{3}\quad\mbox{and\quad}
\hat{\sigma}^{2}_{i}=\frac{1}{n}\sum_{k=1}^{n}(\hat{X}_{ki}-\hat{X}_{i})^{2}.\end{aligned}$$ Also, $$\begin{aligned}
\pr(|T_{i}|\geq t)=\frac{1}{2}G(t)\Big{[}\exp\Big{(}\frac{t^{3}}{\sqrt{n}}\kappa_{i}\Big{)}
+\exp\Big{(}-\frac{t^{3}}{\sqrt{n}}\kappa_{i}\Big{)}\Big{]}(1+o(1)),\end{aligned}$$ uniformly for $0\leq t\leq o(n^{1/4})$, where $\kappa_{i}={\epsilon}Y_{i}^{3}$. A good choice of $\lambda_{ni}$ is to make $\hat{\kappa}_{i}(\lambda_{ni})$ get close to $\kappa_{i}$. As $\kappa_{i}$ is unknown, we propose the following cross-validation method.
[**Data-driven choice of $\lambda_{ni}$.**]{} We propose to choose $\hat{\lambda}_{ni}=|\bar{X}_{i}|+\hat{s}_{ni}\lambda$, where $\lambda$ will be selected as follow. Split the samples into two parts $\mathcal{I}_{0}=\{1,\ldots, n_{1}\}$ and $\mathcal{I}_{1}=\{n_{1}+1,\ldots, n\}$ with sizes $n_{0}=[n/2]$ and $n_{1}=n-n_{0}$ respectively. For $\mathcal{I}=\mathcal{I}_{0}$ or $\mathcal{I}_{1}$, let $$\begin{aligned}
\hat{\kappa}_{i,\mathcal{I}}=\frac{1}{|\mathcal{I}|\hat{s}_{ni,\mathcal{I}}^{3}}\sum_{k\in\mathcal{I}}(X_{ki}-\bar{X}_{i,\mathcal{I}})^{3},\quad
\hat{s}^{2}_{ni,\mathcal{I}}=\frac{1}{|\mathcal{I}|}\sum_{k\in\mathcal{I}}(X_{ki}-\bar{X}_{i,\mathcal{I}})^{2},\quad
\bar{X}_{i,\mathcal{I}}=\frac{1}{|\mathcal{I}|}\sum_{k\in\mathcal{I}}X_{ki}.\end{aligned}$$ Let $\hat{\kappa}_{i,\mathcal{I}}(\lambda_{ni})$, with $\lambda_{ni}=|\bar{X}_{i,\mathcal{I}}|+\hat{s}_{ni,\mathcal{I}}\lambda$, be defined as in (\[a00\]) based on $\{\hat{X}_{ki}, k\in\mathcal{I}\}$. Define the risk $$\begin{aligned}
R_{j}(\lambda)=\sum_{i=1}^{m}(\hat{\kappa}_{i,\mathcal{I}_{j}}(\lambda_{ni})-\hat{\kappa}_{i,\mathcal{I}_{1-j}})^{2}.\end{aligned}$$ We choose $\lambda$ by $$\begin{aligned}
\label{a000}
\hat{\lambda}={\mathop{\rm arg\min}}_{0<\lambda<\infty}\{R_{0}(\lambda)+R_{1}(\lambda)\}.\end{aligned}$$ The final regularized parameter is $\hat{\lambda}_{ni}=|\bar{X}_{i}|+\hat{s}_{ni}\hat{\lambda}$.
It is important to investigate the theoretical property of $\hat{\lambda}_{ni}$ and to see whether Theorem 3.1 still hold when $\hat{\lambda}_{ni}$ is used. We leave this as a future work.
FDR control under dependence
============================
To generalize the results to the dependent case, we introduce a class of correlation matrices. Let $\A=(a_{ij})$ be a symmetric matrix. Let $k_m$ and $s_m$ be positive numbers. Assume that for every $1\leq j\leq m$, $$\begin{aligned}
\label{a50}
\text{Card}\{1\leq i\leq m: |a_{ij}|\geq k_{m}\}\leq s_{m}.\end{aligned}$$ Let $\mathcal{A}(k_{m},s_{m})$ be the class of symmetric matrices satisfying (\[a50\]). Let $\R=(r_{ij})$ be the correlation matrix of $\X$. We introduce the following two conditions.
- Suppose that $\max_{1\leq j<j\leq m}|r_{ij}|\leq r$ for some $0<r<1$ and $\R\in \mathcal{A}(k_{m},s_{m})$ with $k_{m}=(\log m)^{-2-\delta}$ and $s_{m}=O(m^{\rho})$ for some $\delta>0$ and $0<\rho<(1-r)/(1+r)$.
- Suppose that $\max_{1\leq j<j\leq p}|r_{ij}|\leq r$ for some $0<r<1$. For each $X_{i}$, assume the number of variables $X_{j}$ which are dependent with $X_{i}$ is no more than $s_{m}$.
(C1) and (C1$^{*}$) impose the weak dependence between $X_{1},\ldots,X_{m}$. In (C1), each variable can be highly correlated with other $s_{m}$ variables and weakly correlated with the remaining variables. (C1$^{*}$) is stronger than (C1). For each $X_{i}$, (C1$^{*}$) requires the independence between $X_{i}$ and other $m-s_{m}$ variables.
Recall that $m_{1}\leq \gamma m$ for some $\gamma<1$.
\[th21\] Assume that $\max_{1\leq i\leq m}{\epsilon}Y_{i}^{4}\leq {b}_{0}$ for some constant ${b}_{0}>0$ and (\[c1\]) holds.
- Under $\log m=O(n^{\zeta})$ for some $0<\zeta<3/23$ and (C1), we have $$\begin{aligned}
\label{th3}
\lim_{(n,m)\rightarrow\infty} \frac{FDR_{\Phi}}{\frac{m_{0}}{m}\alpha}=1,\quad \lim_{(n,m)\rightarrow\infty} \frac{FDR_{\Psi}}{\frac{m_{0}}{m}\alpha}=1\end{aligned}$$
- Under $\log m=o(n^{1/3})$ and (C1$^{*}$), we have (\[th3\]) holds.
For the bootstrap and regularized procedures, we have the similar results.
\[th22\]Suppose that $\max_{1\leq i\leq m}{\epsilon}e^{tY_{i}^{2}}\leq K$ and (\[c1\]) holds.
- Under the conditions of (i) or (ii) in Theorem \[th21\], we have $\lim_{(n,m)\rightarrow\infty} \frac{FDR_{B}}{\frac{m_{0}}{m}\alpha}=1$
- Under (C1$^{*}$), $\log m=o(n^{1/2})$ and $m_{1}\leq m^{\eta}$ for some $\eta<1$, we have $$\lim_{(n,m)\rightarrow\infty} \frac{FDR_{B}}{\frac{m_{0}}{m}\alpha}=1 .$$
\[th222\]Suppose that $\max_{1\leq i\leq m}{\epsilon}X_{i}^{6}\leq K$ for some constant $K>0$, $\min_{1\leq i\leq m}\sigma_{ii}\geq c_{0}$ for some $c_{0}>0$ and (\[c1\]) holds. Let $c_{1}(n/\log m)^{1/6}\leq \lambda_{ni}\leq c_{2}(n/\log m)^{1/6}$ for some $c_{1},c_{2}>0$.
- Under the conditions of (i) or (ii) in Theorem \[th21\], we have $\lim_{(n,m)\rightarrow\infty} \frac{FDR_{RB}}{\frac{m_{0}}{m}\alpha}=1$
- Under (C1$^{*}$), $\log m=o(n^{1/2})$ and $m_{1}\leq m^{\eta}$ for some $\eta<1$, we have $$\lim_{(n,m)\rightarrow\infty} \frac{FDR_{RB}}{\frac{m_{0}}{m}\alpha}=1 .$$
Theorems \[th21\]-\[th222\] imply that B-H method remains valid asymptotically for weak dependence. As the phase transition phenomenon caused by the growth of the dimension, it would be interesting to investigate when will B-H method fail to control the FDR as the correlation becomes strong.
Numerical Study
===============
In this section, we first carry out a small simulation to verify the phase transition phenomenon. Let $$\begin{aligned}
\label{a5-0}
X_{i}=\mu_{i}+(\varepsilon_{i}-{\epsilon}\varepsilon_{i}),\quad 1\leq i\leq m,\end{aligned}$$ where $(\varepsilon_{1},\ldots,\varepsilon_{m})^{'}$ are i.i.d. random variables. We consider two models for $\varepsilon_{i}$ and $\mu_{i}$.
[**Model 1.**]{} $\varepsilon_{i}$ is the exponential random variable with parameter 1. Let $\mu_{i}=2\sigma\sqrt{\log m/n}$ for $1\leq i\leq m_{1}$ with $m_{1}=0.05 m$ and $\mu_{i}=0$ for $m_{1}<i\leq m$, where $\sigma^{2}=\Var(\varepsilon_{i})$.
[**Model 2.**]{} $\varepsilon_{i}$ is the Gamma random variable with parameter (0.5,1). Let $\mu_{i}=4\sigma\sqrt{\log m/n}$ for $1\leq i\leq m_{1}$ with $m_{1}=0.05 m$ and $\mu_{i}=0$ for $m_{1}<i\leq m$.
In both models, the average of skewnesses $\tau>0$. We generate $n=30, \ 50$ independent random samples from (\[a5-0\]). In our simulation, $\alpha$ is taken to be $0.1,0.2,0.3$ and $m$ is taken to be $500$, $1000$, $3000$. In the usual bootstrap approximation and the regularized bootstrap approximation, the resampling time $N$ is taken to be 200. The simulation is replicated 500 times and the empirical FDR and power are summarized in Tables 1 and 2. The empirical power is defined by the average ratio between the number of correct rejections and $m_{1}$. As we can see, due to the nonzero skewnesses and $m\gg\exp(n^{1/3})$, the empirical FDR$_{\Phi}$ and FDR$_{\Psi}$ are much larger than the target FDR. The bootstrap method and the regularized bootstrap method provide more accurate approximations for the true p-values. So the empirical FDR$_{B}$ and FDR$_{RB}$ are much closer to $\alpha$ than FDR$_{\Phi}$ and FDR$_{\Psi}$ do. For Models 1 and 2, the bootstrap method and the proposed regularized bootstrap method perform quite similarly. All of four methods perform better as the sample size $n$ grows from 30 to 50, although the empirical FDR$_{\Phi}$ and FDR$_{\Psi}$ still have seriously departure from $\alpha$.
Next, we consider the following two models to compare the performance between the four methods when the distributions are symmetric and heavy tailed.
[**Model 3.**]{} $\varepsilon_{i}$ is Student’s $t$ distribution with 4 degrees of freedom. Let $\mu_{i}=2\sqrt{\log m/n}$ for $1\leq i\leq m_{1}$ with $m_{1}=0.1 m$ and $\mu_{i}=0$ for $m_{1}<i\leq m$.
[**Model 4.**]{} $\varepsilon_{i}=\varepsilon_{i1}-\varepsilon_{i2}$, where $\varepsilon_{i1}$ and $\varepsilon_{i1}$ are independent lognormal random variables with parameters $(0,1)$. Let $\mu_{i}=4\sqrt{\log m/n}$ for $1\leq i\leq m_{1}$ with $m_{1}=0.1 m$ and $\mu_{i}=0$ for $m_{1}<i\leq m$.
For these two models, the normal approximation performs the best on the control of FDR; see Tables 3 and 4. FDR$_{B}$ is much smaller than $\alpha$ so the bootstrap method is quite conservative. This is mainly due to the heavy tails of the $t(4)$ and lognormal distributions. The regularized bootstrap method works much better than the bootstrap method on the FDR control. From Table 4, we see that it also has the higher powers (power$_{RB}$) than the bootstrap method (power$_{B}$). Hence, the proposed regularized bootstrap is more robust than the commonly used bootstrap method.
------ --------------------- -------- -------- -------- -------- -------- --------
$m$ $ |\quad { \alpha}$ 0.1 0.2 0.3 0.1 0.2 0.3
500 FDR$_{\Phi}$ 0.3746 0.4670 0.5422 0.2898 0.3913 0.4738
FDR$_{\Psi}$ 0.3081 0.4085 0.4863 0.2482 0.3501 0.4357
FDR$_{B}$ 0.0649 0.1730 0.2778 0.0912 0.1869 0.2845
FDR$_{RB}$ 0.0675 0.1761 0.2860 0.0885 0.1877 0.2851
1000 FDR$_{\Phi}$ 0.3762 0.4717 0.5461 0.2916 0.3962 0.4810
FDR$_{\Psi}$ 0.3097 0.4113 0.4919 0.2488 0.3561 0.4404
FDR$_{B}$ 0.0695 0.1771 0.2860 0.0916 0.1934 0.2906
FDR$_{RB}$ 0.0675 0.1765 0.2864 0.0919 0.1921 0.2909
3000 FDR$_{\Phi}$ 0.3811 0.4785 0.5517 0.2944 0.3987 0.4818
FDR$_{\Psi}$ 0.3129 0.4178 0.4978 0.2510 0.3580 0.4432
FDR$_{B}$ 0.0703 0.1810 0.2865 0.0931 0.1942 0.2938
FDR$_{RB}$ 0.0692 0.1775 0.2850 0.0936 0.1928 0.2922
500 FDR$_{\Phi}$ 0.4973 0.5780 0.6339 0.3963 0.4903 0.5601
FDR$_{\Psi}$ 0.4436 0.5333 0.5981 0.3593 0.4567 0.5301
FDR$_{B}$ 0.0738 0.1751 0.2827 0.0842 0.1853 0.2939
FDR$_{RB}$ 0.0755 0.1758 0.2943 0.0883 0.1882 0.2941
1000 FDR$_{\Phi}$ 0.5019 0.5810 0.6368 0.3992 0.4929 0.5617
FDR$_{\Psi}$ 0.4480 0.5382 0.6019 0.3624 0.4605 0.5322
FDR$_{B}$ 0.0753 0.1758 0.2867 0.0879 0.1883 0.2932
FDR$_{RB}$ 0.0688 0.1740 0.2823 0.0859 0.1902 0.2926
3000 FDR$_{\Phi}$ 0.5025 0.5813 0.6375 0.4023 0.4952 0.5634
FDR$_{\Psi}$ 0.4483 0.5386 0.6021 0.3647 0.4636 0.5351
FDR$_{B}$ 0.0737 0.1769 0.2873 0.0864 0.1909 0.2948
FDR$_{RB}$ 0.0723 0.1741 0.2847 0.0854 0.1878 0.2911
------ --------------------- -------- -------- -------- -------- -------- --------
: Comparison of FDR (FDR=$\alpha$)
\[tb:simu3\]
------ --------------------- -------- -------- -------- -------- -------- --------
$m$ $ |\quad { \alpha}$ 0.1 0.2 0.3 0.1 0.2 0.3
500 power$_{\Phi}$ 0.9998 1.0000 1.0000 0.9999 1.0000 1.0000
power$_{\Psi}$ 0.9995 0.9998 1.0000 0.9997 1.0000 1.0000
power$_{B}$ 0.7473 0.9852 0.9990 0.9831 0.9986 0.9998
power$_{RB}$ 0.7371 0.9848 0.9989 0.9839 0.9981 0.9994
1000 power$_{\Phi}$ 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{\Psi}$ 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{B}$ 0.8873 0.9943 0.9991 0.9945 0.9998 1.0000
power$_{RB}$ 0.8880 0.9936 0.9995 0.9942 0.9996 0.9999
3000 power$_{\Phi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{\Psi}$ 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{B}$ 0.9642 0.9984 0.9999 0.9987 1.0000 1.0000
power$_{RB}$ 0.9650 0.9983 0.9999 0.9989 0.9999 1.0000
500 power$_{\Phi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{\Psi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{B}$ 0.9986 0.9999 1.0000 1.0000 1.0000 1.0000
power$_{RB}$ 0.9982 0.9950 0.9994 1.0000 1.0000 1.0000
1000 power$_{\Phi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{\Psi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{B}$ 0.9988 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{RB}$ 0.9584 0.9978 0.9998 1.0000 1.0000 1.0000
3000 power$_{\Phi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{\Psi}$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{B}$ 0.9994 1.0000 1.0000 1.0000 1.0000 1.0000
power$_{RB}$ 0.9822 0.9988 0.9999 1.0000 1.0000 1.0000
------ --------------------- -------- -------- -------- -------- -------- --------
: Comparison of power (FDR=$\alpha$)
\[tb:simu3\]
------ --------------------- -------- -------- -------- -------- -------- --------
$m$ $ |\quad { \alpha}$ 0.1 0.2 0.3 0.1 0.2 0.3
500 FDR$_{\Phi}$ 0.1147 0.2129 0.3082 0.1006 0.1958 0.2900
FDR$_{\Psi}$ 0.0704 0.1536 0.2442 0.0741 0.1600 0.2514
FDR$_{B}$ 0.0358 0.1112 0.1991 0.0438 0.1214 0.2022
FDR$_{RB}$ 0.0612 0.1435 0.2348 0.0693 0.1565 0.2448
1000 FDR$_{\Phi}$ 0.1170 0.2153 0.3083 0.1014 0.1968 0.2905
FDR$_{\Psi}$ 0.0705 0.1571 0.2472 0.0756 0.1618 0.2532
FDR$_{B}$ 0.0341 0.1072 0.1904 0.0511 0.1333 0.2242
FDR$_{RB}$ 0.0593 0.1432 0.2324 0.0718 0.1584 0.2507
3000 FDR$_{\Phi}$ 0.1166 0.2150 0.3093 0.1014 0.1964 0.2908
FDR$_{\Psi}$ 0.0724 0.1572 0.2485 0.0756 0.1623 0.2539
FDR$_{B}$ 0.0369 0.1090 0.1944 0.0547 0.1343 0.2225
FDR$_{RB}$ 0.0609 0.1433 0.2337 0.0722 0.1599 0.2512
500 FDR$_{\Phi}$ 0.0810 0.1693 0.2667 0.0761 0.1617 0.2560
FDR$_{\Psi}$ 0.0432 0.1123 0.1964 0.0519 0.1297 0.2144
FDR$_{B}$ 0.0005 0.0103 0.0425 0.0059 0.0384 0.0960
FDR$_{RB}$ 0.0300 0.0919 0.1697 0.0466 0.1187 0.2086
1000 FDR$_{\Phi}$ 0.0799 0.1701 0.2657 0.0760 0.1628 0.2572
FDR$_{\Psi}$ 0.0433 0.1133 0.1962 0.0521 0.1296 0.2165
FDR$_{B}$ 0.0004 0.0137 0.0504 0.0064 0.0418 0.1032
FDR$_{RB}$ 0.0339 0.0953 0.1748 0.0485 0.1237 0.2083
3000 FDR$_{\Phi}$ 0.0805 0.1704 0.2654 0.0749 0.1629 0.2578
FDR$_{\Psi}$ 0.0442 0.1142 0.1982 0.0523 0.1283 0.2179
FDR$_{B}$ 0.0008 0.0151 0.0507 0.0070 0.0432 0.1052
FDR$_{RB}$ 0.0319 0.0952 0.1766 0.0488 0.1239 0.2129
------ --------------------- -------- -------- -------- -------- -------- --------
: Comparison of FDR (FDR=$\alpha$)
\[tb:simu3\]
------ --------------------- -------- -------- -------- -------- -------- --------
$m$ $ |\quad { \alpha}$ 0.1 0.2 0.3 0.1 0.2 0.3
500 power$_{\Phi}$ 0.8305 0.8890 0.9190 0.8266 0.8853 0.9173
power$_{\Psi}$ 0.7782 0.8576 0.9007 0.7968 0.8684 0.9058
power$_{B}$ 0.6901 0.8190 0.8746 0.7582 0.8574 0.8984
power$_{RB}$ 0.7554 0.8439 0.8916 0.7908 0.8676 0.9072
1000 power$_{\Phi}$ 0.8633 0.9113 0.9369 0.8648 0.9144 0.9403
power$_{\Psi}$ 0.8200 0.8869 0.9208 0.8389 0.8998 0.9315
power$_{B}$ 0.7472 0.8477 0.8977 0.8050 0.8838 0.9219
power$_{RB}$ 0.8021 0.8788 0.9161 0.8357 0.8992 0.9305
3000 power$_{\Phi}$ 0.9078 0.9413 0.9589 0.9091 0.9434 0.9605
power$_{\Psi}$ 0.8768 0.9249 0.9485 0.8915 0.9339 0.9549
power$_{B}$ 0.8305 0.9053 0.9384 0.8755 0.9293 0.9533
power$_{RB}$ 0.8651 0.9203 0.9455 0.8913 0.9350 0.9555
500 power$_{\Phi}$ 0.7916 0.8453 0.8796 0.7789 0.8390 0.8764
power$_{\Psi}$ 0.7424 0.8165 0.8561 0.7507 0.8209 0.8615
power$_{B}$ 0.3216 0.6267 0.7404 0.5426 0.7275 0.8037
power$_{RB}$ 0.7217 0.8044 0.8486 0.7479 0.8203 0.8623
1000 power$_{\Phi}$ 0.8240 0.8703 0.8989 0.8156 0.8669 0.8975
power$_{\Psi}$ 0.7842 0.8444 0.8795 0.7899 0.8506 0.8859
power$_{B}$ 0.4340 0.6978 0.7898 0.6320 0.7749 0.8379
power$_{RB}$ 0.7647 0.8343 0.8715 0.7869 0.8499 0.8859
3000 power$_{\Phi}$ 0.8634 0.9003 0.9224 0.8610 0.9021 0.9257
power$_{\Psi}$ 0.8314 0.8805 0.9079 0.8415 0.8895 0.9169
power$_{B}$ 0.5880 0.7688 0.8386 0.7192 0.8300 0.8780
power$_{RB}$ 0.8140 0.8711 0.9018 0.8374 0.8865 0.9149
------ --------------------- -------- -------- -------- -------- -------- --------
: Comparison of power (FDR=$\alpha$)
\[tb:simu3\]
Proof of Main Results
=====================
By Theorem 1.2 in Wang (2005) and equation (2.2) in Shao (1999), we have for $0\leq t\leq o(n^{1/4})$, $$\begin{aligned}
\label{le7}
\pr(|T_{i}-\sqrt{n}\mu_{i}/\hat{s}_{n}|\geq t)=\frac{1}{2}G(t)\Big{[}\exp\Big{(}-\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}+
\exp\Big{(}\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}\Big{]}(1+o(1)),\end{aligned}$$ where $o(1)$ is uniformly in $1\leq i\leq m$, $G(t)=2-2\Phi(t)$ and $\kappa_{i}={\epsilon}Y_{i}^{3}$.
For any $b_{m}\rightarrow\infty$ and $b_{m}=o(m)$, we first prove that, under (C1$^{*}$) and $\log m=o(n^{1/2})$ (or (C1) and $\log m=O(n^{\zeta})$ for some $0<\zeta<3/23$), $$\begin{aligned}
\label{aa133}
\sup_{0\leq t\leq G^{-1}_{\kappa}(b_{m}/m)}\Big{|}\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq t\}}{m_{0}G_{\kappa}(t)}-1\Big{|}\rightarrow 0\end{aligned}$$ in probability, where $$\begin{aligned}
G_{\kappa}(t)=\frac{1}{2m_{0}}G(t)\sum_{i\in\mathcal{H}_{0}}\Big{[}\exp\Big{(}-\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}+
\exp\Big{(}\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}\Big{]}=:G(t)\hat{\kappa}_{\Phi}(t)\end{aligned}$$ and $G^{-1}_{\kappa}(t)=\inf\{y\geq 0: G_{\kappa}(y)=t\}$ for $0\leq t\leq 1$. Note that for $0\leq t\leq o(\sqrt{n})$, $G_{\kappa}(t)$ is a strictly decreasing and continuous function. Let $z_{0}<z_{1}<\cdots<z_{d_{m}}\leq 1$ and $t_{i}=G^{-1}_{\kappa}(z_{i})$, where $z_{0}=b_{m}/m$, $z_{i}=b_{m}/m+b^{2/3}_{m}e^{i^{\delta}}/m$, $d_{m}=[\{\log ((m-b_{m})/b^{2/3}_{m})\}^{1/\delta}]$ and $0<\delta<1$ which will be specified later. Note that $G_{\kappa}(t_{i})/G_{\kappa}(t_{i+1})=1+o(1)$ uniformly in $i$, and $t_{0}/\sqrt{2\log (m/b_{m})}=1+o(1)$. Then, to prove (\[aa133\]), it is enough to show that $$\begin{aligned}
\label{aa13}
\sup_{0\leq j\leq d_{m}}\Big{|}\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq t_{j}\}}{m_{0}G_{\kappa}(t_{j})}-1\Big{|}\rightarrow 0\end{aligned}$$ in probability. Under (C1), define $$\begin{aligned}
\mathcal{S}_{j}=\{i\in\mathcal{H}_{0}: |r_{ij}|\geq (\log m)^{-1-\gamma}\},\quad \mathcal{S}_{j}^{c}=\mathcal{H}_{0}-\mathcal{S}_{j},\end{aligned}$$ and under (C1$^{*}$), define $$\begin{aligned}
\mathcal{S}_{j}=\{i\in\mathcal{H}_{0}: \mbox{$X_{i}$ is dependent with $X_{j}$}\}.\end{aligned}$$ We claim that, under (C1$^{*}$) and $\log m=o(n^{1/2})$ (or (C1) and $\log m=O(n^{\zeta})$ for some $0<\zeta<3/23$), for any $\varepsilon>0$ and some $\gamma_{1}>0$, $$\begin{aligned}
\label{a1}
I_{2}(t)&:=&{\epsilon}\Big{(}\sum_{i\in \mathcal{H}_{0}}\{I\{T_{i}\geq t\}-\pr(|T_{i}|\geq t)\}\Big{)}^{2}\cr
&\leq& Cm^{2}_{0}G^{2}_{\kappa}(t)\Big{(}\frac{1}{m_{0}G_{\kappa}(t)}+\frac{\exp\Big{(}(r+\varepsilon)t^{2}/(1+r)\Big{)}}{m^{1-\rho}}+(\log m)^{-1-\gamma_{1}}\Big{)}\end{aligned}$$ uniformly in $t\in[0, K\sqrt{\log m}]$ for all $K>0$. Take $(1+\gamma_{1})^{-1}<\delta<1$. By (\[a1\]) and $G^{-1}_{\kappa}(b_{m}/m)\sim \sqrt{2\log (m/b_{m})}$, for any $\varepsilon>0$, we have $$\begin{aligned}
&&\sum_{j=0}^{d_{m}}\pr\Big{(}\Big{|}\frac{\sum_{i\in \mathcal{H}_{0}}I\{T_{i}\geq t_{j}\}}{m_{0}G_{\kappa}(t_{j})}-1\Big{|}\geq \varepsilon\Big{)}\cr
&&\leq \sum_{j=0}^{d_{m}}\pr\Big{(}\Big{|}\frac{\sum_{i\in \mathcal{H}_{0}}(I\{T_{i}\geq t_{j}\}-\pr(|T_{i}|\geq t_{j})}{m_{0}G_{\kappa}(t_{j})}\Big{|}\geq
\varepsilon/2\Big{)}\cr
&&\leq C\Big{(}\frac{1}{m_{0}G_{\kappa}(t_{0})}+\sum_{j=1}^{d_{m}}\frac{1}{m_{0}G_{\kappa}(t_{j})}+d_{m}m^{-1+\rho+\frac{2r+2\varepsilon}{1+r}+o(1)}
+d_{m}(\log m)^{-1-\gamma_{1}}\Big{)}\cr
&&\leq C\Big{(}b_{m}^{-1}+b_{m}^{-2/3}\sum_{j=1}^{d_{m}}e^{-j^{\delta}}+o(1)\Big{)}=o(1).\end{aligned}$$ This prove (\[aa13\])
To prove (\[a1\]), we need the following lemma which will be proved in the supplementary file.
\[le1\] (i). Suppose that $\log m=O(n^{1/2})$. For any $\varepsilon>0$, $$\begin{aligned}
\label{a8}
\max_{j\in\mathcal{H}_{0}}\max_{i\in \mathcal{S}_{j}\setminus j}\pr\Big{(}|T_{i}|\geq t,|T_{j}|>t\Big{)}\leq C\exp(-(1-\varepsilon)t^{2}/(1+r))\end{aligned}$$ uniformly in $t\in[0,o(n^{1/4}))$.
(ii). Suppose that $\log m=O(n^{\zeta})$ for some $0<\zeta<3/23$. We have for any $K>0$ $$\begin{aligned}
\label{a9}
\pr\Big{(}|T_{i}|>t,|T_{j}|>t\Big{)}=(1+A_{n})\pr(|T_{i}|>t)\pr(|T_{j}|>t)\end{aligned}$$ uniformly in $0\leq t\leq K\sqrt{\log m}$, $j\in\mathcal{H}_{0}$ and $i\in \mathcal{S}_{j}^{c}$, where $
|A_{n}|\leq C(\log m)^{-1-\gamma_{1}}
$ for some $\gamma_{1}>0$.
Set $f_{ij}(t)=\pr\Big{(}|T_{i}|\geq t,|T_{j}|\geq t\Big{)}-\pr\Big{(}|T_{i}|\geq t)\pr\Big{(}|T_{j}|\geq t\Big{)}$. Note that under (C1$^{*}$) $f_{ij}=0$ when $j\in\mathcal{H}_{0}\backslash\mathcal{S}_{i}$. We have $$\begin{aligned}
I_{2}(t)&\leq& \sum_{i\in\mathcal{H}_{0}}\sum_{j\in\mathcal{S}_{i}}\pr\Big{(}|T_{i}|\geq t,|T_{j}|\geq t\Big{)}
+\sum_{i\in\mathcal{H}_{0}}\sum_{j\in\mathcal{H}_{0}\backslash\mathcal{S}_{i}}f_{ij}(t)\cr
&\leq& Cm_{0}G_{\kappa}(t)+C\frac{\exp\Big{(}(r+2\varepsilon)t^{2}/(1+r)\Big{)}}{m^{1-\rho}}m^{2}_{0}G^{2}_{\kappa}(t)+A_{n}m^{2}_{0}G^{2}_{\kappa}(t),\end{aligned}$$ where the last inequality follows from Lemma 6.1 and $G_{\kappa}(t)=G(t)e^{o(1)t^{2}}$ for $t=o(\sqrt{n})$. This proves (\[a1\]).
Proof of Theorem \[th1\] and Corollary \[co1\]
----------------------------------------------
We only prove the theorem for $\hat{p}_{i,\Phi}$. The proof for $\hat{p}_{i,\Psi}$ is exactly the same by replacing $G(t)$ with $2-2\Psi(t)$. By Lemma 1 in Storey, Taylor and Siegmund (2004), we can see that B-H method with $\hat{p}_{i,\Phi}$ is equivalent to the following procedure: reject $H_{0i}$ if and only if $\hat{p}_{i,\Phi}\leq \hat{t}_{0}$, where $$\begin{aligned}
\hat{t}_{0}=\sup\Big{\{}0\leq t\leq 1:~ t\leq \frac{\alpha\max(\sum_{1\leq i\leq m}I\{\hat{p}_{i,\Phi}\leq t\},1)}{m}\Big{\}}.\end{aligned}$$ It is equivalent to reject $H_{0i}$ if and only if $|T_{i}|\geq \hat{t}$, where $$\begin{aligned}
\hat{t}=\inf\Big{\{}t\geq 0:~ 2-2\Phi(t)\leq \frac{\alpha\max(\sum_{1\leq i\leq m}I\{|T_{i}|\geq t\},1)}{m}\Big{\}}.\end{aligned}$$ By the continuity of $\Phi(t)$ and the monotonicity of the indicator function, it is easy to see that $$\begin{aligned}
\frac{mG(\hat{t})}{\max(\sum_{1\leq i\leq m}I\{|T_{i}|\geq \hat{t}\},1)}=\alpha,\end{aligned}$$ where $G(t)=2-2\Phi(t)$. Let $\mathcal{M}$ be a subset of $\{1,2,\ldots,m\}$ satisfying $\mathcal{M}\subset \Big{\{}i: |\mu_{i}/\sigma_{i}|\geq 4\sqrt{\log m/n}\Big{\}}$ and Card$(\mathcal{M})\leq \sqrt{n}$. By $\max_{1\leq i\leq m}{\epsilon}Y_{i}^{4}\leq K$ and Markov’s inequality, for any $\varepsilon>0$, $$\begin{aligned}
\pr(\max_{i\in\mathcal{M}}|\hat{s}^{2}_{ni}/\sigma_{i}^{2}-1|\geq \varepsilon)=O(1/\sqrt{n}).\end{aligned}$$ This, together with (\[c1\]) and (\[le7\]), implies that there exist some $c>\sqrt{2}$ and some $b_{m}\rightarrow\infty$, $$\begin{aligned}
\label{tv2}
\pr\Big{(}\sum_{i=1}^{m}I\{|T_{i}|\geq c\sqrt{\log m}\}\geq b_{m}\Big{)}\rightarrow 1.\end{aligned}$$ This implies that $
\pr\Big{(}\hat{t}\leq G^{-1}(\alpha b_{m}/m)\Big{)}\rightarrow 1
$ and $\pr(\hat{m}\geq b_{m})\rightarrow 1$. By (\[aa133\]) and $G_{\kappa}(t)\geq G(t)$, it follows that $\pr(\hat{t}\leq G_{\kappa}^{-1}(\alpha b_{m}/m))\rightarrow 1$. Therefore, by (\[aa133\]) $$\begin{aligned}
\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq \hat{t}\}}{m_{0}G_{\kappa}(\hat{t})}\rightarrow 1\end{aligned}$$ in probability. Note that $$\begin{aligned}
G(\hat{t})=\frac{\alpha\hat{m}}{m}+\frac{\alpha m_{0}}{m}\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq \hat{t}\}}{m_{0}},\end{aligned}$$ where $\hat{m}=\sum_{i\in\mathcal{H}_{1}}I\{|T_{i}|\geq \hat{t}\}$. With probability tending to one, $$\begin{aligned}
\label{prf3}
G(\hat{t})=\frac{\alpha\hat{m}}{m}+\frac{\alpha m_{0}}{m}G(\hat{t})\hat{\kappa}_{\Phi}(1+o(1))\geq \frac{\alpha m_{0}}{m}G(\hat{t})\hat{\kappa}_{\Phi}(1+o(1)).\end{aligned}$$ So $\pr(\hat{\kappa}_{\Phi}\leq m/(\alpha m_{0})+\varepsilon)\rightarrow 1$ for any $\varepsilon>0$. Let $\hat{\kappa}^{*}_{\Phi}=\hat{\kappa}_{\Phi}I\{\hat{\kappa}_{\Phi}\leq 2(\alpha(1-\gamma))^{-1})\}$. Note that $m/(\alpha m_{0})+\varepsilon\leq 2(\alpha(1-\gamma))^{-1}$. We have $$\begin{aligned}
\frac{FDP_{\Phi}}{\frac{m_{0}}{m}\alpha \hat{\kappa}_{\Phi}^{*}}=\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq \hat{t}\}}{m_{0}G_{\kappa}(\hat{t})}\frac{\hat{\kappa}_{\Phi}}{\hat{\kappa}_{\Phi}^{*}}(1+o(1))\rightarrow 1\end{aligned}$$ in probability, where $FDP_{\Phi}$ is the false discovery proportion $\text{V}/(\text{R}\vee 1)$. Then for any $\varepsilon>0$, $$\begin{aligned}
FDR_{\Phi}\leq (1+\varepsilon)\frac{m_{0}}{m}\alpha{\epsilon}\hat{\kappa}_{\Phi}^{*}+\pr\Big{(}FDP_{\Phi}\geq (1+\varepsilon)\frac{m_{0}}{m}\alpha \hat{\kappa}_{\Phi}^{*}\Big{)}\end{aligned}$$ and $$\begin{aligned}
FDR_{\Phi}\geq (1-\varepsilon)\frac{m_{0}}{m}\alpha{\epsilon}\hat{\kappa}_{\Phi}^{*}-2(\alpha(1-\gamma))^{-1}\pr\Big{(}FDP_{\Phi}\leq (1-\varepsilon)\frac{m_{0}}{m}\alpha \hat{\kappa}_{\Phi}^{*}\Big{)}.\end{aligned}$$ This proves the Theorem 2.1. Corollary \[co1\] (1) follows directly from Theorem 2.1 and $\pr(\hat{t}\leq \sqrt{2\log m})\rightarrow 1$.
To prove Corollary \[co1\] (2), we first assume that $\frac{\alpha m_{0}}{m}\hat{\kappa}_{\Phi}\leq 1-\eta$ for some $(1-\eta)/\alpha>1$. So, by (\[prf3\]) and the condition $m_{1}=\exp(o(n^{1/3}))$, with probability tending to one, $G(\hat{t})\leq 2\alpha\eta^{-1}\hat{m}/m\leq 2\alpha\eta^{-1}m^{-1+o(1)}$. Hence $\hat{t}\geq c\sqrt{\log m}$ for any $c<\sqrt{2}$. Recall that $\tau=\liminf_{m\rightarrow\infty}m_{0}^{-1}\sum_{i\in\mathcal{H}_{0}}|{\epsilon}Y_{i}^{3}|>0$. Set $$\begin{aligned}
\mathcal{H}_{01}=\{i\in\mathcal{H}_{0}: |{\epsilon}Y_{i}^{3}|\geq \tau/8\}.\end{aligned}$$ By the definition of $\tau$ and $|{\epsilon}Y_{i}^{3}|\leq ({\epsilon}(Y_{i}^{4})^{3/4}\leq b_{0}^{3/4}$, $
m_{0}^{-1}|\mathcal{H}^{c}_{01}|\tau/8+b_{0}^{3/4}m_{0}^{-1}|\mathcal{H}_{01}|\geq \tau/2.
$ This implies that $|\mathcal{H}_{01}|\geq \tau b_{0}^{-3/4}m_{0}/4$. Hence we can get $m_{0}^{-1}\sum_{i\in\mathcal{H}_{0}}|{\epsilon}Y_{i}^{3}|^{2}\geq c_{\tau}$ for some $c_{\tau}>0$. It follows from Taylor’s expansion of the exponential function and $\hat{t}\geq c\sqrt{\log m}$ that $\hat{\kappa}_{\Phi}\geq 1+\epsilon$ for some $\epsilon>0$. On the other hand, if $\frac{\alpha m_{0}}{m}\hat{\kappa}_{\Phi}> 1-\eta$, then $\hat{\kappa}_{\Phi}\geq 1+\epsilon$ for some $\epsilon>0$. This yields that $\pr(\hat{\kappa}_{\Phi}\geq 1+\epsilon)\rightarrow 1$ for some $\epsilon>0$. So we have $\kappa_{\Phi}\geq 1+\epsilon$ for some $\epsilon>0$. Note that $m_{0}/m\rightarrow 1$. We prove Corollary \[co1\] (2).
We next prove Corollary \[co1\] (3). By the inequality $e^{x}+e^{-x}\geq |x|$, $\pr(\hat{\kappa}_{\Phi}\leq m/(\alpha m_{0})+\varepsilon)\rightarrow 1$, we obtain that $$\begin{aligned}
\frac{\sum_{i\in\mathcal{H}_{0}}\frac{\hat{t}^{3}}{\sqrt{n}}|{\epsilon}Y_{i}^{3}|}{2m_{0}}\leq m/(\alpha m_{0})+\varepsilon\end{aligned}$$ with probability tending to one. By $\tau>0$, we have $\pr(\hat{t}\leq cn^{1/6})\rightarrow 1$ for some constant $c>0$. So $\pr(G(\hat{t})\geq \exp(-2cn^{1/3})\rightarrow 1$. Since $\hat{m}/m\leq \exp(-Mn^{1/3})$ for any $M>0$, we have by (\[prf3\]) $$\begin{aligned}
\frac{\alpha m_{0}}{m}\hat{\kappa}_{\Phi}\rightarrow 1.\end{aligned}$$ in probability. Hence $\kappa_{\Phi}\rightarrow 1/\alpha$ since $m_{0}/m\rightarrow 1$. The proof is finished.
Proof of Theorems \[th2-2\] and \[th22\]
----------------------------------------
Let $\hat{\kappa}_{i}=\frac{1}{n\hat{s}_{ni}^{3}}\sum_{k=1}^{n}(X_{ki}-\bar{X}_{i})^{3}$. Define the event $$\begin{aligned}
\F=\{\max_{1\leq i\leq m}\frac{1}{n\hat{s}_{ni}^{4}}\sum_{k=1}^{n}(X_{ki}-\bar{X}_{i})^{4}\leq K_{1},\max_{1\leq i\leq m}|\hat{\kappa}_{i}-\kappa_{i}|\leq K_{2}\sqrt{\log m/n}\}\end{aligned}$$ for some large $K_{1}>0$ and $K_{2}>0$. We first suppose that $\pr(\F)\rightarrow 1$. Let $G^{*}_{i}(t)=\pr^{*}(|T^{*}_{ki}|\geq t)$ be the conditional distribution of $T^{*}_{ki}$ given $\mathcal{X}=\{\mathcal{X}_{1},\cdots,\mathcal{X}_{m}\}$. Note that, given $\mathcal{X}$ and on the event $\F$, $$\begin{aligned}
G^{*}_{i}(t)&=&\frac{1}{2}G(t)\Big{[}\exp\Big{(}-\frac{t^{3}}{3\sqrt{n}}\hat{\kappa}_{i}\Big{)}+
\exp\Big{(}\frac{t^{3}}{3\sqrt{n}}\hat{\kappa}_{i}\Big{)}\Big{]}(1+o(1))\cr
&=&\frac{1}{2}G(t)\Big{[}\exp\Big{(}-\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}+
\exp\Big{(}\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}\Big{]}(1+o(1))\end{aligned}$$ uniformly in $0\leq t\leq o(n^{1/4})$. Hence, given $\mathcal{X}$ and on the event $\F$, $$\begin{aligned}
\label{bap}
\frac{G^{*}_{i}(t)}{\pr(|T_{i}-\sqrt{n}\mu_{i}/\hat{s}_{n}|\geq t)}=1+o(1)\end{aligned}$$ uniformly in $1\leq i\leq m$ and $0\leq t\leq o(n^{1/4})$. Put $$\begin{aligned}
\hat{G}_{\kappa}(t)=\frac{1}{2m}G(t)\sum_{1\leq i\leq m}\Big{[}\exp\Big{(}-\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}+
\exp\Big{(}\frac{t^{3}}{3\sqrt{n}}\kappa_{i}\Big{)}\Big{]}.\end{aligned}$$ Set $\hat{c}_{m}=\hat{G}^{-1}_{\kappa}(b_{m}/m)$. Note that, given $\mathcal{X}$, $T_{ki}^{*}$, $1\leq k\leq N$, $1\leq i\leq m$, are independent. Hence, as (\[aa133\]), we can show that for any $b_{m}\rightarrow\infty$, $$\begin{aligned}
\label{aa14}
\sup_{0\leq t\leq \hat{c}_{m}}\Big{|}\frac{G^{*}_{N,m}(t)}{\hat{G}_{\kappa}(t)}-1\Big{|}\rightarrow 0\end{aligned}$$ in probability. For $t=O(\sqrt{\log m})$, under the conditions of Theorem 3.2, we have $\hat{G}_{\kappa}(t)/G_{\kappa}(t)=1+o(1)$. So, it is easy to see that (\[aa133\]) still holds when $G^{-1}_{\kappa}(b_{m}/m)$ is replaced by $\hat{G}^{-1}_{\kappa}(b_{m}/m)$. This implies that for any $b_{m}\rightarrow\infty$, $$\begin{aligned}
\label{aa16}
\sup_{0\leq t\leq \hat{c}_{m}}\Big{|}\frac{\sum_{i\in\mathcal{H}_{0}}I\{|T_{i}|\geq t\}}{m_{0}G^{*}_{N,m}(t)}-1\Big{|}\rightarrow 0\end{aligned}$$ in probability.
Let $$\begin{aligned}
\hat{t}_{0}=\sup\Big{\{}0\leq t\leq 1:~ t\leq \frac{\alpha\max(\sum_{1\leq i\leq m}I\{\hat{p}_{i,B}\leq t\},1)}{m}\Big{\}}.\end{aligned}$$ Then we have $$\begin{aligned}
\hat{t}_{0}=\frac{\alpha\max(\sum_{1\leq i\leq m}I\{\hat{p}_{i,B}\leq \hat{t}_{0}\},1)}{m}.\end{aligned}$$ By (\[le7\]) and (\[bap\]), we have, given $\mathcal{X}$ and on the event $\F$, $G^{*}_{i}(c\sqrt{\log m})=m^{-c^{2}/2+o(1)}$ for any $c>\sqrt{2}$ uniformly in $i$. So, by Markov’s inequality, for any $\varepsilon>0$, we have $\pr\Big{(}G^{*}_{N,m}(c\sqrt{\log m})\leq m^{-c^{2}/2+\varepsilon}\Big{)}\rightarrow 1$. By (\[c1\]) and (\[tv2\]), we have $\pr(\hat{t}_{0}\geq \alpha b_{m}/m)\rightarrow 1$ for some $b_{m}\rightarrow\infty$. It follows from (\[aa16\]) that $$\begin{aligned}
\frac{\sum_{i\in\mathcal{H}_{0}}I\{\hat{p}_{i,B}\leq \hat{t}_{0}\}}{m_{0}\hat{t}_{0}}\rightarrow 1\end{aligned}$$ in probability. This finishes the proof of Theorem \[th2-2\] if we can show that $\pr(\F)\rightarrow 1$. Without loss of generality, we can assume that $\mu_{i}=0$ and $\sigma_{i}=1$. We first show that for some constant $K_{1}>0$, $$\begin{aligned}
\label{aa17}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\sum_{k=1}^{n}(X_{ki}^{4}-{\epsilon}X_{ki}^{4})\Big{|}\geq K_{1}n\Big{)}=o(1).\end{aligned}$$ For $1\leq i\leq n$, put $$\begin{aligned}
\hat{X}_{ki}=X_{ki}I\{|X_{ki}|\leq \sqrt{n/\log m}\},\quad \breve{X}_{ki}=X_{ki}- \hat{X}_{ki}.
\end{aligned}$$ Then, for large $n$, $$\begin{aligned}
&&\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\sum_{k=1}^{n}(\breve{X}_{ki}^{4}-{\epsilon}\breve{X}_{ki}^{4})\Big{|}\geq K_{1}n/2\Big{)}\cr
&&\quad\leq nm\max_{1\leq i\leq m}\pr(|X_{1i}|\geq \sqrt{n/\log m})\cr
&&\quad\leq C\exp(\log m+\log n-tn/\log m)\cr
&&\quad=o(1).\end{aligned}$$ Let $Z_{ki}=\hat{X}_{ki}^{4}-{\epsilon}\hat{X}_{ki}^{4}$. By the inequality $|e^{s}-1-s|\leq s^{2}e^{\max(s,0)}$ and $1+s\leq e^{s}$, we have for $\eta=2^{-1}t(\log m)/n$ and some large $K_{1}$ $$\begin{aligned}
&&\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\sum_{k=1}^{n}Z_{ki}\Big{|}\geq K_{1}n/2\Big{)}\cr
&&\quad\leq \sum_{i=1}^{m}\pr\Big{(}\sum_{k=1}^{n}Z_{ki}\geq K_{1}n/2\Big{)}+ \sum_{i=1}^{m}\pr\Big{(}-\sum_{k=1}^{n}Z_{ki}\geq K_{1}n/2\Big{)}\cr
&&\quad\leq \sum_{i=1}^{m}\exp(-\eta K_{1}n/2)\Big{[}\prod_{k=1}^{n}\exp(\eta Z_{ki})+\prod_{k=1}^{n}\exp(-\eta Z_{ki})\Big{]}\cr
&&\quad\leq 2\sum_{i=1}^{m}\exp(-\eta K_{1}n/2+\eta^{2}n{\epsilon}Z_{1i}^{2}e^{\eta|Z_{1i}|})\cr
&&\quad\leq C\exp(\log m-t K_{1}(\log m)/4)\cr
&&\quad=o(1).\end{aligned}$$ This proves (\[aa17\]). By replacing $X_{ki}^{4}$, $\eta=2^{-1}t(\log m)/n$ and $K_{1}n/2$ with $X_{ki}^{3}$, $\eta=2^{-1}t\sqrt{(\log m)/n}$ and $K_{1}\sqrt{n\log m}/2$ respectively in the above proof, we can show that $$\begin{aligned}
\label{aa18}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(X_{ki}^{3}-{\epsilon}X_{ki}^{3})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1).\end{aligned}$$ Similarly, we have $$\begin{aligned}
\label{aa19}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(X_{ki}^{2}-{\epsilon}X_{ki}^{2})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1)\end{aligned}$$ and $$\begin{aligned}
\label{aa199}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(X_{ki}-{\epsilon}X_{ki})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1).\end{aligned}$$ Combining (\[aa17\])-(\[aa199\]), we prove that $\pr(\F)\rightarrow 1$.
Proof of Theorems \[th2-222\] and \[th222\]
-------------------------------------------
Let $$\begin{aligned}
\hat{\F}=\{\max_{1\leq i\leq m}\frac{1}{n\hat{\sigma}_{i}^{4}}\sum_{k=1}^{n}(\hat{X}_{ki}-\hat{X}_{i})^{4}\leq K_{1},\max_{1\leq i\leq m}|\hat{\kappa}_{i}(\lambda_{ni})-\kappa_{i}|\leq K_{2}\sqrt{\log m/n}\}\end{aligned}$$ By the proof of Theorems \[th2-2\] and \[th22\], it is enough to show $\pr(\hat{\F})\rightarrow 1$. Recall that $\hat{X}_{ki}=X_{ki}I\{|X_{ki}|\leq \lambda_{ni}\}$ and put $Z_{ki}=\hat{X}^{4}_{ki}-{\epsilon}\hat{X}^{4}_{ki}$. Take $\eta=(\log m)/n$. We have $$\begin{aligned}
&&\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\sum_{k=1}^{n}Z_{ki}\Big{|}\geq K_{1}n/2\Big{)}\cr
&&\quad\leq 2\sum_{i=1}^{m}\exp(-\eta K_{1}n/2+\eta^{2}n{\epsilon}Z_{1i}^{2}e^{\eta|Z_{1i}|})\cr
&&\quad\leq C\exp(2\log m-K_{1}(\log m)/4)\cr
&&\quad=o(1).\end{aligned}$$ Similarly, by replacing $\hat{X}_{ki}^{4}$, $\eta=(\log m)/n$ and $K_{1}n/2$ with $\hat{X}_{ki}^{3}$, $\eta=\sqrt{(\log m)/n}$ and $K_{1}\sqrt{n\log m}/2$ respectively in the above proof, we can show that $$\begin{aligned}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(\hat{X}_{ki}^{3}-{\epsilon}\hat{X}_{ki}^{3})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1).\end{aligned}$$ Also, using the above arguments, it is easy to show that $$\begin{aligned}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(\hat{X}_{ki}^{2}-{\epsilon}\hat{X}_{ki}^{2})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1)\end{aligned}$$ and $$\begin{aligned}
\pr\Big{(}\max_{1\leq i\leq m}\Big{|}\frac{1}{n}\sum_{k=1}^{n}(\hat{X}_{ki}-{\epsilon}\hat{X}_{ki})\Big{|}\geq K_{1}\sqrt{(\log m)/n}\Big{)}=o(1).\end{aligned}$$ Note that $$\begin{aligned}
\max_{1\leq i\leq m}{\epsilon}|X_{1i}|^{3}I\{|X_{1i}|\geq \lambda_{ni}\}
\leq C\sqrt{\frac{\log m}{n}}\max_{1\leq i\leq m}{\epsilon}X_{1i}^{6}\end{aligned}$$ and $$\begin{aligned}
\max_{1\leq i\leq m}{\epsilon}|X_{1i}|^{2}I\{|X_{1i}|\geq \lambda_{ni}\}\leq C\Big{(}\frac{\log m}{n}\Big{)}^{2/3}\max_{1\leq i\leq m}{\epsilon}X_{1i}^{6}.\end{aligned}$$ This proves $\pr(\hat{\F})\rightarrow 1$.
Proof of Theorem \[th21\]
-------------------------
Recall that $$\begin{aligned}
\frac{mG(\hat{t})}{\max(\sum_{1\leq i\leq m}I\{|T_{i}|\geq \hat{t}\},1)}=\alpha.\end{aligned}$$ From (\[tv2\]), we have $
\pr\Big{(}\hat{t}\geq G^{-1}(\alpha b_{m}/m)\Big{)}\rightarrow 1.
$ The theorem follows from (\[aa133\]) and the fact $G_{\kappa}(t)/G(t)=1+o(1)$ uniformly in $t\in[0,o(n^{1/6}))$.
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[^1]: Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University. Research supported by NSFC, Grant No.11201298 and No.11322107, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Shanghai Pujiang Program, Foundation for the Author of National Excellent Doctoral Dissertation of PR China and Program for New Century Excellent Talents in University.
[^2]: Department of Statistics, The Chinese University of Hong Kong. Research partially supported by Hong Kong RGC GRF 603710 and 403513.
|
---
abstract: 'The mechanisms for strong electron-phonon coupling predicted for hydrogen-rich alloys with high superconducting critical temperature ($T_c$) are examined within the Migdal-Eliashberg theory. Analysis of the functional derivative of $T_c$ with respect to the electron-phonon spectral function shows that at low pressures, when the alloys often adopt layered structures, bending vibrations have the most dominant effect. At very high pressures, the H-H interactions in two-dimensional (2D) and three-dimensional (3D) extended structures are weakened, resulting in mixed bent (libration) and stretch vibrations, and the electron-phonon coupling process is distributed over a broad frequency range leading to very high $T_c$.'
author:
- 'K. Tanaka'
- 'J. S. Tse'
- 'H. Liu'
title: 'Electron-phonon coupling mechanisms for hydrogen-rich metals at high pressure'
---
Hydrogen is the lightest element and thus, if molecular hydrogen can be compressed into a metal in the solid state, it is expected to become superconducting at a very high transition temperature $T_c$ due to exceptionally strong electron-phonon coupling [@Ashcroft1968]. Although this proposal has not been verified experimentally, calculation based on modern electronic structure theory has predicted high $T_c$ for metallic hydrogen [@Cudazzo2010; @McMahon2011]. According to recent studies, metallization of solid hydrogen may require pressure in excess of 400 GPa, which is difficult to achieve with today’s experimental techniques. Although very recently metallic hydrogen has been claimed to be observed under extremely high pressure of nominal 495 GPa [@Dias2017], the result is contentious [@metallicH] and further experiments are required. On the other hand, it has been realised that the electron density required to metallize molecular hydrogen may be achieved by compression of group-IVa hydrides, in which the hydrogen content is already high [@Ashcroft2004]. This insightful suggestion has stimulated various theoretical studies and experimental investigations. The first prediction of superconductivity in group-IVa hydrides was made using density functional theory for silane (SiH$_4$) [@Feng2006]. First-principles calculation has predicted a monoclinic $C2/c$ metallic solid composed of SiH$_4$ layers bridged by Si-H-Si bonds that is stable between 65 and 150 GPa and has $T_c$ of 45-55 K at 90-125 GPa [@Yao2008]. Subsequent experiment has revealed that insulating molecular silane transforms to a metal at 50 GPa and becomes superconducting with $T_c=17$ K at 96 and 120 GPa [@Eremets2008]. However, the measured diffraction pattern of the superconducting phase did not match the monoclinic structure [@Eremets2008] and the high-pressure structure of silane remains controversial [@Strobel2011].
In the past decade, numerous theoretical predictions have been made on the structure and superconducting behaviour of stoichiometric and hydrogen-enriched hydrides with a variety of elements at high pressures. Most notable examples are the high-pressure polymorphs of CaH$_6$ and YH$_6$, both of which have a novel cagelike structure formed by monatomic hydrogens and have been predicted to have $T_c$ higher than 200 K [@Wang2012; @Li2015]. A major experimental breakthrough has been reported recently with the observation of superconductivity with a critical temperature above 200 K in hydrogen sulfide (H$_2$S) compressed to $\sim 200$ GPa [@Drozdov2015]. Isotopic, magnetic and Meissner-effect measurements have shown that superconductivity is driven by electron-phonon interactions. Electronic structure and electron-phonon coupling calculations [@Bernstein2015; @Duan2015; @Errea2015] and x-ray diffraction experiment [@Einaga2016] have determined that the superconducting phase consists of the decomposed product H$_3$S with a cubic structure. The surprisingly high observed $T_c$ raises the possibility that even higher transition temperatures may be attainable in hydrides. So far, the predictions of superconductivity and $T_c$ in hydrides have entirely relied on calculation for selected structures. It is desirable that general rules can be established to understand the underlying mechanisms of high-$T_c$ superconductivity in hydrogen-rich materials.
In this work, by solving the Eliashberg equations [@Eliashberg1960; @Carbotte1990; @Marsiglio2008], we analyse the functional derivative of $T_c$ with respect to the electron-phonon spectral function $\alpha^2F(\omega)$ of several representative hydride systems to characterise the most effective vibrational modes for enhancement of superconductivity. Our goal is to develop a strategy for synthesising new compounds with high critical temperatures. The functional derivative $\delta T_c/\delta \alpha^2F(\omega)$ [@Bergmann1973] enables us to identify the frequency regions where phonons are most effective in raising $T_c$ [@Mitrovic1981; @Yao2009; @Nicol2015]. We evaluate the functional derivative from the electron-phonon spectral function calculated from linear response theory and density functional perturbation theory, either performed for the present study or taken from reports of previous theoretical studies.
A survey of theoretically predicted hydrogen-dominant main group metallic alloys under high pressures reveals that most alloys adopt a layered structure. For this group of compounds, the calculated $T_c < 100$ K. We have chosen to examine SnH$_4$ in detail as the superconducting metallic phase that is stable between 70 and 160 GPa has a novel layered structure intercalated by “H$_2$” units with high $T_c$ [@Tse2007]. A remarkable feature of this structure is that the phonon band structure and the spectral function can be separated into three distinct frequency regions, corresponding to lattice (L), Sn-H, and H$_2$ vibrations with increasing frequency, as can be seen in Fig. \[fig:SnH4\_SiH4\](a). This unique property facilitates the analysis of the contributions of different vibrational bands to superconductivity, as the spectral function $\alpha^2F(\omega)$ can be decomposed as $\alpha^2F(\omega)_{\rm L}+\alpha^2F(\omega)_{\text{Sn-H}}+\alpha^2F(\omega)_{\text{H-H}}$. The electron-phonon coupling (EPC) parameter $\lambda$ is twice the first inverse moment of the spectral function, $\lambda = 2\int d\omega\,\frac{\alpha^2F(\omega)}{\omega}$. The EPC parameter $\lambda(\omega)$ integrated up to frequency $\omega$ of SnH$_4$ has shown that the lattice and Sn-H librations contribute most to the process of electron-phonon coupling, while there is only very minor contribution from the H-H vibrations [@Tse2007]. We have calculated $T_c$ from the individual vibrational bands by solving the Eliashberg equations at 120 GPa, using the Coulomb pseudopotential $\mu^*(\omega_{\text{max}})$ scaled to $\omega_{\text{max}}$, where $\omega_{\text{max}}$ is six times the maximum phonon frequency. For $\mu^*(\omega_{\text{max}})=0.1$, the entire spectrum yields $T_c\simeq 98$ K. Removing the low-frequency lattice vibrations from the spectrum only reduces $T_c$ roughly by 4 K, while eliminating the high-frequency H-H contribution results in $T_c\simeq 72$ K. In fact, the Sn-H vibrational band by itself yields $T_c\simeq 66$ K. Thus, the Sn-H vibrations are the most dominant contribution to $T_c$. The functional derivative shown in Fig. \[fig:SnH4\_SiH4\](a) is maximum at about 65 meV. This optimal frequency, $\omega_{\text{opt}}$, is close to the onset of the Sn-H bending vibrations. Therefore, it is plausible that variation of the bending vibrations can affect the critical temperature significantly. The optimal frequency is known to be related to the critical temperature by $\omega_{\text{opt}}\sim 7k_BT_c$ [@Carbotte1987; @Carbotte1990], where $k_B$ is the Boltzmann constant. Using this relation, $T_c$ is estimated to be roughly 108 K, in good agreement with 98 K from solving the Eliashberg equations.
The functional derivative for the monoclinic SiH$_4$ structure at 125 GPa is presented in Fig. \[fig:SnH4\_SiH4\](b). For this compound, no H-H species are present and partition of the phonon spectrum to lattice, Si-H bent (libration) and stretch vibrations is less distinctive. For $\mu^*(\omega_{\text{max}})=0.1$, the optimal vibrational frequency is found to be 38 meV, in the frequency range between the lattice and low-frequency Si-H bending vibrations. $T_c$ estimated from $\omega_{\text{opt}}\sim 7k_BT_c$ is about 63 K, again comparable to 53 K obtained from the Eliashberg equations. The above results on SnH$_4$ and SiH$_4$ highlight the importance of lattice and bending vibrations on the electron-phonon coupling process. It is surprising, however, that the low-frequency lattice vibrations in SnH$_4$ contribute substantially to the EPC parameter [@Tse2007], but not so much to $T_c$. This implies that a higher critical temperature cannot necessarily be achieved by simply increasing the mass of the heavier element in hydrides.
[p[ 2 ]{}]{}
Study of a series of high-pressure yttrium hydrides (YH$_n$, $n = 3$, 4 and 6) offers useful insight into the evolution of crystal structure and the superconducting properties as the hydrogen concentration is increased beyond what is required to satisfy the normal covalency (i.e., YH$_3$). YH$_3$ has been predicted to have a face-centered cubic structure formed from monoatomic H situated in the tetrahedral and octahedral interstitial sites and separated by long distances [@Li2015]. This structure is stable from 17.7 GPa to 140 GPa. Similarly to SnH$_4$, the spectral function can clearly be separated into regions for lattice, Y-H and H-H vibrations. At 17.7 GPa, the H-H vibration energy of 165 meV is much lower than that of normal H$_2$ at the same pressure. The compound is predicted to be superconducting with maximum $T_c$ of 40 K at 17.7 GPa. Upon compression, $T_c$ decreases and superconductivity vanishes between 35 and 44 GPa, only to reappear at a lower value of around 6 K at higher pressure. Two energetically competing hydrogen-rich polymorphs YH$_4$ (YH$_3$+H$_2$) and YH$_6$ (2YH$_3$ + 3H$_2$) have been predicted to be thermodynamically more stable than the product of YH$_3$ and solid H$_2$ above 140 GPa. Both are superconductors with maximum $T_c$ of 85 K and 235 K for YH$_4$ and YH$_6$, respectively. As mentioned above, YH$_3$ has a cubic structure composed of atomic hydrogens. YH$_4$ has tetragonal space group and consists of atomic H and molecular “H$_2$”. YH$_6$ has a novel cubic cage structure with Y located in the sodalite cages formed by H atoms. The same structure has also been found in CaH$_6$ which too is predicted to be a good superconductor with $T_c$ of 205 K at 150 GPa [@Wang2012]. Changes in the H network topology in these structures can be rationalised by a charge transfer model proposed earlier for Ca and Sr hydrides [@Wang2012; @Wang2015]. Y is trivalent with a valence shell electron configuration $4d^15s^2$. Since $4d$ and $5s$ orbitals are shielded by their respective core orbitals, the valence electrons can be removed easily. Assuming full charge transfer as expected in YH$_3$, the effective electron number (EEN) of each hydrogen is $-1e$ and thus leads to the formation of monatomic hydrogens in the crystal structure. On the other hand, in YH$_4$, the EEN is $(3/4)e$ and hence some of the “molecular H$_2$” are preserved, albeit with a long H-H bond length of 1.33 ${\textup{\AA}}$ at 120 GPa. In YH$_6$, the EEN is further reduced to $(1/2)e$, but to maintain maximum overlaps of the H orbitals, the cage structure is preferred. The features of H-H interactions in these crystal structures suggest the existence of high-frequency H-H vibrons in YH$_4$, YH$_6$, and YH$_3$ with frequency in descending order. This trend has indeed been confirmed by phonon band structure calculations [@Li2015]. The predicted maximum $T_c$, however, does not follow this sequence. Therefore, the mean vibrational frequency, $\langle \omega \rangle$, often used in estimating the critical temperature in terms of the McMillan [@Mcmillan68] or Allen-Dynes [@Allen75prb; @Allen75] equation, is not necessarily the only factor to be considered for raising $T_c$.
To gain insight into the role of phonons in the superconducting state, the functional derivative $\delta T_c/\delta \alpha^2F(\omega)$ has been computed for YH$_3$, YH$_4$, and YH$_6$ for $\mu^*(\omega_{\text{max}})=0.1$ and the results are compared in Fig. \[fig:YH\_CaH\]. The respective optimal frequencies are 29, 61 and 150 meV with $T_c\sim \omega_{\text{opt}}/7k_B$ of 48, 101 and 249 K, respectively, which compare well with $T_c$ of 43, 92 and 247 K calculated from the Eliashberg equations. An interesting aspect of the optimal vibrations is that in YH$_3$ it is maximised at the lattice acoustic translational branch, while in YH$_4$, it originates from the soft vibrational branch of the localised “molecular H$_2$” units. In YH$_6$, there is no clear distinction between stretch and bent modes as the H atoms form a 3D connected open sodalite framework. Coupling of these vibrations results in the continuous spectral distribution. These vibrations all participate strongly in the electron-phonon interaction and shift the optimal frequency to higher energy, yielding a higher $T_c$. Moreover, the functional derivative curve is broad and does not taper off as rapidly at higher frequencies as in YH$_3$ and YH$_4$. For comparison, the functional derivative of the isostructural CaH$_6$ is also examined \[Fig. \[fig:YH\_CaH\](d)\]. Once again a continuous distribution of H-dominated vibrations is observed. In this case, the maximum of the functional derivative is located at 143 meV. Although the functional derivative profiles for YH$_6$ and CaH$_6$ are broadly similar, it is important to note that the H-H distance in YH$_6$ of 1.31 ${\textup{\AA}}$ is significantly larger than 1.24 ${\textup{\AA}}$ in CaH$_6$ under similar pressure. A larger H-H distance in YH$_6$ can be understood as due to the fact that there is one more valence electron provided by Y than by Ca (three vs. two). Thus, the EEN of the H atom is higher in YH$_6$, leading to a weaker and longer “molecular H$_2$” bond. The cutoff frequency for the H-H vibrons in CaH$_6$ (245 meV) is therefore higher than in YH$_6$ ($\sim 189$ meV), but the optimal frequency is lower.
![\[fig:H2S\] (Colour online) Crystal structure of H$_3$S. ](H3S_cage.pdf){width="\columnwidth"}
The above observation clearly shows that the signature of a high-$T_c$ hydrogen-rich material is the presence of strongly coupled hydrogen-dominant libration and stretch vibrations. As demonstrated above, weak H-H interactions with bond length of 1.2-1.3 ${\textup{\AA}}$ is desirable. At this bond separation, there is no longer clear distinction between stretch and bent vibrations and all H vibrations participate effectively in the electron-phonon coupling process. Another example is the recently discovered high critical temperature in compressed H$_2$S at $\sim 200$ GPa [@Drozdov2015]. The functional derivative of $T_c$ with respect to the spectral function of the candidate H$_3$S has been analysed [@Nicol2015] and the major feature in the spectral function is again found to be the hydrogen libration and stretch vibrations being strongly mixed. However, an important point is that the body centered cubic structure may be viewed as the sulfur atoms being enclathrated at the center of a cubic box created by a 3D hydrogen network (Fig. \[fig:H2S\]). Therefore, based on the proposed charge transfer model [@Wang2012; @Wang2015], the design of materials possessing this unique structural property requires consideration of both the hydrogen concentration and the nature and number of available valence electrons from the donor atom. For example, the sodalite structure is severely distorted when Ca is replaced by the similarly divalent Sr in SrH$_6$, as Sr can no longer be accommodated in the cage due to the large atomic size.
Our findings are consistent with a very recent study of La-H and Y-H systems in terms of density functional theory, where LaH$_{10}$ and YH$_{10}$ have been found to adopt a sodalite-like face-centered cubic structure and have $T_c$ in the range of room temperature [@Liu2017]. We have also solved the Eliashberg equations for these systems and have found that $\delta T_c/\delta \alpha^2F(\omega)$ has a broad distribution and decays rather slowly beyond $\omega_{\rm opt}$, similarly to that for YH$_6$ and CaH$_6$ shown in Fig. \[fig:YH\_CaH\](c) and (d). In Table \[table\] we summarise key quantities for superconductivity for the systems studied in this work, including YH$_{10}$ and LaH$_{10}$ [@note]. It can be seen that $\omega_{\rm opt}$ and hence $T_c$ are the highest for YH$_{10}$ at 250 GPa, even though $\lambda$ is smaller compared to YH$_6$ or SrH$_{10}$ (see below).
[p[ 2 ]{}]{} (a)(c)\
\
(b)\
The sodalite cage structure is not the only structural motif that can support strong electron-phonon coupling. Dense molecular hydrogen with the orthorhombic structure (*Cmca*) has been predicted from superconductivity density functional theory to have a critical temperature of 242K at 450 GPa [@Cudazzo2008]. The crystal structure at 300 GPa shown in Fig. \[fig:SrH10\](a) is composed of staggered 2D puckered honeycomb layers with interatomic distances alternating between 0.78 and 1.10 ${\textup{\AA}}$. The closest H-H distance between two layers is 1.27 ${\textup{\AA}}$. The calculated electron-phonon spectral function is almost continuous up to 494 meV with lattice, libration and molecular vibrations all strongly coupled to the electrons. One may ask, is it possible to construct a similar structural morphology in hydrogen-rich alloys? In a survey of hydrogen-rich strontium hydrides, a high-pressure polymorph, SrH$_{10}$, a rhombohedral crystal with planes of Sr sandwiched between every two puckered honeycomb H layers and H-H bonds alternating between 0.998 and 1.011 ${\textup{\AA}}$ has been found to be stable above 300 GPa \[Fig. \[fig:SrH10\](b)\]. The similarity in the structure to the *Cmca* metallic phase of solid hydrogen is striking and suggests potential superconductivity with a high transition temperature. To examine this possibility, the electronic and phonon band structure and electron-phonon coupling for the 300 GPa structure have been calculated using density functional theory. The phonon band structure presented in Fig. \[fig:SrH10\](c) exhibits strong electron-phonon coupling from the librational phonon branches between 50 and 160 meV along the $X\rightarrow \Gamma\rightarrow T$ symmetry direction. SrH$_{10}$ is indeed a superconductor and the calculated isotropic EPC parameter $\lambda$ is 3.08 and the $T_c$ calculated from the Eliashberg equations is 259 K with $\mu^*(\omega_{\text{max}})=0.1$. The functional derivative is presented in Fig. \[fig:SrH10\](d). As in metallic hydrogen, the distribution of the phonon modes is almost continuous with the functional derivative maximised at 159 meV. In this case, the high-frequency H-H stretch vibrations centered around 270 meV contribute very little to the overall coupling with electrons. These results confirm the expectation and show that as the sodalite structure, the unique layer H-network morphology is relevant to high-temperature superconductivity. So far, the sodalite and puckered honeycomb layer H-networks are the only two structural features found in hydrogen-rich materials possessing very high $T_c$ ($> 200$ K) by theoretical calculations.
[lllllllll]{}\
System & Pressure (GPa) & $T_c$ (K) & $\lambda$ & $\omega_{\text{opt}}$ (meV) & $\omega_{\text{opt}}/7k_B$ (K) & $\langle \omega \rangle$ (meV) & $\omega_{\rm log}$ (meV) & H-H distance (${\textup{\AA}}$)\
\
SiH$_4$ & 125 & 53 & 0.89 & 38 & 63 & 111 & 79 & –\
SnH$_4$ & 120 & 98 & 1.20 & 65 & 108 & 111 & 76 & 0.841\
YH$_6$ & 120 & 247 & 3.19 & 150 & 249 & 87 & 63 & 1.306\
YH$_{10}$ & 250 & 291 & 2.67 & 177 & 293 & 97 & 95 & 1.132\
YH$_{10}$ & 300 & 275 & 2.00 & 170 & 282 & 147 & 125 & 1.029\
LaH$_{10}$ & 300 & 231 & 1.74 & 144 & 239 & 142 & 123 & 1.076\
CaH$_6$ & 150 & 235 & 2.71 & 143 & 237 & 96 & 87 & 1.238\
SrH$_{10}$ & 300 & 259 & 3.08 & 159 & 264 & 96 & 66 & 0.997\
In summary, we have shown from analysis of the structures and functional derivative of selected high-pressure hydrides that the metallic-H bent (or librational) vibrations are most effective in enhancing the superconducting transition temperature. Furthermore, since the vibration profile (or the electron-phonon spectral function) is intimately related to the crystal structure, two types of H networks, the sodalite and puckered honeycomb layer structures with strong mixing of stretch and bent vibrations are most likely to lead to strong electron-phonon coupling for all the modes. Can room-temperature superconductivity be achieved if the phonon-mediated Eliashberg theory is valid? From the relation $\omega_{\text{opt}}\sim 7k_BT_c$, the optimal frequency should be 180 meV (1450 cm$^{-1}$) for a critical temperature $T_c = 300$ K. Among the results presented in Table \[table\], YH$_{10}$ at 250 GPa is close to ideal. The key is to prepare a system with a broad vibration distribution and efficient electron-phonon coupling close to this frequency. As illustrated above, the functional derivative of the sodalite structures (CaH$_6$ and YH$_6$) does not decrease as quickly above the optimal frequency as the other structures, indicating that all the modes, with the exception of lattice vibrations, are very efficient in enhancing $T_c$. Moreover, the spectral function shows that the electron-phonon coupling strength (the area under the spectrum) is uniformly large for these structures. In principle, it is plausible to design materials with such characteristics, perhaps by choosing a di- or trivalent element with a valence electron ionization energy (electron donating property) intermediate between Ca and Y such that the H-H distance in the sodalite structure is about halfway ($\simeq 1.27 {\textup{\AA}}$). This will decrease the stretch frequency but maintain strong electron-phonon coupling. Another possibility is to increase the H$_2$ concentration in compounds with electron donating atoms. As observed in SrH$_{10}$ above, the metallic atoms help to reduce the pressure required to form the puckered “molecular” H$_2$ layers in superconducting solid hydrogen.
The research was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Foundation for Innovation. H. L. acknowledges support by EFree, an Energy Frontier Research Center funded by the DOE, Office of Science, Basic Energy Sciences under Award No. DE-SC-0001057.
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abstract: |
At the ILC, the luminosity-weighted average polarization at the IP needs to be determined at the permille-level. In order to reach this goal, the combined information from the polarimeter and the collision data is required. In this study, a unified approach will be presented, which for the first time combines the cross section measurements with the expected constraints from the polarimeters. Hereby, the statistical and systematical uncertainties are taken into account, including their correlations.\
This study shows that a fast spin flip frequency is required because it easily reduces the systematic uncertainty, while a non-perfect helicity reversal can be compensated for within the unified approach. The final goal is to provide a realistic estimation of the luminosity-weighted average polarization at the IP to be used in the physic analyses.
author:
- 'Robert Karl[^1]'
- 'Jenny List[^2]'
date: '06.12.16'
title: Polarimetry at the ILC
---
Introduction
============
The usage of polarized beams provides great advantages for the International Linear Collider (ILC). It allows deep insights into the chiral structure of the weak-interaction for known and unknown particle as well as a sensitivity to additional observables (e.g. left-right-asymmetry). Furthermore, with polarized beams it is possible to suppress background processes and simultaneously increase signal processes, providing an enhancement of the signal to noise ratio.\
Thus, the ILC[@ILCProjectVol1] beams will be polarized to a degree of $\left|80\%\right|$ for the electron beam and $\left|30\%\text{ - }60\%\right|$ for the positron beam. The sign of the polarization is individually adjustable for each beam, providing a choice of different spin configurations. However, since all event rates depend linearly on the polarization, it is important to provide a determination of the actual beam polarization at the permille-level in order to fully exploit the physics potential of the ILC[@PolarizationRequirement]. This requirement can only be fulfilled by combining the fast time-resolved measurements of the laser-Compton polarimeters with an absolute scale calibration of the luminosity-weighted average polarization at the interaction point (IP) calculated from collision data.
As shown in Fig. \[fig:PolarimeterConcept\], the time-resolved polarization is measured by two polarimeters per beam from the differential Compton cross section of a particle bunch with a circular polarized lasers [@polarimeter]. However, those polarimeters are 1.65km before and 150m after the IP. Thus, the measurement has to be extrapolated from the polarimeters to the IP via spin tracking [@spintracking], considering all uncertainties including beam collision effects. The absolute scale calibration is determined by the luminosity-weighed averaged polarization calculated from the cross section measurement from different, well known standard model processes.\
In the past, such a polarization determination from collision data has been studied for different standard model processes individually. In previous studies, the polarization was determined using the information from W-pair production [@wpairstudy] and using the information from single $W$, $\gamma$, $Z$ events [@singlebosonstudy]. However, the current goal is to find a general strategy for the polarization determination which yields the best precision per measurement time. Thus, the following criteria have to be considered:\
- **Combining all relevant processes**\
By combining all suitable processes, the statistical precision can be increased and different systematic uncertainties give a better control. Suitable processes exhibit a large left-right-asymmetry and a high cross section. For a precise determination all uncertainties and their correlations have to be taken into account. This also provides a more robust polarization determination against possible BSM effects of single processes.\
- **Compensating for a non-perfect helicity reversal**\
In realistic running condition, the absolute polarization will also slightly change when reversing the sign. This has to be considered for the polarization determination in order to increase its precision.\
- **Including constraints from the polarimeter measurement**\
Since the polarimeter already yields a polarization measurement, it can be used as a constraint for the calculation of the cross section measurement to make it more robust against large statistical fluctuations, especially for low luminosities. For a precise determination, all sources of uncertainties, e.g. from spin tracking, have to be taken into account.\
Polarization Measurement Using Collision Data
=============================================
In order to determine the polarization from collision data, the theoretically predicted cross section of each process is compared to the corresponding measurement within the uncertainty by variating the beam polarization equally for all four spin configurations. For this purpose, a $\chi^{2}$-minimization method is used.
Combination of Cross Section Measurements
-----------------------------------------
In order to combine the cross section measurement, the $\chi^{2}$ values of the individual processes are summed up and share a common polarization parameter set. To consider all systematic uncertainties and their correlation, a full covariance matrix $\Xi$ was used, shown in eq. \[eq:chi2formula\]. This $\chi^{2}$-function will be refered to *the unified approach*.
$$\begin{aligned}
\chi^{2} &:=
\sum_{\text{process}}{
\left(\vec{\sigma}_{\text{data}} - \vec{\sigma}_{\text{theory}}\right)^{T}\Xi^{-1}
\left(\vec{\sigma}_{\text{data}} - \vec{\sigma}_{\text{theory}}\right)
};&\qquad
\vec{\sigma} &:= \begin{pmatrix}
\sigma_{-+} & \sigma_{+-} & \sigma_{--} & \sigma_{++}\\
\end{pmatrix}^{T}
\label{eq:chi2formula}\end{aligned}$$
Here, $\Xi^{-1}$ refers to the inverse of the covariance matrix. The covariance matrix is derived from the error propagation of the cross section measurement, as shown in eq. \[eq:covariancematrix\].
$$\begin{aligned}
\left(\vec{\sigma}_{\text{data}}\right)_{i} &= \frac{D_{i} - \mathfrak{B}_{i}}{\varepsilon_{i}\cdot\mathcal{L}_{i}}
&\Rightarrow\quad\textbf{e.g.: }\left(\Xi_{\varepsilon}\right)_{ij} &=
\text{corr}\left(\vec{\sigma}_{i}^{\varepsilon},\ \vec{\sigma}_{j}^{\varepsilon}\right)
\frac{\partial\vec{\sigma}_{i}}{\partial\varepsilon_{i}}\frac{\partial\vec{\sigma}_{j}}{\partial\varepsilon_{j}}
\Delta\varepsilon_{i}\Delta\varepsilon_{j}
&\Rightarrow\quad\Xi &:= \Xi_{D} + \Xi_{\mathfrak{B}} + \Xi_{\varepsilon} + \Xi_{\mathcal{L}};
\label{eq:covariancematrix}\end{aligned}$$
Here, $D_{i}$ is the number of signal events, $\mathfrak{B}_{i}$ is the background expectation value, $\varepsilon_{i}$ is the selection efficiency of the detector and $\mathcal{L}_{i}$ is the integrated luminosity of the data set. The index $i$ corresponds to the different helicity configurations $\left(-+,\ +-,\ --,\ ++\right)$. The final covariance matrix is the sum of the individual covariance matrices of the four quantities ($D_{i}$, $\mathfrak{B}_{i}$, $\varepsilon_{i}$, $\mathcal{L}_{i}$). The correlation factors (e.g. $\text{corr}\left(\vec{\sigma}_{i}^{\varepsilon},\ \vec{\sigma}_{j}^{\varepsilon}\right)$) can vary for each quantity but is equal for each process. Furthermore, only correlations between the different spin configurations are considered individually for each quantity. Correlations between the different quantities are neglected. The correlation factors are fixed and have to determined externally. Since the signal $D_{i}$ is always uncorrelated, $\Xi_{D}$ is here always a diagonal matrix.\
Compensation for a non-perfect helicity reversal is achieved by treating the polarization value for both helicities and beams as independent. This results in 4 free parameters defined with their nominal values in eq. \[eq:freeparameters\].
$$\begin{aligned}
\underbrace{P_{e^{-}}^{-} = -80\%,}_{\text{"left"-handed }e^{-}\text{-beam}}\qquad
\underbrace{P_{e^{-}}^{+} = 80\%,}_{\text{"right"-handed }e^{-}\text{-beam}}\qquad
\underbrace{P_{e^{+}}^{-} = -30\%,}_{\text{"left"-handed }e^{+}\text{-beam}}\qquad
\underbrace{P_{e^{+}}^{+} = 30\%,}_{\text{"right"-handed }e^{+}\text{-beam}}
\label{eq:freeparameters}\end{aligned}$$
An alternative parametrization, which was used in previous studies, is to use an absolute average polarization between both helicities and the deviations, as defined in eq. \[eq:alternativeparameters\]. Both parametrizations are equivalent but for this study the parametrization defined in eq. \[eq:freeparameters\] was used because it has an advantage by using the polarimeter constraint, as described later.
$$\begin{aligned}
P_{e^{\pm}}^{-} & = -\left|P_{e^{\pm}}\right| + \tfrac{1}{2}\delta_{e^{\pm}} &
P_{e^{\pm}}^{+} & = \quad\left|P_{e^{\pm}}\right| + \tfrac{1}{2}\delta_{e^{\pm}}
\label{eq:alternativeparameters}\end{aligned}$$
The processes used for the polarization calculation are listed in tab. \[tab:Consideredprocesses\]. They were selected due to their relatively large left-right-asymmetry and unpolarized cross section in order to gain the most sensitive processes to the polarization with the highest event rate. Thereby, they are the same processes as used for physics analyses (DBD) with respect to their classification, labeling and cross section values. The chiral cross section of those processes were calculated on tree-level but including ISR to consider the reduced center-of-mass energy. Furthermore, any combination of processes can be used to study the effect of different combinations and the sensitivity of the different processes on the polarization precision. Thereby, further processes can easily be added.
**Process** single$W^{\pm}$ $WW$ $ZZ$ $ZZWW$Mix $Z$
------------- ------------------------------ ------------------------------------------------- ----------------------------------------- --------------------------------- ------------------
**Channel** $e\nu q\bar{q}$, $e\nu l\nu$ $q\bar{q}q\bar{q}$, $q\bar{q}l\nu$, $l\nu l\nu$ $q\bar{q}q\bar{q}$, $q\bar{q}ll$,$llll$ $q\bar{q}q\bar{q}$, $l\nu l\nu$ $q\bar{q}$, $ll$
: Currently considered processes which are suitable for minimization due to their left-right-asymmetry and unpolarized cross section.[]{data-label="tab:Consideredprocesses"}
In order to test the theoretical limit on the polarization precision, a perfect $4\pi$ detector is assumed with a zero background estimation value and no systematic uncertainties. The statistical precision limit for such a scenario is shown in tab. \[tab:statisticalprecision\](*left*) for the H-20[@ILCrunningscenario] scenario using all implemented processes listed in tab. \[tab:Consideredprocesses\]. For the data sets with larger integrated luminosity, the permille-level is clearly achievable but for lower integrated luminosity data sets (e.g. top-threshold scan at 350GeV), this does not apply. However, in order to still achieve the permille-level precision in particular for lower integrated luminosities, the polarimeter constraint will become important, as described in sec. \[sec:polarimeterconstraint\].
[|c|c|c|c||c|c|]{}\
$E$\[GeV\] & $\textbf{500}$ & $\textbf{350}$ & $\textbf{250}$ & $\textbf{500}$ & $\textbf{250}$\
$\mathcal{L}$\[fb$^{-1}$\] & $\textbf{500}$ & $\textbf{200}$ & $\textbf{500}$ & $\textbf{3500}$ & $\textbf{1500}$\
$\Delta P_{e^{-}}^{-}/P$ & & & $0.1$ & $0.08$ & $0.09$\
$\Delta P_{e^{-}}^{+}/P$ & $0.05$ & $0.06$ & $0.03$ & $0.02$ & $0.02$\
$\Delta P_{e^{+}}^{-}/P$ & $0.1$ & $0.1$ & $0.06$ & $0.04$ & $0.04$\
$\Delta P_{e^{+}}^{+}/P$ & & & $0.1$ & $0.08$ & $0.08$\
[|c|c|]{}\
$E$\[GeV\] & $\textbf{500}$\
$\mathcal{L}$\[fb$^{-1}$\] & $\textbf{500}$\
&\
&\
&\
&\
[|c|c|]{}\
$E$\[GeV\] & $\textbf{500}$\
$\mathcal{L}$\[fb$^{-1}$\] & $\textbf{2000}$\
$\Delta P_{e^{-}}/P$ & $0.085$\
$\Delta \delta_{e^{-}}/P$ & $0.12$\
$\Delta P_{e^{+}}/P$ & $0.22$\
$\Delta \delta_{e^{+}}/P$ & $0.32$\
In order to get a more realistic point of view on the achievable statistical precision, the results are shown in comparison to two previous studies. Both of them also use only statistical uncertainties and a $\chi^{2}$ minimization. The first is the polarization calculation from $W$-pairs, shown in the middle of tab. \[tab:statisticalprecision\]. It additionally uses the information from the production angle and includes fiducial cuts and a complete background calculation. However, for this study, only a 2 parameter fit was performed because a constant absolute polarization was assumed for the helicity reversal. The second study used a combined cross section measurement from single $W^{+}$, $W^{-}$, $Z$, $\gamma$ processes. No angular information was used and no background was considered for this study. But it included fiducial cuts on the total cross section and considered a deviation on the absolute polarization as free parameter. However, $\delta_{e^{\pm}}$ was constraint to $10^{-3}$ in this study. In comparison to the ideal case of the unified approach, the precision is in the same order of magnitude. It shows that even a more realistic scenario can get close on the statistic precision limit.\
In order to reach the goal of 0.1% on the total uncertainty, the systematical uncertainties also need to be controlled at the permille-level. The uncertainties on the integrated luminosity $\Delta\mathcal{L}$ and the selection efficiency $\Delta\varepsilon$ are influenced by the machine performance and detector calibration and alignment, respectively. Thus, $\Delta\mathcal{L}$ and $\Delta\varepsilon$ are time dependent. If the switch between the different spin configuration (helicity reversal) is faster than a change in the calibration, alignment, etc. of the detector and accelerator, $\Delta\mathcal{L}$ and $\Delta\varepsilon$ become correlated, as introduced in eq. \[eq:covariancematrix\]. This correlation leads to cancellation of the systematic uncertainties reducing the impact on the polarization precision. Otherwise, the polarization precision would saturate and it would be limited by the systematic uncertainties, as shown in Fig. \[fig:correlateduncertainty\].\
Therefore, a fast helicity reversal (e.g. train-by-train) is required to reduce the influence of the time dependent systematic uncertainties on the polarization precision. Note that this still applies even if the helicity reversal is not perfect because a non-perfect helicity reversal has close to no influence on the precision due to compensation of the unified approach.\
![The effects of the fast helicity reversal on the polarization precision because of correlated uncertainties.[]{data-label="fig:correlateduncertainty"}](CorrelatedLefthandedElectronPolarization500GeVsingleWLsL_WWHLsL_ZZHLsL_ZHL.pdf){width="50.00000%"}
Frequency and Accuracy of the Fast Helisity Reversal
----------------------------------------------------
The advantage of a fast helicity reversal was demonstrated in the last section. For the realization, it is important that the switch between the four different spin configuration is performed during normal operation with no additional breaks. As seen in Fig. \[fig:bunchstrcture\] for the ILC bunch structure, there are in principle two possible options:
![The ILC structure of a bunch train[]{data-label="fig:bunchstrcture"}](figures_ILCtimeslicing.png){width="50.00000%"}
- **Bunch-by-bunch:**\
The helicity reversal is faster than the duration the 337ns time gap between two bunches.
- **Train-by-train:**\
The helicity reversal is performed within the break between two trains. In the ILC design, an approximately 199ms break is foreseen between the trains. This time slot, which is inter alia used for readout of the detector, can also be used to switch between the different spin configurations.
It is assumed that the switch train-by-train yields a sufficient correlations to suppress systematic uncertainty but a precise determination of the correlation factor is still an open topic. For a realistic scenario, a non-perfect helicity reversal has to be considered. This is particularly important for the positron beam. As shown in Fig. \[fig:positronflip\], the helicity reversal for the positron beam is accomplished by switching between two beam lines, which also enables the possibility of a fast helicity reversal. The polarized electron beam is generated by shooting a circular polarized laser onto a photo cathode. The sign of the electron beam polarization corresponds to the sign of the laser polarization. Thus, the helicity reversal for the electron beam is achieved by a switch of the laser polarization, which can easily performed train-by-train. Here, the deviation from a perfect helicity reversal are assumed to be very small. In principle, a non-perfect helicity reversal is no issue as long as it can be accurately measured.
![The schematic layout of positron transport to Damping Ring with a two parallel lines spin rotator section.[@positronflip][]{data-label="fig:positronflip"}](PositronSpinFlip.jpg){width="80.00000%"}
To show that the new unified approach can handle variations in the absolute polarization, toy measurements with 5 different polarization discrepancies for both beams were performed. However, the nominal absolute polarization values $\left|P_{e^{-}}\right| = 80\%$, $\left|P_{e^{+}}\right| = 30\%$ were used as initial parameters for each toy measurements. Only statistical uncertainties were assumed.
![Results for 5 different absolute polarization deviations after helicity reversal. *Left*: the correlation between the two helicity configuration of the electron beam from the toy measurements. The magenta triangles corresponds to actual polarization values. *Right*: The residual of the measured polarization from the actual polarization combined for all 5 polarization discrepancy.[]{data-label="fig:nonperfecthelicityreversal"}](Pvariation_LvsR_electron500GeV.pdf "fig:"){width="49.00000%"} ![Results for 5 different absolute polarization deviations after helicity reversal. *Left*: the correlation between the two helicity configuration of the electron beam from the toy measurements. The magenta triangles corresponds to actual polarization values. *Right*: The residual of the measured polarization from the actual polarization combined for all 5 polarization discrepancy.[]{data-label="fig:nonperfecthelicityreversal"}](Pvariation_display500GeV.pdf "fig:"){width="49.00000%"}
As seen in the left plot of Fig. \[fig:nonperfecthelicityreversal\], the results of $\chi^{2}$-fit fluctuating within their uncertainties around the actual beam polarization values, displayed as magenta triangles. As seen in the right plot of Fig. \[fig:nonperfecthelicityreversal\], all deviations from the actual polarization only occur due to the statistical uncertainty. A change in the uncertainty in comparison to equal absolute polarization is not noticeable. Therefore, the assumption that a non-perfect helicity reversal has no influence on the polarization determination is justified because the absolute polarizations are correctly determined and there is also no change in the uncertainty. Thus, the new unified approach compensates for a non-perfect helicity reversal.\
Improvement by Constraints from Polarimeter Measurement {#sec:polarimeterconstraint}
=======================================================
In a simplified approach, the effects of using the polarimeter measurement as an additional constraint were studied. This is just a first step to demonstrate the proof of principle. In this approach, spin transport was neglected and it was pretended that the polarimeter could measure the polarization direct at the IP with the relative nominal precision of $\Delta P/P = 0.25\%$. Furthermore, a Gaussian distribution of the polarimeter measurement is assumed with the nominal absolute polarization values $\left|P_{e^{-}}\right| = 80\%,\left|P_{e^{+}}\right| = 30\%$ as mean value and the nominal uncertainty $\Delta P$ as deviation.\
For the implementation in the $\chi^{2}$ minimization, the squared pull terms of the 4 polarization parameters and the polarimeter measurement were additionally added to the existing $\chi^{2}$ function, as seen in Fig. \[eq:chiwithconstraint\]
$$\begin{aligned}
\chi^{2} &=
\underbrace{\sum_{\text{process}}{
\left(\vec{\sigma}_{\text{data}} - \vec{\sigma}_{\text{theory}}\right)^{T}\Xi^{-1}
\left(\vec{\sigma}_{\text{data}} - \vec{\sigma}_{\text{theory}}\right)
}}_{\text{Contribution from the cross section measurement}} + \underbrace{\sum_{P}{\left[\frac{\left(P_{e^{\pm}}^{\pm} - \mathcal{P}_{e^{\pm}}^{\pm}\right)^{2}}{\Delta\mathcal{P}^{2}}\right]}}_{\text{Polarimeter term}}
\label{eq:chiwithconstraint} \end{aligned}$$
Here, $P_{e^{\pm}}^{\pm}$ are the 4 polarization parameters used also in the other part of the $\chi^{2}$ function and $\mathcal{P}_{e^{\pm}}^{\pm}$ are the corresponding polarimeter measurement. Here the advantage of the parametrization becomes clear because the free polarization parameter can be directly compared with the polarimeter measurement without any further conversion of the parameters. $\Delta\mathcal{P}$ is the uncertainty on the polarimeter measurement.\
The results for the H-20[@ILCrunningscenario] scenario can be seen in Fig. \[figtab:polarimeterconstraint\]. With the constraint, the overall polarization precision can be improved for all runs. This is of particular importance for runs with less integrated luminosity. As described earlier, the precision goal of a permille-level can not be reached in such runs, which is now possible due to the polarimeter constraint.\
![The improvement on the polarization precision for the left-handed electron beam by using the polarimeter constraint displayed for the H-20[@ILCrunningscenario] scenario.[]{data-label="figtab:polarimeterconstraint"}](Chi_Running_Senarios_constraint_H-20electronleft.pdf){width="\textwidth"}
[|c|c|c|c||c|c|]{}$E$ & $\textbf{500}$ & $\textbf{350}$ & $\textbf{250}$ & $\textbf{500}$ & $\textbf{250}$\
$\mathcal{L}$ & $\textbf{500}$ & $\textbf{200}$ & $\textbf{500}$ & $\textbf{3500}$ & $\textbf{1500}$\
\
$P_{e^{-}}^{-}$ & & & $0.1$ & $0.08$ & $0.09$\
\
$P_{e^{-}}^{-}$ & & & $0.1$ & $0.07$ & $0.07$\
As a next step, the constraint has to be modified to a more realistic scenario. First of all, the polarimeters do not measure the polarization at the IP, thus the measurement has to be extrapolated. However, the information of the polarimeters can also be included to compensate for misalignments in the BDS (Spin tracking) and for beam collision effects.[@spintracking]\
Conclusion and Outlook
======================
Polarization provides a deep insight in the chiral structure of the standard model and beyond. Therefore, a permille-level precision of the luminosity-weighted average polarization at the IP is required. With the new unified approach, all suitable cross section measurements as well as the constraints of the polarimeter measurement are combined by using a overall $\chi^{2}$ minimization\
With this approach a statistical precision of a permille-level is achievable. Furthermore, the impact of time-dependent systematic uncertainties can be reduced due to a fast helicity reversal, while a non-perfect helicity reversal has no impact on the statistical precision. Thereby, the further improvement due to polarimeter constraints is particular important for the low integrated luminosity.\
A further topic will attend to the time-dependence of the beam polarization. Currently there is always the assumption of a time-independent beam polarization. But with the H-20[@ILCrunningscenario] scenario in mind, it is not realistic to assume that the polarization stays constant over a time of up to 7 years. Thus, potential time-dependencies also have to be corrected in the polarimeter constraint. An important point is that it is straight-forward to include such effects in the new unified approach.\
In order to gain a realistic and precise description of the polarization uncertainty, it is important to consider an accurate estimation of the systematic quantities (selection efficiency $\varepsilon$ and background estimation $B$), their uncertainty ($\Delta B$, $\Delta\varepsilon$, $\Delta\mathcal{L}$) and correlations. Although systematic quantities are fully implemented in the $\chi^{2}$ algorithm, their actual value still has to be determined.\
Using the angular information of a process, the precision on the polarization can be further improved. This is achieved by implementing differential cross sections within the $\chi^2$-method.\
[99]{} **ILC Project**, T. Behnke et al., eds., *International Linear Collider Reference Design Report Volume 1: Executive Summary*. 2013 Annika Vauth, Jenny List. *Beam Polarization at the ILC: Physics Case and Realization*,\
Spin Physics (SPIN2014), International Journal of Modern Physics: Conference Series, 29 February 2016,\
<http://www.worldscientific.com/doi/pdf/10.1142/S201019451660003X> Jenny List, Annika Vauth, and Benedikt Vormwald:\
*A Quartz Cherenkov Detector for Compton-Polarimetry at Future $e^{+}e^{-}$Colliders*\
(<https://bib-pubdb1.desy.de/record/221054>)\
*A Calibration System for Compton Polarimetry at $e^{+}e^{-}$Colliders*\
(<https://bib-pubdb1.desy.de/record/289025>) Moritz Beckmann, Jenny List, Annika Vauth, and Benedikt Vormwald:\
*Spin transport and polarimetry in the beam delivery system of the international linear collider*\
(<http://iopscience.iop.org/article/10.1088/1748-0221/9/07/P07003/pdf> Theses Ivan Marchesini, *Triple Gauge Couplings and Polarization at the ILC and Leakage in a Highly Granular Calorimeter* (<http://pubdb.xfel.eu/record/94888>) Talk Graham W. Wilson, *Beam Polarization Measurement Using Single Bosons with Missing Energy*\
(<https://agenda.linearcollider.org/event/5468/contributions/24027/>) Jenny List, *Running Scenarios for the ILC*,\
LC Forum / Terascale Annual Meeting November 17-18, 2015, DESY\
(<http://pubdb.xfel.eu/record/289187/files/jlist_lcforum_1511.pdf?version=1>) L.I. Malysheva, O.S. Adeyemi,V. Kovalenko, G.A. Moortgat-Pick, *THE SPIN-ROTATOR WITH A POSSIBILITY OF HELICITY SWITCHING FOR POLARIZED POSITRON AT THE ILC*, 18.04.2013
[^1]: email: robert.karl@desy.de
[^2]: email: jenny.list@desy.de
|
---
abstract: 'High resolution magnetoresistance data in highly oriented pyrolytic graphite thin samples manifest non-homogenous superconductivity with critical temperature $T_c \sim 25~$K. These data exhibit: i) hysteretic loops of resistance versus magnetic field similar to Josephson-coupled grains, ii) quantum Andreev’s resonances and iii) absence of the Schubnikov-de Haas oscillations. The results indicate that graphite is a system with non-percolative superconducting domains immersed in a semiconducting-like matrix. As possible origin of the superconductivity in graphite we discuss interior-gap superconductivity when two very different electronic masses are present.'
author:
- 'P. Esquinazi'
- 'N. García'
- 'J. Barzola-Quiquia'
- 'J. C. González'
- 'M. Muñoz'
- 'P. Rödiger'
- 'K. Schindler'
- 'J.-L. Yao'
- 'M. Ziese'
title: Intrinsic Superconductivity at 25 K in Highly Oriented Pyrolytic Graphite
---
The standard way to ascribe superconductivity to materials is by observing the screening of an external applied magnetic field, the Meissner effect, below a critical field $B_{c1}$ and, although less important from the physical point of view, by measuring the drop of resistance to practically zero below a critical temperature $T_c$. These phenomena are observed for percolative or homogenous superconductors where a macroscopic wave function of the Cooper pairs exists [@thinkam]. It is well known that in inhomogeneous superconducting samples, as for example the well-known ceramic high $T_c$ oxides, sometimes superconductivity does not percolate, then the resistance does not drop to zero and the Meissner effect is small. In this case the criteria to assign non-percolative inhomogeneous superconductivity to a material is much less obvious. In addition, we would like to discuss here a superconducting high-$T_c$ material with a very low density of free electrons or quasiparticles $n \lesssim 10^{18}~$cm$^{-3}$, with very different effective masses $m^\star$. We think that this is the case of highly oriented pyrolytic graphite (HOPG), the material studied in this work.
Untreated HOPG samples manifest large electronic mean free path and Fermi wavelength of order of microns [@gon07]. On the other hand the same samples reveal that the surface is not an equipotential with metallic and insulating regions that can move [@lu06; @gom07]. It seems clear that the view of graphite as a more or less ordered, homogeneous system and with a homogeneous density of carriers cannot be hold and it does not represent the interesting piece of the physics of HOPG. Although resistance $R(T)$ data can be fitted, in some cases, with an homogeneous two band model (TBM) using two mobilities and two carrier concentrations (all temperature dependent parameters) [@kelly], there are other observations as a function of the applied magnetic field reported here that cannot be explained within this model. In this work we treat HOPG as a non-uniform electronic system and as such it will be discussed.
To aboard this hard problem we have obtained over $10^6$ high resolution magnetoresistance (MR) data points in a range of temperatures. These data exhibit: (i) irreversible hysteretic loops of resistance versus magnetic field similar to those observed in granular superconductors with Josephson-coupled grains [@ji93; @kope01] that can be assigned to superconducting fluxons, (ii) quantum Andreev’s resonances in the MR [@gar07] and (iii) absence of Schubnikov-de Haas (SdH) oscillations. The experimental data indicate the existence of energy gaps at the Fermi level and that HOPG is a non-percolative superconductor with “granular" domains immersed in a semiconductor-like matrix. The origin of the superconductivity in graphite may be assigned to interior-gap superconductivity that predicts a gapless stability when two different masses are present, a problem that has been discussed by Liu and Wilczeck [@liu03].
The high-resolution, low-noise four-wires MR measurements have been performed by AC technique (Linear Research LR-700 Bridge with 8 channels LR-720 multiplexer) with ppm resolution and in some cases also with a DC technique (Keithley 2182 with 2001 Nanovoltmeter and Keithley 6221 current source). The temperature stability achieved was $\sim 0.1~$mK and the magnetic field, always applied normal to the graphene planes, was measured by a Hall sensor just before and after measuring the resistance, and located at the same sample holder inside a superconducting-coil magnetocryostat. We used currents between $1 \ldots 100~\mu$A.
To start with our strategy we have prepared different samples of HPOG that just differ in its ordering and size and they exhibit apparently different behaviors with T. Figure \[rt\] shows $R(T)$ for the samples indicated in the figure caption. Usually one tends to fit these curves with the TBM. In particular the $R(T)$ of sample (3) can be fitted approximately. However, carriers in HOPG have two different masses and one of them is practically zero corresponding to Dirac electrons [@luky04]. Furthermore, there are other important aspects described below that undoubtedly cannot be put into accord with the TBM. We concentrate in the very thin and micrometer small sample because it should have less number of fluctuating domains and this should provide more clear superconducting-related effects. Note that this sample shows a semiconducting like behavior that levels off at $T
\simeq 25~$K; its in-plane resistivity $\rho_{ab} (10~$K$) \simeq
(50 \pm 10)~\mu\Omega$cm is similar to the one of sample (1) from which it has been obtained by careful exfoliation.
. The Pd-electrodes (to avoid Schottky barriers) were prepared using conventional electron lithography. The sample (2) $(R(275~$K) = 6.5 m$\Omega$) was obtained from HOPG grade ZYC bulk sample ($3.5^\circ$). The inset shows schematically the energy dispersion relations for two carriers in graphite, massive and massless (Dirac fermions). Following Ref. , the instability region lies around the Fermi wavevector of the light particles.[]{data-label="rt"}](rt.eps){width="85mm"}
Figure \[osc\](a) shows the MR of sample (3) at 4 K in detail and in the region 4 T to 8 T with larger resolution using a magnetic field step of $\simeq 1$ Oe. The first surprise is that the MR is very small compared with the MR of larger samples of HOPG. In these samples the ordinary MR of HOPG between 0 T and 8 T is $\sim 10000\%$ while in the small sample measured here is only $\lesssim 300\%$. This difference is discussed in Ref. . In addition, SdH oscillations are absent in sample (3) (in other samples of similar size we measured they appear very weak). This might imply that the Fermi level lies in a gap. Notice that we decided to perform experiments with very small field increment. This was not done accidentally. The reason is that we expected to have weak quantum oscillation resonances – compared with the classical SdH oscillations – due to the small number of potential fluctuations (note that the sample is small, of the order or smaller than the mean free path and Fermi wave length) and these fluctuations will induce an oscillating transmissivity through the potential wells. These quantum oscillations were proposed theoretically to interpret observed structures that were over seen or consider noise in graphene samples [@gar07]. And of course the sample of Fig. \[osc\] shows the expected quantum oscillations. These quantum oscillations have a two period spectrum indicating that in the sample one has at least two characteristic potential wells. Figure \[osc\](b) shows the oscillation amplitude of the two harmonics (see also the inset) as a function of $T$, which remain constant below 10 K and vanish at a critical temperature $T_c
\simeq 25~$K.
![(a) Resistance of sample (3) between two adjacent voltage electrodes as a function of magnetic field. A close inspection of the MR of this sample at fields above 0.5 T reveals an anomalous behavior, namely the MR oscillates. The oscillations shown in the insets were obtained after subtracting a quadratic field dependence around 5.3 T and 6.4 T. These small-field-period oscillations in the resistance are superposed to oscillations of larger amplitude and field period, see inset in (b). Further measurements indicate that the overall shape, field positions and period are independent of the field sweeping rate, field step and field sweep direction. The oscillations are observed at low as well as high fields, as expected because the slope of $R$ vs. $B$ does not depend appreciable with field [@gar07]. Different periods as well as oscillation amplitudes are observed for other samples. The inset shows an optical microscope picture of the sample with the Pd-electrodes. Counting clockwise from input current electrode 1 at the right, the data shown were taken between electrodes 3 and 4 (Ch.2). (b) Temperature dependence of the voltage amplitude of the two oscillations taken from the Fourier fit, see inset. The continuous lines are a guide. The inset shows the data at 2 K after subtraction of a linear field background and the continuous line is the Fourier fit with periods 0.1 T and 0.387 T. These periods are independent of temperature within experimental resolution.[]{data-label="osc"}](osc.eps){width="85mm"}
We claim that these oscillations, given their small amplitude of $\sim 100~$nV$ \ldots 400~$nV (much smaller than the corresponding values in temperature for the used range $T \eqslantgtr 2~$K) are due to the interference of wave functions that suffer Andreev’s reflections at the potential walls matching low-gap semiconducting with superconducting regions. From the period of the oscillations in field we can estimate that there are superconducting “granular" domains of size around $1~\mu$m separated by small-gap semiconducting matrix of similar size, which couples the superconducting grains. If this picture is realized one expects to see pinning and dissipation effects due to fluxons, as discussed by Ji et al. in Ref. , with circumvent superconducting currents between the superconducting grains through the semiconducting regions. One may argue against the physical ground of the model we are proposing: how is it possible that superconducting pairs can be kept in a micron-size semiconducting-like regions connecting the superconducting ones? This should not be a problem. By using nano-fabricated constrictions and measuring the transition from ohmic to ballistic transport we have observed that the mean free path of the carriers in HOPG at 10 K is $\gtrsim 10~\mu$m. Therefore, it should be perfectly possible that the pairs travel $\sim 1~\mu$m distance without breaking out. In other words the proximity effect in graphite may extend to microns.
If there are fluxons then one should have irreversible hysteretic loops of the kind observed in granular superconductors [@ji93; @kope01]. Figure \[irr\](a) shows this irreversibility that cannot be explained by ferromagnetism, ferroelectricity due to motion of charges or by usual Abrikosov vortices, since no sign of irreversibility has been seen within experimental error for magnetic fields applied parallel to the planes. We have a huge anisotropy in an otherwise a small spin-orbit coupling material. Note that the two minima in $R$ are observed at the positive and negative fields coming from high fields from the same direction. Only by fluxons running between the superconducting and the semiconducting-like regions these hysteresis loops can be explained. For a better appreciation of the hysteresis the inset in Fig. \[irr\](a) shows the difference between the two curves, i.e. the resistance curve obtained by starting at a negative field and sweeping to positive fields is subtracted from the resistance curve measured when starting at a positive field and sweeping to negative fields. The height of the extreme as well as their fields $B_m(T)$ depend on $T$. The $T$-dependence of this irreversibility $\Delta R$ as well as $B_m(T)$ vanish at $T_{i} \sim 11~$K. The reason why the irreversible behaviour shown in Fig. \[irr\] vanishes at $\sim
11$ K in contrast to the $\sim 25$ K observed from the oscillatory behavior of Fig. \[osc\], can be easily related to the pinning of the fluxons inside the grains. The temperature dependence of the irreversibility in field, continuous lines in Fig. \[irr\](b), follows $(1-(T/T_i))^{1.5}$ a similar dependence as for the irreversibility line of vortices observed in high-temperature superconductors.
![(a) Strongly enhanced MR curve near zero field. A weak hysteresis appears similar to butterfly MR loops for superconductors with Josephson-coupled grains [@ji93]. For a clear observation of the hysteresis we present in the inset the difference of the resistance curves (see text). (b) The height of the irreversibility maximum $\Delta R$ as well as their field positions $B_m$ (see inset) vs. temperature. The continuous lines follow the function $\propto (1 - (T/11))^{1.5}$.[]{data-label="irr"}](irr.eps){width="85mm"}
Because it is just graphite, the superconducting regions have a very small number of free electrons, say $\lesssim 10^{-4}$ electrons per carbon atom [@kelly]. A simple estimate shows that the London penetration length is larger than microns and therefore the Meissner effect should be unnoticeable. Also the resistance does not drop to zero because the superconducting regions do not percolate, in additions to the resistance due to the motion of fluxons. The observed hysteresis is a very strong fingerprint of superconducting fluxons, difficult to rule out.
The density of carriers in HOPG samples is very probably highly inhomogeneous, and upon region in the sample it may be much smaller than $10^{-4}$/C-atom. What might be the physical origin of this superconductivity? Graphite contains two carrier families with very different $m^{\star}$, one with a negligible mass called Dirac fermions. Therefore, the ratio between masses may be very large, 100 or larger. These different masses establish large instabilities if the number of the different carriers is not that different. Then we have a large extended band with a large Fermi energy corresponding to the light carriers and a lower Fermi energy for the heavy carriers, see inset in Fig. \[rt\]. A large density of the heavy carriers pinned at the Fermi energy of the light particles has strong electron interaction and creates instabilities that will be discussed in other work. In particular for this situation Liu and Wilczek [@liu03] have predicted a condensed superfluid state called interior-gap superconductivity or breached superconductivity. Graphite might be a good candidate where some concepts of this theory could be useful. In fact the picture they describe for their theory [@liu03] is similar to that of the inset in Fig. \[rt\]. In this theory no gap exists and the material may exhibit $p$-type superconductivity, which has been also discussed for graphite [@gon01] as a more robust state in a non-homogeneous system.
The results of Figs. \[osc\] and \[irr\] belong to a micrometer size sample (parallel to the planes) and 12 nm thickness in order to have few potential fluctuations. Measurements in two other samples of similar size show similar behavior but slightly different $T_c$’s. In larger samples, as for example the other two reported in Fig. 1, the same type of effects should be seen but more in terms of universal conductance fluctuations. In fact we have observed in these and other larger samples fluctuations in the resistance up to room temperature, however they are difficult to tackle down and their amplitudes change with time, an effect that is probably related to the motion of charges with current and applied magnetic field. Superconductivity in graphite should by no means limited to the 25 K here obtained for the small sample, but depends on the charge density, defect density and the related instabilities at Fermi level.
We note that hints for superconductivity in HOPG samples from SQUID measurements have been invoked in the past [@kopejltp07]. However, resolution limits of the magnetometer and the partial admixture of ferromagnetic-like signals casted doubts on the origin of those signals. Other studies [@yakovadv03] claimed superconductivity in graphite based on the metal-insulator transition observed under a magnetic field, although superconductivity does not necessarily need to be invoked to understand this transition. There is also a theoretical work that claims high-$T_c$ $d$-wave superconductivity in graphite based on resonating valence bonds [@doni07].
Concluding, in this work we have obtained evidence that supports the existence of intrinsic superconductivity in HOPG based on the irreversibility of the MR and on the quantum oscillations. We think that interior-gap – breached superconductivity [@liu03] is an interesting starting concept to understand the observed as well as other phenomena in the transport properties of graphite.
We gratefully thank Y.Kopelevich for fruitful discussions on the superconductivity of graphite. This work was done with the support of the DFG under ES 86/11, the Spanish CACyT and Ministerio de Educación y Ciencia. J.-L. Yao acknowledges the support from the A. von Humboldt foundation.
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abstract: 'We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple $Z(N)$ model for $N \ge 3$ for which the model hamiltonian is non-hermitian. For $N=2$ the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the $Z(N)$ model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing $N$. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.'
author:
- 'Francisco C. Alcaraz'
- 'Murray T. Batchelor'
title: 'Anomalous bulk behaviour in the free parafermion $Z(N)$ spin chain'
---
Introduction
============
It is well known that non-hermitian systems are expected to behave differently to hermitian systems. This is because non-hermitian hamiltonians describe the dynamics of physical systems that are not conservative. Specifically, hermiticity guarantees that the energy spectrum is real and that time evolution is probability-preserving. Although there are many examples of integrable hermitian hamiltonians, integrable non-hermitian spin chain hamiltonians are relatively rare. An important exception is the class of non-hermitian spin chains whose hamiltonians are ${\cal P} {\cal T}$ symmetric, ensuring a real eigenspectrum [@Bender; @others].
Arguably the simplest of all exactly solved hermitian hamiltonians are those described by free fermions. Indeed, the concept of free fermions plays an all pervasive and enduring role in the description of interacting classical and quantum spin systems. Recently it has become apparent that there is a simple exactly solved non-hermitian $Z(N)$ hamiltonian $$H_{\mathrm{open}}(L) = - \sum_{j=1}^{L-1} \sigma_j \sigma_{j+1}^{\dagger} - \lambda \sum_{j=1}^L \tau_j
\label{ham}$$ which displays the remarkable property of free parafermions [@Baxter1989a; @Baxter1989b; @Baxter2004; @Fendley2014; @ABL2017], with a complex eigenspectrum. This model is an $N$-state generalisation of the widely studied (hermitian) quantum Ising chain in a transverse field. Here $\sigma_j$ and $\tau_j$ are the usual $Z(N)$ operators, which in matrix form are defined by $$\begin{aligned}
\sigma_j &=& I \otimes I \otimes \cdots \otimes I \otimes \sigma \otimes I \otimes \cdots \otimes I\\
\tau_j &=& I \otimes I \otimes \cdots \otimes I \otimes \tau \otimes I \otimes \cdots \otimes I\end{aligned}$$ where $I$, $\sigma$ and $\tau$ are each $N \times N$ matrices, with $\sigma$ and $\tau$ in position $j$. Here $I$ is the identity, with $\sigma$ and $\tau$ having components $$\sigma_{m,n} = \omega^{m-1}\delta_{m,n}, \quad \tau_{m,n} = \delta_{m,n+1} %\, (n < N)$$ with $\omega = {\mathrm e}^{2\pi {\mathrm i}/N}$ and $\tau_{m,N}=\delta_{m,1}$. These are the clock and shift matrices satisfying $$\sigma \tau = \omega \tau \sigma, \quad
\sigma^\dagger = \sigma^{N-1}, \quad \tau^\dagger = \tau^{N-1}$$ with $\sigma^N = \tau^N = I$. For $N=2$ they are the usual Pauli matrices $\sigma^z$ and $\sigma^x$.
The parameter $\lambda$ plays the role of temperature. Following [@ALC], for the duality transformation for general $Z(N)$ quantum chains, it is simple to verify that hamiltonian (1) is self dual, namely $H(\lambda) = \lambda H(1/\lambda)$. We then expect, by usual arguments, that the model is critical at the self dual point $\lambda=\lambda_c =1$. This is verified in the open boundary case, where the finite-size gaps are exactly known [@ABL2017].
Generalizations of the hamiltonian (\[ham\]) with the hermitian conjugate term included have been the subject of recent studies [@hcon], mostly for $N=3$, in the context of parafermionic edge modes [@review]. The unique property of hamiltonian (\[ham\]) is that the energy eigenspectrum has the simple form $$-E/\lambda = \omega^{s_1} \epsilon_1 + \omega^{s_2} \epsilon_2 + \cdots + \omega^{s_L} \epsilon_L
\label{spec}$$ for any choice of the integers $s_k = 0, \ldots, N-1$. This covers all $N^L$ eigenvalues in the spectrum. Just as the fact that the special $N=2$ case $E/\lambda= \pm \epsilon_1 \pm \epsilon_2 \pm \cdots \pm \epsilon_L$ can be taken as the basic property of a free fermion system, the form (\[spec\]) is the basic property of a free parafermion system.
The quasi energy levels $\epsilon_j$ $(j=1,\ldots,L)$ appearing in (\[spec\]) are functions of $\lambda$. Defining $g=1/\lambda^{N/2}$, the values $\epsilon_j^N$ are determined by the eigenvalues of the $L \times L$ matrices $C^\dagger C$ or $C C^\dagger$, where $$C=\begin{bmatrix} 1 & & & &\\ g &1&&&\\ & g & 1 & &\\ & & \ddots & \ddots & \\& & & g & 1
\end{bmatrix}%, \quad
%C^\dagger=\begin{bmatrix} 1 & \lambda & 0 & 0\\ 0 &1&\lambda&0\\ 0 & 0 & 1 & \lambda \\ 0 & 0 & 0 & 1
%\end{bmatrix}$$ with $$\epsilon_{j} = \left( 1 + g^2 + 2 g \cos {k_j} \right)^{1/N}.$$ The roots $k_j$, $j=1,\ldots,L$, satisfy the equation [@ABL2017] $${\sin(L+1) k = - g \sin Lk}.
\label{eqn}$$ Using this solution a number of exact results have been derived for this model [@ABL2017]. Although having a simpler hamiltonian than the free fermionic superintegrable chiral Potts model, the free parafermionic model is seen to share some critical properties with it, namely the specific heat exponent $\alpha=1-2/N$ and the anisotropic correlation length exponents $\nu_\parallel =1$ and $\nu_\perp=2/N$.
Here we consider the more general hamiltonian $$H(L,a) = H_{\mathrm{open}}(L) - a \, \sigma_L \sigma_1^\dagger
\label{gen}$$ where $H_{\mathrm{open}}(L)$ is as defined in (\[ham\]) and $a$ is a [real]{} parameter interpolating between periodic boundary conditions (PBC) ($a=1$) and anti-periodic boundary conditions ($a=-1$). Obviously $a=0$ recovers the model with open boundary conditions (OBC). The motivation for the present study is to investigate the role of boundary conditions on the properties of the free parafermion $Z(N)$ model for $N \ge 3$ [@footnote]. As discussed for the chiral Potts model from the perspective of conformal field theory [@Cardy1993], several of the usual properties of hermitian systems, such as insensitivity of bulk thermodynamic quantities to boundary conditions, can fail in the non-hermitian case. As foreshadowed, this note of caution applies even more so for the model under consideration [@ABL2017]. We report here that the role of boundary conditions is seen to have a profound effect on the bulk properties of the non-hermitian free parafermion $Z(N)$ hamiltonian.
Bulk ground state energy per site
=================================
Periodic boundary conditions
----------------------------
As remarked above, the $Z(N)$ model defined in Eq. (\[gen\]) is solved exactly for general $N$ and finite $L$ for the case of OBC ($a=0$). For PBC ($a=1$) we resort to numerical diagonalization to calculate the ground-state energy per site $e_L=E_0(L)/L$ for the $Z(N)$ model for chain sizes $L=2,3,\ldots,L_{\mathrm{max}}$. For comparison we also consider OBC in the same way. From the energy expression (\[spec\]) it is evident that the ground-state energy is real for OBC, corresponding to the integers $s_k = 0$ for all $k$. For PBC, although no similar such exact solution has been obtained for PBC, we observe that the ground-state energy is also real. A proof of this observation, based on symmetries of these quantum chains is still missing.
In the present study, we concentrate on the value $\lambda=1$. The values for the ground-state energy per site are plotted for some fixed chain sizes and different values of $N$ in Fig. \[figurea\]. We clearly see that for a given size $L$, the difference between $e_L$ for PBC and OBC increases with $N$. Moreover, while $e_L$ increases with $N$ for OBC, it decreases with $N$ for PBC. Extrapolated estimates for $e_\infty$ are shown in Table \[table1\]. The extrapolations were performed using van der Broeck-Schwartz extrapolants with $\epsilon$-extension (VBS) [@VBS]. In each case the error indicated is an evaluation taking into account the stability as $\epsilon$ is changed in the extrapolation. The estimates for $e_\infty$ are visualized in Fig. \[figureb\], which shows the striking dependence of the bulk ground-state energy per site on the boundary conditions. The known exact result for $e_\infty$ with OBC is given further below in Eq. (\[eq-3-1\]), with $e_\infty=-1$ in the limit $1/N \to 0$.
![ The ground-state energy per site for the $Z(N)$ spin chain with periodic boundary conditions (PBC) and open boundary conditions (OBC) for $N=3,4,5,6,7,8,10$ and $20$. The data points (see legend) are the values for the $Z(N)$ model for chain sizes $L=7$, $L=10$ and $L=11$.[]{data-label="figurea"}](figure1.eps){width="45.00000%"}
$L_{\mbox{max}}$ Extrap. PBC Extrap. OBC Exact OBC
--------- ------------------ ---------------------- ---------------------- ------------------
$Z(3)$ 21 $-1.1544 \pm 0.0002$ $-1.1321 \pm 0.0002$ $-1.13209336...$
$Z(4)$ 17 $-1.2219 \pm 0.0002$ $-1.0787 \pm 0.0001$ $-1.07870520...$
$Z(5)$ 14 $-1.3280 \pm 0.0002$ $-1.0524 \pm 0.0001$ $-1.05246524...$
$Z(6)$ 13 $-1.4192 \pm 0.0002$ $-1.0375 \pm 0.0001$ $-1.03754819...$
$Z(7)$ 12 $-1.4913 \pm 0.0002$ $-1.0282 \pm 0.0001$ $-1.02823144...$
$Z(8)$ 11 $-1.5482 \pm 0.0001$ $-1.0220 \pm 0.0001$ $-1.02201332...$
$Z(10)$ 10 $-1.6312 \pm 0.0002$ $-1.0145 \pm 0.0001$ $-1.01447454...$
$Z(20)$ 7 $-1.8080 \pm 0.0004$ $-1.0038 \pm 0.0001$ $-1.00384106...$
![Depiction of the contrast between the extrapolated estimates for the ground-state energy per site for the $Z(N)$ model with periodic boundary conditions (PBC) and open boundary conditions (OBC) for $N=3,4,5,6,7,8,10$ and $20$. These results are the values shown in Table \[table1\].[]{data-label="figureb"}](figure2.eps){width="45.00000%"}
General boundary conditions
---------------------------
In order to further investigate the effect of the boundary conditions, we now consider the general boundary hamiltonian $H(L,a)$ given in Eq. (\[gen\]). Here the parameter $a$ interpolates between the open and periodic cases. In Fig. \[fig3\] we show the values of $e_L(a)=E_0(L,a)/L$ for the $Z(6)$ model for chain sizes $L=2-9$. We see in this figure the existence of peaks as a function of the parameter $a$. As $L$ becomes larger the peaks tend to the position $a=0$, i.e., the OBC case, and become sharper as the chain size grows. In Fig. \[fig4\] we show the curves of Fig. \[fig3\] in a larger scale around $a=0$, at which the exact result is known. These figures appear to indicate that, except for the OBC $a=0$, all the closed boundaries $a\neq 0$ have the same value for the ground-state energy per site in the infinite size limit. In Fig. \[fig3\] we also show the values obtained from the VBS-extrapolations using the lattice sizes $L=2-9$. Here the errors shown in the extrapolation are not errors in the strict sense, but rather subjective evaluations taking into account the behavior of the extrapolations.
![ The ground-state energy per site for the $Z(6)$ model (\[gen\]) for the general boundary conditions defined by the parameter $a$. The VBS-extrapolated results are also shown (the deviations in the extrapolations are subjective).[]{data-label="fig3"}](figure3.eps){width="45.00000%"}
![ The ground-state energy per site for the $Z(6)$ model (\[gen\]) for the general boundary conditions defined by the parameter $a$. The exact value for the open boundary case is shown.[]{data-label="fig4"}](figure4.eps){width="45.00000%"}
In order to confirm the abnormal behavior at $a=0$ we compute numerically the derivative $e'_L(L,a)=de_L(a)/da |_{a=0}$. Specifically, we compute the right-derivative $$\begin{aligned}
\label{eq4-4}
\frac{df(x)}{dx} &=& \frac{-3f(x) +4 f(x+\Delta x) -f(x+2\Delta x)}{2\Delta x} \nonumber \\
&& + \, O((\Delta x)^2). \end{aligned}$$ The results for this derivative up to $L=9$ are shown in Table II for the $Z(6)$ model. These values are shown in a log-log plot in Fig. \[fig5\]. We clearly see that the derivatives diverge to $-\infty$ polynomially with $L$. A fit for the $Z(6)$ model, obtained from the chain sizes $L=6-9$ (dashed rectangle in Fig. 5), gives $de_L(a)/da |_{a=0} \approx -0.00025 L^{6.25}$. The tendency for an infinite derivative can also be seen in Fig. 6, where we plot the inverse of the derivative as a function of $1/L$. Here the tendency is clearly towards the value zero as $L\to \infty$.
$L$ $e'_L(L,a) |_{a=0}$
----- ---------------------
2 $-0.91763825$
3 $-1.58769897$
4 $-3.32838276$
5 $-7.67373638$
6 $-18.7154986$
7 $-46.7908356$
8 $-112.234431$
9 $-233.157167$
![ A log-log plot of the derivative $de_L(a)/da |_{a=0}$ as a function of $1/L$ for the $Z(6)$ model (\[gen\]) with boundaries specified by the parameter $a$.[]{data-label="fig5"}](figure5.eps){width="45.00000%"}
![ The inverse of the derivative $de_L(a)/da |_{a=0}$ as a function of $1/L$ for the $Z(6)$ model (\[gen\]) with boundary conditions specified by the parameter $a$.[]{data-label="fig6"}](figure6.eps){width="45.00000%"}
Leading finite-size corrections
-------------------------------
In the open boundary case the leading finite-size corrections to the ground-state energy are known to be given exactly by [@ABL2017] $$\label{eq3}
E_0(L) = Le_{\infty} + f_{\infty} + \frac{b_N}{L^{\nu}} + O(\frac{1}{L^{1+\nu}})$$ where $$\label{eq-3-1}
e_{\infty} = -\frac{2^{\nu}}{\sqrt{\pi}}\frac{\Gamma(\frac{1}{2}+\frac{1}{N})}{\Gamma(1+\frac{1}{N})}, \quad f_{\infty} = \frac{1}{2}e_{\infty} + 2^{\nu -1}$$ and $\nu = 2/N$. The amplitude $b_N$ is also known. In the periodic case we would expect the leading behavior to be of the form $$\label{eq-fit}
\frac{E_0(L)}{L} = e_{\infty} + \frac{b}{L^{\gamma}} +o(1/L^{\gamma})$$ with the exponent value $\gamma = 1+ \nu$. To test this we have evaluated the exponent $\gamma$ in two distinct ways. Firstly we have made a fit where $e_{\infty}$, $b$ and $\gamma$ are free parameters. Secondly we take the extrapolated values shown in Table \[table1\] for the ground-state energy per site $e_{\infty}$ and then perform a fit of the form $$\label{eq4}
\frac{E_0(L)}{L} - e_{\infty} = \frac{b}{L^{\gamma}}$$ with $b$ and $\gamma$ taken as free parameters. For the sake of illustration we show in Fig. \[fig7\] the various fittings for the $Z(N)$ model for values $N=5, 7, 8, 10$ and 20.
![ The fittings, following Eq. (\[eq-fit\]), for the ground-state energy per site $E_0(L)/L$ as a function of $1/L$ for the $Z(N)$ model ($N=5,7,8,10$ and $20$) with periodic boundaries. The values $e_{\infty}$ in the bulk limit are shown in the inset.[]{data-label="fig7"}](figure7.eps){width="45.00000%"}
The values obtained by the two procedures are shown in Table \[table3\]. In columns 2-4 of Table \[table3\] we show the results obtained for the exponent via the first method, with the results obtained via the second method shown in column 5. We believe that the second method is more reliable since it takes into account the extrapolated values of $e_{\infty}$, given in Table \[table1\]. Taking into account both methods we give the estimate shown in column 6, where the error is an indication of the expected precision (clearly subjective).
$N$ $e_{\infty} \mbox{(fit)}$ $b \mbox{(fit)}$ $\gamma \mbox{(fit)}$ $\gamma \mbox{(extr)}$ $ \gamma$ $\gamma_{\mathrm{open}}$
----- --------------------------- ------------------ ----------------------- ------------------------ ----------------- --------------------------
3 $-1.15355$ $-0.68$ 1.68 1.70 1.68 $\pm$ 0.02 1.67
4 $-1.22118$ $-0.72$ 1.89 1.92 1.90 $\pm$ 0.03 1.50
5 $-1.32810$ $-0.63$ 2.02 2.02 2.02 $\pm$ 0.02 1.40
6 $-1.41952$ $-0.53$ 2.05 2.01 2.03 $\pm$ 0.03 1.33
7 $-1.49135$ $-0.46$ 2.06 2.02 2.04 $\pm$ 0.03 1.29
8 $-1.54849$ $-0.40$ 2.06 2.03 2.04 $\pm$ 0.03 1.25
10 $-1.63144$ $-0.33$ 2.06 2.02 2.04 $\pm$ 0.03 1.20
20 $-1.80820$ $-0.17$ 2.07 2.03 2.05 $\pm$ 0.03 1.10
We clearly see from the results of Table \[table3\] that the leading finite-size correction for the ground-state energy is governed by the exponent values $\gamma \approx 2$ for $N \geq 4$, which are quite distinct from the corresponding values with OBC, namely $\gamma = 1 + 2/N$. For comparison of the methods, we also show, up to two decimal digits, the values obtained in this way for the exponent $\gamma$ in the OBC case, using the same lattice sizes as in the periodic case. They are in close agreement with the known result.
Gap exponent
============
The excitation energies above the ground-state, and consequently the energy gaps of the parafermionic models have complex values, irrespective of whether the boundary conditions are open or periodic. Although some energy levels are real, those with lowest real part are complex. In this section we consider the gap with lowest real part. The model (\[gen\]) has a $Z(N)$ symmetry, due to the commutation relation $$\label{eq8}
[H,{\cal{P}}] =0, \quad {\cal{P}}=\prod_{j=1}^L \tau_j.$$ The ground-state belongs to the $Z(N)$ charge ${\cal P} =0$, with the first gap to the sector of charge ${\cal P}=1$. The correlation length exponent $\nu$ can be estimated from the leading finite-size behavior of the first gap, with $$\label{eq9}
G_L = \mbox{Re}\{ E_1(L) - E_0(L)\}=\frac{A}{L^{\nu}} +o(1/L^{\nu})$$ where $A$ is a constant. We consider the finite-size estimator for the exponent $\nu$ defined by $$\label{eq9p}
\nu_{L,L+1} = \frac{ \ln (G_L/G_{L+1})}{\ln((L+1)/L)}.$$
In Table \[table4\] we show the results obtained from VBS-extrapolants of the data for $\nu_{L,L+1}$. We show in the third column the results with our subjective evaluation of the errors. We also show in this table the results obtained for the exponents for OBC, using the same chain sizes. In the last column we show the known exact results for OBC. We clearly see that the values of the gap exponent $\nu$ are quite distinct for PBC vs OBC. It seems that the exponent for the periodic case is close to (if not exactly) the value $\nu=1$, in distinction to OBC where $\nu=2/N$. To illustrate this difference we show in Fig. \[fig8\] the extrapolated results for PBC together with the exact results for OBC.
$N$ $\nu$(extr.) $\nu$ (predicted) $\nu_{\mathrm{open}}$(extr.) $\nu_{\mathrm{open}}$(exact)
----- -------------- -------------------- ------------------------------ ------------------------------
3 1.080 $1.080 \pm 0.005$ 0.667 $2/3=0.666\ldots$
4 1.005 $1.005 \pm 0.003$ 0.500 $2/4=0.5$
5 1.001 $1.001 \pm 0.002$ 0.400 $2/5=0.4$
6 1.002 $1.002 \pm 0.002$ 0.333 $2/6=0.333\ldots$
7 1.000 $1.000 \pm 0.001$ 0.288 $2/7=0.2857\ldots$
8 1.000 $1.000 \pm 0.001$ 0.250 $2/8=0.25$
10 1.000 $1.000 \pm 0.001$ 0.200 $2/10=0.2$
20 1.000 $1.000 \pm 0.001$ 0.100 $1/10=0.1$
![ The results for the exponent $\nu$ obtained from the VBS extrapolations of the estimator (\[eq9p\]) for the periodic $Z(N)$ model and the corresponding exact results for OBC.[]{data-label="fig8"}](figure8.eps){width="45.00000%"}
Specific heat exponent
======================
We calculate in this section the specific heat of the $Z(N)$ model with PBC at the critical point $\lambda =\lambda_c =1$. This quantity is given by $$\label{eq14}
C(\lambda,L) = -\frac{1}{L} \frac{d^2E_0(L)}{d\lambda}.$$ At the critical point we should expect the leading finite-size behavior $$\label{eq15}
C(\lambda=1,L) \sim A \, L^{\alpha/\nu_{\parallel}}$$ where $A$ is a constant. In the case of OBC, $\alpha = 1-2/N$ and $\nu_{\parallel}=1$ [@ABL2017]. In the periodic case the finite-size values of (\[eq15\]) are given in Table \[table5\] for the $Z(N)$ model with $N=3,5,6,7$ and 8. Surprisingly, we see that the data saturates as $L$ increases with a clear indication that the specific heat exponent $\alpha =0$ for the periodic case, as for the $N=2$ Ising model. Actually the results we have obtained show that the periodic case, at least for $N>4$ exhibits a similar behavior as the standard Ising model. This fact should be explored further in subsequent studies.
$L$ $N=3$ $N=5$ $N=6$ $N=7$ $N=8$
----- ---------- ---------- ---------- ---------- ----------
2 0.433013 0.248680 0.175466 0.117594 0.092118
3 0.629961 0.278889 0.189414 0.130737 0.105481
4 0.755042 0.278853 0.191451 0.135007 0.110214
5 0.840759 0.276337 0.192507 0.137145 0.112457
6 0.901140 0.274801 0.193252 0.138354 0.113684
7 0.943967 0.274056 0.193770 0.139095 0.114426
8 0.974148 0.273712 0.194129 0.139580 0.114908
9 0.995022 0.273552 0.194384 0.139914 0.115238
10 1.008975 0.273475 0.194570 0.140154 0.115475
11 1.017767 0.273437 0.194710 0.140331 0.115650
12 1.022719 0.273417 0.194816 0.140466 -
13 1.024835 0.273406 - - -
14 1.024883 0.273401 - - -
15 1.023453 - - - -
16 1.020994 - - - -
17 1.017848 - - - -
18 1.014273 - - - -
19 1.010465 - - - -
20 1.006565 - - - -
Summary and Discussion
======================
The bulk properties of the $Z(N)$ model defined by the non-hermitian hamiltonian (\[gen\]) have been demonstrated here to exhibit a striking dependence on boundary conditions. For illustrative purposes we have focussed on the critical point $\lambda=1$. For $N=2$, the widely studied hermitian quantum Ising chain in a transverse field, the bulk properties are well known to be independent of the boundary conditions. As can be seen clearly in Fig. \[figureb\], the difference between the values obtained for the bulk ground-state energy per site $e_\infty$ with OBC ($a=0$) and PBC ($a=1$) increases with increasing $N$ for $N\ge 3$. As a function of the boundary condition parameter $a$, the bulk ground-state energy per site is a singular point at $a=0$, as can be seen for the $Z(6)$ model in Fig. \[fig3\] and Fig. \[fig4\]. We observed the divergence of the derivative with respect to the parameter $a$ at $a=0$. This is precisely the open boundary case.
The finite-size corrections to the bulk ground-state energy per site are also dependent on the boundary conditions. We found that for PBC the leading finite-size correction to the bulk ground-state energy is of the form (\[eq-fit\]) governed by the exponent values $\gamma \approx 2$ for $N \geq 4$, which are distinct from the corresponding exactly known values for OBC, namely $\gamma = 1 + 2/N$.
The first mass gap exponent has also been numerically estimated for PBC, with values for all $N$ close to the Ising $N=2$ value $\nu=1$. This result is again strikingly different to the known value $\nu=2/N$ for OBC, recall Fig. \[fig8\]. Moreover, the analysis of the specific heat in Section IV indicates that for PBC the values of the specific heat exponent $\alpha$ are also suggestive, at least for $N>4$, of the Ising model value $\alpha=0$. The fact that for the periodic case, for large $N$, the exponent $\gamma$ in (\[eq4\]) is close to 2 suggests we have a relativistic energy-momentum dispersion relation, and possibly an underlying conformal invariance in the bulk limit. Since for large $N$ the exponents $\nu\approx 1$ and $\alpha\approx 0$, the natural possibility would be the Ising universality class with central charge $c=1/2$. In order to test this possibility we have calculated the mass gaps with lowest real part in the eigensectors labeled by the momentum ${2\pi} p/{L}$ ($p=0,1,\ldots,N-1$) and $Z(N)$ charges ($Q=0,\ldots,N-1$) of the $Z(8)$ quantum chain with $L=10$. Exploring the well known consequences of conformal invariance, the mass gap amplitudes of finite lattices give us predictions for the conformal dimensions in clear contradiction with the expected results of an Ising conformal field theory.
At this stage we can only begin to speculate on the reasons for why the boundary conditions have such a profound effect on the bulk properties of this simple $Z(N)$ model. Systems for which the boundary conditions affect the finite-size corrections are usual, normally producing an additional surface term of $O(1/L)$ in the energy. There also exist systems where the mass gap and critical behavior may change or even vanish under change of boundary conditions. An example is the non-hermitian hamiltonian associated with the time-evolution operator of the asymmetric exclusion process where the open problem is gapped (the hamiltonian is related to the XXZ quantum chain in the gapped ferromagnetic regime), but the closed system is gapless and critical (in the KPZ universality class) [@ASEP1; @ASEP2; @AR1]. However, the ground-state energies (with value zero in this example) are the same for both boundary conditions. Systems for which the bulk energy changes with the boundary conditions are surprising exceptions. A prominent example for two-dimensional classical systems is the six-vertex model with domain wall boundary conditions, for which the bulk free energy differs from the well known result obtained using periodic or open boundary conditions [@Korepin]. For the model under consideration here it took some time for us to be fully convinced by our numerical results. For the periodic $Z(N)$ model at $\lambda=1$ the ground-state energy per site decreases with increasing $N$, in contrast to the open case where it increases. The ordinary $Z(N)$ hermitian quantum chains like the Potts or the $Z(N)$ parafermionic models [@FZ] give a bulk ground-state energy which is independent of the boundary conditions and decreases with increasing $N$ [@FCA1; @FCA2]. This suggests that the ground-state of the $Z(N)$ model with open ends is constrained (probably topologically restricted), but by insertion of a single link connecting both sides of the chain, and thereby changing the lattice topology, the energy of the ground-state is decreased enormously (by $O(L)$). Conversely, the physics of the $Z(N)$ model defined on a ring changes drastically by cutting a single link. In this sense it is the $Z(N)$ model with OBC which is the exceptional case.
Here we can also throw into the mix the fact that the $Z(N)$ model with OBC is described by the physics of free parafermions. The free parafermion description works perfectly for this model when subject to OBC, but there is of course no guarantee of a solution in terms of free parafermions for PBC. The underlying reason may thus again be topological and related to the ordering of the parafermionic operators.
[*Acknowledgments.*]{} The work of FCA is supported in part by the Brazilian agencies FAPESP and CNPq. The work of MTB is supported by The 1000 Talent Program of China, National Natural Science Foundation of China Grant No. 11574405 and Australian Research Council Discovery Project DP180101040.
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|
---
abstract: 'We report on a lattice investigation of heavy meson interactions and of tetraquark candidates with two very heavy quarks. These two quarks are treated in the static limit, while the other two are up, down, strange or charm quarks of finite mass. Various isospin, spin and parity quantum numbers are considered.'
address:
- |
$^1$ Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,\
$\phantom{xxx}$ Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
- |
$^2$ Dep. Física and CFTP, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa,\
$\phantom{xxx}$ Portugal
- '$^3$ European Twisted Mass Collaboration (ETMC)'
author:
- 'Björn Wagenbach$^1$, Pedro Bicudo$^2$, Marc Wagner$^{1,3}$'
title: Lattice investigation of heavy meson interactions
---
Introduction
============
We study the potential of two static quarks in the presence of two quarks of finite mass. While in [@Wagner:2010ad; @Wagner:2011ev; @Bicudo:2012qt] we have exclusively considered two static antiquarks and two light quarks ($\bar{Q}\bar{Q}ll$), where $l \in \{ u,d \}$, here we also use $s$ and $c$ quarks, i.e. investigate $\bar{Q}\bar{Q}ss$ and $\bar{Q}\bar{Q}cc$, to obtain certain insights regarding the quark mass dependence of the static antiquark-antiquark interaction. We also discuss first steps regarding the static quark-antiquark case, i.e. $\bar{Q}Q\bar{l}l$, $\bar{Q}Q\bar{s}s$ and $\bar{Q}Q\bar{c}c$.
$\bar{Q}\bar{Q}qq$ systems as well as $\bar{Q}Q\bar{q}q$ systems have been studied also by other groups (cf. e.g. [@Stewart:1998hk; @Michael:1999nq; @Cook:2002am; @Bali:2005fu; @Doi:2006kx; @Detmold:2007wk; @Bali:2010xa; @Bali:2011gq; @Brown:2012tm]).
Creation operators and trial states
===================================
The $\bar{Q}\bar{Q}qq$ and $\bar{Q}Q\bar{q}q$ potentials $V(r)$ are extracted from correlation functions $$C(t) \ \ \equiv \ \ {{\left\langle \Omega \right|}} \mathcal{O}^\dagger(t) \mathcal{O}(0) {{\left| \Omega \right\rangle}}$$ according to $$V(r) \ \ =_{\textrm{large }t} \ \ V_{\textrm{eff}}(r,t) \quad , \quad V_{\textrm{eff}}(r,t) \ \ \equiv \ \ \frac{1}{a} \ln\bigg(\frac{C(t)}{C(t+a)}\bigg) ,$$ where $a$ is the lattice spacing and $\mathcal{O}$ denote suitable creation operators, which are discussed in detail below. For an introduction to lattice hadron spectroscopy cf. e.g. [@Weber:2013eba].
Static-light mesons (“$B$ and $\bar{B}$ mesons”)
------------------------------------------------
The starting point are static-light mesons, which either consist of a static quark $Q$ and an antiquark $\bar{q}$ or of a static antiquark $\bar{Q}$ and a quark $q$ with $q \in \{u,d,s,c\}$. These mesons can be labeled by parity $\mathscr{P} = \pm$, by the $z$-component of the light quark spin $j_z=\pm 1/2$ ($j=1/2$, because we do not consider gluonic excitations) and in case of $q \in \{u,d\}$ by the $z$-component of isospin $I_z = \pm 1/2$ ($I = 1/2$). The lightest static-light meson has $\mathscr{P} = -$ and is commonly denoted by $S$, its heavier parity partner with $\mathscr{P} = +$ by $P_-$. The static-light meson $S$ is an approximation for $B/B^*$, $B_s/B_s^*$ and $B_c$ listed in [@PDG].
We use static-light meson trial states $$\mathcal{O} {{\left| \Omega \right\rangle}} \ \ \equiv \ \ \bar{Q} \Gamma q {{\left| \Omega \right\rangle}}$$ with $\Gamma \in \{\gamma_5,\gamma_0\gamma_5,\gamma_j,\gamma_0\gamma_j\}$ for the $S$ and $\Gamma \in \{1,\gamma_0,\gamma_j\gamma_5,\gamma_0\gamma_j\gamma_5\}$ for the $P_-$ meson. For a more detailed discussion of static-light mesons cf. [@Jansen:2008si; @Michael:2010aa].
$B \bar{B}$ systems
-------------------
We are interested in the potential of two static-light mesons, i.e. their energy as a function of their separation $r$. W.l.o.g. we separate the mesons along the $z$-axis, i.e. their static antiquark $\bar{Q}$ and quark $Q$ are located at $\vec{r}_1 = (0,0,+r/2)$ and $\vec{r}_2 = (0,0,-r/2)$, respectively. The corresponding $B \bar{B}$ trial states are $$\label{eq_BBbartrial} \mathcal{O} {{\left| \Omega \right\rangle}} \ \ \equiv \ \ \Gamma_{AB} \tilde{\Gamma}_{CD} \Big(\bar{Q}_C^a(\vec{r}_1) q_A^{(f_1)a}(\vec{r}_1)\Big) \Big(\bar{q}_B^{(f_2)b}(\vec{r}_2) Q_D^b(\vec{r}_2)\Big) {{\left| \Omega \right\rangle}}$$ ($A,B,\ldots$ are spin indices, $a,b$ color indices and $(f_1),(f_2)$ flavor indices). Since there are no interactions involving the static quark spins, one should not couple static spins and spins of finite mass, but contract the static spin indices with $\tilde{\Gamma} \in \{ \gamma_5, \gamma_0\gamma_5, \gamma_3, \gamma_0\gamma_3, \gamma_1, \gamma_2, \gamma_0\gamma_1, \gamma_0\gamma_2 \}$. This results in a non-vanishing correlation function independent of $\tilde{\Gamma}$.
The separation of the static quark and the static antiquark restricts rotational symmetry to rotations around the axis of separation, i.e. the $z$-axis. Therefore, and since there are no interactions involving the static quark spins, we can label states by the $z$-component of the light quark spin $j_z = -1,0,+1$. For $j_z=0$, i.e. for rotationally invariant states, spatial reflections along an axis perpendicular to the axis of separation are also a symmetry operation (w.l.o.g. we choose the $x$-axis). The corresponding quantum number is $\mathscr{P}_x = \pm$. $\mathscr{P}_x$ can be used as a quantum number also for $j_z \neq 0$ states, if we use $|j_z|$ instead of $j_z$. Parity $\mathscr{P}$ is not a symmetry, since it exchanges the positions of the static quark and the static antiquark. However, parity combined with charge conjugation, $\mathscr{P} \circ C$ is a symmetry and, therefore, a quantum number. When $q,\bar{q} \in \{u,d\}$, isospin $I \in \{0,1\}$ and its $z$-component $I_z \in \{-1,0,+1\}$ are also quantum numbers. In summary, there are up to five quantum numbers, which label $B\bar{B}$ states, $(I, I_z, |j_z|, \mathscr{P} \circ C, \mathscr{P}_x)$.
$B B$ systems (and $\bar{B} \bar{B}$ systems)
---------------------------------------------
We use $B B$ trial states $$\label{eq_BBtrial} \mathcal{O} {{\left| \Omega \right\rangle}} \ \ \equiv \ \ (\mathcal{C}\Gamma)_{AB} \tilde{\Gamma}_{CD} \Big(\bar{Q}_C^a(\vec{r}_1) \psi_A^{(f_1)a}(\vec{r}_1)\Big) \Big(\bar{Q}_D^b(\vec{r}_2) \psi_B^{(f_2)b}(\vec{r}_2)\Big) {{\left| \Omega \right\rangle}}$$ with $\tilde{\Gamma} \in \{1, \gamma_0, \gamma_3\gamma_5, \gamma_1\gamma_2, \gamma_1\gamma_5, \gamma_2\gamma_5, \gamma_2\gamma_3, \gamma_1\gamma_3 \}$ ($\mathcal{C} \equiv \gamma_0 \gamma_2$ denotes the charge conjugation matrix). Arguments similar to those of the previous subsection lead to quantum numbers $(I, I_z, |j_z|, \mathscr{P}, \mathscr{P}_x)$. For a more detailed discussion cf. [@Wagner:2010ad; @Wagner:2011ev].
\[sec\_setup\]Lattice setup
===========================
We use three ensembles of gauge link configurations generated by the European Twisted Mass Collaboration (ETMC) (cf. Table \[tab\_ensembles\]). For the $\bar{Q}\bar{Q}qq$ potentials we use $N_f = 2$ ensembles with lattice spacing $a \approx 0.079 \, \textrm{fm}$ for $q \in \{ u,d \}$ and an even finer lattice spacing $a \approx 0.042 \, \textrm{fm}$ for $q \in \{ s,c \}$, because in the latter case the potentials are quite narrow. Existing $\bar{Q}Q\bar{q}q$ results are rather preliminary and have been obtained exclusively with $q = c$ and the $N_f = 2+1+1$ ensemble with $a \approx 0.086 \, \textrm{fm}$. For details regarding these ETMC gauge link ensembles cf. [@Boucaud:2008xu; @Baron:2009wt; @Baron:2010bv; @Jansen:2011vv; @Cichy:2012is].
[ccccccccc]{} Ensemble & $N_f$ & $\beta$ & $(L/a)^3 \times (T/a)$ & $a\mu_l$ & $a\mu_\sigma$ & $a\mu_\delta$ & a & $m_\pi$\
A40.24 & 2 & 3.90 & $24^3 \times 48$ & 0.00400 & - & - & $0.079 \, \textrm{fm}$ & $340 \, \textrm{MeV}$\
E17.32 & 2 & 4.35 & $32^3 \times 64$ & 0.00175 & - & - & $0.042 \, \textrm{fm}$ & $352 \, \textrm{MeV}$\
A40.24 & 2+1+1 & 1.90 & $24^3 \times 48$ & 0.00400 & 0.15 & 0.19 & $0.086 \, \textrm{fm}$ & $332 \, \textrm{MeV}$\
Correlation functions have been computed using around 100 gauge link configurations from each of the three ensembles. We have checked that these correlation functions transform appropriately with respect to the symmetry transformations (1) twisted mass time reversal, (2) twisted mass parity, (3) twisted mass $\gamma_5$-hermiticity, (4) charge conjugation and (5) cubic rotations. In a second step we have averaged correlation functions related by those symmetries to reduce statistical errors.
Numerical results
=================
$\bar{Q}\bar{Q}qq$ potentials
-----------------------------
In the following we focus on the attractive channels between ground state static-light mesons ($S$ mesons). For $q \in \{ u,d \}$ there is a more attractive scalar isosinglet ($qq = (ud - du)/\sqrt{2}$, $\Gamma = \gamma_5 + \gamma_0 \gamma_5$ corresponding to quantum numbers $(I, |j_z|, \mathscr{P}, \mathscr{P}_x) = (0,0,-,+)$) and a less attractive vector isotriplet ($qq \in \{ uu,(ud + du)/\sqrt{2},dd \}$, $\Gamma = \gamma_j + \gamma_0 \gamma_j$ corresponding to quantum numbers $(I, |j_z|, \mathscr{P}, \mathscr{P}_x) = (1,\{0,1\},-,\pm)$). For $qq = ss$ there is only a single attractive channel, the equivalent of the vector isotriplet. To study also the scalar isosinglet with $s$ quarks, we consider two quark flavors with the mass of the $s$ quark, i.e. $qq = (s_1 s_2 - s_2 s_1)/\sqrt{2}$. Similarly we consider $qq = (c_1 c_2 - c_2 c_1)/\sqrt{2}$ to study a charm scalar isosinglet.
Proceeding as in [@Bicudo:2012qt] we perform $\chi^2$ minimizing fits of $$\label{eq_potfit} V(r) \ \ = \ \ -\frac{\alpha}{r} \exp\bigg(-\bigg(\frac{r}{d}\bigg)^p\bigg)$$ with respect to the parameters $d$ (light isotriplet), $(d,\alpha)$ ($q = s$ or $q = c$) or $(d,\alpha,p)$ (light isosinglet) to the lattice results for the $\bar{Q}\bar{Q}qq$ potentials. The resulting functions $V(r)$ are shown in Figure \[FIG001\].
![\[FIG001\]$\bar{Q}\bar{Q}qq$ potentials (\[eq\_potfit\]) for $q = u/d$, $q = s$ and $q = c$ (error bands are not shown). **(a)** Scalar isosinglet. **(b)** Vector isotriplet.](V_phys_A.eps "fig:"){width="13.0pc"}![\[FIG001\]$\bar{Q}\bar{Q}qq$ potentials (\[eq\_potfit\]) for $q = u/d$, $q = s$ and $q = c$ (error bands are not shown). **(a)** Scalar isosinglet. **(b)** Vector isotriplet.](V_phys_E.eps "fig:"){width="13.0pc"}
To determine, whether the investigated mesons may form a bound state, i.e. a tetraquark, we insert the potentials shown in Figure \[FIG001\] into Schrödinger’s equation with reduced mass $\mu \equiv m(S)/2$ and solve it numerically (cf. [@Bicudo:2012qt] for details). While there is strong indication for a bound state in the light scalar isosinglet channel, there seems to be no binding for the light vector isotriplet, or when $q=s$ or $q=c$. To quantify these statements, we list in Table \[binding\] the factor by which the reduced mass $\mu$ has to be multiplied to obtain a bound state with confidence level $1 \, \sigma$ and $2 \, \sigma$, respectively (the factors $\leq 1.0$ in the light scalar isosinglet indicate binding). These results clearly show that meson-meson bound states are more likely to exist for $B$ mesons than for $B_s$ or $B_c$ mesons. In other words it seems to be essential for a tetraquark to have both heavy quarks (leading a large reduced mass $\mu$) and light quarks (resulting in a deep and wide potential).
[l|cc|cc|cc]{} flavor & & &\
confidence level for binding & $1 \, \sigma$ & $2 \, \sigma$ & $1 \, \sigma$ & $2 \, \sigma$ & $1 \, \sigma$ & $2 \, \sigma$\
scalar isosinglet & 0.8 & 1.0 & 1.9 & 2.2 & 3.1 & 3.2\
vector isotriplet & 1.9 & 2.1 & 2.5 & 2.7 & 3.4 & 3.5\
$\bar{Q}Q\bar{q}q$ potentials
-----------------------------
At the moment there are only preliminary results for $\bar{Q}Q\bar{q}q$ potentials corresponding to isospin $I=1$ and $q = c$, i.e. $\bar{q}q = (\bar{c}_1 c_2 - \bar{c}_2 c_1) / \sqrt{2}$. Interestingly we observed that all these potentials are attractive, while in the $\bar{Q}\bar{Q}qq$ case only half of them are attractive and the other half is repulsive. This can be understood in a qualitative way by comparing the potential of $\bar{Q}Q$ and of $\bar{Q}\bar{Q}$ generated by one-gluon exchange. For $\bar{Q}\bar{Q}$ the Pauli principle applied to $qq$ implies either a symmetric (sextet) or an antisymmetric (triplet) color orientation of the static quarks corresponding to a repulsive or attractive interaction, respectively. For $\bar{Q}Q$ no such restriction is present, i.e. all channels contain contributions of the attractive color singlet, which dominates the repulsive color octet.
$I=0$ requires the computation of an additional diagram and $u/d$ and $s$ quarks are more demanding with respect to HPC resources than $c$ quarks. We expect corresponding results to be available soon.
Conclusions
===========
We have obtained insights regarding the quark mass dependence of $\bar{Q}\bar{Q}qq$ potentials, which suggest that tetraquark states with two heavy $\bar{b}$ antiquarks seem to be more likely to exist, when there are also two light $u/d$ quarks involved but not $s$ or $c$ quarks.
Preliminary results for $\bar{Q}Q\bar{q}q$ potentials indicate that there are only attractive channels, which is in contrast to the $\bar{Q}\bar{Q}qq$ case.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Joshua Berlin, Owe Philipsen, Annabelle Uenver-Thiele and Philipp Wolf for helpful discussions. M.W. acknowledges support by the Emmy Noether Programme of the DFG (German Research Foundation), grant WA 3000/1-1. This work was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.
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abstract: 'A brief summary of the theoretical and experimental knowledge of the spin structure of the proton is presented. The helicity distributions of quark and gluons are discussed, together with their related sum rules. The transversity distribution is also introduced with possible strategies for its measurement. Novel spin dependent and $\bfk_\perp$ unintegrated distribution and fragmentation functions are discussed, in connection with a new and rich phenomenology of transverse single spin asymmetries.'
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[**What do we know about the proton spin structure? [^1]**]{}
0.8cm [Mauro Anselmino]{} 0.5cm [*Dipartimento di Fisica Teorica, Università di Torino and\
INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy*]{}\
The spin nucleon structure – as observed in high energy, short distance interactions – is schematically described in Fig. 1.
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The large $Q^2$ probe – typically, a virtual photon – “sees” QCD partons, carrying a longitudinal momentum fraction $x$, and their interactions, with gluon and $q\bar q$ pair creation; the information about such a complicated structure is usually collected via measurements of the Deep Inelastic Scattering (DIS) cross-section and storaged in the structure functions which appear in the most general expression of the cross-section. When neglecting weak, parity violating contributions, there are 2 unpolarized ($F_1$, $F_2$) and 2 polarized ($g_1$, $g_2$) structure functions: perturbative QCD allows a simple partonic interpretation of $F_2$ and $g_1$ ($F_1$ is related to $F_2$ while $g_2$ does not have a partonic interpretation). We only consider here the polarized proton structure trying to summarize in a short time and space the main ideas, the most recent results and the open problems; many detailed and comprehensive reviews on the subject can be found in the literature [@rev].
The main issues and questions we are going to discuss here are:
- our knowledge about the polarized structure functions $g_1(x,$ $Q^2)$ and $g_2(x,Q^2)$ and about quark and gluon helicity distributions, $\Delta q(x,Q^2)$ and $\Delta g(x,Q^2)$; how well do we know them?
- are fundamental sum rules satisfied and what do we know about quark and gluon orbital angular momentum, $L_q$ and $L_g$?
- $\Delta q(x,Q^2)$, $\Delta g(x,Q^2)$, $L_q$ and $L_g$ are not the whole story: how and where do we learn about the transversity distribution $h_1(x,Q^2)$?
- could we learn more and understand more from intrinsic $\bfk_\perp$ [*unintegrated*]{} distribution and fragmentation functions?
At NLO in the QCD parton model the structure function $g_1$ is given by g\_1(x, Q\^2) = 12 \_q e\_q\^2 { C\_q + C\_g g } \[g1evol\] where $\Delta q(x, Q^2)$ and $\Delta g(x, Q^2)$ are respectively the quark (of flavour $q$) and gluon helicity distributions; we have, as usual, defined the convolution C q \_x\^1 C ( xy, \_s ) q(y, Q\^2) \[conv\] and the coefficients functions $\Delta C_i$ have a perturbative expansion C\_i(x, \_s) = C\_i\^0(x) + C\_i\^[(1)]{}(x) + \[coeff\] The LO terms are simply C\_q\^0 = (1-x) C\_g\^0 = 0 , \[lo\] and the NLO corrections are scheme dependent; typical choices differ in the amount of gluon contribution to the quark singlet distributions, while quark non-singlet distributions are scheme independent [@rev]. Finally, the $Q^2$ evolution of the parton densities obeys the DGLAP evolution equations [@dglap], and, if known at an initial scale $\mu^2$, the r.h.s. of Eq. (\[g1evol\]) can be computed at any perturbative $Q^2$ value.
By comparing data on $g_1(x, Q^2)$ with Eq. (\[g1evol\]) one obtains information on the quark and gluon helicity distributions; the more data one has and the wider the $x$ and $Q^2$ range is, the more stringent the comparison is. The normal procedure is that of using a simple ansatz for the unknown distribution functions at the initial scale $\mu^2$, with some assumptions regarding the sea quark densities (for example, whether $SU(3)_F$ symmetric or not) and some constraints from $SU(3)_F$ hyperon decay sum rules on the first moments $\Delta q(1,Q^2) \equiv \int_0^1 \Delta q(x,Q^2) \> dx$.
In Fig. 2 a most recent analysis of the world data on $xg_1(x)$ is shown together with a fit from Ref. \[3\], where the resulting helicity distributions can also be seen. Several similar analyses can be found in the literature; a complete list of references is given in Ref. \[3\].
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Some data on $xg_2(x)$ are also available and the most recent ones [@e155] are shown in Fig. 3.
=3.5in
Let us shortly comment on these experimental results and the information which they offer.
- We have now good data on $g_1$ and $g_2$, although not yet comparable with the amount and quality of similar data obtained on the unpolarized structure functions.
- $g_1$ \[see Eq. (\[g1evol\])\] allows to obtain information on linear combinations of ($\Delta q + \Delta \bar q$). We still need a better flavour resolution; this might come from semi-inclusive DIS which gives information on $\sum_q \, \Delta q \, D_q^h$ rather than $\sum_q e_q^2
(\Delta q + \Delta \bar q)$, where $D_q^h$ is the quark $q$ fragmentation function into the observed hadron $h$. Flavour separation in inclusive DIS would naturally be possible in neutrino iniziated charged current processes [@fmr].
- Eq. (\[g1evol\]) also offers indirect (via QCD evolution) information, on $\Delta g$. This is not stringent enough and a more direct measurement of the gluon helicity distribution is needed. This might come from the study of spin dependences in processes like $\ell p \to \ell +
2\,jets$, $\ell p \to \ell + c + \bar c + X$, $p N \to \gamma + X$, [*etc.*]{} which could be performed at HERMES, COMPASS, RHIC.
In extracting information from experimental data – or in testing theories – a special role is plaid by sum rules. Let us mention a few of them.
The Bjorken sum rule ($g_A/g_V = 1.2670 \pm 0.0035$), && \_0\^1 \[g\_1\^p(x,Q\^2) - g\_1\^n(x,Q\^2)\]dx \[bj\]\
&=& 16 { 1 - - - 20.2 + } , is used in many ways. One can simply assume the validity of Eq. (\[bj\]) and deduct from data on $g_1$ the value of $\alpha_s$ [@abfr]; or one can use in it a value of $\alpha_s$ otherwise obtained, to check whether data on $g_1$ obey the sum rule or not (the answer is yes); or, also [@smallx], one can assume to know the r.h.s. of Eq. (\[bj\]), and see – among the poorly known behaviours of $g_1(x)$ at small $x$ – which one best satisfies the sum rule.
Another, more debated, sum rule is the so called Burkhardt-Cottingham sum rule, according to which $\int_0^1 g_2(x,Q^2)\,dx = 0$, [*provided*]{} the integral exists. The recent E155 data [@e155] of Fig. 3 seem to indicate $\int_{0.02}^{0.8} g_2^p(x,Q^2)\,dx = - 0.042 \pm 0.008$, which, taking into account uncertainties in the extrapolation to $x=0$ and $x=1$, might be the first indication of a violation of the sum rule, assuming that no $\delta$-function contributes at the origin [@del].
The last, fundamental sum rule which we mention is the spin sum rule: 12 = 12 (1) + g(1) + L\_q + L\_g \[spin\] where $\Delta \Sigma(1)$ is the first moment of $\sum_q
[\Delta q(x) + \Delta \bar q(x)]$ and $L_{q,g}$ is the third component of the orbital angular momentum carried by quarks, gluons. This last quantity is unavoidable in a picture of the proton like that of Fig. 1: a spin 1/2 massless quark can emit a spin 1 massless gluon, via a helicity conserving coupling, only if some orbital angular momentum restores the total angular momentum conservation. However, there is little agreement at the moment both about the proper formal definition of a $\hat L_{q,g}$ operator and about a possible measurement of its expectation value between proton states [@jaf].
The trasverse polarization of quarks inside a trasversely polarized nucleon, denoted by $h_1$, $\delta q$ or $\Delta_T q$, is a fundamental twist-2 quantity, as important as the unpolarized distributions $q$ and the helicity distributions $\Delta q$. It is given by h\_1(x, Q\^2) = q\_\^(x, Q\^2) - q\_\^(x,Q\^2) , that is the difference between the number density of quarks with transverse spin parallel and antiparallel to the nucleon spin. It is the same as the helicity distribution only in a non relativistic approximation, but it is expected to differ from it for a relativistic nucleon.
When represented in the helicity basis (see Fig. 4) $h_1$ relates quarks with different helicities, revealing its chiral-odd nature. This is the reason why this important quantity has never been measured in DIS: the electromagnetic or QCD interactions are helicity conserving, there is no perturbative way of flipping helicities and $h_1$ decouples from inclusive DIS dynamics, as shown in Fig. 4a.
However, it can be accessed in semi-inclusive DIS, where some non perturbative chiral-odd effects may take place in the non perturbative fragmentation process, Fig. 4b. Indeed, a serious program to measure $h_1$ in semi-inclusive DIS at HERMES, where a transversely polarized proton target is now available, is in progress. A similar program, in different, complementary, kinematical regions, is planned at COMPASS.
=3.5in
The transversity distribution is also accessible at RHIC, where transversely polarized proton beams are available; by measuring double transverse spin asymmetries in Drell-Yan processes one obtains an observable which depends on the convolution of two transversity distributions, which might make the overall effect rather tiny [@rhic]. In general, $h_1$ must appear in a physical observable coupled to another chiral-odd quantity, which is either the transversity itself or a new unknown function.
We conclude by mentioning a new phenomenological approach to the description of many single transverse spin asymmetries which have been measured and keep being measured, with unexpected and interesting results [@me]. The apparent problem with these asymmetries is related to the fact that, within perturbative QCD and the collinear factorization scheme, they should be vanishing, which is not true experimentally.
Recently, a series of papers [@asy] have shown how single spin asymmetries may occurr at the level of parton distributions and fragmentations, provided one takes into account the intrinsic motion of partons inside hadrons and of hadrons relatively to the fragmenting parton. For example, there might be a correlation between the transverse spin of a quark and the $\bfk_\perp$ of a resulting hadron, say a pion. This is the so-called Collins effect [@col], pictorially shown in Fig. 5.
=3.5in
Similar spin-$\bfk_\perp$ correlations may occurr also in the fragmentation of an unpolarized quark into a polarized hadron (the so-called polarizing fragmentation functions [@polff]), in the distribution of unpolarized quarks inside polarized nucleons (the Sivers effect [@siv]) and in the distribution of polarized quarks inside an unpolarized hadron [@dan].
When generalizing the factorization scheme with the inclusion of intrinsic $\bfk_\perp$, both in the distribution/fragmentation functions and in the elementary interactions, single transverse spin asymmetries appear immediately as possible and even sizeable. A phenomenological approach can be developed in which experimental information on the new functions is obtained from some processes and then used to make predictions in other cases.
The spin structure of the nucleon is subtle and challenging. Enormous progress has been achieved in the last years; yet, new surprising experimental results keep beeing obtained and fresh, interesting ideas keep being suggested. A lot more good work, both experimental and theoretical, is in progress.
12 pt I would like to thank the organizers of the Symposium for the invitation and for the beautiful and stimulating organization.
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[^1]: Talk delivered at the 2nd International Symposium on the Gerasimov-Drell-Hearn sum rule and the spin structure of the nucleon, GDH 2002, July 3-6 2002, Genova, Italy
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abstract: 'We present 3D hydrodynamic simulations aimed at studying the dynamical and chemical evolution of the interstellar medium in dwarf spheroidal galaxies. This evolution is driven by the explosions of Type II and Type Ia supernovae, whose different contribution is explicitly taken into account in our models. We compare our results with avaiable properties of the Draco galaxy. Despite the huge amount of energy released by SNe explosions, in our model the galaxy is able to retain most of the gas allowing a long period ($> 3$ Gyr) of star formation, consistent with the star formation history derived by observations. The stellar \[Fe/H\] distribution found in our model matches very well the observed one. The chemical properties of the stars derive from the different temporal evolution between Type Ia and Type II supernova rate, and from the different mixing of the metals produced by the two types of supernovae. We reproduce successfully the observed \[O/Fe\]-\[Fe/H\] diagram.'
address:
- 'Institute of Astronomy, Vienna University, Türkenschanzstrasse 17, 1180 Vienna, Austria'
- 'INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy'
- 'Dipartimento di Astronomia, Università di Bologna, via Ranzani 1, 40127 Bologna, Italy'
- 'INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131, Trieste, Italy'
author:
- Andrea Marcolini
- 'Annibale D’Ercole'
- Fabrizio Brighenti
- 'Simone Recchi$^{ 1,}$'
title: About the evolution of Dwarf Spheroidal Galaxies
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Introduction
============
Due to their proximity, galaxies of the Local Group (see Mateo 1998, Grebel 2006 for a review) offer an unique opportunity to study in detail their structural properties, formation and chemical evolution.
In particular, the distribution of the local galaxies shows the clustering of dwarf ellipticals and dwarf spheroidal (dSphs) around the dominant spirals galaxies (Milky Way and Andromeda). Dwarf spheroidals are the least massive galaxies known, but yet, their velocity dispersions imply mass to light ratios as large as 100 $\rm
M_{\odot}/L_{\odot}$. This is usually explained assuming that these systems are dark matter dominated. Actually, in the past few years both observational evidences (e.g. Kleyna et al. 2002, Lokas 2002, Walker et al. 2006) and theoretical works (e.g. Kazantzidis et al. 2004, Mashchenko et al. 2005) confirm the possibility that these galaxies are relatively massive bounded system with virial masses in the range $10^8-5\times10^9$ M$_{\odot}$. Such galaxies are very metal poor and lack of neutral hydrogen and recent star formation. Thus they were initially believed to be very similar to Galactic globular clusters and to have a very simple star formation history (SFH). Recent studies have shown, instead, that these systems are much more complex, with varied and extended SFHs. High resolution spectroscopy of several dSphs showed the presence of a wide range in metallicity (Harbeck et al. 2001). For example, abundance analyses of stars belonging to Draco and Ursa Minor have shown values of \[Fe/H\] in the range $-3 \leq$\[Fe/H\]$\leq -1.5$ (Shetrone et al. 1998, 2001) with a mean value in the interval $-2.0 \leq \langle$\[Fe/H\]$\rangle \leq
-1.6$, depending on the authors (e.g. Shetrone et al. 2001, Bellazzini et al. 2002).
The above ranges are consistent, for some dSphs, with a single period of star formation extended in time for a few Gyr (e.g. Mateo 1998, Dolphin 2002). As a further hint of long SFH Shetrone et al. (2001) found that their observed dSphs have \[$\alpha$/Fe\] abundances that are $\sim 0.2$ dex lower than those of Galactic halo field stars in the same \[Fe/H\] range. This suggests that the stars in these systems were formed in gas pre-enriched by Type II supernovae (SNe II) as well as by Type Ia supernovae (SNe Ia), and star formation must thus continue over a relatively long timescale in order to allow a sufficient production of iron by SNe Ia.
Given the small dynamical mass inferred for dSphs, the interstellar medium (ISM) binding energy is small when compared to the energy released by the SNe II explosions occurring during the star formation period; for instance, as shown in the next section, in a dSph like Draco the baryonic matter has a binding energy of $\sim 10^{53}$ erg, while the expected number of SN explosions in the past was $10^3-10^4$, realising an energy much larger than the binding energy. It is thus quite puzzling how the ISM can remain bound long enough to allow such a long star formation duration. Infact, contrary to SNe II, SNe Ia are poor producers of oxigen, great producers of iron, and start to explode after longer time scales; thus stars with low \[O/Fe\] indicate long SFHs.
Motivated by the above arguments, in this paper we explore the possibility that dSphs formed stars at a low SFR for a long period. To compare our results with observations, we have tailored our models on the Draco galaxy: this galaxy is supposed to have experienced a star formation lasting for 3-4 Gyr, and which essentially ceased 10 Gyr ago (e.g. Mateo 1998). Obviously, our results may be confronted with other dSphs which are strongly dark matter dominated and have similar SFHs as, e.g., Ursa Minor (Mateo 1998).
We run a number of three-dimensional (3D) hydrodynamical simulations to study the dynamical and chemical evolution of this system, following an assumed SFH. A special attention is paid to the influence of both SNe Ia and SNe II on the chemical enrichment of the new forming stars.
Model and discussions
=====================
We start our simulations with the ISM in hydrostatic equilibrium in the dark matter halo potential well. Although the stellar contribution to the gravitational potential well is neglected, we approximate the observed stellar distribution in Draco with a King profile with a mass content $M_{*}=5.6 \times 10^5$ M$_{\odot}$. The stellar distribution is important to estimate (from the observed mass-to-light ratio) the dark matter halo properties and properly locate the SNe explosions. More details on the model construction can be found in Marcolini et al 2006. One of the basic assumptions of the model is that the dark matter halo extends beyond the steller component ($R_{*}=650$ pc) of the system (the mass of the dark matter halo at 1.2 kpc is $6.2 \times 10^7$ M$_{\odot}$). The initial gas mass is $M_{\rm ISM} = 0.18 M_{\rm h}$, which corresponds to the baryonic fraction given by Spergel et al. (2006).
{width="13cm"}
Here we focus on a model in which we assumed that stars form in a sequence of 25 instantaneous bursts separated in time by 120 Myr. We further assume that a single SN II explodes for each 100 M$_{\odot}$ of formed stars, raching the total number of 5600 at the end of the simulation (3 Gyr). The SNe II explode at a constant rate for 30 Myr (the lifetime of a 8 M$_{\odot}$ star, the less massive SN II progenitor) after the occurrence of each burst, while SNe Ia rate follows the prescription of Matteucci & Recchi 2001 (see Marcolini et al. 2006 for further details).
![Left panel: final \[Fe/H\] distribution function of the long-lived stars. Right panel: abundance ratio \[O/Fe\] plotted against \[Fe/H\] of 1000 sampled stars. The solid line represents the mean value of the \[O/Fe\] distribution for any fixed \[Fe/H\]. We also show the observative values obtained by Shetrone et al (2001) for Draco.](marcolini_figure2.ps){width="11cm"}
Figure 1 shows that as the SNe II start to explode, a large fraction of the central volume is filled by the hot rarefied gas of the SNRs’ interior, while the dense SNRs’ shells form dense cold filaments after colliding one with another. Once the SNe II stop to explode the global cavity collapses and the ISM goes back into the potential well; this happens nearly 30-40 Myr after the last SN II explosion. Note that the initial binding energy of the gas $E_{\rm bind} \sim 8.3 \times
10^{52}$ erg is lower than the total energy $2.24 \times 10^{53}$ erg released by the SNe II after a single burst. The simulation thus shows that the radiative losses are substantial and prevent the evacuation of the gas, as shown in Fig. 1.
After 120 Myr a central high density gas region of the same size of the stellar volume is recovered, although turbulences and inhomogeneities are now present (see Fig. 1, fourth panel). A second burst of star formation then occurs leading to a second sequence of SN II explosions. The gas undergoes a new cycle of merging bubbles which eventually collapse again.
The influence of the SN Ia explosions on the general hydrodynamical behaviour of the ISM is not very important because during a cycle of SN II explosions no more than 8-9 SNe Ia occur, only $\sim 4\%$ of the SN II number. Despite their little importance from a dynamical point of view, the role of SNe Ia is very relevant for the chemical evolution of the stars. The simulation shows that while the SN II ejecta become more and more homogeneous with time as the turbulence diffuse it, the SN Ia ejecta appear to be distributed less homogeneously; the reason for this is the low SNe Ia rate.
We point out that during the entire evolution the fraction of the SN ejecta present inside the stellar region remains very low ($\sim 18\%$ after 3 Gyr). This is the amount of metals which contributes to the metallicity of the forming stars. A large fraction of the ejecta is pushed at larger distances by the continuous action of the SN explosions. Figure 2 shows the \[Fe/H\] distribution function (MDF) of our simulated long-lived stars (with mass $\le$ 0.9 M$_{\odot}$), i.e. the mass fraction of these stars as a function of their \[Fe/H\]. At the end of the simulation we obtain a mean value of $\langle$\[Fe/H\]$\rangle$=-1.7 with a spread of $\sim$ 1.5 dex, in reasonable agreement with observations (e.g. Shetrone et al. 2001, Bellazzini et al. 2002), while the distribution maximum occurs at \[Fe/H\]$\sim$-1.6. Note that stars with \[Fe/H\] $\ge$ -1.4 (the high metallicity tail in Fig. 2) are particularly enriched by SN Ia ejecta and formed in the (relatively) small volume occupied by SN Ia renmants. This is particular evident in Fig. 2 (right panel) where we show the final \[O/Fe\]-\[Fe/H\] diagram. The open circles form a statistically representative sample of the stellar distributions in the \[O/Fe\]-\[Fe/H\] diagram. The plateau at \[O/Fe\]$\sim$0.35 at low \[Fe/H\] is representative of the \[O/Fe\] value in SNe II ejecta, because the contribution of SNe Ia becomes important after a longer time scale. Indeed the small negative gradient of the plateau is due to the slowly growing contribution in the Fe enrichment by SNe Ia (which contribute only marginally to the Oxigen production). The sharply decreasing branch at higher \[Fe/H\] is due to stars formed in the regions of ISM recently polluted (mostly by iron) by SNe Ia. A glance at Fig. 2 shows that the stars on the decreasing branch populate the MDF high \[Fe/H\] tail, while the majority of the stars occupies the high \[Fe/H\] edge of the plateau in the \[O/Fe\]-\[Fe/H\] diagram. We point out that our representative stellar sample is in reasonable agreement with the stars observed by Shetrone et al. (2001).
Other models
------------
Here we describe the evolution of two models quite similar to the reference model (described above). Model B has the same SFH but differs in the dark matter content ($2.2 \times 10^7$ M$_{\odot}$) and ISM mass ($4 \times 10^6$ M$_{\odot}$) in order to preserve the cosmological ratio between the amount of baryonic and non-baryonic matter. Model C has the same properties of the reference model but differs in the duration ($\le$ 1 Gyr) and intensity (10 bursts) of SFH.
Model B loses all its gas in a period too short ($\le$ 250 Myr) to be consistent with the longer SFH of Draco. We point out that this effective gas removal is mainly due to a less efficient radiative cooling (due to the lower density of the gas) rather than to the shallower galactic potential.
Model C, instead, retains its gas for a longer time, and is able to form stars up to 900 Myr, consistently with recent cosmological simulations (e.g. Ricotti & Gnedin 2005, Kawata et al. 2006) before loosing the ISM via a galactic wind. This model shows $\alpha$/Fe $\sim$ 0.1 dex higher than the value of the reference model because of the lower number of SN Ia explosions. However, although its chemical properties (both the AMD and the \[O/Fe\]-\[Fe/H\] diagram) are still in marginal agreement with observations, we belive that the star formation duration must be longer than $\sim 1$ Gyr. Infact, Fenner et al (2006) find that only long SFHs (of the order of several Gyr) are able to reproduce the Ba/Y ratio because the stars must form over an interval long enough for the low-mass stars to pollute the ISM with $s$-elements.
Conclusion
==========
We presented 3D simulations of a forming dSph resembling the Draco galaxy. With our assumptions, in our reference model the galaxy never gets rid of its gas (due to the huge efficiency of radiative cooling despite the low metallicity of the gas) and the star formation can last for several Gyr (as suggested by observations). This in turn implies the need of an external mechanism to remove the gas and stop the star formation, such as gas stripping (e.g. Marcolini et al 2003, only stripping, and 2004, stripping+SN feedback) and/or tidal interaction with the Galaxy (Mayer et al. 2006). Indeed, ram pressure due to gaseous haloes of the Milky Way and M31 is belived to explain the observed correlation between stellar content and galactocentric distance of dwarf galaxies (van den Bergh 1993). We are now running simulations of forming dSphs interacting with the Milky Way halo in order to understand whether the combined action of SNe feedback and ram pressure stripping can help in depriving these systems of gas.
Although the SN ejecta remain gravitationally bounded during the star formation, yet only a low fraction ($\sim 18$%) stays in the region where star forms. This effect mimics the assumption of metal removal by galactic winds in chemical evolution models. Our model succeeds in reproducing the \[Fe/H\] distribution of the stars. In agreement with observations, we find a mean value $\langle$\[Fe/H\]$\rangle = -1.7$ with a spread of $\sim 1.5$ dex. We can also satisfactory reproduce the observed \[O/Fe\] vs \[Fe/H\] diagram. The origin of the break in this diagram, in our interpretation, is due to the low value of the porosity of SN Ia remnants. Indeed, given the low SN Ia rate, these remnants are located quite apart one from another, and the iron ejected by SNe Ia is distributed rather inhomogeneously through the stellar volume. As a consequence, stars forming in the (relatively small) volume occupied by SN Ia remnants have a ratio \[O/Fe\] lower than those forming elsewhere.
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abstract: 'The second Bianchi identity can be recast as an evolution equation for the Riemann curvatures. Here we will report on such a system for a vacuum static spherically symmetric spacetime. This is the first of two papers. In the following paper we will extend the ideas developed here to general vacuum spacetimes. In this paper we will demonstrate our ideas on a Schwarzschild spacetime and give detailed numerical results. For suitable choices of lapse function we find that the system gives excellent results with long term stability.'
author:
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Leo Brewin\
School of Mathematical Sciences\
Monash University, 3800\
Australia
date:
- '10-Nov-2010'
- '16-Jan-2011'
- '26-Jan-2011'
title: |
An Einstein-Bianchi system for\
Smooth Lattice General Relativity. I.\
The Schwarzschild spacetime.
---
=1
Introduction {#sec:intro}
============
Despite a slow start, hyperbolic formulations of the Einstein equations have in recent times become the system of choice for numerical relativity.
The confidence afforded to hyperbolic systems is borne out not just by the recent success in numerical relativity [@pretorius:2005-01; @campanelli:2006-01; @baker:2006-01] but also from their strict mathematical underpinnings (which guarantees that future evolutions exist and that they depend smoothly on the initial data, these are key aspects of the theorems that demonstrate the stability of the system, see [@reula:1998-01] for details).
One of the earlier hyperbolic formulations was given by Friedrich [@friedrich:1996-01] in which he used the second Bianchi identities to evolve the Weyl curvatures in-situ with the metric. This idea has been extended by many other authors [@anderson:1999-01; @estabrook:1997-01; @van-elst:1997-01; @jantzen:2009-01] and the resulting equations are commonly referred to as an Einstein-Bianchi system.
Yet despite their mathematical elegance and the virtues that this would bestow upon a numerical code there seems to be have been very few numerical applications employing an Einstein-Bianchi system (though see [@van-putten:1996-01; @van-putten:1997-01]).
In this paper we will report on a simple Einstein-Bianchi system adapted to a discrete lattice for static spherically symmetric spacetimes. We were lead to this formulation not by way of Friedrich’s paper but rather as a direct extension of our own ideas developed in an earlier series of papers [@brewin:2009-05; @brewin:2009-04; @brewin:2002-01; @brewin:1998-02]. In those papers we used the spatial form of the second Bianchi identities (the second Bianchi identity for the 3-metric) to compute the 3-Riemann curvatures across a Cauchy surface. This device proved to be the key element in obtaining accurate and stable evolutions of the initial data.
Our longer term intention is to employ an Einstein-Bianchi system to evolve a 3-dimensional lattice. This will require not only evolution equations for the legs of the lattice, such as those given in [@brewin:2009-04], but also evolution equations for the curvatures. This will be the subject of the second paper in this series.
For the simple case presented here we find that the system works very well. The evolutions are stable, though this depends on the choice of the lapse function, see sections (\[sec:ArtVisco\]) and (\[sec:Slicing\]). We also find that the constraints are well behaved (they appear to grow linearly with time and converge to zero as the lattice is refined, see section (\[sec:CodeTests\])).
As this paper borrows heavily from two of our previous papers, which we refer to as and , we will skip over many of the derivations and arguments assuming instead that the reader is familiar with the material in and .
Spherically symmetric spacetimes {#sec:SphericalSpacetime}
================================
In this paper we will be constructing lattice approximations to the Schwarzschild spacetime in various slicings. In each case the continuum metric can be written in the form $$ds^2 = -N(r,t)^2 dt^2 + A(r,t)^2 dr^2 + B(r,t)^2 d\Omega^2
\label{eqn:SphericalMetric}$$ for some set of functions $N(r,t)$, $A(r,t)$, $B(r,t)$ and where $d\Omega^2=d\theta^2 +
\sin^2\theta d\phi^2$ is the metric of the unit 2-sphere. We have introduced this coordinate form of the continuum metric simply as a precursor to the introduction of the lattice. As we shall soon see, we will use the coordinate lines and their local tangent vectors as a scaffold on which to build the lattice, after which we will have no further need for the coordinates (indeed we could dispense with the coordinates altogether at the possible expense of the clarity of the exposition).
Consider a local orthonormal tetrad built from the future pointing unit normal $t^\mu$ to a typical Cauchy surface and three unit vectors $m^\mu_x$, $m^\mu_y$ and $m^\mu_z$ where $m^\mu_z$ is parallel to the radial axis (see figure (\[fig:LatticeLocal\])). These basis vectors are also tangent vectors to the coordinate axes of a local Riemann normal frame. We will use this tetrad to record the frame components of the extrinsic and Riemann curvatures on the lattice. Our notation, which we borrow from , will be to use script characters to denote frame quantities, thus $\Kxx:=K_{\mu\nu} m^\mu_x m^\nu_x$ while $\Rtztz:=R_{\mu\alpha\nu\beta} t^\mu m^\alpha_z t^\nu m^\beta_z$. Also, to avoid an overflow of symbols, we will allow $\Lxx$ and $\Lzz$ to represent both the length of the corresponding leg as well the leg itself.
In this class of spacetimes, and on this tetrad, we know that the extrinsic curvature is diagonal and that a basis for the non-trivial Riemann curvatures is given by $$\begin{gathered}
\Rx,\quad \Rz,\quad \Rtxtx,\quad \Rtztz,\quad \Rtxxz\end{gathered}$$ Now using $R_{\mu\nu}=R^{\alpha}{}_{\mu\alpha\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$ we find $$\begin{aligned}
\Rtz &= -2\Rtxxz\\[2pt]
\Rtt &= \Rtztz + 2\Rtxtx\\[2pt]
\Rzz &= -\Rtztz + 2\Rz\\[2pt]
\Rxx &= -\Rtxtx + \Rx + \Rz\\[2pt]
\R &= -4\Rtxtx - 2\Rtztz + 2\left(\Rx+2\Rz\right)\end{aligned}$$ while the non-trivial vacuum Einstein equations yield $$\Rtz = \Rtt = \Rzz = \Rxx = 0$$ Combining the above shows that we can express all of the non-trivial Riemann curvatures solely in terms of $\Rx$ and $\Rz$, namely $$\begin{aligned}
\Rtxxz & = 0\label{eqn:Rtxxz}\\[2pt]
\Rtztz & = 2\Rz\label{eqn:Rtztz}\\[2pt]
\Rtxtx & = \Rx + \Rz\label{eqn:Rtxtx}\end{aligned}$$ In obtaining these relations we used $g^{\mu\nu} = -t^\mu t^\nu + m^\mu_x m^\nu_x + m^\mu_y m^\nu_y + m^\mu_z m^\nu_z$.
Note that $\Rx$ and $\Rz$ are not independent for the simple equation $R=0$ leads to $$0 = \Rx + 2\Rz$$ We will use this equation as a check on our numerical integrations (see section (\[sec:CodeTests\]) for more details).
The lattice {#sec:LatticeDescribe}
===========
The symmetries in the Schwarzschild spacetime allows us to use a very simple ladder-like structure for the lattice, as indicated in figure (\[fig:SchwarzLattice\]). One way to imagine the construction of the lattice is to consider the coordinate mesh generated by setting $t={}$constant and $\theta=\pi/2$ in the coordinate form of the metric in (\[eqn:SphericalMetric\]). Then the rungs of the ladder are generated by small increments in $\phi$ leading to $\Lxx\approx B\Delta\phi$ while the side rails would coincide with two radial curves ($\phi={}$constant) with $\Lzz\approx A\Delta r$. Clearly, specifying all of the $\Lxx$ and $\Lzz$ is equivalent to specifying the metric components $A(r,t)$ and $B(r,t)$. Note that throughout this paper we treat the $\Lxx$ and $\Lzz$ as continuous functions of time.
We will label the nodes from $0$ to $\Nnode$ and on the few occasions where we need to discuss more than one leg at a time we will write $\iLxx_i$ to denote an $\Lxx$ leg at node $i$. In the same way $\iLzz_i$ will denote the $\Lzz$ that joins the nodes $i$ and $i+1$. Similar notation will be used for other data on the lattice.
The initial data (as described in section (\[sec:InitialData\])) are constructed in a way that guarantees reflection symmetry at the throat (which is always tied to node 0).
In our computer code we extend our lattice a small way over the throat, by including the nodes -3 to -1, so that we can readily impose the reflection symmetries (by simply copying data across the throat, at no point do we independently evolve any of the data to the left of the throat).
The evolution equations {#sec:EvolveEqtns}
=======================
Our present task is to develop evolution equations for the leg-lengths, the extrinsic curvatures and, the principle innovation in this paper, evolution equations for the Riemann curvatures.
A simple derivation of the evolution equations for our lattice can be obtained from a general pair of equations developed in . There it was shown that the first and second variations of arc lengths can be written in a form remarkably similar to the ADM equations, namely $$\begin{aligned}
\DLsqDt &= -2 N K_{\mu\nu} \Delta x^\mu_{ij} \Delta x^\nu_{ij} + \BigO{L^3}
\label{eqn:ADMDLijb}\\[10pt]
\DDLsqDt &= 2 N_{|\alpha\beta} \Dxij^\alpha \Dxij^\beta
\label{eqn:ADMDLijc}\\
& \quad+ 2 N\left( K_{\mu\alpha}K^\mu{}_\beta
- R_{\mu\alpha\nu\beta} t^\mu t^\nu \right)
\Dxij^\alpha \Dxij^\beta + \BigO{L^3}\notag\end{aligned}$$ Note that in the following we will ignore the leading error terms $\BigO{L^3}$. Applying these equations to the two legs $\Lxx$ and $\Lzz$ of our spherically symmetric lattice leads immediately to $$\begin{aligned}
\DLxx & = -N \Kxx \Lxx\label{eqn:DLxx}\\[5pt]
\DLzz & = -N \Kzz \Lzz\label{eqn:DLzz}\\[5pt]
\DKxx & = -\dNdxx + N\left( \Rtxtx + \Kxx^2\right)\label{eqn:DKxx}\\[5pt]
\DKzz & = -\dNdzz + N\left( \Rtztz + \Kzz^2\right)\label{eqn:DKzz}\end{aligned}$$ The last part of the picture is to provide evolution equations for the Riemann curvatures, $\Rx$ and $\Rz$. The basic idea is to rearrange the terms in the Bianchi identities to isolate the time derivatives while estimating the spatial derivatives from data imported from neighbouring cells. The calculations are straight-forward but a bit tedious to present here so we defer the full details to the Appendix. This leads to the following evolution equations $$\begin{aligned}
\DRxDt &= 2N\Kxx\left( 2\Rx + \Rz\right)\label{eqn:EvolveRx}\\[5pt]
\DRzDt &= 3N\Kxx\Rz + N\Kzz\left(\Rx + 2\Rz\right)\label{eqn:EvolveRz}\end{aligned}$$
The Riemann curvatures $\Rx$ and $\Rz$ would normally not be *evolved* but rather *derived* from the lattice data such as the leg lengths $\Lxx$ and $\Lzz$. In we used (discrete versions of) the geodesic deviation equation and the spatial Bianchi identity[^1] $$\begin{aligned}
0 &= \DDLxxDz + {}^{3}\Rz \Lxx\label{eqn:GeodDev}\\[5pt]
0 &= \frac{d\left(\LLxx {}^{3}\Rx\right)}{dz} - {}^{3}\Rz\DLLxxDz\label{eqn:Bianchi}\end{aligned}$$ to compute the 3-dimensional Riemann curvatures ${}^{3}\Rx$ and ${}^{3}\Rz$ on the lattice. In raising the $\Rx$ and $\Rz$ to dynamical variables on the lattice we are forced to view equations (\[eqn:GeodDev\],\[eqn:Bianchi\]) as constraints on the lattice data. In section (\[sec:CodeTests\]) we shall present discretised versions of these constraints which we will later use to check the quality of our numerical results.
The one remaining constraint is the standard momentum constraint (see for details) $$0 = \DLKxxDz - \Kzz{\DLxxDz}
\label{eqn:Momentum}$$
Artificial viscosity {#sec:ArtVisco}
--------------------
Our numerical experiments (which we will present shortly) showed that the future evolutions can be subject to high-frequency instabilities. This was seen to occur only in the cases where the lapse function was controlled by its own evolution equation (as in Harmonic slicing). For such cases we found that stability could be recovered with the addition of an artificial viscosity term to the evolution equations.
Let $W$ be any one of the dynamical variables, $\Lxx$, $\Lzz$, $\Kxx$, $\Kzz$. Then the artificial viscosity is introduced by the addition of a simple dissipation term to the evolution equation for $W$. After some experimentation we settled on the following form $$\frac{dW_i}{dt} = \frac{d{\bar W}_i}{dt}
+ \mu N_i\left( \frac{W_{i+1}-W_i}{\iLzz_i}
-\frac{W_i-W_{i-1}}{\iLzz_{i-1}}\right)$$ where $d{\bar W}_i/dt$ is the right hand side of the original evolution equation (\[eqn:DLxx\]–\[eqn:DKzz\]) and $\mu$ is a (small) constant. Other choices were tried but this form seemed to produce stable evolutions for the longest periods of time. Note that we do not add the dissipation terms to the evolution equations for the curvatures (doing so seemed to make no difference to the evolutions and had no effect in controlling the instabilities).
How should $\mu$ be chosen? We need to choose it large enough to ensure that the evolution is stable over a given time interval while also keeping it sufficiently small so as to not effect the large scale features of the numerical solution. By trial and error we found that setting $\mu=\SchwOnePLogArtViscosity$ worked well for evolutions to $t=\SchwOnePLogTimeMax$ using $\Nnode=\SchwOnePLogNumNode$ nodes. We also found that as the number of nodes was increased we had to make a proportionate increase in $\mu$ to maintain the same quality of the evolution over the same time interval. That is $\mu =\BigO{\Nnode}$.
The dissipation term is easily seen to be a finite difference approximation to $\mu \Lzz d^2
W/dz^2$ and thus it may appear to be like a Kreiss-Oliger term that vanishes in the continuum limit. However, since we are forced to set $\mu=\BigO{\Nnode}$ and as $\Lzz=\BigO{1/\Nnode}$ we see that the term $\mu\Lzz$ is approximately constant, say $\mu'$, and thus the dissipation term is actually of the form $\mu' d^2W/dz^2$. This is a standard dissipation term commonly used in hydrodynamic simulations and it does not vanish in the continuum limit.
Initial data {#sec:InitialData}
============
The initial data on the lattice are the $\Lxx$, $\Lzz$, $\Kxx$, $\Kzz$, $\Rx$ and $\Rz$ at each node of the lattice. Their time symmetric initial values were set by a combination of the Hamiltonian constraint, the geodesic deviation equation and the Bianchi identities. A full account of the choices made in coming to the equations described below can be found . Here we will just quote the relevant equations simply to provide explicit details of how we constructed our initial data.
To ensure that the initial data is time symmetric we set $\Kxx=0$ and $\Kzz=0$.
The $\Lzz$ were set according to the method of Bernstein, Hobill and Smarr [@bernstein:1989-01] using $n=\SchwGeodNumNode$ on a grid of length $\SchwGeodLength{}m$. The ADM mass, $m$, was set to be $\SchwGeodMass$ and the $\Lxx$, $\Rx$ and $\Rz$, for $i=1,2,3\dots n$, were set according to $$\begin{aligned}
\iLxx_{i} &= \iLxx_{i-1} + \frac{\iLzz_{i-1}}{\iLzz_{i-2}}
\left(\iLxx_{i-1}-\iLxx_{i-2}\right)\notag\\
&\quad - \frac{1}{2}\iLzz_{i-1}\left(\iLzz_{i-1}+\iLzz_{i-2}\right)
\left(\Lxx\Rz\right)_{i-1}\label{eqn:LxxInit}\\[5pt]
\iRz_i &= \iRz_{i-1}\left(\frac{5\iLLxx_{i-1}
-\iLLxx_{i}}{5\iLLxx_{i}
-\iLLxx_{i-1}}\right)\label{eqn:SchwRzInit}\\[5pt]
\iRx_i &= -2\iRz_i\label{eqn:SchwRxInit}\end{aligned}$$ At the reflection symmetric throat (at node 0) we set $\iLxx_0 =\SchwGeodLxx$ and $\iRx_0=-2\iRz_0=\SchwGeodRx$.
Results {#sec:Results}
=======
In all of our results we used a 4-th order Runge-Kutta integrator with the time step set equal to $1/2$ the smallest $\Lzz$ on the lattice (which happens to be $\iLzz_0$).
Slicing conditions {#sec:Slicing}
------------------
We ran our code for eight distinct slicing conditions, some were set by simple algebraic expressions while others involved differential operators.
We made four choices for the algebraic slicings, $$\begin{aligned}
N&=\exp(-2\Kxx)\label{eqn:LapseAlgebraicA}\\[5pt]
N&=\frac{20\Lxx}{1+20\Lxx}\label{eqn:LapseAlgebraicB}\\[5pt]
N&=\exp(-\Rx)\label{eqn:LapseAlgebraicC}\\[5pt]
N&=\frac{1}{1+\Rx}\label{eqn:LapseAlgebraicD}\end{aligned}$$ and three choice for the differential slicings, $$\begin{aligned}
&\text{$1+\log$ slicing}&&\frac{dN}{dt} = -2N K\label{eqn:LapseOnePLog}\\[5pt]
&\text{Harmonic slicing}&&\frac{dN}{dt} = -N^2 K\label{eqn:LapseHarmonic}\\[5pt]
&\text{Maximal slicing}&&\nabla^2 N = {}^3 RN\label{eqn:LapseMaximal}\end{aligned}$$
The eighth slicing condition was the simple case of geodesic slicing $N=1$.
The algebraic slicings were introduced after our early explorations with the differential lapses, all of which developed high-frequency instabilities after a short time (well before $t=100$). The algebraic slicings did not require any artificial viscosity and performed remarkably well, showing no signs of instabilities to at least $t=1000$ (excluding the lapse (\[eqn:LapseAlgebraicB\]) which hits the singularity at $t\approx32$). We have not run our codes beyond $t=1000$ so we can not comment its stability for $t>1000$.
Code tests and results {#sec:CodeTests}
----------------------
We subjected our code to many of the tests used in , such as the time at which geodesic slicing hits the singularity, the rate at which the lapse at the throat collapses in maximal slicing and the constancy of $\Lxx$ on the horizon. The results for these various slicings are shown in figures (\[fig:GeodesicLapse\]–\[fig:Lx3RxErr\]). All of the results are as expected. For the geodesic slicing the code crashes at approximately one time step short of the singularity. The familiar exponential collapse of the lapse for maximal slicing is evident in figure (\[fig:LapseProfiles\]). In this case it is known that the lapse at the throat should behave as $N\sim\beta\exp(\alpha t)$ for $t\rightarrow\infty$ with $\alpha=-(2/3)^{(3/2)}\approx\LapseSlopeExact$, see [@beig:1998-01]. We estimated the slope of $\ln N$ vs $t$ from our numerical data to be $\LapseSlope$ which agrees with the exact value to within $\LapseSlopeError$ percent.
We also have a new test obtained by a simple combination of the evolution equations. From equations (\[eqn:EvolveRx\],\[eqn:EvolveRz\]) we find that $$\frac{d\left( \Rx + 2\Rz \right)}{dt}= 2N\left(2\Kxx+\Kzz\right)\left( \Rx + 2\Rz \right)
\label{eqn:ConserveHamiltonian}$$ and as $0=\Rx+2\Rz$ on the initial slice (by construction, see (\[eqn:SchwRxInit\])) we conclude that $0=\Rx+2\Rz$ for all time. This is not surprising, our evolution equations for the curvatures are based on the Bianchi identities and these are guaranteed to preserve the constraints. If we now set $0=\Rx+2\Rz$ in (\[eqn:EvolveRz\]) and combine the result with (\[eqn:DLxx\]) we find $$0=\frac{d\Lxx^3 \Rx}{dt}
\label{eqn:ConserveLx3Rx}$$ This gives us a new test of our code, that the quantity $\Lxx^3 \Rx$ should be constant throughout the evolution. Importantly this applies to all slicing conditions. In figure (\[fig:Lx3RxErr\]) we have plotted the fractional variations in $\Lxx^3 \Rx$ for two choices of slicings. We see that the errors for the $1+\log$ slicing are much larger than those for the algebraic slicing which we attribute to the use of an artificial viscosity. This last claim is easily checked by varying the artificial viscosity parameter $\mu$. We find that the errors in $\Lxx^3 \Rx$ varies linearly with $\mu$. Note that in obtaining equation (\[eqn:ConserveLx3Rx\]) we have ignored the higher order error terms that would arise if we had carried through the $\BigO{L^3}$ truncation error from (\[eqn:ADMDLijb\]). Thus even if we set $\mu=0$ we can expect some variation of $\Lxx^3 \Rx$ over time (though this variation should vanish more rapidly than $\BigO{L^3}$).
We also have three constraint equations, namely the geodesic deviation equation (\[eqn:GeodDev\]), the 3-dimensional Bianchi identity (\[eqn:Bianchi\]) and the momentum constraint (\[eqn:Momentum\]). The discrete form of these equations are $$\begin{aligned}
P &= \frac{D^2\Lxx}{Dz^2} + {}^{3}\Rz \Lxx\label{eqn:DiscreteGeodDev}\\[5pt]
Q &= \frac{{\tilde D}\left(\LLxx {}^{3}\Rx\right)}{Dz}
- {}^{3}{\tilde{\cal R}_{xzxz}}
\frac{{\tilde D}\Lxx^2}{Dz}\label{eqn:DiscreteBianchi}\\[5pt]
M &= \frac{D\left(\Lxx\Kxx\right)}{Dz} - \Kzz\frac{D\Lxx}{Dz}\label{eqn:DiscreteMomentum}\end{aligned}$$ where ${\tilde{\cal R}_{xzxz}}$ is the average of $\Rz$ across $\Lzz$ while $D/Dz$ and ${\tilde D}/Dz$ are discrete derivative operators defined as follows. For a typical smooth function $f(z)$ sampled at the grid points $z_i$ we define ø $$\begin{aligned}
\left(\frac{{\tilde D}f}{Dz}\right)_i &:=\frac{\fp-\fo}{\Lzzo}\\[5pt]
\left(\frac{Df}{Dz}\right)_i &:=
\frac{1}{\Lzzo+\Lzzm}\left( \Lzzm\left(\frac{\fp-\fo}{\Lzzo}\right)
+\Lzzo\left(\frac{\fo-\fm}{\Lzzm}\right) \right)\\[5pt]
\left(\frac{D^2f}{Dz^2}\right)_i &:=
\frac{2}{\Lzzo+\Lzzm}\left( \frac{\fp-\fo}{\Lzzo}
-\frac{\fo-\fm}{\Lzzm} \right)\end{aligned}$$ where we have introduced the superscripts $\p$, $\o$ and $\m$ to denote quantities at the grid points $z_{i+1}$, $z_i$ and $z_{i-1}$ respectively. Note that the sample points $z_i$ are constructed from the lattice $\Lzz$ by the recurrence relation $z_{i+1}=z_i + (\Lzz)_i$ with $z_0=0$. In this notation we have ${\tilde{\cal R}}_{xyxy}:=(\Rx^{\p}+\Rx^{\o})/2$. Finally we note that the 3-curvatures can be computed from the 4-curvatures by way of the Gauss equation, $$\begin{aligned}
{}^{3}\Rx &= \Rx - \Kxx^2\\
{}^{3}\Rz &= \Rz - \Kxx \Kzz\end{aligned}$$ Ideally we would like to see $P=Q=M=0$ but in reality we expect $P_i$, $Q_i$ and $M_i$ to be non-zero but small. This is indeed what we observe, see figure (\[fig:Lx3RxErr\]). We also computed a crude estimate of the rate of convergence (of $Q$, $P$ and $M$ to zero at a fixed time) by running our code twice, once with $\Nnode=2048$ and once with $\Nnode=1024$ and then forming suitable ratios of the constraints at the horizon. In this manner we estimated, in the absence of artificial viscosity, that $P=\BigO{\Nnode^{-4}}$, $Q=\BigO{\Nnode^{-2}}$ and $M=\BigO{\Nnode^{-3}}$ while the addition of artificial viscosity degraded the convergence to $P=\BigO{\Nnode^{-1}}$, $Q=\BigO{\Nnode^{-1}}$ and $M=\BigO{\Nnode^{-2}}$.
We also tried setting ${\tilde D}/Dz:= D/Dz$ and ${\tilde{\cal R}}_{xyxy}:=\Rx^{\o}$ in the discrete Bianchi constraint but this lead to a reduction in the rate of convergence. The form of the discrete Bianchi constraint as given above (\[eqn:DiscreteBianchi\]) is readily seen [@brewin:2002-01] to be a second-order accurate estimate to the continuum Bianchi identity at the centre of the leg $\Lzz$.
One might ask why we have not included the Hamiltonian constraint in our code tests. The simple answer is that it is trivially satisfied by our discrete equations. This follows from the discussion surrounding equation (\[eqn:ConserveHamiltonian\]) where we showed that $0=\Rx + 2\Rz$ for all time. It follows that the Hamiltonian $H:=G_{\mu\nu}t^\mu t^\mu$ will also vanish for all time. Note that this analysis was based on our discrete equations, not on the continuum equations. We did indeed check that our code maintained $0=\Rx+2\Rz$ throughout the evolution.
Bianchi identities {#sec:Bianchi}
==================
Here we will use the Bianchi identities to obtain evolution equations for the two curvatures $\Rxyxy$ and $\Rxzxz$. We will follow the method given in our earlier paper [@brewin:2002-01] in which we used data imported from the neighbouring computational cells to estimate (by a finite difference approximation) the various derivatives required in the Bianchi identities. We will employ Riemann normal coordinates[^2], one for each computational cell, with the origin centred on the central vertex and the coordinate axes aligned with those described in section (\[sec:SphericalSpacetime\]), see also figure (\[fig:LatticeLocal\]). In these coordinates, the metric in a typical computational cell is given by $$g_{\mu\nu}(x) = g_{\mu\nu}
- \frac{1}{3} \rmanb x^\alpha x^\beta
- \frac{1}{6} \drmanbg x^\alpha x^\beta x^\gamma
+ \BigO{L^4}\\[5pt]$$ where $L$ is a typical length scale for the computational cell and $\gmn$ and $\rmanb$ are constant throughout the computational cell. A convenient choice for $\gmn$ is $\diag(-1,1,1,1)$ (such a choice can always be made by suitable gauge transformations within the class of Riemann normal frames). In this case the frame components $\Rxyxy$ and $\Rxzxz$ reduce to the coordinate components $\rxyxy$ and $\rxzxz$ respectively. A further advantage of using Riemann normal coordinates is that at the origin, where the connection vanishes, covariant derivatives reduce to partial derivatives.
The two Bianchi identities that we need are $$\begin{aligned}
0 &= \dRxyxyt - \dRtyxyx + \dRtxxyy\label{eqn:RiemEvolA}\\
0 &= \dRxzxzt - \dRtzxzx + \dRtxxzz\label{eqn:RiemEvolD}\end{aligned}$$ This pair of equations contains 4 spatial derivatives each of which we will estimate by a finite difference approximation. But in order to do so we must first have a sampling of the 4 curvatures at a cluster of points near and around the central vertex. Our simple ladder-like lattice, with its collection of computational cells along one radial axis, would allow us to compute only the $z$ partial derivatives. For the $x$ and $y$ derivatives we will need to extend the lattice along the $x$ and $y$ axes. In short we need a truly 3 dimensional lattice. Fortunately this is rather easy to do for this spacetime. We can use the spherical symmetry of the Schwarzschild spacetime to clone copies of the ladder (by spherical rotations) so that a typical central vertex of the parent ladder-lattice becomes surrounded by 4 copies of itself. It has two further nearby vertices, fore and aft along the radial axis, that are themselves central vertices of neighbouring cells in the original ladder-like lattice. In figure (\[fig:LatticeCloned\]) we display an $xz$ slice of the cloned lattice.
We now need the coordinates of all six of the neighbouring vertices. This would require a solution of $$% \Lsqij = g_{\mu\nu} \Dxij^\mu \Dxij^\nu
\Lsqij = g_{\mu\nu} \left(x_j^\mu-x_i^\mu\right) \left(x_j^\nu-x_i^\nu\right)
- \frac{1}{3} \rmanb x^\mu_i x^\nu_i x^\alpha_j x^\beta_j
+ \BigO{L^5}
\label{eqn:RNCLsq}$$ for the $x^\mu_i$ for given values for the $\Lij$ and $\rmanb$. However, as we are only going to use these coordinates to construct transformation matrices which will in turn multiply the Riemann curvatures, it is sufficient to solve (\[eqn:RNCLsq\]) using a flat metric. Note that the above equations can only be used to compute (in fact estimate) the spatial coordinates of the vertices. For the time coordinates we can appeal to the smoothness of the underlying metric[^3] to argue that for each vertex $t=\BigO{L^2}$. The result is that the typical central vertex, with coordinates $(0,0,0,0)$, will have 6 neighbouring central vertices with coordinates as per Table (\[tbl:Coords\]).
\#1[to 1.0cm[$#1$]{}]{} \#1[to 1.0cm[$#1$]{}]{} \#1[to 1.0cm[$#1$]{}]{}
height 14pt depth 7pt width 0pt
Vertex $t$ $x$ $y$ $z$
-- -------- -- ----- ----- ----- ----- --
0 (, , , )
1 (, , , )
2 (, , , )
3 (, , , )
4 (, , , )
5 (, , , )
6 (, , , )
: Riemann normal coordinates, to $\BigO{L^2}$, of the central vertex and its 6 immediate neighbours. These coordinates were computed using a flat space approximation.[]{data-label="tbl:Coords"}
This accounts for the structure of our lattice but what values should we assign to the curvatures at the newly created vertices? Let $(A)_{PQ}$ denote the value of a quantity $A$ at the vertex $P$ in the local Riemann normal frame for vertex $Q$. Since our spacetime is spherically symmetric we can assert that $$(A)_{00} = (A)_{11} = (A)_{22} = (A)_{33} = (A)_{44}$$ Then the idea that we will import data from neighbouring cells can be expressed as $$(A)_{PQ} = (U)_{PQ} (A_{PP})$$ where $(U)_{PQ}$ is the transformation matrix, evaluated at $P$, from the Riemann normal frame of $P$ to that of $Q$. This matrix will be composed of spatial rotations and boosts.
To get the correct estimates for the first partial derivatives we need only compute $U$ to terms linear in the leg-lengths.
As an example, let us suppose we wished to compute $v^\mu{}_{,x}$ for a spherically symmetric vector field $v$ on the lattice. We start with $(v)_{10}=(U)_{10}(v)_{11}$ and $$(U)_{10} = (B)_{10}(R)_{10}$$ where $(R)_{10}$ represents a rotation in the $x-y$ plane and $(B)_{10}$ a boost in the $t-x$ plane. Note that as we are working only to linear terms in the lattice scale the order in which we perform the rotation and boost does not matter. Thus we have $$\begin{aligned}
(R)_{10} &=
\begin{bmatrix}
1&0&0&0\\
0&\phantom{-}\cos\alpha&\sin\alpha&0\\
0&-\sin\alpha&\cos\alpha&0\\
0&0&0&1
\end{bmatrix}\\[5pt]
(B)_{10} &=
\begin{bmatrix}
\cosh\beta&\sinh\beta&0&0\\
\sinh\beta&\cosh\beta&0&0\\
0&0&1&0\\
0&0&0&1
\end{bmatrix}\end{aligned}$$ The columns in the above matrices are labelled $(t,x,y,z)$ from left to right and likewise for the rows. As we will latter be forming products of these matrices with the curvatures it is sufficient to compute these matrices as if we were working in flat spacetime. Thus to leading order in the lattice spacing we find[^4] $$\begin{gathered}
\cos\alpha = 1 + \BigO{L^3}\>,\quad \sin\alpha = \frac{d\Lxx}{dz} + \BigO{L^2}\\[5pt]
\cosh\beta = 1 + \BigO{L^3}\>,\quad \sinh\beta = -K_{xx}\Lxx + \BigO{L^2}\end{gathered}$$ and thus ¶[K\_[xx]{}]{} $$(U)_{10} = (B)_{10}(R)_{10} =
\begin{bmatrix}
1&-\P&0&0\\[5pt]
-\P&1&0&\Q\\[10pt]
0&0&1&0\\[5pt]
0&-\Q&0&1
\end{bmatrix} + \BigO{L^2}$$ In a similar manner we find For the remaining two matrices, $(U)_{50}$ and $(U)_{60}$, the job is quite simple, these matrices are built solely on boosts. This leads to
Returning now to the construction of $(v)_{10}$, we have ¶[K\_[xx]{}]{} $$\begin{aligned}
(v^\mu)_{10} &= (U^\mu{}_\nu)_{10} (v^\nu)_{11}\\
&= (v^\mu)_{11}
+ \left[-\P v^x ,-\P v^t+\Q v^z,0,-\Q v^x\right]^\mu_{11}\\
\intertext{and}
(v^\mu)_{30} &= (U^\mu{}_\nu)_{30} (v^\nu)_{33}\\
&= (v^\mu)_{33}
+ \left[\P v^x,\P v^t-\Q v^z,0,\Q v^x\right]^\mu_{33}\end{aligned}$$ We are now in a position to finally compute $(v^t_{,x})_{00}$, to wit $$\begin{aligned}
(v^t_{,x})_{00} &= \frac{(v^t)_{10}-(v^t)_{30}}{2\Lxx} + \BigO{L^a}\\
&= \frac{(v^t)_{11}-(v^t)_{33}}{2\Lxx}
- K_{xx}\frac{(v^x)_{11}+(v^x)_{33}}{2} + \BigO{L^a}\end{aligned}$$ Here we have written the truncation errors as $\BigO{L^a}$ with $a>0$ for it is not clear, at this level of analysis, what the exact nature of this term is (save that it vanishes as $L\rightarrow0$). Since our spacetime is spherically symmetric we have $$(v)_{00} = (v)_{11} = (v)_{22} = (v)_{33} = (v)_{44}$$ and thus $$(v^t_{,x})_{00} = -K_{xx}(v^x)_{00} + \BigO{L^a}$$ Similar calculations can be used to compute all of the spatial derivatives of $v^\mu$ at the central vertex.
We can now return to the principle objective of this section – to compute the various partial derivatives of the curvatures. We proceed exactly as above but with a minor change in that we will no longer carry the truncation errors within the calculations. Thus we have $$(\rmanb)_{i0} = (U_{\mu}{}^{\tau})_{i0}(U_{\alpha}{}^{\rho})_{i0}
(U_{\nu}{}^{\delta})_{i0}(U_{\beta}{}^{\lambda})_{i0}(\rtrdl)_{ii}$$ for $i=1,2,3,4,5,6$ and $(U_{\mu}{}^{\nu})_{i0} =
g_{\mu\rho}g^{\nu\tau}(U^{\rho}{}_{\tau})_{i0}$ with $g_{\mu\nu} = \diag(-1,1,1,1)$. And, as before, $$\begin{gathered}
(\rmanb)_{00} = (\rmanb)_{11} = (\rmanb)_{22} = (\rmanb)_{33} = (\rmanb)_{44}\end{gathered}$$ due to spherical symmetry.
Using the above expressions for the $(U)_{i0}$ and the following finite difference approximations $$\begin{aligned}
(\dRtyxyx)_{00} &= \frac{(\rtyxy)_{10}-(\rtyxy)_{30}}{2\Lxx}\\[5pt]
(\dRtxxyy)_{00} &= \frac{(\rtxxy)_{20}-(\rtxxy)_{40}}{2\Lyy}\\[5pt]
(\dRtzxzx)_{00} &= \frac{(\rtzxz)_{10}-(\rtzxz)_{30}}{2\Lxx}\\[5pt]
(\dRtxxzz)_{00} &= \frac{(\rtxxz)_{50}-(\rtxxz)_{60}}{2\Lzz}\end{aligned}$$ we find that $$\begin{aligned}
\dRtyxyx &= \phantom{-}\kxx\left(\rxyxy+\rtyty\right)
+ \frac{1}{\Lxx}\DLxxDz \rtyyz\label{eqn:DerivRa}\\[5pt]
\dRtxxyy &= -\kyy\left(\rxyxy+\rtxtx\right)
- \frac{1}{\Lyy}\DLyyDz \rtxxz\label{eqn:DerivRb}\\[5pt]
\dRtzxzx &= \phantom{-}\kxx\left(\rxzxz+\rtztz\right)
+ \frac{1}{\Lxx}\DLxxDz \rtxxz\label{eqn:DerivRc}\\[5pt]
\dRtxxzz &= -\kzz\left(\rxzxz+\rtxtx\right)\label{eqn:DerivRd}\end{aligned}$$ We have dropped the $00$ subscript as we no longer need to distinguish between the neighbouring frames. By spherical symmetry we have $$\begin{gathered}
\Lxx = \Lyy\>,\quad \kxx=\kyy\>,\quad \rtxxz=\rtyyz\>,\quad \rxzxz=\ryzyz\end{gathered}$$ while from the vacuum Einstein equations we have $$\begin{aligned}
0 &= \rtz = -\rtxxz - \rtyyz\\
0 &= \rxx = \rxyxy + \rxzxz - \rtxtx\\
0 &= \ryy = \rxyxy + \ryzyz - \rtyty\\
0 &= \rzz = \rxzxz + \ryzyz - \rtztz\end{aligned}$$ Combining the last few equations leads to $$\begin{gathered}
\rtxtx = \rtyty = \rxyxy+\rxzxz\\
\rtxxz = \rtyyz = 0\>,\quad \rtztz = 2\rxzxz\end{gathered}$$ Substituting these into the above equations (\[eqn:DerivRa\]–\[eqn:DerivRd\]) and subsequently into the previous expressions for the Bianchi identities (\[eqn:RiemEvolA\],\[eqn:RiemEvolD\]) leads to the following pair of equations $$\begin{aligned}
\DrxDt &= 2\kxx\left( 2\rxyxy + \rxzxz\right)\\[5pt]
\DrzDt &= 3\kxx\rxzxz + \kzz\left(\rxyxy + 2\rxzxz\right)\end{aligned}$$ Our job is almost complete, but we still have two tasks ahead of us i) to introduce a lapse function and ii) to account for the limited time interval over which a single Riemann normal frame can be used. The first task is rather easy, we simply make the coordinate substitution $t\rightarrow Nt$ leading to $$\begin{aligned}
\DrxDt &= 2N\kxx\left( 2\rxyxy + \rxzxz\right)\label{eqn:EvolveA}\\[5pt]
\DrzDt &= 3N\kxx\rxzxz + N\kzz\left(\rxyxy + 2\rxzxz\right)\label{eqn:EvolveB}\end{aligned}$$ and where we now have $(\gmn)_o=\diag(-N^2,1,1,1)$. The lapse $N$ can be freely chosen at each vertex of the lattice (but subject to the obvious constraint that $N>0$). The second task is a bit more involved. We know that each Riemann normal frame is limited in both space and time. Thus no single Riemann normal frame can be used to track the evolution for an extended period of time. We will have no choice but to jump periodically to a new frame. This can be elegantly handled in the moving frame formalism. Thus our task reduces to finding a new set of evolution equations for the frame components $\Rxyxy$ and $\Rxzxz$ based on the equations given above for $\rxyxy$ and $\rxzxz$.
Let $e^\mu{}_a$, $a=t,x,y,z$ be an orthonormal tetrad[^5], tied to the worldline of the central vertex and aligned to the coordinate axes. Thus we have $e^\mu{}_t$ as the future pointing tangent vector to the worldline while $e^\mu{}_z$ points along the $z$-axis. Then $$\begin{aligned}
\DRxDt =& \frac{d}{dt}\left(\rmanb \emx \eay \enx \eby \right)\\[5pt]
\DRzDt =& \frac{d}{dt}\left(\rmanb \emx \eaz \enx \ebz \right)\end{aligned}$$ Since our spacetime is spherically symmetric it is not hard to see that the tetrads of two consecutive cells (on the vertex worldline) are related by a boost in the $t-z$ plane (arising from gradients in the lapse function). A simple calculation shows that $$\begin{gathered}
\DemxDt = 0\>,\quad \DemyDt = 0\>,\quad
\DemtDt = \dNdz \emz\>,\quad \DemzDt = \dNdz \emt\end{gathered}$$ which when combined with the above leads to $$\begin{aligned}
\DRxDt =& \left(\DrmanbDt\right) \emx \eay \enx \eby\\[5pt]
\DRzDt =& \left(\DrmanbDt\right) \emx \eaz \enx \ebz
-2 \frac{N_{,z}}{N} \rmanb \emt \eax \enx \ebz\end{aligned}$$ In our frame we have chosen $\left(\gmn\right)_{o}=\diag(-N^2,1,1,1)$, $e^\mu{}_{a}=\delta^\mu_a$ for $a=x,y,z$ and $e^\mu{}_{t}=1/N$, thus we see that the last term in the previous equation is proportional to $\Rtxxz$. But for the Schwarzschild spacetime we know that $\Rtxxz=0$ and thus we have $$\begin{aligned}
\DRxDt =& \left(\DrmanbDt\right) \emx \eay \enx \eby\\[5pt]
\DRzDt =& \left(\DrmanbDt\right) \emx \eaz \enx \ebz\end{aligned}$$ which, when combined with (\[eqn:EvolveA\],\[eqn:EvolveB\]), leads immediately to the evolution equations (\[eqn:EvolveRx\],\[eqn:EvolveRz\]) quoted in section (\[sec:EvolveEqtns\]).
\#1\#2
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[plots/schwarz/geodesic/short/91]{}
[plots/schwarz/1+log/short/04]{} [plots/schwarz/algebraic/01/short/04]{} [plots/schwarz/algebraic/02/short/04]{}
[plots/schwarz/1+log/short/11]{} [plots/schwarz/algebraic/01/short/11]{} [plots/schwarz/algebraic/02/short/11]{}
[plots/schwarz/algebraic/04/long/04]{} [plots/schwarz/algebraic/04/short/11]{} [plots/schwarz/algebraic/04/long/11]{}
[plots/schwarz/algebraic/02/short/93]{} [plots/schwarz/maximal/short/93]{} [plots/schwarz/1+log/short/93]{}
[plots/schwarz/1+log/short/29]{} [plots/schwarz/algebraic/04/short/30]{} [plots/schwarz/1+log/short/30]{}
[^1]: Here $z$ is the proper distance measured up the middle of the lattice along a trajectory that passes through the mid-points of each $\Lxx$
[^2]: For more details on Riemann normal coordinate see [@brewin:2009-03] and the references cited therein.
[^3]: If $(t,x^i)$ are the coordinates for a local Riemann normal frame, then a smooth Cauchy surface through $(0,0,0,0)$ is described locally by $2t = -K_{ij} x^i
x^j$ and as each $x^i=\BigO{L}$ we also have $t=\BigO{L^2}$.
[^4]: For the rotations we use standard Euclidian trigonometry, for the boost we use the definition $n^\mu_i-n^\nu_j =
-K^\mu{}_\nu(x^\nu_i-x^\nu_j)$ where $n^\mu_a$ is the future pointing unit normal to the Cauchy surface at the point $a$.
[^5]: This tetrad is identical to that used in section (\[sec:SphericalSpacetime\]), the change of notation introduced here is simply to avoid unwanted clutter in the following equations.
|
---
abstract: 'The acoustic-to-word model based on the connectionist temporal classification (CTC) criterion was shown as a natural end-to-end (E2E) model directly targeting words as output units. However, the word-based CTC model suffers from the out-of-vocabulary (OOV) issue as it can only model limited number of words in the output layer and maps all the remaining words into an OOV output node. Hence, such a word-based CTC model can only recognize the frequent words modeled by the network output nodes. Our first attempt to improve the acoustic-to-word model is a hybrid CTC model which consults a letter-based CTC when the word-based CTC model emits OOV tokens during testing time. Then, we propose a much better solution by training a mixed-unit CTC model which decomposes all the OOV words into sequences of frequent words and multi-letter units. Evaluated on a 3400 hours Microsoft Cortana voice assistant task, the final acoustic-to-word solution improves the baseline word-based CTC by relative 12.09% word error rate (WER) reduction when combined with our proposed attention CTC. Such an E2E model without using any language model (LM) or complex decoder outperforms the traditional context-dependent phoneme CTC which has strong LM and decoder by relative 6.79%.'
address: 'Microsoft AI and Research, One Microsoft Way, Redmond, WA 98052\'
bibliography:
- 'strings.bib'
- 'refs.bib'
title: 'Advancing Acoustic-to-Word CTC Model'
---
CTC, OOV, acoustic-to-Word, end-to-end training, speech recognition
Introduction {#sec: Introduction}
============
As one of the most popular end-to-end (E2E) methods, the connectionist temporal classification (CTC) approach [@Graves-CTCFirst; @Graves-E2EASR] was introduced to map the speech input frames into an output label sequence [@Hannun-DeepSpeech; @sak2015learning; @sak2015fast; @miao2015eesen; @kanda2016maximum; @soltau2016neural; @Zweig-AdvancesNeuralASR; @liu2017gram; @audhkhasi2017direct; @Li17CTCnoOOV; @Yu-RecentProgDeepLearningAcousticModels; @Li2018Speaker]. To deal with the issue that the number of output labels is smaller than that of input speech frames in speech recognition tasks, CTC introduces a special blank label and allows for repetition of labels to force the output and input sequences to have the same length.
CTC outputs are usually dominated by blank symbols and the output tokens corresponding to the non-blank symbols usually occur with spikes in their posteriors. Thus, an easy way to generate ASR outputs using CTC is to concatenate the non-blank tokens corresponding to the posterior spikes and collapse those tokens into word outputs if needed. This is a very attractive feature for E2E modeling as there is neither LM nor complex decoding involved. We refer this decoding strategy as greedy decoding, and our E2E models studied in this paper all use greedy decoding.
As the goal of ASR is to generate a word sequence from speech acoustics, word is the most natural output unit for network modeling. A big challenge in the word-based CTC is the out-of-vocabulary (OOV) issue [@bazzi2002modelling; @decadt2002transcription; @yazgan2004hybrid; @bisani2005open]. In [@sak2015fast; @soltau2016neural; @audhkhasi2017direct], only the most frequent words in the training set were used as targets whereas the remaining words were just tagged as OOVs. All these OOV words can neither be further modeled nor be recognized during evaluation. For example in [@sak2015fast], the CTC with up to 27 thousand (k) word output targets was explored but the ASR accuracy is not very good, partially due to the high OOV rate when using only around 3k hours training data.
To solve this OOV issue in the word-based CTC, we proposed a hybrid CTC [@Li17CTCnoOOV] which uses the output from the word-based CTC as the primary ASR result and consults a letter-based CTC at the segment level where the word-based CTC emits an OOV token. A shared-hidden-layer structure is used to align the word segments between the word-based CTC and the letter-based CTC so that the OOV token lookup algorithm can work. However, the shared-hidden-layer structure still cannot guarantee a perfect alignment between the word and letter based CTCs. It also hurts the modeling accuracy of the auxiliary CTC model. In [@audhkhasi2017building], a spell and recognize model is used to learn to first spell a word and then recognize it. Whenever an OOV is detected, the decoder consults the letter sequence from the speller. In [@Li17CTCnoOOV; @audhkhasi2017building], the displayed hypothesis is more meaningful than OOV to users. However, both methods cannot improve the overall recognition accuracy too much.
In this study, we propose a solution to the OOV issue in the acoustic-to-word modeling by decomposing the OOV word into a mixed-unit sequence of frequent words and letters at the training stage. We use attention CTC to address the inherent CTC modeling issue. During testing, we do greedy decoding for the whole E2E system in a single step without the need of using the two-stage (OOV-detection and then letter-sequence-consulting) process as in [@Li17CTCnoOOV; @audhkhasi2017building]. With all these components, the final acoustic-to-word solution improves the baseline acoustic-to-word CTC by relative 12.09% word error rate (WER) reduction and also outperforms the traditional context-dependent-phoneme CTC with strong LM and decoder by relative 6.79%.
Advance Acoustic-to-Word CTC
============================
\[sec: E2E\]
Word-based Connectionist Temporal Classification (CTC)
------------------------------------------------------
\[ssec: CTC\] A CTC network uses an recurrent neural network (RNN) and the CTC error criterion [@Graves-CTCFirst; @Graves-E2EASR] which directly optimizes the prediction of a transcription sequence. As the length of output labels is shorter than the length of input speech frames, a CTC path is introduced to have the same length as the input speech frames by adding the blank symbol as an additional label and allowing repetition of labels.
Denote $\bf{x}$ as the speech input sequence, $\bm\pi$ as the CTC path, $\bf{l}$ as the original label sequence (transcription), and $B^{-1}(\bf{l})$ as the preimage mapping all possible CTC paths $\bm\pi$ resulting from $\bf{l}$. Then, the CTC loss function is defined as the sum of negative log probabilities of correct labels as, $$L_{CTC} = - \ln P( {\bf{l}|\bf{x}} ) = - \ln \sum_{{\bm\pi} \in B^{-1}(\bf{l})} P( {\bm\pi} | \bf{x} ).$$ With the conditional independence assumption, $P( {\bm\pi} | \bf{x} )$ can be decomposed into a product of posteriors from each frame as, $$P( {{\bm\pi} | \bf{x}} ) = \prod_{t=1}^T P( \pi_{t}| \bf{x}).$$
As the goal of ASR is to generate a word sequence from the speech waveform, the word unit is the most natural output unit for network modeling. The recently proposed acoustic-to-word models [@soltau2016neural; @audhkhasi2017direct], a.k.a. word-based CTC models, build multiple layer long short-term memory (LSTM) [@Hochreiter1997long; @Graves2013speech; @Sak2014long] networks and use words as the network output units, optimized with the CTC training criterion. It is very simple to generate the word sequence with this word-based CTC model using greedy decoding: pick the words corresponding to posterior spikes to form the output word sequence. There is neither language model nor complex decoding process involved.
However, when training a word-based CTC model, only the most frequent words in the training set were used as targets whereas the remaining words were just tagged as OOVs. All these OOV words cannot be modeled by the network and cannot be recognized during evaluation. For example, if the transcription of an utterance is “have you been to newyorkabc” in which newyorkabc is an infrequent word, the training token or recognition output sequence for this utterance will be “have you been to OOV”.
Hybrid CTC
----------
\[ssec: hybCTC\]
To solve the OOV issue in the acoustic-to-word model, the hybrid CTC model uses a word-based CTC as the primary model and a letter-based CTC as the auxiliary model. The word-based CTC model emits a word sequence, and the output of the letter-based CTC is only consulted at the segment where the word-based CTC emits an OOV token. The detailed steps for building the hybrid CTC model are described as follows:
- Build a multi-layer LSTM-CTC model with words as its output units. Map all the words occurring less than $N$ times in the training data as the OOV token. The output units in this LSTM-CTC model are all the words occurring at least $N$ times in the training data, together with OOV, blank, and silence tokens.
- Freeze the bottom $L-1$ hidden layers of the word-CTC, add one LSTM hidden layer and one softmax layer to build a new LSTM-CTC model with letters as its output units.
- During testing, generate the word output sequence using greedy decoding. If the output word sequence contains an OOV token, replace the OOV token with the word generated from the letter-based CTC that has the largest time overlap with the OOV token.
CTC with Multi-letter Units
---------------------------
\[ssec: multiCTC\]
In [@Li17CTCnoOOV], the letter-based CTC uses single-letter units as the output units. Inspired by gram CTC [@liu2017gram] and multi-phone CTC [@siohan2017ctc], we extend the output units with double-letter and triple-letter units to benefit from long temporal units which are more stable. We hope to improve the hybrid CTC system as the OOV token may be replaced by more precise words generated by the CTC with multi-letter units.
Gram CTC and multi-phone CTC are based on letter and phoneme respectively, but allow to output variable number of letters (i.e., gram) and phonemes at each time step. The units in gram CTC and multi-phone CTC are learned automatically with the modified forward-backward algorithm to take care of all the decompositions. Both of them need much more complicated decoding than greedy decoding when generating outputs. In contrast, we just simply decompose every word into a sequence of one or more letter units, with examples shown in the first three rows of Table \[Tab:units\]. This decomposition is much simpler, without changing the CTC forward-backward process and can use the same greedy decoding as the CTC with single-letter units.
Decomposition Type newyork newyorkabc
------------------------------- --------------- ---------------------
All words: single-letter n e w y o r k n e w y o r k a b c
All words: double-letter ne wy or k ne wy or ka bc
All words: triple-letter new yor k new yor kab c
All words: word newyork OOV
OOVs only: single-letter newyork n e w y o r k a b c
OOVs only: word+single-letter newyork newyork a b c
OOVs only: word+triple-letter newyork newyork abc
: Examples of how words are represented with different output units. “Newyork" is a frequent word while “newyorkabc" is an infrequent word. The word-based CTC treats “newyork" as a unique output node and “newyorkabc" as the OOV output node.
\[Tab:units\]
Acoustic-to-Word CTC with Mixed Units
-------------------------------------
\[ssec: mixCTC\]
In hybrid CTC, the shared-hidden-layer constraint is used to help the time synchronization of word outputs between the word-based and letter-based CTC models. However, the blank symbol dominates most of the frames, and therefore the time synchronization is not very reliable. The ideal case should be when the spoken word is in the frequent word list the system emits a word output. And when the spoken word is an OOV (infrequent) word, the system emits a letter sequence from which a word is generated by collapsing all those letters. This cannot be done with the hybrid CTC because the two CTCs are running in parallel without a perfect time synchronization. A direct solution is to train a single CTC model with mixed units. If the word is a frequent word, then we just keep it in the output token list. If the word is an infrequent word, then we decompose it into a letter sequence. As shown in the fifth row of Table \[Tab:units\], the infrequent word “newyorkabc" is decomposed into “n e w y o r k a b c” for single letter decompositions. However, the frequent word “newyork" is not decomposed because it is a frequent word. Therefore, the output units of the CTC are mixed units, with both words (for frequent words) and letters (for OOV words).
However we note that artificially decomposing OOVs only into single-letter sequences may confuse CTC training because the network output modeling units are frequent words and letters. To solve such a potential issue, we decompose the OOV words into a combination of frequent words and letters. For example, in the last two rows of Table \[Tab:units\], “newyorkabc" is decomposed into “newyork a b c” if we use single-letter units with words or “newyork abc” if we use triple-letter units with words. In the CTC with mixed units, we use “\$” to separate each word in the sentence. For example, the sentence “have you been to newyorkabc” is decomposed into “\$ have \$ you \$ been \$ to \$ newyork abc \$”. If \$ is not used to separate words, we don’t know how to collapse the mixed units (words+letters) into output word sequences. Now, because during training the OOV words are decomposed into mixed units from words and letters, there is no OOV output node in the mixed unit CTC model. Consequently, during testing the model is very likely to emit OOV words as a sequence of frequent words and letters while still emitting frequent words when frequent words are spoken.
CTC with Attention
------------------
\[ssec: CTCAttn\] We present a brief outline of modeling attention directly within CTC proposed by us in [@Das18CTCAttention]. One drawback of standard CTC training is the hard alignment problem. This is because CTC relies only on one hidden feature to make the current prediction. CTC Attention overcomes the hard alignment problem by producing a context vector which is a weighted sum of the most relevant hidden features within a context window. The resulting context vector can then be used to make the current prediction. Thus, the main components of proposed CTC Attention are: (a) the generation of context vectors as time convolution (TC) features, and (b) the computation of the weights of the hidden features using an attention mechanism. In this section, we use indices $t$ and $u$ to denote the time step for input and output sequences respectively. However, it is understood that in CTC every input frame ${\mathbf{x}}_{t}$ generates output ${\mathbf{y}}_{t} = {\mathbf{y}}_{u}$.
The context vector ${\mathbf{c}}_{u}$ can be computed as a TC feature by convolving the hidden feature ${\mathbf{h}}_{t}$ with learnable weight matrices ${\mathbf{W}}^{\prime}$ across time as, $$\begin{aligned}
{\mathbf{c}}_{u} &= {\mathbf{W}}^{\prime} \ast {\mathbf{h}} = \sum_{t =u-\tau}^{u+\tau} {\mathbf{W}}^{\prime}_{u - t} {\mathbf{h}}_{t} \nonumber \\
&\stackrel{\Delta}{=} \sum_{t =u-\tau}^{u+\tau} {\mathbf{g}}_{t} = \gamma \sum_{t =u-\tau}^{u+\tau} \alpha_{u,t} {\mathbf{g}}_{t}. \label{eq:CTCAttn-TimeConvolution}\end{aligned}$$ The duration $[u-\tau,\ u+\tau]$ represents a context window of length $C = 2\tau + 1$ and ${\mathbf{g}}_{t}$ represents the $filtered$ signal at time $t$. The last step in Eq. holds when $\alpha_{u,t} = \frac{1}{C}$ and $\gamma = C$. The term $\alpha_{u,t}$ is the attention weight determining the relevance of ${\mathbf{h}}_{t}$ in generating ${\mathbf{c}}_u$. The context vector ${\mathbf{c}}_{u}$ is related to the output ${\mathbf{y}}_u$ using the softmax operation as, $$\begin{aligned}
{\mathbf{z}}_{u} &= {\mathbf{W}}_{\text{soft}}{\mathbf{c}}_{u} + {\mathbf{b}}_{\text{soft}}, \nonumber \\
{\mathbf{y}}_{u} &= \text{Softmax}({\mathbf{z}}_{u}). \label{eq:CTCAttn-generate}\end{aligned}$$ To include non-uniform attention weights $\alpha_{u,t}$ instead of uniform weights ($\alpha_{u,t} = \frac{1}{C} $ in Eq. ), we use the Attend(.) function, $$\begin{aligned}
\bm\alpha_{u} &= \text{Attend}({\mathbf{z}}_{u-1}, \bm{\alpha}_{u-1}, {\mathbf{g}}). \label{eq:CTCAttn-attend}\end{aligned}$$ Thus, Eq. represents hybrid attention (HA) as it encodes both content (${\mathbf{z}}_{u-1}$) and location ($\bm{\alpha}_{u-1}$) information. In the absence of $\bm{\alpha}_{u-1}$, Eq. would represent content attention (CA).
The performance of the attention model can be improved further by providing more reliable content information. This is possible by introducing another recurrent network that can utilize content from several time steps in the past. This network, in essence, would learn an implicit language model (LM) and can be represented as, $$\begin{aligned}
{\mathbf{z}}^{\text{LM}}_{u-1} &= \mathcal{H}({\mathbf{x}}_{u-1}, {\mathbf{z}}^{\text{LM}}_{u-2}), \quad
{\mathbf{x}}_{u-1} =
\begin{bmatrix}
{\mathbf{z}}_{u-1} \\
{\mathbf{c}}_{u-1}
\end{bmatrix}, \label{eq:CTCAttnLM-LSTM} \\
\bm\alpha_{u} &= \text{Attend}({\mathbf{z}}^{\text{LM}}_{u-1}, \bm{\alpha}_{u-1}, {\mathbf{g}}), \label{eq:CTCAttn-attendLM}\end{aligned}$$ where $\mathcal{H}(.)$ is a LSTM unit.
In the final step to improve attention, each of the $n$ components of ${\mathbf{g}}_{t} \in {\mathbb{R}}^{n}$ in Eq. could be weighted distinctively. This is possible by replacing the scalar attention weight $\alpha_{u,t} \in [0, 1]$ with a vector attention weight $\bm\alpha_{u,t} \in [0, 1]^{n}$ for each $t \in [u-\tau,\ u+\tau]$. Under this formulation, the context vector ${\mathbf{c}}_{u}$ can be computed using, $$\begin{aligned}
{\mathbf{c}}_{u} &= \gamma \sum_{t=u-\tau}^{u+\tau} \bm\alpha_{u,t} \odot {\mathbf{g}}_{t}, \label{eq:CTCAttn-Comp-annotate}\end{aligned}$$ where $\odot$ is the Hadamard product.
Comparison with Other End-to-end Methods
----------------------------------------
\[ssec: CompareE2E\]
In addition to CTC, there are also popular E2E methods in ASR, such as RNN encoder-decoder (RNN-ED) [@Chan-LAS; @lu2016training] and RNN transducer (RNN-T) [@rao2017exploring]. Initially working on letter units, these methods recently got significant improvement when working on word-piece units [@schuster2012japanese], either pre-trained [@chiu2017state; @rao2017exploring] or automatically derived [@chan2016latent] during training. In all these works, all the words are decomposed into word-piece units which range from single letter all the way up to entire words. In contrast, our acoustic-to-word model directly uses frequent words as basic units, and only decomposes infrequent words into a sequence of frequent words and multi-letters. The majority units are still words. Therefore, our units are more stable and natural for the E2E system outputting word hypotheses. In [@lu2016training], words were also used as the basic units with the RNN-ED structure. However, the reported WER was much higher than the one obtained with traditional systems.
As extensions of CTC, both RNN-T and RNN aligner [@sak2017recurrent] either change the objective function or the training process to relax the frame independence assumption of CTC. The proposed attention CTC in Section \[ssec: CTCAttn\] is another solution by working on hidden layer representation with more context information without changing the CTC objective function and training process.
Experiments
===========
\[sec: Expts\]
The proposed methods were evaluated using the Microsoft’s Cortana voice assistant task. The training dataset contains approximately 3.3 million short utterances ($\sim$ 3400 hours) in US-English. The test set contains about 5600 utterances ($\sim$ 6 hours). The base feature vector for every 10 ms is a 80-dimensional vector containing log filterbank energies. The base feature vectors in three continuous frames are stacked together as the 240-dimension input feature to the CTC models [@sak2015fast]. All CTC models are bi-directional LSTM models.
We first built a phoneme-based bi-directional 6-layer LSTM model trained with the CTC criterion, modeling around 9000 tied context-dependent (CD) phonemes. Every layer of the bi-directional LSTM has 512 memory units in each direction. Unless otherwise stated, all CTC models except attention CTC models in this study use the same structure as this model. This CD-phone CTC model has 9.28% WER when decoding with a 5-gram LM with totally around 100 million (M) n-grams. In this study, except this CD-phone CTC model, all the other CTC models are E2E models using greedy decoding which generate the final output sequence without using any LM or complicated decoding process.
Next, we built an acoustic-to-word CTC model with the same model structure as the CD-phone CTC by modeling around 27k most frequent words in the training data. These frequent words occurred at least 10 times in the training data. All other infrequent words were mapped to an OOV output token. We have also tried other word-based CTCs with varying number of output units. However, the model using 27k word outputs performs the best. This word-based LSTM-CTC model yields 9.84% WER, among which the OOV tokens contribute 1.87% WER. It significantly improves the WER of uni-directional word-based CTC reported in [@Li17CTCnoOOV] which indicates the bi-directional modeling is critical to the E2E system.
Letter CTC with Attention
-------------------------
\[ssec: exp\_letter\] As the word output in the letter-based CTC is used to replace the OOV token from the word-based CTC model during testing, the letter-based CTC should be as accurate as possible. In this set of experiments, we first evaluate the impact of using different size of letter units for the vanilla CTC [@Graves-CTCFirst]. All the letter-based CTC models are 6-layer bi-directional LSTM models. The single-letter set has 30 symbols, including 26 English characters \[a-z\], ’, \*, \$, and blank. The double-letter and triple-letter sets have 763 and 8939 symbols respectively, covering all the double-letter and triple-letter occurrence in the training set. As shown in the second column of Table \[Tab:WER\_letterCTC\], the WER reduces significantly when the output units become larger, i.e., more stable. The letter-based CTC using triple-letter as output units achieves 13.28% WER, reducing 24.29% relative WER from the letter-based CTC using single-letter as output units.
The attention CTC presented in Section \[ssec: CTCAttn\] is then trained with $\tau$ empirically set as 4 (context window size $C$ = 9). As shown in the third column of Table \[Tab:WER\_letterCTC\], attention CTC improves the vanilla CTC hugely, obtaining 18.47%, 20.88%, and 14.46% relative WER reduction for single-letter, double-letter, and triple-letter CTC models, respectively. The best letter-based E2E CTC model is the one with triple-letter outputs and attention modeling, which can obtain 11.36% WER.
The hybrid CTC model described in Section \[ssec: hybCTC\] has both word-based CTC and letter-based CTC, which share 5 hidden LSTM layers. On top of the shared hidden layers, we add a new LSTM hidden layer and a softmax layer to model letter (single, double, or triple-letters) outputs. Attention modeling is applied to boost the performance. As shown in the fourth column of Table \[Tab:WER\_letterCTC\], the WER of letter-based CTC with such shared-hidden-layer constraint performs worse than its counterpart. This indicates one shortcoming of the hybrid CTC – it sacrifices the accuracy of the letter-based CTC because of the shared-hidden-layer constraint used to synchronize the word outputs between the word-based and letter-based CTC.
--------------- --------- ----------- ------------------
E2E Model
Vanilla Attention Attention
5 layers sharing
single-letter 17.54 14.30 16.74
double-letter 15.37 12.16 14.00
triple-letter 13.28 11.36 12.81
--------------- --------- ----------- ------------------
: WERs of letter-based CTC models with single, double, and triple-letter output units. Three structures are evaluated: vanilla CTC [@Graves-CTCFirst], attention CTC, and attention CTC sharing 5 hidden layers with the word CTC.
\[Tab:WER\_letterCTC\]
Hybrid CTC
----------
\[ssec: exp\_hybrid\]
As the CTC models with double-letter and triple-letter output units worked very well in Table \[Tab:WER\_letterCTC\], we use them to build the hybrid CTC models with the OOV lookup process described in Section \[ssec: hybCTC\]. Both hybrid models achieved 9.66% WER as shown in Table \[Tab:WER\_HybCTC\]. Several factors contribute to such small improvement (from 9.84% WER of the word-based CTC) of the hybrid CTC. First, the shared-hidden-layer constraint degrades the performance of the letter-based CTC, potentially affecting the final hybrid system performance. Second, although the shared-hidden-layer constraint helps to synchronize the word outputs from the word and letter based CTC, we still observed that the time synchronization can fail sometimes. In such cases, the OOV token is replaced with its neighboring frequently occurring word because of word segments misalignment. Because of these factors, although the triple-letter CTC is better than double-letter CTC in Table \[Tab:WER\_letterCTC\], there is no difference when they are combined with the baseline word CTC in the hybrid CTC setup in which they only handle the small portion of OOV words.
E2E Model
---------------------------------------------- ------
Word-based CTC 9.84
Word-based CTC + double-letter Attention CTC 9.66
Word-based CTC + triple-letter Attention CTC 9.66
: WERs of vanilla word-based CTC and hybrid CTC models. All Hybrid CTC models have a word-based CTC and a letter-based attention CTC, sharing 5 hidden layers.
\[Tab:WER\_HybCTC\]
CTC with Mixed Units
--------------------
\[ssec: exp\_mix\]
We evaluate the CTC with mixed units in Table \[Tab:WER\_mixCTC\]. In the first experiment, the mixed units contain single-letters and 27k frequent words. During training, OOV words are decomposed into single-letter sequence. As analyzed in Section \[ssec: mixCTC\], artificially decomposing OOV words into letter sequence while keeping the frequent words confuses CTC training for these types of words. Therefore, the trained CTC model achieved 20.10% WER. When looking at the posterior spikes of this model, we observed that the word spikes and letter spikes are scattered into each other which proves our hypothesis.
Next, we decompose OOV words into frequent word and single-letter sequences, and train the CTC network with the mixed units (around 27k). Immediately, the WER improved to 10.17%, but still a little worse than the baseline word-based CTC. This is because the single-letter sequence brings instability to the modeling. When we decompose the OOV words into frequent words and double-letters (totally 27k units), the situation becomes better, and the resulting WER is 9.58%. When the triple-letters and frequent words are used (totally 33k units), the WER reaches 9.32%, beating the baseline word-based CTC by 5.28% relative WER reduction.
Finally, we improve the final E2E CTC model by applying attention CTC. To save computational cost with large number of output units, we didn’t integrate the implicit LM in Eq.. The WER becomes 8.65%, which is about relative 12.09% WER reduction from the 9.84% WER of vanilla word-based CTC. Such a model without using LM and complex decoder also outperforms the traditional context-dependent-phoneme CTC with strong LM and decoder which obtained 9.28% WER. Note that the proposed method not only reduces the WER of the word-based CTC, but also improves the user experience. The proposed method provides more meaningful output without outputting any OOV token to distract users. Most of the time, even if the proposed method cannot get the OOV word right, it comes out with a very close output. For example, the proposed method recognize “text fabine” as “text fabian” and “call zubiate” as “call zubiat”, while the vanilla word-based CTC can only output “text OOV” and “call OOV”.
E2E Model
------------------------------------------------- -------
word-based CTC 9.84
mixed (OOV: single-letter) CTC 20.10
mixed(OOV: word + single-letter) CTC 10.17
mixed (OOV: word + double-letter) CTC 9.58
mixed (OOV: word + triple-letter) CTC 9.32
mixed (OOV: word + triple-letter) attention CTC 8.65
: WERs of the vanilla word-based CTC and the CTC with mixed units.
\[Tab:WER\_mixCTC\]
Conclusions
===========
\[sec: Conclusions\] We advance acoustic-to-word CTC model with a mixed-unit CTC whose output units are frequent words combined with sequences of multi-letters. For the frequent word, we just model it with a unique output node. For the OOV word, we decompose it into a sequence of frequent words and multi-letters. We present the attention CTC which significantly improves the modeling power of CTC. The proposed method is simpler and more effective than the hybrid CTC which has to rely on shared-hidden-layer to maintain the time synchronization of word outputs between the word-based and letter-based CTCs. We evaluate all these methods on a 3400 hours Microsoft Cortana voice assistant task. The proposed acoustic-to-word CTC with mixed-units reduces relative 5.28% WER from the vanilla word-based CTC, and reduces relative 12.09% WER if combined with the proposed attention CTC. Such an acoustic-to-word CTC is a pure end-to-end model without any LM and complex decoder. It also outperforms the traditional context-dependent-phoneme CTC with strong LM and decoder by relative 6.79% WER reduction.
|
---
author:
- 'S. Andersson'
- 'E. F. van Dishoeck'
date: 'Received date /Accepted date'
subtitle: A molecular dynamics study
title: Photodesorption of water ice
---
Introduction
============
Ices are a major reservoir of the heavy elements in a variety of astrophysical environments, ranging from cold and dense molecular clouds [e.g., @wil82; @whi88; @smi89; @mur00; @pon04] and protoplanetary disks [@pon05; @ter07] to the icy bodies in our own solar system such as comets [e.g., @mum93] and Kuiper Belt Objects [@jew04]. In star-forming clouds, the fraction of carbon and oxygen locked up in ice is comparable to that in gaseous CO [@pon06; @vand96], whereas at the centers of cold pre-stellar cores more than 90% of the heavy elements can be frozen out [e.g., @cas99; @ber02]. Similarly, the cold midplanes of protoplanetary disks around young stars are largely devoid of gaseous molecules other than H$_2$, H$_3^+$ and their isotopologues [e.g., @aik02; @cec05]. Thus, a good understanding of how molecules adsorb and desorb from the grains is critical to describe the chemistry in regions in which stars and planets are forming.
The importance of ultraviolet (UV) radiation in affecting interstellar ices is heavily debated in the literature. On the one hand, the large extinctions of 100 mag or more along the lines of sight where ices are detected prevent UV radiation from penetrating deep into the clouds [e.g., @ehr01; @sta04], unless there are cavities through which the stellar UV photons can escape [@spa95]. Thus, the bulk of the ices are thought to be shielded from both external and internal radiation sources in which case photodesorption is thought to be unimportant [e.g., @leg85; @har90]. On the other hand, UV photons are produced locally throughout the cloud by the interaction of cosmic rays with the gas, albeit at a level about $10^4$ times less than that of the general interstellar radiation field [@pra83; @she04]. In addition, X-rays from young stars penetrate much further into their surroundings than UV and can produce local UV photons through a similar process [@dal99; @sta05]. Moreover, the observed emission of optically thick millimeter lines from gaseous molecules is often dominated by the outer layers of the cloud where UV photons play a role. These UV photons can be important not only in the desorption of ices but also in the creation of reactive photo-products such as energetic H atoms and radicals which can move through the ice and encounter other species leading to the formation of more complex molecules [e.g., @dhe82; @gar06].
Water ice is the dominant consituent of interstellar ices [e.g., @gib00; @pon06] with an abundance at least three orders of magnitude larger than that of gaseous water in cold clouds [@sne00; @boo03]. Thus, evaporation of water ice, even at a low fraction, can significantly affect the gaseous water abundance. Recent models of translucent and dense clouds invoke photodesorption of water ice in the outer regions to explain the gaseous water emission observed by the Submillimeter Wave Astronomy Satellite (SWAS) [@ber05]. Photodesorption is also used to interpret the tentative detections of HDO and other gaseous species in the surface layers of protoplanetary disks [@wil00; @dom05]. The adopted desorption efficiencies in these models, about 0.1% per incident UV photon, are based on a single experiment by @wes95a [@wes95b] exposing ices at 35–100 K to Lyman-$\alpha$ radiation. The authors did not detect any water photodesorption in the limit of low UV photon fluence (integrated flux). Therefore it was suggested that water photodesorption only occurs from ices that have been subject to large doses of UV photons and not directly upon the first exposure to UV radiation. This is in contrast with recent experiments reported by Öberg et al. (submitted to ApJ), where there is a clear component of the water photodesorbed from amorphous ices at 18–100 K that is detected directly upon the first exposure to a UV lamp. Apart from that the results remain quite similar to the ones by Westley et al.
Other experiments on UV irradiation of water ices have also been performed. @gho71 observed H, OH, and H$_2$O$_2$ following UV irradiation of crystalline ice at 263 K, while @ger96 found production of OH, HO$_2$, and H$_2$O$_2$ in the ice upon exposing amorphous ice at 10 K to UV light covering mainly the first and second electronic absorption bands of H$_2$O. @wat00 irradiated amorphous D$_2$O ice at 12 K with UV photons and observed substantial amounts of D$_2$ after irradiation at $\lambda$ = 126 nm, but very little at $\lambda$ = 172 nm. In the experiments by @yab06 H atoms were found to desorb from the ice after UV irradiation at $\lambda$ = 157 nm and $\lambda$ = 193 nm. There are also a few reports on two- and multi-photon excitation of water ice leading to photodesorption of H$_2$O molecules [@nis84; @ber06]. In these cases the photon energies are below the threshold for absorption in the ice, but upon multiple absorptions the excitation energies fall between 9 and 10 eV, between the first and second absorption band in ice. Clearly, there is a need for more quantitative information on the processes induced by UV photons in ices, even for the simplest cases such as pure water ice.
We present here the results of the first theoretical study of the dissociation of H$_2$O molecules in pure water ice following absorption by UV photons. In addition to providing probabilities for desorption to be used in astrochemical models, these simulations provide insight into the mechanisms leading to desorption as well as the movements of the energetic photoproducts in the ice before they become trapped. In Sect. 2, we present the methods used in this study, in Sect. 3 the main results, and in Sect. 4 a short discussion and astrochemical implications. In Sect. 5 the results are summarized and some concluding remarks are given.
Method
======
All our calculations have been performed using classical Molecular Dynamics (MD) methods [@all87] with analytical potentials. Details of the computational procedure have been described in @and06; here only a brief outline of the methods will be presented.
Amorphous water ice
-------------------
To create an amorphous ice slab, the procedure outlined in @alh04a was used. In brief, a slab of 8 bilayers (16 monolayers) of crystalline ice was first created consisting of a cell containing 480 H$_2$O molecules. The cell has the dimensions $x$: 22.4 [Å]{}, $y$: 23.5 [Å]{}, and $z$: 29.3 [Å]{}. Periodic boundary conditions are applied in the $x$- and $y$-directions, the $z$ coordinate being parallel to the surface normal. Thus an infinite ice surface is created. The H$_2$O molecules are treated as rigid rotors and their interactions are governed by the TIP4P potential [@jor83], which describes the interaction as a sum of pair interactions (electrostatic and Lennard-Jones potentials). The two bottom bilayers are kept fixed to simulate bulk ice and the molecules in the other 6 bilayers are allowed to move without any dynamical constraints other than that they remain rigid. The dynamics are at all times governed by classical Newtonian mechanics. To force the transition to amorphous ice, the surface is initially allowed to equilibrate for 5 ps at 10 K but then the temperature is increased to 300 K using a computational equivalent of a thermostat [@ber84]. In this way the top bilayers form a liquid. The system is left to equilibrate for 100 ps after which it is rapidly cooled to 10 K. Then it is once again equilibrated for 100 ps. The resulting amorphous ice structure most closely resembles the structure of compact amorphous ice obtained experimentally and is thought to be representative of the structure of interstellar water ice [@alh04a; @alh04b]. See Fig. 4 of @alh04a and Fig. 2 of @and06 for images. It does not exhibit the microporous structure that is obtained in vapor deposited ice [@may86; @kim01a; @kim01b]. Given the dimensions of the simulation cell such a structure is simply not possible to obtain, since the pores should have a size on the same order as the cell we use. However, we believe that it is a good representation of an amorphous ice surface on a local scale, whether that be at the “outside" of the surface or inside a void deeper in the ice. Implications of this for the obtained results are discussed in Sect. \[DiscAstro\].
In the rest of the paper we will discuss the depth into the ice in terms of [*monolayers*]{}. To avoid confusion our definition of monolayer is the thickness of ice corresponding to half a crystalline bilayer, i.e., if the ice were crystalline each bilayer would consist of two monolayers. For ease of definition the monolayers have been taken to be divided according to the $z$ values of the centers-of-mass of the molecules, e.g., the 30 molecules with the largest values of $z$ constitute the top (“first") monolayer.
Initial conditions
------------------
Once the ice surface is set up, one H$_2$O molecule is chosen to be photodissociated. This molecule is then made completely flexible and its intramolecular (internal) interactions are governed by an analytic potential energy surface (PES) for the first electronically excited state (the Ã$^1{\mathrm B}_1$ state) of gas-phase H$_2$O based on high-quality *ab initio* electronic structure calculations [@dob97]. This excited potential is fully repulsive so that absorption into this state leads to dissociation of the H$_2$O molecule into H + OH. The intermolecular interactions of the excited state H$_2$O with the surrounding H$_2$O molecules are governed by specially devised partial charges for describing the electrostatic interactions. In short, a charge of -0.2$e$ is put on the O atom and charges of +0.1$e$ on the H atoms. This gives a smaller dipole moment than that of the ground state H$_2$O potential. The effect of this is that a less favorable interaction is obtained with the surrounding H$_2$O molecules, giving higher excitation energies than with the uncorrected gas-phase potential energy surface. This leads to the blue-shift of about 1 eV of the ice UV spectra seen in Fig. \[FigSpect\], which agree well with the first UV absorption band in amorphous and crystalline ices. If the ground state partial charges were to be used for the excited state the excitation spectrum would coincide with the gas-phase UV spectrum. For more details on the potentials see @and06. Similar procedures using potential energy surfaces for higher excited states (B, C, etc.) should in principle give reasonable representations of higher-lying absorption bands in water ice.
The initial internal coordinates and momenta of the atoms in the selected molecule are sampled by a Monte Carlo procedure using a semi-classical (Wigner) phase-space distribution [@sch93] that has been fitted to the ground-state vibrational wave function of H$_2$O [@vanh01]. This procedure gives initial conditions that are very similar to those found in fully quantum mechanical methods and has been shown to work well for the description of photodissociation processes of gaseous molecules. The transition dipole moment function, which governs the strength of the absorption, is taken from the calculation for gaseous H$_2$O by @vanh00.
Dissociation of molecules in the top six monolayers has been considered. For each monolayer all 30 molecules have been dissociated, one molecule at a time. For each molecule 200 configurations and momenta were sampled from the Wigner distribution. This gives 6000 trajectories per monolayer and 36000 trajectories in total.
Calculation of spectra
----------------------
The excitation energy is computed by taking the energy difference between an ice slab with an excited state H$_2$O and one with a ground state H$_2$O (with the same coordinates). Each excitation is assigned a weight calculated as the square of the coordinate-dependent transition moment. By summing the weights of the excitation energies binned in 0.05 eV-wide energy intervals, “intensities" are obtained. Taken together these intensities form a UV absorption spectrum for the ice. The monolayers 5 to 6 were found to be converged to a “bulk behavior" [@and05] and could therefore be used to compare the calculated spectra to experimental data. The gas-phase spectrum presented in Sect.\[ResDisc\] was obtained using the same intramolecular potential surfaces as above, but without the surrounding molecules and with 1000 sampled configurations.
Dynamics of the dissociating molecule
-------------------------------------
After putting the molecule in the excited state, the dissociating trajectory is integrated with a timestep of 0.02 fs. A maximum time of 20 ps has been used before terminating the trajectory. Most of the trajectories (99.6%) were terminated before that because the system was found in one of the final outcomes (see Sect. \[ResDisc\]) with negligible probability of transforming into a different state.
When the excited H$_2$O dissociates, the intermolecular interactions are smoothly switched into separate interactions between the photoproducts and water ice, i.e., H-H$_2$O and OH-H$_2$O potentials. All details of the potentials and the functions used to switch between different potentials are given in @and06. The switching functions connect the partial charges, the dispersion interactions and repulsive potentials between the H$_2$O-H$_2$O potentials and the OH-H$_2$O and H-H$_2$O potentials affecting the dissociating molecule. The switches are functions of the OH distances ($R_{\mathrm{OH}}$) within this molecule and will give the interaction parameters as continuous functions in the range 1.1 – 1.6 Å in $R_{\mathrm{OH}}$. The intramolecular potential is switched to the ground-state PES, which allows H and OH to recombine to form H$_2$O. This switch is done in an analogous way as for the intermolecular interactions above, but here the range of $R_{\mathrm{OH}}$ where the switch is made is 3.0 – 3.5 Å. In this range the excited-state and ground-state PES are near-degenerate, so a high transition probability between the two states is quite probable. Once $R_{\mathrm{OH}}$ becomes larger than 3.5 Å the system will remain on the ground-state PES, even if $R_{\mathrm{OH}}$ again becomes smaller than 3.5 Å. This is what allows for recombination of H and OH. The intermolecular interactions for the recombined ground state H$_2$O with the surrounding H$_2$O molecules are taken from the TIP3P potential [@jor83].
A slightly different stop criterion has been used in these calculations compared to the results presented previously [@and05; @and06]. When an H atom or OH is accommodated to the ice surface (“trapped") the trajectory is run until its translational energy equals $k_{\rm B} T$ or lower [*and*]{} the binding energy to the surface is 0.02 eV (H atom) or 0.1 eV (OH) or stronger. In the older version of the code, the stop criterion was based on the kinetic energy of the individual [*atoms*]{} in relation to the potential energy [@and06]. The introduction of the new termination scheme led to a reduction by about 50% of the number of trajectories exceeding 20 ps.
Although most of the results presented here focus on amorphous ice, calculations have been performed for crystalline ice as well for comparison. Details can be found in @and06.
Results and Discussion {#ResDisc}
======================
Ice UV absorption spectrum
--------------------------
As presented in Fig. \[FigSpect\], our calculated spectra of the first UV absorption bands in amorphous and crystalline ice match very well the experimentally obtained spectra, both in general shape as well as in the peak and threshold energies. The calculated gas-phase spectrum of the first absorption band shown in Fig. \[FigSpect\] also matches the experimental peak energy (7.4–7.5 eV) quite well. This is naturally to be expected, since the potential surfaces used are based on very high-quality *ab initio* calculations of the energy. The success in reproducing the measured spectra leads us to believe that the amount of excess energy released into the ice is basically correct.
The ice spectrum is blue-shifted with respect to that of gaseous H$_2$O and has significant cross section only in the 7.5–9.5 eV range. Thus, the photodesorption probabilities computed here are appropriate for photons in the 1300–1500 Å range. Dissociation of H$_2$O can also occur following absorption into higher excited states (e.g., the equivalent of the B state of gaseous H$_2$O) but these generally contribute less than 20% of the total absorption in a dense cloud.
Photoprocesses and desorption probabilities {#PhotoDesProb}
-------------------------------------------
### Overall probabilities {#OverProb}
Photodissociation of a water ice molecule can have several outcomes, with the H and OH photoproducts either becoming trapped in the ice, recombining back to an H$_2$O molecule, or desorbing from the ice surface. Fig. \[FigMainOut\] shows the probabilities (as fractions of the number of absorbed photons) of the main processes as functions of how deep into the ice the dissociating molecule initially is located. Note that these probabilities are given per [*absorbed*]{} UV photons and [*not*]{} per incident photon. In the first monolayer the dominant outcome is that the hydrogen atom desorbs and the OH radical is trapped in or on the ice. The probability of this event drops rapidly from about 0.9 in the first monolayer to just over 0.1 in the sixth monolayer. This reflects the effect of the ice on the motion of the H atom. At the surface there are very few molecules to stop the H atom from desorbing, but starting from deeper into the ice there are more obstacles on the way to the gas phase.
For the same reasons the probabilities of the other two major outcomes steadily increase as one moves deeper into the ice. These are the events when either both H and OH become separately trapped in the ice or when H and OH recombine to form H$_2$O to subsequently become trapped in the ice. Except for the top two monolayers the probabilities of these two events are roughly equal and in the sixth monolayer the probabilities are up to 0.4. The reason for the lower probability of the H$_2$O molecule being recombined and trapped in the top monolayers can be understood from the open structure of the uppermost layers, which more easily allows for the photofragments to escape the region of the ice where they were initially formed.
ML H atoms OH H$_2$O
---- --------- -------------------- --------------------
1 0.92 0.024 7.3$\times10^{-3}$
2 0.70 0.015 7.7$\times10^{-3}$
3 0.51 4.0$\times10^{-3}$ 4.8$\times10^{-3}$
4 0.30 0.00 3$\times10^{-4}$
5 0.21 0.00 3$\times10^{-4}$
6 0.12 0.00 0.00
: Total probabilities of H atom, OH, and H$_2$O desorption (per absorbed UV photon) as functions of monolayer[]{data-label="TabDes1"}
The probability of photodesorption of H$_2$O is seen to be low compared with the above processes, 0.7% in the top layer and 0.8% in the second layer, and then decreases with distance from the surface (see Table \[TabDes1\]). However, this is only considering the desorption of [*intact*]{} H$_2$O molecules. If one is interested in the removal of H$_2$O from the surface without considering what enters the gas phase, the dominant mechanism in the top two monolayers is actually desorption of separate H and OH fragments. Desorption of OH is in most cases accompanied by the desorption of an H atom, but a minor fraction of OH desorption occurs with the H atom being trapped in the ice (see Sect. \[MechProb\]). In summary, the desorption probabilities of OH and H$_2$O are about 2 orders of magnitude lower than that of H atoms with OH desorption being about twice as probable as H$_2$O desorption if one sums over the probabilities from all monolayers (see also Sect. \[DiscAstro\]).
### Mechanisms and their probabilities {#MechProb}
Analysis of the trajectories shows that there are three distinct mechanisms for H$_2$O removal (see Fig. \[FigMovie\]). Note that these snapshots are taken from calculations on crystalline ice [@and06] for ease of visualization: (a) An H atom released from photodissociation of H$_2$O is able to transfer enough momentum to one of the other H$_2$O molecules to “kick" it off the surface, (b) H and OH recombine to form H$_2$O and subsequently desorb, and (c) the H and OH both desorb from the surface separately. In Table \[TabDes2\] the absolute probability of H$_2$O desorption is given along with the relative probabilities of the three different mechanisms for each monolayer. With the additional mechanism (c) the probability of H$_2$O removal is 2.7% and 1.9% per absorbed UV photon in the first and second monolayer, respectively.
[c c c c c]{} ML & H$_2$O$^a$ & (a) & (b) & (c)\
& desorption & H$_2$O& H$_2$O& H + OH\
& probability & intact & intact & fragments\
1 & 0.027 & 0.10 & 0.17 & 0.73\
2 & 0.019 & 0.16 & 0.24 & 0.60\
3 & 7.0$\times10^{-3}$ & 0.43 & 0.26 & 0.31\
4 & 3$\times10^{-4}$ & 1.00 & 0.00 & 0.00\
5 & 3$\times10^{-4}$ & 1.00 & 0.00 & 0.00\
In the top two monolayers the direct desorption of H and OH fragments is the dominant desorption mechanism, but in the third layer the three distinct desorption mechanisms (“a", “b", and “c") are roughly equally probable. Following UV absorption in monolayers 4 and 5 only the indirect “kick-out" mechanism is effective. In Fig. \[FigDetOut\] the probabilities of all mechanisms of desorption of H atoms, OH radicals, and H$_2$O molecules are presented for the top five monolayers. The desorption of OH is possible either together with the H atom as shown above or separately with the H atom remaining trapped. The former mechanism is found to have a higher probability. In total, the desorption of OH is about a factor of 2 more probable than the desorption of H$_2$O. Below the third monolayer the released OH radicals do not have sufficient kinetic energy to make it to the top of the surface [*and*]{} desorb. The rightmost three categories constitute a further division of the category of indirect H$_2$O desorption. Here “H$_2$O indirect desorption" only refers to the cases where a molecule is kicked out by an H atom, which subsequently remains in the ice. The category “H$_2$O desorption induced by recombination" refers to the rare occurence where it is the excess energy from the recombination of the H and OH fragments that kicks the molecule off the surface. The case “H + H$_2$O desorb" refers to when the H atom kicks the molecule off the surface and subsequently desorbs itself. This last category dominates the indirect desorption in the first two monolayers and also consitutes the maximum amount of matter that has been observed to desorb following photoexcitation. For H$_2$O photoexcited below the fifth monolayer there is no evidence of photodesorption of H$_2$O molecules.
\[FigDetOut\]
The division into monolayers in Table \[TabDes2\] refers to where the photoexcited H$_2$O is situated. The H$_2$O molecules that actually desorb upon being expelled by an H atom or recombining H$_2$O molecule all originate in monolayers 1 (84%) and 2 (16%). In most of these cases it is not only the transfer of momentum that is effective, but also the repulsive interaction from the photoexcited molecule, which most often is in the near vicinity of the desorbing molecule. To illustrate this one can consider the lowering in binding energy of the molecule about to be desorbed. In monolayer 1 the average binding energy of all molecules is 0.9 eV (calculated using the TIP4P potential) and of the desorbing molecules prior to excitation it is 0.8 eV. However, the photoexcitation lowers the binding energy of these molecules by on average 0.3 eV. About 25% of the desorbing molecules do not have their binding energy significantly lowered by excitation, but are kicked out solely by momentum transfer. If these molecules are excluded then the binding energy is on average lowered by 0.4 eV. In the second monolayer only about 10% of the desorbing molecules do not have their binding energy significantly lowered. There the average binding energy is 1.1 eV and this is lowered by 0.3 eV on average upon photoexcitation.
There is no sign of molecules being electronically excited and then desorbing intact [*directly*]{}, i.e., expelled by the repulsive interaction of the excited state molecule with its surroundings. The molecules dissociate very quickly (on the order of 10 fs) and that is not sufficient time for the molecule to desorb before it is dissociated. As discussed above the photofragments can however recombine and [*then*]{} subsequently desorb as H$_2$O.
### Effects of product energies {#EffProdEn}
The H atoms that are released have an average energy of 1.5-2.5 eV depending on the excitation energy and to a lesser extent in which monolayer they originate (see Fig. \[FigEtvvEexc\]). As can be seen the average initial H atom translational energy increases with increasing excitation energy, but above an excitation energy of 9 eV it drops to somewhat lower energies. The average vibrational energy of OH has a minimum value of 0.3 eV around $E_{\mathrm{exc}}$ = 8 eV but increases strongly to about 2 eV at $E_{\mathrm{exc}}$ = 9.5 eV. This implies that at lower excitation energies the vast majority of the OH molecules are formed in the vibrational ground state, since the experimental zero-point energy of OH is 0.23 eV [@hub79] (see also @and06). When the excitation energy is increased above 9 eV, large fractions of vibrationally excited OH are produced. The average initial translational energy of OH is only weakly dependent on excitation energy and lies around 0.2 eV with only a slight increase with increasing $E_{\mathrm{exc}}$.
In Fig. \[FigOutvEtH\] the H atom desorption probability is plotted alongside the probability of H and OH both becoming trapped as functions of initial H atom translational energy. The plotted probabilities are somewhat higher than they should be because the probability of recombination of H$_2$O has been excluded in the set of outcomes. The reason for this is that during recombination the translational energy of the H atom becomes very high. If recombination occurs immediately after dissociation it is quite difficult to distinguish the maximum translational energy the H atom normally would have after photodissociation and the maximum translational energy it gets during recombination. However, the trend is clear that the H atom desorption probability increases with increasing initial translational energy, as one would intuitively expect. The reason for the unexpectedly high desorption probability at $E_{\mathrm{trans,i}}$(H) = 0.7 eV is not quite clear and it could simply be an effect of insufficient sample size, given that the error bars are fairly large.
Similarly, the desorption probability of OH has been plotted in Fig. \[FigOutvEtOH\] as function of initial OH translational energy in the top three monolayers. Also in this case the desorption probability increases with increasing initial kinetic energy. The effect is much stronger than for the case of H atoms, which reflects the much stronger binding energy of OH to its surroundings compared to that of the H atom. Interestingly, if the desorption probability is weighted with the initial distribution of translational energies, the total OH desorption probability (for the top three monolayers) increases rapidly at 0.2 eV (the average initial translational energy) and attains a basically constant value of 0.1% for all energies above that.
The dependence of the indirect H$_2$O desorption probability on the translational energy of the H atom is found to be weak (see Fig. \[FigDesH2OvEtH\]). There is not much evidence of any variation with translational energy and the desorption probability is fairly constant at 0.1% (averaged over the top six monolayers) over the whole energy range. It would be natural to expect that there could be a strong dependence on translational energy, but as discussed above the desorption is in most cases a combined effect of a repulsive force from the excited molecule, a lowered binding energy, and the momentum transfer from the H atom. Apparently, this allows also H atoms with relatively low translational energies to kick out H$_2$O molecules.
\[FigDesH2OvEtH\]
### Dependence on photon energy {#DepPhoEn}
Considering the desorption probabilities as functions of the excitation energy (Fig. \[FigDesPvEexc\]) it is interesting to note that H atom desorption becomes [*less*]{} probable with increasing excitation energy (0.8 at 7.3 eV and 0.3 at 9.5 eV). Since it was shown that the average initial translational energy mainly increases with increasing excitation energy (Fig. \[FigEtvvEexc\]) and that the desorption probability increases with increasing translational energy (Fig. \[FigOutvEtH\]) this seems like a paradox. The simple explanation of this behavior is that the lower excitation energies dominate in the top monolayers while the more energetic UV photons are mainly absorbed towards the bulk of the ice [see, e.g., Fig. 5 of @and05]. The desorption probability summed over the whole excitation energy range decreases rapidly with depth into the ice (Table \[TabDes1\]) and therefore this unexpected behavior is found. The desorption of OH seems to increase with increasing excitation energy. This is a reflection of the fact that even though the [*average*]{} initial translational energy of OH varies only slightly over the excitation energy range (Fig. \[FigEtvvEexc\]), there is a high-energy tail of the OH translational energy distribution (see Fig. \[FigOutvEtOH\]) that becomes larger with higher excitation energies. The desorption probability of H$_2$O does not show strong dependence on excitation energy, but considering the rather large error bars some energy dependence cannot be entirely ruled out.
\[FigDesPvEexc\]
Mobility of photoproducts {#MobProd}
-------------------------
The H atoms produced in the photodissociation event are found to be quite mobile in the ice. On average the H atoms that become trapped move 8 [Å]{} from their original locations. In extreme cases distances over 70 [Å]{} are recorded. The OH radicals formed in the ice move only about 1 [Å]{} with maximum distances moved of 5 [Å]{}. However, OH radicals formed from photodissociation in the top three monolayers are able in some cases to move tens of [Å]{} *on top* of the surface (up to more than 60 [Å]{}). The fact that some of the photofragments are able to move large distances implies an increased probability of reactions with other species in or on the ice, than if they would remain in the immediate vicinity of their point of origin. For more details see @and06.
Comparison to experiments {#CompExp}
-------------------------
In the experiments by @yab06 on UV irradiation of polycrystalline and amorphous ices at 100 K absorption at $\lambda$ = 157 nm ($E_{\mathrm{exc}}$ = 7.9 eV) was found to result in H atom desorption. The translational energy distribution of the desorbing atoms was observed to consist of three components at 0.61 eV, 0.081 eV, and 0.014 eV, respectively. The lowest-energy component seems to consist of atoms that have thermalized prior to desorption. Thermal desorption occurs on a time scale that is likely much longer than would be feasible to do with molecular dynamics simulations. Therefore, it is not likely that we would see this third component in our calculations, but the other two should be possible to reproduce. In Fig. \[FigTrans79\] the calculated distributions of initial translational energies and desorption energies of the released H atoms following excitation at 7.9 eV are shown. It is seen that the initial translational energy has a peak at around 1.9 eV and the desorption energy peaks at about the same energy. This clearly is much higher than found in the experiments, so it seems the energy of the desorbing H atoms is overestimated in our calculations. There could be three explanations to this behavior: (i) either the H atoms lose more energy prior to desorption or (ii) they are initially formed with less translational energy or (iii) a combination of these two effects. The possible sources of loss of highly energetic H atoms in the ice that cannot be treated by our calculations are: (a) the loss of energy by excitation of intramolecular modes in collisions with H$_2$O molecules and (b) reactions with H$_2$O molecules to form, e.g., H$_2$ and OH. For a thorough discussion see @and06. Since we cannot directly tell how effective mechanism (i) is it is hard to speculate how much the desorption energy distribution is cooled through energy transfer and reaction. It is easier to speculate about mechanism (ii), since it is possible to monitor the dependence of the average desorption energy on the initial translational energy. If the initial translational energy is around 1 eV the average desorption energies lie in the range 0.4–0.8 eV, which would be in much better accord with the experimentally measured desorption energies. If this is the most important mechanism for cooling the desorption energy distriubution, then at $E_{\mathrm{exc}}$ = 7.9 eV the initial H atom translational energies are overestimated by roughly 1 eV. As discussed in Sect. \[PhotoDesProb\] this would probably have little importance for the H$_2$O desorption probability (Fig. \[FigDesH2OvEtH\]). However, the desorption probability of H atoms could be somewhat lower than predicted in our calculations, but it is still quite likely to be of the same order of magnitude (Fig. \[FigOutvEtH\]). If the average H atom translational energy is overestimated it is also likely that the OH translational energy is somewhat overestimated. Considering the weak dependence on excitation energy (Fig. \[FigEtvvEexc\]) the [*average*]{} OH translational energy might not be highly overestimated, but the high energy tail could be smaller, meaning that less OH radicals desorb than predicted here. If the initial kinetic energies of the photofragments are overestimated, the only possibility to account for the blue shift of the excitation energy is that the intermolecular repulsion is underestimated. This could have as an interesting effect that the indirect desorption of the surrounding molecules could be [*underestimated*]{}, since they would experience an even larger repulsive force from the excited molecule than predicted by our calculations. This remains to be investigated.
The high probabilities of H atom desorption is also in accordance with the finding of @ger96 that their UV-irradiated water ice was most likely depleted of H atoms, since a large amount of oxygen rich products was found. Our results on mobility of the released photofragments [Sect. 3.3, @and06] have been supported by recent experiments by @ell07, who observed an average separation of H and OH of 7$\pm$2 $\AA$ immediately following photodissociation of H$_2$O upon UV irradiation of liquid water. This is in excellent agreement with our calculated value of 8 $\AA$ for the average distance from the site of photodissociation of the H atoms in amorphous ice. Even though the temperatures are quite different in the two cases, it is expected that liquid water and compact amorphous ice are quite similar on the short time scales during which photodissociation takes place.
A direct comparison with the water ice photodesorption results by @wes95a [@wes95b] is difficult to make since they used Lyman-$\alpha$ radiation, which leads to excitation to a higher absorption band than considered here [@kob83]. Their findings of basically no photodesorption of intact H$_2$O at low temperatures in the limit of single-photon absorption can therefore be neither refuted nor confirmed by our results. However, the detection of desorbing H$_2$ and O$_2$, and possibly OH and H$_2$O$_2$ actually agree with the present results [@wes95a] (see below). In the experiments of Öberg et al. (submitted to ApJ) the photodesorbing material is detected in the form of OH, H$_2$O, H$_2$, and O$_2$. This is the first time a positive detection of OH photodesorption has been reported. The detection of OH agrees nicely with our simulations (see also Sect. \[DiscAstro\]).
It is important to bear in mind that even though only H, OH, and H$_2$O would desorb upon absorption of a *single* UV photon, in the experimental setup a large amount of UV photons may be absorbed in the ice during a relatively short time interval. This makes it possible to produce significant amounts of H and OH in the ice, and possibly also O atoms from for instance the photodissociation of the formed OH radicals. Given that thermal diffusion is effective in moving these reactive species into close contact, all the aforementioned desorbing species can be accounted for through recombination followed by desorption.
Discussion and astrophysical implications {#DiscAstro}
=========================================
Based on the results presented in this paper some important conclusions can be drawn on the possible outcomes of UV irradiation of water ice in interstellar environments. First, it is important to realize that by far the most likely species to desorb are H atoms, followed by OH radicals. In addition, when there is removal of H$_2$O from the surface it seems likely that most of it comes off in the form of separate H and OH fragments. Therefore, there is not a one-to-one correspondence between H$_2$O molecules removed from a surface and H$_2$O appearing in the gas phase.
So far, this paper has been concerned with probabilities of photoinduced processes following absorption of one UV photon in a specific layer in the ice. However, not all incident UV photons are absorbed by molecules in the top six monolayers. To estimate desorption probabilities per *incident* photon rather than per *absorbed* photon one needs information about the absorption cross section. In our semiclassical simulations we are not able to calculate the absolute absorption cross section. However, @mas06 have measured the absorption cross section of water ice at 25 K and found the peak absorption cross section in the first absorption band to be about $6\times10^{-18}$ cm$^2$ at an excitation energy of 8.61 eV. This leads to an absorption probability of 0.007 photons ML$^{-1}$ (see Appendix \[AppA\] for an outline of this calculation).
From the above estimate of the absorption probability it is possible to calculate photodesorption probabilities per incident UV photon. This is done by weighting the desorption probabilities per absorbed photon for each monolayer by the absorption probability for the specific monolayer with the absorption probabilities in any upper monolayers subtracted from the incoming photon flux. Using the information in Table \[TabDes2\] one arrives at a probability of removal of H$_2$O from the ice of $3.7 \times 10^{-4}\ \mathrm{photon}^{-1}$. About 60% of the removed H$_2$O comes off in the form of H + OH, 20% desorbs as recombined H$_2$O, and the remaining 20% consists of H$_2$O “kicked" out from the surface. The total photodesorption yield of intact H$_2$O molecules is $1.4 \times 10^{-4}\ \mathrm{photon}^{-1}$. For OH desorption the probability is $3 \times 10^{-4}\ \mathrm{photon}^{-1}$, which includes both desorption with and without H atoms. The H atom photodesorption probability is relatively high, $0.02\ \mathrm{photon}^{-1}$. The above ratios of photodesorption of OH and H$_2$O are in excellent agreement with the recent experiments by Öberg et al. (submitted to ApJ), which inferred that roughly equal amounts of OH and H$_2$O photodesorb at low surface temperature (18 K). They obtain a total photodesorption yield of about $1.3 \times 10^{-3}$ in the low-temperature limit, which is about 3 times higher than what is found from our simulations. Given the experimental uncertainties and the approximations made in the simulations this can be considered as good agreement. Commonly used estimates of H$_2$O photodesorption probabilities in the range $1\times 10^{-4}$–$3.5\times 10^{-3}$ have been used to model different environments in agreement with observations [@ber95; @wil00; @sne05; @dom05; @ber05]. Our results indicate that these estimates are reasonable. However, the finding that less than half of the desorbed material leaves the grain in the form of intact H$_2$O molecules is something that should be included in the models.
The results presented here are only for photoinduced processes that are followed until the photoproducts desorb or are thermalized within the ice. This is all happening on a picosecond time scale. For longer time scales there is the possibility of thermal desorption of especially the H atoms. These are relatively weakly bound to the ice surface and it is quite probable that some fraction of the released H atoms desorb after thermalization in the ice. This contribution to the desorption probability is therefore not included in the above estimate.
It is interesting to discuss the possible effects of the overall morphology of the ice surface. It is clear that the vast majority of water ice surfaces in the interstellar medium are amorphous [@hag81]. However, it is debated whether the ice is mostly porous, as found in vapor deposited ice [@may86; @kim01a; @kim01b], or more compact [@fra04; @pal06]. A complicating factor is that most likely the ice is formed through chemical reactions on grains rather than accretion of H$_2$O molecules from the gas phase [@one99]. Therefore, the exact ice morphology that results from such a chemical build-up of the ice is not yet clear (see however @cup07). If a porous ice surface is subjected to UV irradiation one could have release of H, OH, or H$_2$O into a void in the ice rather than directly into the gas phase. This is inferred to happen in the experiments by @yab06 where a large component of the desorbing H atoms released after UV irradiaton of amorphous ice are thermalized, likely due to H atoms being accommodated within a void in the ice and then desorbing thermally. Some OH and H$_2$O could photodesorb in a similar way. However, if the path to reach the gas phase from inside a pore is restricted, these species have a high probability of being trapped inside a pore because of their strong attractive interactions with the ice. In practice, the OH and H$_2$O released in this way might not show up in desorption. Therefore, if a porous ice is considered it is necessary to distinguish between the surface area that is directly exposed to the gas phase and that which is within a void with restricted access to the outside. In conclusion, the H atom desorption from a porous ice is likely to be different from that of compact non-porous ice, but the desorption yields of OH and H$_2$O are not necessarily very different in the two types of ice.
The cosmic ray induced UV flux inside dense clouds is about $10^4\ \mathrm{cm}^{-2}\mathrm{s}^{-1}$ [@she04]. For an ice-coated grain with a typical size of 0.1 $\mu$m, this would give an arrival rate of about 1 UV photon per day. The case of a single UV absorption event as described in our simulations therefore gives a realistic picture of photodesorption in dense clouds. In the laboratory, the UV flux is many orders of magnitude higher and multiple absorptions of UV photons within the ice surface in a short time interval may drive secondary reactions of photodissociation products from different H$_2$O molecules. Experiments with different UV flux levels down to low levels will be needed to provide quantitative data on photodesorption yields relevant for astrophysical applications.
As has also been noted in our previous work another important aspect is the release of reactive species into the ice. That would have implications for reactivity in the ice with, e.g., CO that could react with energetic H or OH to form HCO or CO$_2$. This could be one clue to unraveling the mystery of CH$_3$OH and CO$_2$ formation in the interstellar medium. Indeed, formation of CO$_2$ is readily observed when a mixed H$_2$O:CO ice is photolysed [@dhe86; @wat02; @wat07].
Conclusions
===========
We have shown that it is possible to have H$_2$O photodesorption upon UV absorption to the first absorption band in the top five monolayers of an amorphous ice surface. The main mechanisms for this photodesorption are either photodissociation followed by recombination of H and OH and subsequent desorption of the recombined H$_2$O molecule or a “kick-out" of another H$_2$O molecule in the ice by the energetic H atom released from photodissociation or, less likely, by the energy released from a recombined H$_2$O molecule. In most cases, however, removal of an H$_2$O molecule from the ice is in the form of separate H and OH fragments. An estimate of the photodesorption yield per incident UV photon from our calculations agrees well with the H$_2$O photodesorption yields that are commonly used in modeling astrophysical environments.
UV absorption leads in most cases to desorption of H atoms or the trapping of H and OH in the ice either as separate fragments or as recombined H$_2$O. The desorption of H atoms is about 2 or 3 orders of magnitude more probable than desorption of OH and H$_2$O. The OH desorption probability is about twice the H$_2$O desorption probability. The high mobility of H atoms inside the ice and OH radicals on the ice surface will facilitate formation of other molecules such as CO$_2$.
We thank Karin Öberg, Herma Cuppen, and Geert-Jan Kroes for stimulating discussions. Some of the calculations reported here were performed at Chalmers Centre for Computational Science and Engineering (C3SE) computing resources. This research was funded by a Netherlands Organization for Scientific Research (NWO) Spinoza grant \[for one of the authors (E.F.v.D)\] and a NWO-CW Top grant.
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Estimate of absorption probability {#AppA}
==================================
To calculate an absorption probability per monolayer in an interstellar ice surface one needs an estimate of the effective area taken up by one molecule. Since the angle of incidence of the photon is arbitrary, all possible incidence angles have to be taken into account, not only normal incidence. The surface area of our simulation cell is $22.4\ \mathrm{\AA}\ \times\ 23.5\ \mathrm{\AA}\ =\ 526\ \mathrm{\AA}^2\ =\ 5.26\ \times10^{-14}\ \mathrm{cm}^2$. If one assumes a flat surface, the average effective surface area seen by a photon from any incidence angle between 0$^{\circ}$ (normal incidence) and 90$^{\circ}$ (parallel to the surface) is the actual surface area divided by two. This is arrived upon by the following expression:
$$\langle A_\mathrm{eff} \rangle = \frac{\int^{\pi / 2}_0 A \cos\theta \sin\theta d\theta}
{\int^{\pi / 2}_0 \sin\theta d\theta} = \frac{A}{2},$$
with the effective area given by $A_\mathrm{eff} = A\cos\theta$ where $A$ is the actual surface area and $\theta$ is the angle of incidence. The average effective surface area of the simulation cell is 2.63$\times$10$^{-14}$ cm$^2$. Each monolayer consists of 30 H$_2$O molecules. A single molecule will therefore have an average effective area $\langle A_\mathrm{eff}^\mathrm{mol} \rangle = 8.77\times10^{-16}\ \mathrm{cm}^2$. @mas06 measured the peak absorption cross section, $\sigma$, in the first absorption band to be about 6$\times$10$^{-18}$ cm$^2$ around an excitation energy of 8.61 eV. With this value the absorption probability per monolayer becomes $P_\mathrm{abs}^\mathrm{ML} = \sigma / \langle A_\mathrm{eff}^\mathrm{mol} \rangle = 7\times10^{-3}$ (this number may vary somewhat with excitation energy). If one considers an infinitely deep ice surface all photons will be absorbed within the ice. For an ice surface in the interstellar medium consisting of a finite number of layers, the value of $P_\mathrm{abs}^\mathrm{ML}$ may vary depending on the shape of the grain, since the surface is not necessarily flat. However, it is likely that any photon that passes through an ice mantle will be absorbed by the silicate grain, leading to almost total absorption of the incident photons within the grain.
|
---
abstract: |
Two families of orthonormal tapers are proposed for multitaper spectral analysis: minimum bias tapers, and sinusoidal tapers $\{ \vv^{(k)}\}$, where $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, and $N$ is the number of points. The resulting sinusoidal multitaper spectral estimate is $\hat{S}(f)=\frac{1}{2K(N+1)}
\sum_{j=1}^K |y(f+\frac{j}{2N+2}) -y(f-\frac{j}{2N+2})|^2$, where $y(f)$ is the Fourier transform of the stationary time series, $S(f)$ is the spectral density, and $K$ is the number of tapers. For fixed $j$, the sinusoidal tapers converge to the minimum bias tapers like $1/N$. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the $j$th taper is simply $\frac{1}{N}$ centered about the frequencies $\frac{\pm j}{2N+2}$. Thus the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, $\int_{-w}^w |V(f)|^2 df$, of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian tapers. In contrast, the Slepian tapers can have the local bias, $\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$, much larger than of the minimum bias tapers and the sinusoidal tapers.
author:
- |
Kurt S. Riedel and Alexander Sidorenko\
Courant Institute of Mathematical Sciences, New York University\
New York, New York 10012-1185
date: 'EDICS: SP 3.1.1'
title: ' Minimum bias multiple taper spectral estimation [^1] '
---
Introduction
============
We consider a stationary time series, $\{ x_n, n=1\ldots N\}$ with a spectral density, $S(f)$. A common estimator of the spectral density is to smooth the square of the discrete Fourier transform (DFT) locally: $$\label{I1}
\hat{S}(f)=\frac{1}{(2L+1)N}
\sum_{j=-L}^L |y(f+\frac{j}{N})|^2,$$ where $y(f)$ is the Fourier transform (FT) of the stationary time series: $y(f) \; \equiv \; \sum_{n=1}^N x_n e^{-i2\pi nf} \;$. Since (\[I1\]) is quadratic in the FT, $y(f)$, it is natural to consider a more general class of quadratic spectral estimators. We examine quadratic estimators where the underlying self-adjoint matrix has rank $K$, where $K$ is prescribed. Using the eigenvector representation, the resulting quadratic spectral estimator can be recast as a weighted sum of $K$ orthonormal rank one spectral estimators. This class of spectral estimators was originally proposed by Thomson [@T90] under the name of multiple taper spectral analysis (MTSA). We refer the reader to [@B85; @MS90; @PLV87; @PW93; @RST94; @T82; @T90] for excellent expositions and generalizations of Thomson’s theory.
In MTSA, a rank $K$ quadratic spectral estimate is constructed by choosing an orthonormal family of tapers/spectral windows and then averaging the $K$ estimates of the spectral density. In practice, only the Slepian tapers (also known as discrete prolate spheroidal sequences [@S78]) are routinely used for MTSA.
In the present paper, we propose and analyze two new orthonormal families of tapers: minimum bias (MB) tapers and sinusoidal tapers. The MB tapers minimize the local frequency bias, $\int f^2 |V(f)|^2 df$, subject to orthonormality constraints, where $V(f)$ is the DFT of the taper. For continuous time, the MB tapers have simple analytic expressions. The first taper in the family is Papoulis’ optimal taper [@P73]. For discrete time, the MB tapers satisfy a selfadjoint eigenvalue problem and may be computed numerically.
In the case of discrete time, we define the $k$th sinusoidal taper, $\vv^{(k)}$, as $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, where $N$ is the sequence length. The sinusoidal tapers are an orthonormal family that converge to the MB tapers with rate $1/N$ as $N\rightarrow\infty$. These results are given in Section \[MBT\]. Section \[CSL\] compares the local bias, $\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$, and the spectral concentration, $\int_{-w}^w |V(f)|^2 df$, of the MB tapers, the sinusoidal tapers and the Slepian tapers.
In Section 5, we show that the quadratic spectral estimator which minimizes the expected square local error is weighted multitaper estimate using the MB tapers. A local error analysis is given and the optimal number of tapers is determined. At frequencies where the spectral density is changing rapidly, fewer tapers should be used. In Section \[KS\], we show that kernel smoother spectral estimates [@GR57; @P58] are multitaper estimates and we show that smoothing the logarithm of the multitaper estimate significantly reduces the variance in comparison with smoothing athe logarithm of a single taper estimate. We also describe our data adaptive method for estimating the spectrum. In Section \[DATA\], we apply our spectral estimation techniques to real data and show that our tapers outperform the Slepian tapers whenever a variable bandwidth is needed. In the Appendix, we show that the leading principal components of kernel smoother spectral estimates resemble the MB tapers.
Quadratic Estimators of the Power Spectrum
==========================================
Let $N$ discrete measurements, $x_1,x_2,\ldots ,x_N$, be given as a realization of a stationary stochastic process. We normalize the time interval between measurements to unity. The Cramer representation of a discrete stationary stochastic process [@GR57; @PW93] is $$x(t) \; = \; \int_{-1/2}^{1/2} e^{2\pi inf} dZ(f) \; ,$$ where $dZ$ has independent spectral increments: $ E[dZ(f)d\overline{Z}(g)] = S(f) \delta (f-g) df dg $. We assume that the spectral density, $S(f)$, is twice continuously differentiable.
The spectral inverse problem is to estimate the spectral density, $S(f)$, given $\{ x_n\}$. As shown in [@B85; @MS90], every quadratic, modulation-invariant power spectrum estimator has the form: $$\label{E4}
\widehat{S}(f) \; = \;
\sum_{n,m=1}^N q_{nm} e^{2\pi i (m-n)f} x_n x_m \; ,$$ where $\Qbf=[q_{nm}]$ is a symmetric matrix of order $N$ and does not depend on frequency. Consider the eigenvector decomposition: $ \Qbf = \sum_{k=1}^K \mu_k \vv^{(k)}\left(\vv^{(k)}\right)^T $, where $K$ is the rank of $\Qbf$, and $\vv^{(1)},\vv^{(2)},\ldots ,\vv^{(K)}$ is an orthogonal system of eigenvectors. The multitaper representation of the quadratic spectral estimator is $$\label{E6}
\widehat{S}(f) \; = \;
\sum_{k=1}^K \mu_k
\left|\sum_{n=1}^N v_n^{(k)} x_n e^{-2\pi inf}\right|^2 \; .$$ In the case $K=1$, estimator (\[E6\]) turns out a [*tapered periodogram estimator*]{}: $$\label{E2}
\widehat{S}_v(f) \; = \;
\left|\sum_{n=1}^N v_n x_n e^{-2\pi inf}\right|^2
\; ,$$ with a taper $\vv=(v_1,v_2,\ldots ,v_N)^T$. If the tapering is uniform (i.e. $v_1=v_2=\ldots =v_N=\frac{1}{\sqrt{N}}$), we name (\[E2\]) the [*periodogram estimator*]{}. The estimator (\[E6\]) is a linear combination of $K$ orthogonal tapered periodogram estimators. In MTSA, $K$ is normally chosen to be much less than $N$. The multiple taper spectral estimate can be thought of as a low rank, “principal components" approximation of a general quadratic estimator. Multiple taper analysis has also been applied to nonstationary spectral analysis [@R93; @Am94].
In practice, one does not begin the analysis with a given quadratic estimator, $\Qbf$. Instead, one usually [*specifies a family of orthonormal tapers $\{ \vv^{(1)},\ldots ,\vv^{(K)} \}$ with desirable properties.*]{} Previously, only the family of Slepian tapers were used in practice. The goal of this article is to introduce other families of tapers.
We define the $k$th spectral window, $V^{(k)}$, to be the FT of the $k$th taper: $$\label{E40}
V^{(k)}(f) \; = \; \sum_{n=1}^N v_n^{(k)} e^{-i2\pi nf} \; .$$ The tapers are normally chosen to have their spectral density localized near zero frequency. We define two common measures of frequency localization.
The [*local bias*]{} of a spectral window $V$ is $\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$. The term “local bias” is used because it is proportional to the leading order term in the bias error of a taper estimate as $N\rightarrow\infty$.
The [*spectral concentration*]{} in band $[-w,w]$ is defined as $\int_{-w}^w |V(f)|^2 df$. The bandwidth, $w$, is a free parameter. The Slepian tapers are the unique sequences which maximize the spectral concentration subject to the constraint that they form an orthonormal family. Detailed analysis of the Slepian sequences is given in [@S78]. We stress that the Slepian tapers depend on the bandwidth parameter, $w$, and that the first $2Nw$ spectral windows are concentrated in the band $[-w,w]$ while the remaining windows are concentrated outside.
Minimum Bias Tapers {#MBT}
===================
Continuous Time Case
--------------------
We consider time-limited signals; the time interval is normalized to $[0,1]$. In the time domain, the taper $\nu (t)$ is a function in ${\cal L}_2[0,1]$ which we normalize to $\int_0^1 \nu^2(t) dt =1$. The functions $\{\sin (\pi kt),\; k=1,2,\ldots \}$ form a complete orthogonal basis on $[0,1]$. (Completeness can be proven by extending $\nu(t)$ to be an odd function on $[-1,1]$ and using the completeness of the complex exponentials on $[-1,1]$. See [@KF].)
Setting $a_k = 2 \int_0^1 \nu (t) \sin (\pi kt) dt$, then $\sum_{k=1}^{K} a_k \sin (\pi kt) dt$ converges to $\nu (t)$ in ${\cal L}_2[0,1]$ as $K\rightarrow\infty$. The taper normalization is equivalent to $ \frac{1}{2} \sum_{k=1}^\infty a_k^2 = 1 $. The Fourier transform of the taper is the complex-valued, spectral window function: $$\label{E8}
V(f) \; = \; \int_0^1 \nu (t) e^{-i2\pi ft} dt \; .$$ $V(f)$ is defined on the frequency domain $[-\infty ,\infty ]$, belongs to ${\cal L}_2[-\infty ,\infty ]$, and satisfies $ \int_{-\infty}^\infty |V(f)|^2 df =
2\pi \int_0^1 \nu^2(t) dt = 2\pi $.
The local bias of a taper spectral estimate [@GR57; @P73; @P58] is $$\label{E20}
E[\widehat{S}(f)] - S(f)
\; = \;
\int_{-\infty}^\infty |V(g-f)|^2 (S(g)-S(f)) dg
\; \approx \;
\frac{S''(f)}{2} \int_{-\infty}^\infty |V(h)|^2 h^2 dh \; .
$$ We consider tapers which minimize the leading order term: $$\begin{aligned}
\label{E30}
\int_{-\infty}^\infty |V(f)|^2 f^2 df
& = &
\int_0^1 \left( \frac{1}{2\pi} \frac{d}{dt}
\sum_{k=1}^\infty a_k \sin (\pi kt) \right)^2 dt
\nonumber \\
& = &
\frac{1}{4} \int_0^1
\left( \sum_{k=1}^\infty a_k k \sin (\pi kt) \right)^2 dt
\; = \; \frac{1}{8} \sum_{k=1}^\infty a_k^2 k^2 \; .\end{aligned}$$ The last expression attains the global minimum when $a_1^2=2,\; a_2=a_3=\ldots =0$. Hence, the leading order term in the bias expression (\[E20\]) is minimal for the taper $\sqrt{2} \sin (\pi t)$ (This result was obtained by Papoulis [@P73]). In [@P73], Papoulis extends $\nu(t)$ to be zero outside of $(0,1)$, and therefore has a Fourier integral representation of $\nu(t)$. We have extended $\nu(t)$ to be periodic and vanish at each integer value. Since $\nu(t)$ is optimized for $t \in [0,1]$, both representations are valid.
Equation (\[E30\]) implies the more general result:
\[T1\] $v_k(t)=\sqrt{2}\sin (\pi kt)$ $(k=1,2,\ldots )$ is the only system of functions in ${\cal L}_2[0,1]$ which satisfy the requirements:
> \(i) $\;\;\;\int_0^1 v_k^2(t) dt = 1$, and
>
> \(ii) $\;\;\;v_k$ minimizes $\int_{-\infty}^\infty |V^{(k)}(f)|^2 f^2 df$ in the subspace of functions orthogonal to $\;\;\;\;v_1,\ldots ,v_{k-1}$.
The kth minimum value is $
\int_{-\infty}^\infty |V^{(k)}(f)|^2 f^2 df \; = \; \frac{k^2}{4} \; .
$
We name $v_k(t)=\sqrt{2}\sin (\pi kt)$ $(k=1,2,\ldots )$ the [*continuous time minimum bias tapers*]{}. The Fourier transform of $v_k(t)$ is $$\begin{aligned}
V^{(k)}(f)
& = &
\frac{e^{-i\pi\left( f-\frac{k}{2}\right) }}{i\sqrt{2}} \left\{
\frac{\sin\left[ \pi\left( f-\frac{k}{2}\right)\right] }
{\pi\left( f-\frac{k}{2}\right)}
- (-1)^k
\frac{\sin\left[ \pi\left( f+\frac{k}{2}\right)\right] }
{\pi\left( f+\frac{k}{2}\right)}
\right\}
\\
& = &
e^{-i\pi\left( f-\frac{k-1}{2}\right) } \cdot
\frac{k^2 \sin\left(\pi f - \frac{\pi k}{2}\right)}
{4\sqrt{2}\pi\left[ f^2 - \left(\frac{k}{2}\right)^2\right] }
\; .\end{aligned}$$ Thus, $|V^{(k)}(f)|$ decays as $f^{-2}$ for large frequencies.
Discrete Time Case
------------------
We now consider the discrete time domain $\{ 1,2,\ldots ,N\}$ with the corresponding normalized frequency domain $[-\frac{1}{2},\frac{1}{2}]$. A taper is a vector, $\nubm = (\nu_1,\ldots ,\nu_N)$, normalized by $\sum_{n=1}^{N} (\nu_n)^2 = 1$. By the same argument as in the previous section, the leading order term of the bias of a taper spectral estimate is proportional to the local bias, $\int_{-\frac{1}{2}}^{\frac{1}{2}} |V(f)|^2 f^2 df$, where the frequency window, $V(f)$, is defined in Eq. (\[E40\]).
\[T2\] For the frequency window of a discrete time taper, $$\int_{-\frac{1}{2}}^{\frac{1}{2}} |V(f)|^2 f^2 df \; = \;
\nubm \Abf \nubm\ ^* \; ,$$ where $\Abf=[a_{nm}]$ with $$a_{nm} \; = \;
\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{i2\pi (n-m)f} f^2 df \; = \;
\left\{ \begin{array}{l}
\frac{1}{12} \;\;\;\; if \;\; n=m \; ; \\
\frac{(-1)^{n-m}}{2\pi^2(n-m)^2} \;\; if \;\; n\neq m \; .
\end{array}
\right.$$
\[C3\] The tapers $\nubm^{(1)},\ \nubm^{(2)},\ldots , \nubm^{(N)}$, defined by the requirements
> \(i) $\;\;\;\sum_{n=1}^{N} (\nu_n^{(k)})^2 = 1$,
>
> \(ii) $\;\;\;\nubm^{(k)}$ minimizes $\int_{-\frac{1}{2}}^{\frac{1}{2}} |V^{(k)}(f)|^2 f^2 df$ in the subspace of vectors orthogonal to $\;\;\;\;\nubm^{(1)} ,\ldots, \nubm^{(k-1)}$,
are the eigenvectors of the matrix $\Abf$ sorted in the increasing order of the eigenvalues. The integral $\int_{-\frac{1}{2}}^{\frac{1}{2}} |V^{(k)}(f)|^2 f^2 df$ is equal to the kth eigenvalue.
We name $\nubm^{(1)},\nubm^{(2)},\ldots, \nubm^{(N)}$ the [*discrete minimum bias tapers*]{}. They can be approximated by the [*sinusoidal tapers*]{}, $\vv^{(1)},\vv^{(2)},\ldots ,\vv^{(N)}$, which are discrete analogs of the continuous time minimum bias tapers. Namely, we define $\vv^{(k)}=(v_1^{(k)},\ldots ,v_N^{(k)})^T$ with $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, $k=1,2,\ldots ,N$.
\[T4\] The sinusoidal tapers, $\vv^{(1)},\vv^{(2)},\ldots ,\vv^{(N)}$, form an orthonormal basis in ${\bf R}^N$ and the local bias of $\vv^{(k)}$ is $\frac{k^2}{4N^2}\left(1 + {\cal O}(\frac{1}{N})\right)$.
\[C5a\] The multitaper estimate (\[E6\]) using $K$ sinusoidal tapers has local bias equal to $\sum_{k=1}^K\mu_k\frac{k^2}{4N^2}+ {\cal O}(\frac{K^2}{N^3})$ and has the following representation: $$\begin{aligned}
\label{E31a}
\hat{S}(f)= \sum_{j=1}^K \frac{\mu_j}{2(N+1)} |y(f+\frac{j}{2N+2}) -y(f-\frac{j}{2N+2})|^2 .\end{aligned}$$ The uniformly weighted estimate, $\mu_k=\frac{1}{K}$, has local bias equal to $\frac{K^2}{12N^2}+ {\cal O}(\frac{K^2}{N^3})$.
From (\[E31a\]), the reason for the low bias of the sinusoidal tapers is apparent: [*the frequency sidelobe from $y(f+\frac{j}{2N+2})$ cancels the sidelobe of $y(f-\frac{j}{2N+2})$.*]{} As a result, the sidelobe of $y(f+\frac{j}{2N+2})$ minus $y(f-\frac{j}{2N+2})$ is much smaller than that of the periodogram.
Our preferred weighting is the parabolic weighting: $\mu_j = C (1 - j^2/K^2)$ because the parabolic weighting minimizes the expected square error in kernel smoothers as $K$ and $N$ tend to infinity. Since the weights decrease smoothly to zero, the resulting estimate is smooth in frequency.
\[C5b\] The uniformly weighted multitaper estimate using $K$ sinusoidal tapers can be computed in ${\cal O}(N\ln N) + {\cal O}(KN)$ operations while the generic multitaper estimate requires ${\cal O}(KN\ln N)$ operations plus the cost of computing the $K$ tapers.
The following result demonstrates that the $k$th spectral window is concentrated on $\left[\frac{k-1}{2(N+1)},\frac{k+1}{2(N+1)}\right] \cup
\left[-\frac{k+1}{2(N+1)},-\frac{k-1}{2(N+1)}\right]$.
\[T7\] The Fourier transform of $\vv^{(k)}$ equals $$\begin{aligned}
V^{(k)}(f)
& = &
\frac{e^{-i\pi \left( (N+1)f-\frac{k}{2}\right)}}{i\sqrt{2(N+1)}}
\left\{\frac{\sin\left[ N\pi\left(f-\frac{k}{2(N+1)}\right)\right]}
{\sin\left[ \pi\left(f-\frac{k}{2(N+1)}\right)\right]}
- (-1)^k
\frac{\sin\left[ N\pi\left(f+\frac{k}{2(N+1)}\right)\right]}
{\sin\left[ \pi\left(f+\frac{k}{2(N+1)}\right)\right]}
\right\}
\\
\\
& = &
e^{-i\pi \left( (N+1)f-\frac{k-1}{2}\right) } \cdot
\frac{\sin\frac{\pi k}{N+1}}{\sqrt{2(N+1)}} \cdot
\frac{\sin\left[ (N+1)\pi f - \frac{\pi k}{2}\right]}
{\sin^2 (\pi f) - \sin^2 \frac{\pi k}{2(N+1)}}
\; .\end{aligned}$$
Thus $|V^{(k)}(f)|=\sqrt{\frac{N+1}{2}}$ for $|f|=\frac{k}{2(N+1)}$, and $|V^{(k)}(f)|\sim\frac{1}{N^{3/2}}$ for $f={\cal O}(1)$. In particular, $|V^{(k)}(f)|\approx\frac{\pi k}{\sqrt{2} N^{3/2}}$ when $f\rightarrow\frac{1}{2}$. In the intermediate frequencies, $\frac{k}{2(N+1)}<f\ll\frac{1}{2}$, $|V^{(k)}(f)|$ decays as $\frac{1}{f^2}$.
Numerical evaluation (see Table 1) shows that for all $k$. The same rate of convergence is observed in $L_{\infty}$ norm: $\left\| \frac{\vv^{(k)}}{\| \vv^{(k)}\|_{L_{\infty}}}
- \frac{\nubm^{(k)}}{\|\nubm^{(k)}\|_{L_{\infty}}} \right\|_{L_{\infty}}
< \frac{k}{2(N+2)}$. ($\|\cdot\|_{L_{\infty}}$ is the supremum norm in the time domain.) Figure 1 plots the envelopes of $|V^{(k)}(f)|^2$ for both the minimum bias and sinusoidal tapers with $N=200,\; k=1$. The spectral energy of both tapers is nearly identical for $|f|<.25$. Near the Nyquist frequency, the spectral energy of the sinusoidal taper is roughly three times larger than that of the minimum bias taper. In the time domain, the sinusoidal tapers are virtually indistinguishable from the MB tapers.
Comparison of Spectral Localizations {#CSL}
====================================
We now compare the local bias and the spectral concentration of three families of orthonormal tapers: minimum bias (MB) tapers, sinusoidal tapers and Slepian tapers. Both the local bias and the spectral concentration of the Slepian tapers depend on the bandwidth parameter, $w$. For properties of the Slepian tapers, we refer the reader to [@PW93; @RST94; @S78; @T82; @T90].
Since the MB tapers minimize the local bias, clearly the sinusoidal tapers and the Slepian tapers have larger local bias. The only question is whether the difference is large or small. Table 2 gives the local bias, $ \sum_{k=1}^K \int_{-1/2}^{1/2} f^2 |V^{(k)}(f)|^2 df$, of the three families of tapers for $N=50$. [*The sinusoidal tapers come within 0.2% of achieving the optimal local bias.* ]{} In contrast, the local bias of the Slepian tapers can be many times larger. We compute the local bias for three different values of the bandwidth, $w$. The general pattern is that the $k$th Slepian taper has roughly the same local bias as the MB taper does when $Nw<k<2Nw$. The ratio of the local bias of the Slepian tapers to that of the MB tapers is smallest at $k\approx 1.2Nw$. As $|k-1.2Nw|$ increases, the local bias rapidly departs from the optimal value.
Table 3 compares the spectral concentration of the tapers for $N=50$ and $w=.08\:$. Both the MB tapers and the sinusoidal tapers are within 1.7% of the optimal value, except for $k=8,9$. Notice that $2Nw=8$. Although the ratio of the spectral concentration, $\int_{-w}^w |V(f)|^2 df$, for the MB and sinusoidal tapers to that of the Slepian tapers is usually very close to one, the ratio of the spectral energy outside of the frequency band $|f|<w$ can be quite large. Thus our conclusions depend on using $\int_{-w}^w |V(f)|^2 df$ and not $1-\int_{-w}^w |V(f)|^2 df$ as the measure of frequency concentration.
Figures 1-3 compare $|V^{(k)} (f)|^2$ of the Slepian and MB tapers for $N=200$. For Figures 1 and 2, we select the Slepian parameter, $w$, equal to .01 so that $K=2Nw$ equals four. Figure 2 plots $|V_1 (f)|^2$ for the frequencies up to $f=0.14$. The central peak of the MB taper is more concentrated around $f=0$ than the Slepian taper is. The first sidelobe of the MB taper is visible while the first Slepian sidelobe is much smaller.
Figure 1 plots the logarithm of $|V_1 (f)|^2$ over the entire frequency range. The MB taper has smaller range bias in the frequency range $|f|
<0.3w$ and in the frequency range $|f| > 0.13$. In the middle frequency range, the Slepian taper is clearly better. The Slepian penalty function maximizes the energy inside the frequency band, $[-w,w]$, and thus it is natural that the Slepian tapers do better for $f \sim w$. By using a discontinuous penalty function, the Slepian spectral windows experience Gibbs phenomenon and decay only as $\frac{1}{f},\; (|V(f)|^2\sim \frac{1}{f^2})$. The MB spectral windows decay as ${1 \over f^2}$, and thus, it is natural that the MB tapers have lower bias for $f \sim {\cal O}(1)$.
Figure 3 plots $\sum_{k=1}^3 |V^{(k)} (f)|^2$ for $|f| <0.14$ and on this scale, the MB tapers are clearly preferable to the Slepian tapers. For larger frequencies, the energy of the multitaper estimate, $\sum_{k=1}^K |V^{(k)} (f)|^2$, is very similar to Fig. 2 on the logarithmic scale provided that $K<<N$.
In summary, the sinusoidal tapers perform nearly as well as the MB tapers while the Slepian tapers have several times larger local bias (except when $k\approx 1.2Nw$). For $k \ll 2Nw$, the Slepian tapers have better broad-band bias protection than the minimum bias tapers do. For $k \sim 2Nw$, the minimum bias tapers provide both smaller local bias and better broad-band protection due to the Gibbs phenomena which the Slepian tapers experience.
Local Error Analysis and Optimal Multitapering
==============================================
We now give a local error analysis of MTSA and determine the optimal number of tapers. Our results are the multitaper analog of the local error analysis of the smoothed periodogram [@GR57; @P58]. We assume the time series is a Gaussian processes and do not consider frequencies near $f=0$ and $f=1/2$. In this case, the variance of the multitaper estimate is approximately $
{\rm Variance} [ \hat{S} (f)]
\approx S(f)^2 \sum_{k=1}^K \mu_k^2$ due to the orthonormality of the tapers
Asymptotically, the local bias of the multitaper estimate of Eq. (\[E6\]) is $$\begin{aligned}
{\rm Bias} [\hat{S}]
& = &
S(f) \left( \sum_{k=1}^K \mu_k -1 \right) + \frac{1}{2}
S''(f) \sum_{k=1}^K \lambda_k \mu_k
\; ,\end{aligned}$$ where $\lambda_k = \int_{-1/2}^{1/2} f^2 |V^{(k)} (f)|^2 df$. The second term is the MT generalization of (\[E20\]). When $\sum_{k=1}^K \mu_k \ne 1$, the MT estimate has bias even in white noise. When we require $\sum_{k=1}^K \mu_k = 1$, the local expected loss simplifies:
\[T5.4\] For a Gaussian process, away from $f=0$ and $f=1/2$, the expected square error of the multitaper spectral estimate (\[E6\]) with $\sum_{k=1}^K \mu_k = 1$ is asymptotically (to leading order in $K/N$) $$\label{E666}
{\rm Bias}^2 + {\rm Variance} \; \approx \; \left[
\frac{1}{2}S''(f) \sum_{k=1}^K \lambda_k \mu_k \right]^2 +
S(f)^2 \sum_{k=1}^K \mu_k^2
\; .$$
\[D5a\] The multitaper estimate which minimizes the local loss (\[E666\]) (with $\mu_k \ge0$) is constructed with the minimum bias tapers.
Proof: We order the $\mu_k$ such that $\mu_1\ge \mu_2 \ge \ldots \ge \mu_K$ and define $\mu_{K+1}=0$. Since the weights, $\mu_k$ are fixed, we need to minimize $ \sum_{k=1}^K \mu_k \uv_k \Abf \uv_k^* $ over all sets of $K$ orthonormal tapers, $\uv_1, \ldots, \uv_K$. We split the series in $K$ subseries and minimize each subseries separately: $$\begin{aligned}
\label{PR1}
\min_{\uv_1, \ldots, \uv_K} \sum_{k=1}^K \mu_k \uv_k \Abf \uv_k^*
&=& \min_{\uv_1, \ldots, \uv_K} \sum_{k=1}^K
(\mu_k -\mu_{k-1}) \left(\sum_{j=1}^k \uv_j \Abf \uv_j^*\right)
\nonumber \\
&\ge & \sum_{k=1}^K (\mu_k -\mu_{k-1})
\left( \min_{\uv_1^{(k)}, \ldots, \uv_k^{(k)} }
\sum_{j=1}^k \uv_j^{(k)} \Abf \uv_j^{(k)*} \right)
\nonumber \\ & = &
\sum_{k=1}^K (\mu_k -\mu_{k-1}) \left(
\sum_{j=1}^k \lambda_{A,j} \right) \; = \;
\sum_{k=1}^K \mu_k \lambda_{A,j} ,\end{aligned}$$ where the $\lambda_{A,j}$ are the eigenvalues of $\Abf$, given in increasing order. The $\uv_j^{(k)}$ are subject to orthonormality constraints that $\uv_j^{(k)}\cdot\uv_{j'}^{(k)} = \delta_{j,j'}$, but are otherwise independent and minimized separately. In the last line of (\[PR1\]), we use Fan’s Theorem [@MO79]: $\min_{\uv_1^{(k)}, \ldots, \uv_k^{(k)} }
\sum_{j=1}^k \uv_j^{(k)} \Abf \uv_j^{(k)*}
= \sum_{j=1}^k \lambda_{A,j}$, where the $ \uv_j^{(k)}$ are again subject to orthonormality constraints. The theorem is now proved because the MB tapers are precisely the eigenvectors of $\Abf$.
\[D5b\] The uniformly weighted multitaper estimate using $K$ sinusoidal tapers has an asymptotic local loss of $$\label{E66s}
{\rm Bias^2 + Variance}\ \simeq\ \left[ {S^{\prime\prime} (f)K^2 \over 24N^2}
\right]^2 + {S(f)^2 \over K} \; .$$
\[D5c\] The asymptotic local loss of (\[E66s\]) is minimized when the number of tapers is chosen as $$\label{E66K}
K_{opt} \sim \left[ {12 S(f)N^2 \over S^{\prime\prime} (f)}
\right]^{2/5}
\; .$$
Thus, the optimal number of tapers is proportional to $N^{4/5}$ and varies with the ratio of $S(f)$ to $S^{\prime\prime}(f)$. Intuitively (\[E66K\]) shows that fewer tapers should be used when the spectrum varies more rapidly. A key advantage of the MB and sinusoidal tapers is that the tapers need not be recomputed as $K$ is changed. In contrast, the Slepian tapers are most efficient when the bandwidth parameter, $w$, is chosen such that $K \sim 2Nw$. Thus, when the number of tapers is changed, as in (\[E66K\]), the Slepian tapers should be recomputed.
Smoothed Multitaper Estimates {#KS}
=============================
In our own comparison of kernel smoothing and multitaper estimation [@RST94], we found that a smoothed multiple taper estimate worked best. We now evaluate the expected error of the kernel smoothed multitaper estimator and show that smoothing the logarithm of the multitaper estimate is useful for estimating the $logarithm$ of the spectrum.
We begin by evaluating that the quadratic estimator (\[E4\]) which is equivalent to a kernel smoother estimates of the spectrum [@GR57; @P58]. Let $\hat{S}(f)$ be the quadratic spectral estimator (\[E4\]), and smooth it with a kernel $\kappa(\cdot)$ of halfwidth $w$ : $$\label{E90a}
\widehat{\widehat{S}}(f) \; = \;
\int_{-w}^w \kappa({g\over w}) \widehat{S}(f+g)dg \; ,$$ where $w$ is the bandwidth parameter and $\kappa(\cdot)$ is a kernel smoother with domain $[-1,1]$. This can be rewritten as $$\label{E90}
\widehat{\widehat{S}}(f) \; = \;
\sum_{n,m=1}^N \qtl_{nm} e^{i2\pi (m-n)f} x_n x_m \; ,$$ where $ \qtl_{nm} = q_{nm}\hat{\kappa}_{m-n} $ with $\hat{\kappa}_{m} =\ \int_{-w}^w \kappa(g/w) e^{2\pi img} dg $. Thus smoothing replaces the original quadratic estimator with matrix $[q_{nm}]$ by another quadratic estimator with matrix $\Qbtl =[\qtl_{nm}]$. By Theorem 5.2, this hybrid method cannot outperform the pure multitaper method with minimum bias tapers.
We now show that combining kernel smoothing with multitapering does improve the estimation of the $logarithm$ of the spectral density, $\theta (f) = \log
[S(f)]$. One standard approach is to kernel smooth the logarithm of the tapered periodogram. This approach has the disadvantage that $|y (f)|^2$ has a $\chi_2^2$ distribution and $\log [ \chi_2^2 ]$ has a long lower tail of its distribution. As a result, $\log [| y(f)|^2 ]$ has an appreciable bias and its variance is inflated by $\pi^2 /6$. A common alternative is to estimate the spectrum either by kernel smoothing or by multitapering and then to take logarithms. This approach has the disadvantage that the smoothed spectral estimate tends to be more sensitive to nonlocal bias effects than the corresponding smoothed log-spectral estimate.
To robustify the log-spectral estimate while reducing the variance inflation from the long tail, we propose the following hybrid estimate: 1) compute the multitaper estimate using the sinusoidal tapers with $\mu_k = {1 \over K}$ and then 2) smooth $\hat{\theta}_{MT} (f) \equiv \ln [ \hat{S}_{MT} (f)] -
B_K / K$, where $B_K$ is the bias of $\ln [ \chi_{2K}^2 ]$. ($B_K \equiv\ \psi (K) - \ln K$ where $\psi (K)$ is the digamma function). For white noise, the variance of $\hat{\theta}_{MT} (f)\ =\ \psi^{\prime} (K)
\underline{\sim}\ {1 \over K} + {1 \over 2K^2}$, so the variance enhancement from the logarithm tends rapidly to zero.
In [@RS94b], we show that the asymptotic error for this scheme is $$\label{E94}
\theta^{\prime\prime} (f)^2 \left[ b_k w^2 + {K^2 \over 24N^2} \right]^2
+ {C_{\kappa} \over Nw} \left( 1+ {1 \over 2K} \right)^2 \ ,$$ where $b_{\kappa}$ and $C_{\kappa}$ are constants which depend on the kernel shape. In (\[E94\]), we assume uniformly weighted sinusoidal tapers are used and $1\ll K \ll Nw$. In (\[E94\]), one factor of $(1+{1 \over 2K})$ is the variance enhancement from the logarithmic transformation and one factor of $(1+{1 \over 2K})$ arises in the variance calculation of (15) with sinusoidal tapers. Optimizing (\[E94\]) with respect to both $w$ and $K$ yield $w \sim N^{-1/5}$ and $K \sim
N^{8/15}$, thus the smoothing halfwidth $w$ is much larger than $K/N$. The expected error (\[E94\]) depends weakly on $K$ provided that $1
\ll K \ll Nw$.
For simplicity, we set $K = N^{8/15}$ and optimize (\[E94\]) with respect to the halfwidth $w$. The resulting halfwidth depends on $\theta^{\prime\prime}
(f)$: $w_{opt} ( \theta^{\prime\prime } (f))$ with $w_{opt} \sim
| \theta^{\prime\prime} (f)|^{-2/5} N^{-1/5}$. Thus when the log-spectrum varies rapidly, the halfwidth should be reduced as $|\theta^{\prime\prime} (f)|^{-2/5}$.
Since $\theta^{\prime\prime} (f)$ is unknown, we consider two stage estimators which begin by making a preliminary estimate of $\theta^{\prime\prime} (f)$ prior to estimating $\theta (f)$. We then insert the estimate $\widehat{\theta^{\prime\prime} (f)}$ into the expression for $w_{opt}$: $w(f) = w_{opt} ( \widehat{\theta^{\prime\prime}}
(f))$ and use a variable halfwidth kernel smoother with halfwidth $\hat{w}(f)$ to estimate $\theta (f)$. Multiple stage kernel estimators are described in [@BGH94; @MS87; @R93; @RS94a; @RS94b]. These multiple stage schemes have a convergence rate of $N^{-4/5}$ and have a relative convergence rate of at least $N^{-2/9}$. A more detailed description is given in [@BGH94; @R93; @RS94b].
Application {#DATA}
===========
We now compare spectral estimates on an actual data series. We use the microwave scattering data set which is described in [@RST94]. The data measures turbulent plasma fluctuations in the Tokamak Fusion Test Reactor at Princeton. The spectrum is dominated by a 1 MHz peak which is quasicoherent. The spectral density varies by over five orders of magnitude.
The bias versus variance trade-off of Sec. 5 shows that fewer tapers should be used near the peak. To make the spectral estimate smooth, a parabolic weighting of the tapers is used as described in Sec. 3. To determine how many tapers to use locally, we use the multiple stage “plug-in” method as described in the previous section; i.e. we determine the number of tapers using a pre-estimate on the same data. To reduce the fluctuations from the estimate of the optimal number, we use a longer data segment to determine the number of tapers at each frequency. We find the optimal number of sinusoidal tapers is roughly 24 for frequencies in the 200 to 800 kHz range. Near the 1 MHz peak, as few as 12 tapers are used to minimize the local bias error. Between 1300 and 2400 kHz, the spectrum is flatter and we use up to 40 tapers.
The dotted line is the sinusoidal multitaper estimate, and the solid, more wiggly, curve is the corresponding Slepian estimate using 24 tapers with $w= 60$ kHz. The 1 MHz peak is poorly resolved in the Slepian estimate, and the regions of high curvature are artificially flattened. For $ f \ge 1.5$ MHz, the Slepian estimate is artificially bumpy due to statistical noise. The variable taper number estimate suppresses these bumps by averaging over a larger frequency halfwidth. We have also used a variable taper number estimate with the Slepian tapers. Since the Slepian parameter, $w$ was fixed at 100 kHz to allow for forty tapers, the artificial broadening was even more exreme. Comparing with a converged estimate of the spectrum based on $N = 45,000 $ shows that the sinuosidal taper estimate is more accurate. Another significant difference is that the Slepian multitaper estimate requires much more CPU time than the sinusoidal multitaper estimate.
Conclusion
==========
We have proposed and analyzed the minimum bias and the sinusoidal tapers, $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, for multitaper spectral estimation. The resulting sinusoidal multitaper spectral estimate is $\hat{S}(f)=\frac{1}{2K(N+1)}
\sum_{j=1}^K |y(f+\frac{j}{2N+2}) -y(f-\frac{j}{2N+2})|^2$. The sinusoidal tapers have low bias because [*the frequency sidelobe from $y(f+\frac{j}{2N+2})$ cancels the sidelobe of $y(f-\frac{j}{2N+2})$.*]{} The minimum bias tapers minimize the local bias, $\int_{-1/2}^{1/2} f^2 |V^{(k)}b (f)|^2 df$, and have good broad-band bias protection as well. Asymptotically, the quadratic spectral estimate which minimizes the expected local square error is a multiple taper estimate using the minimal bias tapers.
The sinusoidal tapers have a simple analytic form and approximate the minimum bias tapers to ${\cal O}\left(\frac{1}{N}\right)$. The $k$th sinusoidal taper has its spectral energy concentrated in the frequency bands $\frac{k-1}{2(N+1)}\leq |f|\leq\frac{k+1}{2(N+1)}$.
The minimum bias and sinusoidal tapers have no auxiliary bandwidth parameter, and the bandwidth of the spectral estimate is determined solely by the number of used tapers. By adaptively adding and deleting tapers, a multitaper estimate with the optimal convergence properties of kernel smoothers can be constructed. In contrast, the Slepian tapers need to be recomputed with a different bandwidth. Thus the Slepian tapers are only practical for fixed bandwidth estimation and this is inherently inefficient.
Appendix: Multitaper decomposition of kernel estimates
======================================================
In Sec. 6, we showed that kernel smoother estimators (\[E90a\]) have an equivalent multitaper representation (\[E6\]) We now show that the equivalent multitapers of some popular kernel smoother estimates of the spectrum strongly resemble the MB/sinusoidal tapers. In one special case, this corresondence is exact; i.e. the smoothed periodogram can be exactly decomposed into MB tapers.
\[T8\] Let $w=\frac{1}{2}$ and $\kappa(f)$ be the parabolic kernel, $\kappa(f)=\frac{3}{2}-6f^2$. The eigenvectors of the kernel smoothed periodogram are exactly the discrete minimum bias tapers.
Proof: The $\Qbtl$ matrix in [(\[E90\])]{} can be calculated explicitly for this case. We find $\Qbtl=[b_{nm}]=\frac{1}{N}(\frac{3}{2}\Ibf-6\Abf)$ where $\Abf$ is the matrix from Lemma \[T2\]. Thus $\Qbtl$ and $\Abf$ have the same eigenvectors.
To illustrate that this result is typical even when we apply a taper and smooth over a small band, we consider a smoothed tapered periodogram with $N=200$. We use Tukey’s split-cosine taper [@RST94] and then smooth the estimate with a square box kernel with a halfwidth of $.01$. We then evaluate the corresponding $\Qbtl$ matrix and compute its eigenvectors. Figure 5 displays the first 4 eigenvectors. They are very close to the sinusoids $\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$. Table 4 shows that this spectral estimate is virtually a $K=4$ multiple taper spectral estimate. After $k>4$, the eigenvalues decrease sharply, and these higher eigenvectors contribute very little to the overall estimate.
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[**Table Captions:**]{}
Table 1: Convergence of the sinusoidal tapers to the minimum bias tapers.
Table 2: Normalized bias term, $4(N+1)^2 \sum_{k=1}^K\int_{-1/2}^{1/2} f^2 \sum_{k=1}^Kf^2|V^{(k)}(f)|^2 df$, for $N=50$.
Table 3: Spectral concentration, $\int_{-w}^w \sum_{k=1}^K|V^{(k)}(f)|^2 df$, for $N=50$.
Table 4: Eigenvectors of the smooth tapered periodogram estimator.
[**Figure Captions:**]{}
Figure 1: Spectral energy of the minimum bias, sinusoidal and Slepian tapers, .
Figure 2: Spectral energy of the minimum bias and sinusoidal tapers, .
Figure 3 Spectral energy, $\sum_{k=1}^3 \int_{-1/2}^{1/2} f^2 |V^{(k)}(f)|^2 df$, of the minimum bias and Slepian tapers, .
Figure 4: Estimated spectral density of the plasma fluctuations. Dashed line is sinusoidal multitaper estimate and solid line is estimate using Slepian tapers with $w=60$ kHz. Because the Slepian tapers have a fixed bandwidth, the corresponding estimate spectral density at 1 MHz is artificially broadened while being undersmoothed for $f \ge 1.5$ MHz.
Figure 6: First eigenvectors of the smooth tapered periodogram estimator.
Table 1. Convergence of the sinusoidal tapers to the minimum bias tapers
--------------------------------------------------------------------------------------------------------------------------------------------------------------
$N$ $ \;\;\;\; $ \;\;\;\;
\max_k \left\{ \frac{N+2}{k} \|\vv^{(k)}-\nubm^{(k)}\|_{L_2} \right\} \max_k \left\{ \frac{N+2}{k}
\;\;\;\; $ \left\| \frac{\vv^{(k)}}{\| \vv^{(k)}\|_{L_{\infty}}}
- \frac{\nubm^{(k)}}{\|\nubm^{(k)}\|_{L_{\infty}}} \right\|_{L_{\infty}}
\right\}
\;\;\;\; $
----- ------------------------------------------------------------------------ -------------------------------------------------------------------------------
20 0.24750 0.4602
50 0.24844 0.4760
200 0.24852 0.4829
800 0.24844 0.4844
--------------------------------------------------------------------------------------------------------------------------------------------------------------
Table 2. Normalized bias term, $4(N+1)^2 \sum_{k=1}^K \int_{-1/2}^{1/2} f^2 |V^{(k)}(f)|^2 df$, for $N=50$
----- ------------- ------------ ----------- ---------------- ----------
$K$ Minimum Sinusoidal Slepian tapers
bias tapers tapers $w$=0.04 $w$=0.08 $w$=0.16
1 1.0095 1.0116 1.3439 2.6316 5.1039
2 5.0475 5.0580 5.7724 10.5484 20.4670
3 14.1328 14.1622 18.0651 23.8086 46.1953
4 30.2846 30.3475 58.9520 42.5181 82.4018
5 55.5217 55.6366 154.4818 66.9996 129.2087
6 91.8634 92.0528 305.4382 99.1800 186.7507
7 141.3284 141.6185 496.7959 150.0103 255.1797
8 205.9362 206.3570 721.1743 251.8833 334.6717
9 287.7056 288.2899 976.5088 437.5993 425.4379
10 388.6562 389.4409 1262.4251 702.1523 527.7433
----- ------------- ------------ ----------- ---------------- ----------
Table 3. Spectral concentration, $\int_{-w}^w |V(f)|^2 df$, for $N=50,\; w=0.08$
$k$ Minimum bias tapers Sinusoidal tapers Slepian tapers
----- --------------------- ------------------- ----------------
1 .9997 .9997 1.
2 .9988 .9988 .9999999
3 .9972 .9972 .9999989
4 .9940 .9937 .99997
5 .9888 .9887 .9995
6 .9760 .9753 .9928
7 .9381 .9417 .9380
8 .6084 .6247 .7002
9 .1688 .1780 .2981
10 .0637 .0624 .0628
Table 4. Eigenvectors of the smooth tapered periodogram estimator
[|c|ccc|]{} $k$ & Weight of the eigenvector & Normalized local bias & Local bias in comparison with\
& $\lambda_k(B)/{\rm tr}(B)$ & $4(N+1)^2 \int_{-1/2}^{1/2} f^2 |V(f)|^2 df$ & the minimum bias taper (ratio)\
\
1 & .2856 & 1.5138 & 1.509\
2 & .2828 & 4.7371 & 1.181\
3 & .2519 & 9.6254 & 1.067\
4 & .1416 & 19.2095 & 1.198\
5 & .0340 & 33.7118 & 1.345\
6 & .0037 & 51.3616 & 1.423\
7 & .0002 & 72.9747 & 1.486\
[^1]: The authors thank D. J. Thomson and the referees for useful comments. Research funded by the U.S. Department of Energy.
|
---
abstract: |
Locating arrays (LAs) can be used to detect and identify interaction faults among factors in a component-based system. The optimality and constructions of LAs with a single fault have been investigated extensively under the assumption that all the factors have the same values. However, in real life, different factors in a system have different numbers of possible values. Thus, it is necessary for LAs to satisfy such requirements. We herein establish a general lower bound on the size of mixed-level $(\bar{1},t)$-locating arrays. Some methods for constructing LAs including direct and recursive constructions are provided. In particular, constructions that produce optimal LAs satisfying the lower bound are described. Additionally, some series of optimal LAs satisfying the lower bound are presented.
[**Keywords**]{}: combinatorial testing, locating arrays, lower bound, construction, mixed orthogonal arrays
[**Mathematics Subject Classifications (2010)**]{}: 05B15, 05B20, 94C12, 62K15
author:
- |
Ce Shi$^1$, Hao Jin$^2$ and Tatsuhiro Tsuchiya $^2$\
$^1$ School of Statistics and Mathematics\
Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China\
$^2$ Graduate School of Information Science and Technology\
Osaka University,Suita 565-0871, Japan\
title: '**Locating arrays with mixed alphabet sizes [^1]**'
---
Introduction
============
Testing is important in detecting failures triggered by interactions among factors. As reported in [@CMMSSY2006], owing to the complexity of information systems, interactions among components are complex and numerous. Ideally, one would test all possible interactions (exhaustive testing); however, this is often infeasible owing to the time and cost of tests, even for a moderately small system. Therefore, test suites that provide coverage of the most prevalent interactions should be developed. Testing strategies that use such test suites are usually called combinatorial testing or combinatorial interaction testing (CIT). CIT has shown its effectiveness in detecting faults, particularly in component-based systems or configurable systems [@KKL2013; @NL2011].
The primary combinatorial object used to generate a test suite for CIT is covering arrays (CAs). CAs are applied in the testing of networks, software, and hardware, as well as construction and related applications [@Colbourn2004; @KC2019; @Sloane1993]. In a CA, the factors have the same number of values. However, in real life, different factors have different numbers of possible values. Thus, mixed-level CAs or mixed covering arrays (MCAs) are a natural extension of covering array research, which improves their suitability for applications [@CDPP1996; @CDFP1997; @CMMSSY2006; @CSWY2011; @MSSW2003; @Sherwood2008]. A CA or MCA as a test suite can be used to detect the presence of failure-triggered interactions. However, they do not guarantee that faulty interactions can be identified. Consequently, tests to reveal the location of interaction faults are of interest. To address this problem, Colbourn and McClary formalized the problem of non-adaptive location of interaction faults and proposed the notion of locating arrays (LAs) [@CM2008].
LAs are a variant of CAs with the ability to determine faulty interactions from the outcomes of the tests. An LA with parameters $d$ and $t$ is denoted by $(d,t)$-LA, where $d$ and $t$ represent the numbers of faulty interactions and of components or factors in a faulty interaction, respectively. $t$ is often called [*strength*]{}. When the number of faulty interactions is at most, instead of exactly $d$, we use the notation $(\bar{d},t)$-LA to denote it. Generally, testing with a $(d,t)$-LA can not only detect the presence of faulty interactions, but can also identify $d$ faulty interactions. Similarly, using a $(\bar{d},t)$-LA as a test suite allows one to identify all faulty interactions if the number is at most $d$.
LAs have been utilized in measurement and testing [@ACS2015; @CMCPS2016; @CS2016]. Martínez et al. [@MMPS] developed adaptive analogues and established feasibility conditions for an LA to exist. Only the minimum number of tests in $(1,1)$-LA and $(\bar{1},1)$-LA is known precisely [@CFH2017]. The minimum number of rows in an LA is determined when the number of factors is small [@STY2012; @TCY2012]. When $(d,t)=(1,2)$, three recursive constructions are provided, as in [@CF2016]. Beyond these few direct and recursive constructions, computation methods are applied to construct $(1,2)$-LAs using a Constraint Satisfaction Problem (CSP) solver and a Satisfiability (SAT) solver [@KKNT2017; @KKNT2019; @NKNT2014]. Lanus [*et al.* ]{}[@LCM2019] described a randomized computational search algorithm called partitioned search with column resampling to construct $(1,t)$-LAs. Furthermore, column resampling can be applied to construct $(\bar{1},t)$-LA with $\delta\leq 4$ [@SSCS2018]. The first and third authors extended the notion of LAs to expand the applicability to practical testing problems. Specifically, they proposed constrained locating arrays (CLAs), that can be used to detect and locate failure-triggering interactions in the presence of constraints. Computational constructions for this variant of LAs can be found in [@JKCT2018; @JT2018; @JT20183].
Although a few constructions exist for $(1,t)$-LAs and $(\bar{1},t)$-LAs, these methods do not treat cases where different factors have difference values. For real-world applications, it is desirable for LAs to satisfy such requirements. Herein, we will focus on mixed-level $(\bar{1},t)$-LAs, which is equivalent to mixed-level $(1,t)$-LAs, and an MCA by Lemma \[(1,t)-LA and MCA\].
The remainder of the paper is organized as follows. The next section provides the definitions of basic concepts, such as MCAs and LAs. A general lower bound on the size of mixed-level $(\bar{1},t)$-LAs will be established in Section 3, which will be regarded as benchmarks for the construction of optimal LAs with specific parameters. Some methods for constructing LAs including direct and recursive constructions are provided in Section 4. In particular, some constructions that produce optimal LAs satisfying the lower bound will be described in this section. The final section contains some concluding remarks.
Definitions and Notations
=========================
The notation $I_n$ represents the set $\{1,2,\cdots, n\}$, while the notations $N,k$ and $t$ represent positive integers with $t<k$. We herein model CIT as follows. Suppose that $k$ factors denoted by $F_1,F_2,\cdots, F_k$ exist. The $i$th factor has a set of $v_i$ possible values (levels) from a set $V_i$, where $i\in I_k$. A test is a $k$-tuple $(a_1,a_2,\cdots, a_k)$, where $a_i\in V_i$ for $1\leq i\leq k$. A test, when executed, has the following outcome: [*pass*]{} or [*fail*]{}. A test suite is a collection of tests, and the outcomes are the corresponding set of pass/fail results. A fault is evidenced by a failure outcome for a test. Tests are considered to be executed in parallel; therefore, testing is non-adaptive or predetermined.
Let $A=(a_{ij}) (i \in I_N, j\in I_k)$ be an $N\times k$ array with entries in the $j$th column from a set $V_j$ of $v_j$ symbols. A [*$t$-way interaction*]{} is a possible $t$-tuple of values for any $t$-set of columns, denoted by $T=\{(i, \sigma_i): \sigma_i\in V_i, i\in I\subseteq I_k, |I|=t\}$. We denote $\rho (A,T)=\{r: a_{ri}=\sigma_i, i\in I\subseteq I_k, |I|=t\}$ for the set of rows of $A$, in which the interaction is included. For an arbitrary set $\mathcal {T}$ of $t$-way interactions, we define $\rho (A,\mathcal {T})=\cup_{T\in {\cal T}}\rho (A,T)$. We use the notation $\mathcal {I}_t$ to denote the set of all $t$-way interactions of $A$.
The array $A$ is termed MCAs, denoted by MCA$_\lambda(N; t, k, (v_1,v_2,\cdots, v_k))$ if $|\rho (A,T)|\geq \lambda$ for all $t$-way interactions $T$ of $A$. In other words, $A$ is an MCA if each $N \times t$ sub-array includes all the $t$-tuples $\lambda$ times at the least. Here, the number of rows $N$ is called the array size. The number $\lambda$ is termed as the array index. The number of columns $k$ is called the number of factors (or variables), number of components, or degree. The word “strength” is generally accepted for referring to the parameter $t$. When $\lambda=1$, the notation MCA$(N; t, k, (v_1, v_2,\cdots, v_k))$ is used.
When $v_1=v_2=\cdots =v_k=v$, an MCA$_\lambda(N; t, k, (v_1,v_2,\cdots, v_k))$ is merely a CA$_\lambda(N; t,k,v)$. When $\lambda=1$ in a CA, we omit the subscript. Without loss of generality, we often assume that the symbol set sizes are in a non-decreasing order, i.e., $v_1\leq v_2\leq \cdots \leq v_k$. Hereinafter, these assumptions will continue to be used. When $v_i=1$, the presence of the $i$th factor does not affect the properties of the mixed covering arrays; thus, it is often assumed that $v_i\geq 2$ for $1\leq i\leq k$.
Following [@CM2008], if, for any ${\cal T}_1, {\cal T}_2 \subseteq \mathcal {I}_t$ with $|{\cal T}_1|
= |{\cal T}_2| = d$, we have $$\rho(A, {\cal T}_1) = \rho(A, {\cal T}_2)
\Leftrightarrow {\cal T}_1 = {\cal T}_2,$$
then the array $A$ is regarded as a $(d,t)$-LA and denoted by $(d,t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$. Similarly, the definition is extended to permit sets of $d$ interactions at the most by writing $\bar{d}$ in place of $d$ and permitting instead $|{\cal T}_1| \le d$ and $|{\cal T}_2| \le d$. In this case, we use the notation $(\bar{d},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$. Clearly, the condition $\rho(A, {\cal T}_1) = \rho(A, {\cal T}_2)
\Leftrightarrow {\cal T}_1 = {\cal T}_2$ is satisfied if ${\cal T}_1 \not= {\cal T}_2\Rightarrow \rho(A, {\cal T}_1) \not= \rho(A, {\cal
T}_2).$ In the following, we will fully apply this fact.
We herein focus on $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ in this paper. One of the main problems regarding $(\bar{1},t)$-LA$(N; k, (v_1,v_2,\cdots, v_k))$ is the construction of such LAs having the minimum $N$ when its other parameters have been fixed. However, this is a difficult and challenging problem. The larger the strength $t$, the more difficult it is to construct a minimum LA. We use the notations $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))$ to represent the minimum number $N$, for which a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ exists. A $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ is called [*optimal*]{} if $N=(\bar{1},t)\mbox{-LAN}(k, (v_1,v_2,\cdots, v_k))$.
[@KKNT2019]\[(1,t)-LA and MCA\] Suppose that $A$ is an $N\times k$ array. $A$ is a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ if and only if it is a $(1,t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ and an MCA.
Lemma \[(1,t)-LA and MCA\] shows that $A$ is a $(\bar{1},t)$-LA if $A$ is an MCA and $\rho(A,T_1)\not= \rho(A,T_2)$ whenever $T_1$ and $T_2$ are distinct $t$-way interactions. We will use this simple fact hereinafter.
A lower bound on the size of $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$
===========================================================================
A benchmark to measure the optimality for $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ is described in this section. It follows from Lemma \[(1,t)-LA and MCA\] that $A$ is a $(\bar{1},t)$-LA only if $A$ is an MCA, which implies that $|\rho(A,T)|\geq 1$ for any $t$-way interaction $T$ of $A$. Consequently, $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))\geq \prod _{i=k-t+1}^k v_i$, where $2\leq v_1\leq v_2\leq \cdots \leq v_k$. Specifically, we have the following results.
\[case 1\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_{k-t}, 2v_{k-t}\leq v_{k-t+1}\leq \cdots \leq v_k$. Then, $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots,v_k))\\ \geq \prod _{i=k-t+1}^k v_i$.
It is remarkable that the lower bound on the size of $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ in Lemma \[case 1\] can be achieved. We will present some infinite classes of optimal $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ satisfying the lower bound in the next section. When $v_i=v_{i+1}=\cdots= v_{k-t}= v_{k-t+1}$, where $i\in \{1,2,\cdots, k-t\}$, we can obtain a lower bound on the size of $(\bar{1},t)$-LA by the similar argument as the proof of Theorem 3.1 in [@TCY2012]. We state it as follows.
\[case 2\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_k$. If $v_i=v_{i+1}=\cdots= v_{k-t}= v_{k-t+1}$, where $i\in \{1,2,\cdots, k-t\}$, then $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))\geq \left\lceil\frac{2\sum_{i\leq j_1<\cdots<j_t \le k}
\prod_{s=1}^{t} v_{j_s} }{1+\binom {k-i+1} {t}}\right\rceil$.
[[**Proof.**]{} ]{}Let $A$ be a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$. We can obtain an $N\times (k-i+1)$ array $A'$ by selecting the last $(k-i+1)$ columns of $A$ (if $i=1$, then $A'$ is merely $A$). In the array $A'$, for any $i\leq j_1<\cdots<j_t \le k$, we write $n^{\ell}_{j_1\dots j_t} = |S^{\ell}_{j_1\dots j_t}|$, where $S^{\ell}_{j_1\dots j_t} = \left\{((j_1,x_1), \ldots, (j_t,x_t))\big||\rho(A',((j_1,x_1), \cdots, (j_t,x_t)))|={\ell}\right\}, \\{\ell}=1,2,3,\dots $.
As stated above, $|\rho(A,T)|\geq 1$ for any $t$-way interaction $T$ of $A'$. Consequently, $\sum_{{\ell}\ge 1}n^{\ell}_{j_1\dots j_t} = \prod_{s=1}^{t} v_{j_s}$ and $\sum_{{\ell}\ge 1}({\ell} \times n^{\ell}_{j_1\dots j_t}) = N$ hold. It is deduced that $n^1_{j_1\dots j_t} \ge 2\prod_{s=1}^{t} v_{j_s}-N$. By Lemma \[(1,t)-LA and MCA\] and the proof of Lemma \[trun\], $A'$ is a $(1,t)$-LA. Thus, in any two of $\binom{k-i+1}{t}$ sets, $\rho(A',S^1_{j_1\dots j_t})'$s with $i\le j_1<\cdots<j_t \le k$ share no common elements. Hence, $\sum_{i\leq j_1<\cdots<j_t \le k}n^1_{j_1\dots j_t}\leq N$, which implies that $\sum_{i\leq j_1<\cdots<j_t \le k}(2\prod_{s=1}^{t} v_{j_s} - N )\leq \sum_{i\leq j_1<\cdots<j_t \le k}n^1_{j_1\dots j_t} \leq N$, i.e., $N\geq \left\lceil\frac{2\sum_{i\leq j_1<\cdots<j_t \le k} \prod_{s=1}^{t} v_{j_s} }{1+\binom {k-i+1}{t}}\right\rceil$. Hence, $(\bar{1},t)\mbox{-LAN}(k, (v_1,v_2,\cdots, v_k))\geq \left\lceil\frac{2\sum_{i\leq j_1<\cdots<j_t \le k} \prod_{s=1}^{t} v_{j_s} }{1+\binom {k-i+1}{t}}\right\rceil $.
Based on $i=1$ and $v_{k-t+1}=\cdots=v_k=v$ in Lemma \[case 2\], the following corollary can be easily obtained. It serves as a benchmark for a $(1,t)$-LA$(N;k,v)$, which was first presented in [@TCY2012].
Let $v, t$, and $k$ be integers with $t<k$. Then, $(1,t)-\mbox{LAN}\ (t, k, v) \ge \left\lceil\frac{2\left(^k_t\right) v^t}{1+\left(^k_t\right)}\right\rceil$.
In a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$, we often assume that $2\leq v_1\leq v_2\leq \cdots \leq v_{k-t}\leq v_{k-t+1} \leq \cdots \leq v_k$. Lemma \[case 1\] and Lemma \[case 2\] consider the cases $v_{k-t}=v_{k-t+1}$ and $2v_{k-t}\leq v_{k-t+1}$, respectively. The left case is $v_{k-t}<v_{k-t+1}<2v_{k-t}$, which is considered in the following lemma.
\[case 3\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_k$. If $v_{k-t}< v_{k-t+1}<2v_{k-t}$, then $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))\geq m $, where $$\begin{aligned}
m = \left\{
\begin{array}{ll}
\mbox{max}\{\left\lceil\frac{2\sum_{k-t\leq j_1<\cdots<j_t \le k}
\prod_{s=1}^{t} v_{j_s} }{t+2}\right\rceil, \prod_{i=k-t+1}^k v_i+\prod_{i=k-t+2}^k v_i \}, & \mbox{if }\ t\geq 2;\\
\left\lceil\frac{2 v_{k-1}+2v_k }{3}\right\rceil, & \mbox{if}\ t=1.
\end{array}
\right.\end{aligned}$$
[[**Proof.**]{} ]{}From the above argument, it is known that $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))\geq M=\prod_{i=k-t+1}^k v_i$. Suppose that $A$ is a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$, where $N=M+L$ and $L\geq 0$. Select the last $(t+1)$ columns of $A$ to form an $N\times (t+1)$ array $A'$. By Lemma \[trun\], $A'$ is a $(\bar{1},t)$-LA$(N;t+1,(v_t,v_{t+1},\cdots,v_k))$. Similar to the proof of Lemma \[case 2\], we can prove that $N\geq \left\lceil\frac{2\sum_{k-t\leq j_1<\cdots<j_t \le k} \prod_{s=1}^{t} v_{j_s} }{t+2}\right\rceil$. When $t=1$, we can obtain $m=\left\lceil\frac{2 v_{k-1}+2v_k }{3}\right\rceil$. For $t\geq 2$, we will prove that $N\geq M+\prod_{i=k-t+2}^k v_i$, i.e., $L\geq \prod_{i=k-t+2}^k v_i$. Without loss of generality, suppose that $A'$ contains two parts, the first part is an $M\times (t+1)$ array $B$ containing an $M\times t$ sub-array comprising all $t$-tuples over $V_{k-t+1}\times V_{k-t+2}\times \cdots \times V_k$; the left part is an $L\times (t+1)$ array $C$. (If $L=0$, then $B=A'$).
If $L<\prod_{i=k-t+2}^k v_i$, then at least one $(t-1)$-way interaction $T=\{(i,a_i): i\in I_k\setminus I_{k-t+1},a_i\in V_i\}$ exists such that it is not included by any row of $C$ (If $B=A'$, then all the $(t-1)$-way interactions satisfy the condition. We can choose an arbitrary one). Hence, we have $|\rho(A', T_1)|=1$ for any $t$-way interaction $T_1\in \mathcal{T}_1=\{T\cup(k-t+1,i):i\in V_{k-t+1}\}$. Since $A$ is a $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$, $|\rho(A',T_2)|\geq 1$ for any $t$-way interaction $T_2\in \mathcal{T}_2=\{T\cup(k-t,i):i\in V_{k-t}\}$. It is clear that $\rho(A',\mathcal{T}_1)= \rho(B,T)=\rho(A',T)=\rho(A',\mathcal{T}_2)$ with $|\rho(A',\mathcal{T}_1)|=v_{k-t+1}$.
Because $|\mathcal{T}_2|=v_{k-t}<|\mathcal{T}_1|=v_{k-t+1}<2|\mathcal{T}_2|$, at least one $t$-way interaction $T'\in \mathcal{T}_2$ exists such that $|\rho(A',T')|=1$. Otherwise, $|\rho(A',T')|\geq 2$ for any $t$-way interaction $T'\in \mathcal{T}_2$, which implies that $|\rho(A',\mathcal{T}_2)|\geq 2|\mathcal{T}_2|=2v_{k-t}$, but $|\rho(A',\mathcal{T}_2)|=|\rho(A',\mathcal{T}_1)|=v_{k-t+1}<2v_{k-t}$. It follows that $\rho(A',T')=\rho(A',T_1')$, where $T_1'$ is a certain $t$-way interaction of $\mathcal{T}_1$. It is obvious that $T'\not =T_1'$. Consequently, $A'$ is not a $(1,t)$-LA. Thus, $L\geq \prod_{i=k-t+2}^k v_i$. Consequently, $m=\mbox{max}\{\left\lceil\frac{2\sum_{k-t\leq j_1<\cdots<j_t \le k}
\prod_{s=1}^{t} v_{j_s} }{t+2}\right\rceil, \prod_{i=k-t+1}^k v_i+\prod_{i=k-t+2}^k v_i \}$ if $t\geq 2$.
Combining Lemmas \[case 1\], \[case 2\], and \[case 3\], a lower bound on the size of $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ can be obtained, which serves as a benchmark to measure the optimality.
\[L-bound\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_k$. Then, $(\bar{1},t)$-LAN$(k, (v_1,v_2,\cdots, v_k))\geq$
1. $\prod _{i=k-t+1}^k v_i$, if $2v_{k-t}\leq v_{k-t+1}$;
2. $\left\lceil\frac{2\sum_{i\leq j_1<\cdots<j_t \le k}
\prod_{s=1}^{t} v_{j_s} }{1+\binom {k-i+1} {t}}\right\rceil$, if $v_i=v_{i+1}=\cdots= v_{k-t}= v_{k-t+1}$, where $i\in \{1,2,\cdots, k-t\}$;
3. $\mbox{max}\{\left\lceil\frac{2\sum_{k-t\leq j_1<\cdots<j_t \le k}
\prod_{s=1}^{t} v_{j_s} }{t+2}\right\rceil, \prod_{i=k-t+1}^k v_i+\prod_{i=k-t+2}^k v_i \}$, if $v_{k-t}< v_{k-t+1}<2v_{k-t}$ and $t\geq 2$;
4. $\left\lceil\frac{2 v_{k-1}+2v_k }{3}\right\rceil$, if $v_{k-t}< v_{k-t+1}<2v_{k-t}$ and $t=1$.
Type Minimum Size Stimulation Annealing
----------- -------------- -----------------------
(2,3,4) 16 16
(3,3,4) 17 17
(2,4,4) 16 16
(2,2,3,4) 16 16
(2,2,5,5) 25 25
(2,3,3,4) 17 17
: Lower Bounds on the size of $(\bar{1},2)$-LA[]{data-label="CLBound"}
Table \[CLBound\] presents a lower bound on the size of some certain mixed-level $(\bar{1},2)$-LAs. The first column lists the types, while the second column displays the lower bound on the size of mixed-level $(\bar{1},2)$-LAs with the type. The last column presents the size obtained by simulation annealing [@T2019].
A $(\bar{1},t)$-$(N; k, (v_1,v_2,\dots, v_k))$ is called [*optimal*]{} if its size is $(\bar{1},t)$-$(k, (v_1,v_2,\dots, v_k))$. In what follows, we will focus on some constructions for mixed level LAs from combinatorial design theory. Some constructions that produce optimal LAs satisfying the lower bound in Lemma \[case 1\] will also be provided.
Constructions of $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$
===============================================================
Some constructions and existence results for $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ are presented in this section.
A construction for optimal $(\bar{1},t)$-LA$(\prod_{i=k-t+1}^k v_i;k, (v_1,v_2,\cdots, v_k))$
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Let $2\leq v_1\leq v_2\leq \cdots \leq v_k$. An $N\times k$ array $A$ is called MCA$_2^*(\prod_{i=k-t+1}^k v_i;t,k,(v_1,v_2,\cdots,v_k))$ if $|\rho(A,T)|=1$ for any $t$-way interaction $T\in {\cal T}=\{\{(k-t+1,v_{k-t+1}),\cdots, (k,v_{k})\}: v_i\in V_i\ (k-t+1\leq i\leq k)\}$ and $|\rho(A,T')|\geq 2$ for any $t$-way interaction $T'\not \in {\cal T}$. If an optimal $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$ with $N=\prod_{i=k-t+1}^k v_i$ exists, then the following condition must be satisfied.
\[NC\] Let $2\leq v_1\leq v_2\leq \cdots\leq v_{k-t}, 2v_{k-t}\leq v_{k-t+1}\leq v_{k-t+2}\leq \cdots \leq v_k$. If $A$ is an optimal $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$ with $N=\prod_{i=k-t+1}^k v_i$. Then, $A$ is an MCA$_2^*(N;t,k,(v_1,v_2,\cdots,v_k))$.
[[**Proof.**]{} ]{}Let $A$ be the given optimal $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$ with $N=\prod_{t=k-t+1}^k v_i$. Then, $A$ is an MCA$(N;t,k,(v_1,v_2,\cdots,v_k)$ by Lemma \[(1,t)-LA and MCA\]. Because $N=\prod_{t=k-t+1}^k v_i$, we have $|\rho(A,T)|=1$ for any $t$-way interaction $T\in \mathcal{T}$. It follows that $|\rho(A,T')|\geq 2$ for any $t$-way interaction $T'$ of $A$ from the definition of $(\bar{1},t)$-LA, where $T'\not \in \mathcal{T}$. Hence, $A$ is an MCA$_2^*(\prod_{i=k-t+1}^k v_i;t,k,(v_1,v_2,\cdots,v_k))$, as desired.
Clearly, an MCA$_2^*(N;t,k,(v_1,v_2,\cdots,v_k))$ is not always a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$. Next, we present a special case of MCA$_2^*$, which produces optimal $(\bar{1},t)$-LAs. First, we introduce the notion of mixed orthogonal arrays (MOAs).
An MOA, or MOA$(N;t,k,(v_1,v_2,\cdots,v_k))$ is an $N\times k$ array with entries in the $i$th column from a set $V_i$ of size $v_i$ such that each $N\times t$ sub-array contains each $t$-tuple occurring an equal number of times as a row. When $v_1=v_2=\cdots=v_k=v$, an MOA is merely an [*orthogonal array*]{}, denoted by OA$(N;t,k,v)$.
The notion of mixed or asymmetric orthogonal arrays, introduced by Rao [@R1973], have received significant attention in recent years. These arrays are important in experimental designs as universally optimal fractions of asymmetric factorials. Without loss of generality, we assume that $v_1\leq v_2\leq \cdots \leq v_k$. By definition of MOA, all $t$-tuples occur in the same number of rows for any $N \times t$ sub-array of an MOA. This number of rows is called [*index*]{}. It is obvious that $\binom{k}{t}$ indices exist. We denote it by $\lambda_1,\lambda_2,\cdots, \lambda_{\binom{k}{t}}$. If $\lambda_i\not =\lambda_j$ for any $i\not =j$, then an MOA is termed as a [*pairwise distinct index mixed orthogonal array*]{}, denoted by PDIMOA$(N;t,k,(v_1,v_2,\cdots, v_k))$. Moreover, if $\lambda_i=1$ for a certain $i\in \{1,2,\cdots, \binom{k}{t}\}$ holds, then it is termed as PDIMOA$^*(N;t,k,(v_1,v_2,\cdots,v_k))$. It is clear that $N=\prod_{i=k-t+1}^k v_i$ in the definition of PDIMOA$^*$.
The transpose of the following array is a PDIMOA $^*(24;2,3,(2,4,6))$. $$\left(
\begin{array}{cccccccccccccccccccccccccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 &
0 & 0 & 1 & 0 & 1 & 1 & 0\\
2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 2 & 3 & 4 & 5 & 0 & 1 & 2 & 3 & 4 & 5 & 0 & 1 & 2 & 3 & 4 & 5 & 0 & 1 & 2 & 3 & 4 & 5 & 0
\end{array}
\right)$$
The following lemma can be easily obtained by the definition of PDIMOA$^*$; therefore, we omit the proof herein.
\[NC1\] Suppose that $v_1 \leq v_2 \leq \cdots \leq v_k$. If $A$ is a PDIMOA$^*(\prod_{i=k-t+1}^ k v_i;t,k,(v_1,v_2,\cdots, v_k))$, then $v_1 <v_2 < \cdots < v_k$ and $v_i|v_j$, where $1\leq i\leq k-t$ and $k-t+1\leq j\leq k$.
\[PDIMOA-LA\] Let $2< v_1< v_2< \cdots< v_k$. If a PDIMOA$(N;t,k,(v_1,v_2,\dots, v_k))$ exists, then a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\dots, v_k))$ exists. Moreover, if $N=\prod_{i=k-t+1}^k v_i$, then the derived $(\bar{1},t)$-LA is optimal.
[[**Proof.**]{} ]{}Let $A$ be a PDIMOA$(N;t,k,(v_1,v_2,\dots, v_k))$. Clearly, $A$ is an MCA. By Lemma \[(1,t)-LA and MCA\], we only need to prove that $T_1\not =T_2$ implies $\rho(A,T_1)\not =\rho(A,T_2)$, where $T_1$ and $T_2$ are two $t$-way interactions. In fact, if $\rho(A,T_1) =\rho(A,T_2)$, then $|\rho(A,T_1)|=|\rho(A,T_2)|$, which contradicts the definition of a PDIMOA. The optimality can be obtained by Theorem \[L-bound\].
We will construct an optimal $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\dots, v_k))$ with $N=\prod_{i=k-t+1}^k v_i$ in terms of PDIMOA$^*$. First, we have the following simple and useful construction for PDIMOA$^*$. A similar construction for MOAs was first stated in [@CJL2014].
\[f\] Let $b=r_1r_2\cdots r_m<v_2< \cdots<v_k$ and $r_1<r_2<\cdots <r_m$. If a PDIMOA$^*(\prod_{i=k-t+1}^ k v_i;\\t,k,(r_1r_2\cdots r_m,v_2,v_3,\cdots v_k))$ exists, then a PDIMOA$^*(\prod_{i=k-t+1}^ k v_i;t,k+m-1,(r_1,r_2,\cdots, r_m, v_2,v_3,\cdots, v_k))$ also exists.
[[**Proof.**]{} ]{}Let $A$ be PDIMOA$^*(N;t,k,(b,v_2,v_3,\cdots v_k))$ with $b=r_1r_2,\cdots r_m$. We can form an $N\times (k+m-1)$ array $A'$ by replacing the symbols in $V_b$ by those of $V_{r_1}\times V_{r_2} \times \cdots \times V_{r_m}$. It is easily verified that $A'$ is the required PDIMOA$^*$.
The following construction can be obtained easily; thus, we omit its proof.
\[pc2\] Let $a_1<a_2< \cdots<a_k$ and $b_1<b_2< \cdots<b_k$. If both a PDIMOA$^*(\prod_{i=k-t+1}^ k a_i;t,k,\\(a_1,a_2,\cdots, a_k))$ and a PDIMOA$^*(\prod_{i=k-t+1}^ k b_i;t,k, (b_1,b_2,\cdots, b_k))$ exist, then a PDIMOA$^*(\prod_{i=k-t+1}^ k a_ib_i;\\t,k,(a_1b_1,a_2b_2,\cdots, a_kb_k))$ exists. In particular, if both a PDIMOA$^*(\prod_{i=k-t+1}^ k a_i;t,k,(a_1,a_2,\cdots, a_k))$ and an OA$(t,k,v)$ exist, then a PDIMOA$^*(\prod_{i=k-t+1}^ k a_iv^t;t,k,(a_1v,a_2v,\cdots, a_kv))$ exists.
Methods for constructing $(\bar{1},t)$-LA$(N;k, (v_1,v_2,\cdots, v_k))$
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In this subsection, we modify some constructions for MCAs to the case of $(\bar{1},t)$-LAs. The next two lemmas provide the “truncation” and “derivation” constructions, which were first used to construct mixed CAs.
[(Truncation)]{}\[trun\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_{i-1}\leq v_i\leq v_{i+1}\leq \cdots \leq v_k$. Then, $(\bar{1},t)$-LAN$(k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))\leq$ $(\bar{1},t)$-LAN$(k,(v_1,v_2,\dots,v_{i-1},v_i,v_{i+1},\cdots,v_k))$.
[[**Proof.**]{} ]{}Let $A$ be a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\dots,v_{i-1},v_i,v_{i+1},\cdots,v_k))$ with $N=(\bar{1},t)$-LAN$(k,(v_1,v_2,\dots,\\v_{i-1},v_i,v_{i+1},\cdots,v_k))$. Delete the $i$th column from $A$ to obtain a $(\bar{1},t)$-LA$(N;k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\\\cdots,v_k))$. Thus, $(\bar{1},t)$-LAN$(k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))\leq N=(\bar{1},t)$-LAN$(k,(v_1,v_2,\dots,v_{i-1},v_i,\\v_{i+1},\cdots,v_k))$.
[(Derivation)]{}\[deri\] Let $2\leq v_1\leq v_2\leq \cdots \leq v_{i-1}\leq v_i\leq v_{i+1}\leq \cdots \leq v_k$. Then $v_i \cdot (\bar{1},t-1)$-LAN$(k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))\leq$ $(\bar{1},t)$-LAN$(k,(v_1,v_2,\cdots,v_{i-1},v_i,v_{i+1},\cdots,v_k))$, where $t\geq 2$.
[[**Proof.**]{} ]{}Let $A$ be a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\dots, v_k))$ with $N=(\bar{1},t)$-LAN$(k,(v_1,v_2,\dots, v_k))$. By Lemma \[(1,t)-LA and MCA\], $A$ is an MCA and a $(1,t)$-LA. For each $x \in \{0,1,\cdots, v_i-1\}$, taking the rows in $A$ that involve the symbol $x$ in the $i$th columns and omitting the column yields an MCA$(N_x; t-1,k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))$. We use $A(x)$ to denote the derived array. Next, we prove that $A(x)$ is a $(1,t-1)$-LA$(N_x;k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))$. In fact, for any $(t-1)$-way interaction $T_1$ and $T_2$ with $T_1\not= T_2$, if $\rho(A(x),T_1)=\rho(A(x),T_2)$, we can form two $t$-way interactions $T_1'$ and $T_2'$ by inserting $(i,x)$ into $T_1$ and $T_2$, respectively. Hence, $\rho(A,T_1')=\rho(A,T_2')$, where $|\rho(A,T_1')|= |\rho(A(x),T_1)|$ but $T_1'\not=T_2'$. Consequently, $A$ is not a $(1,t)$-LA. It is clear that $N_i\geq (\bar{1},t-1)$-LAN$(k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))$ for $0\leq i\leq v_i-1$. Thus, $N=N_0+N_1+\cdots+N_{v_i-1}\geq v_i\cdot(\bar{1},t-1)$-LAN$(k-1,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_k))$.
The following product construction can be used to produce a new LA from old LAs, which is a typical weight construction in combinatorial design.
[(Product Construction)]{}\[pc1\] If both a $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\dots, v_k))$ and an MCA$(N_2;t,k,\\(s_1,s_2,\dots, s_k))$ exist, then a $(\bar{1},t)$-LA$(N_1N_2;k,(v_1s_1,v_2s_2,\dots, v_ks_k))$ exists. In particular, if both a $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\dots, v_k))$ and a $(\bar{1},t)$-LA$(N_2;k,(s_1,s_2,\dots, s_k))$ exist, then a $(\bar{1},t)$-LA$(N_1N_2;k,(v_1s_1,\\v_2s_2,\dots, v_ks_k))$ also exists.
[[**Proof.**]{} ]{}Let $A=(a_{ij})\ (i \in I_{N_1}, j\in I_k )$ and $B=(b_{ij})\ (i \in I_{N_2}, j\in I_k )$ be the given $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\\\dots, v_k))$ and MCA$(N_2;t,k,(s_1,s_2,\dots, s_k))$, respectively. We form an $N_1N_2\times k$ array as follows. For each row $(a_{i1}, a_{i2 }, \cdots, a_{ik})$ of $A$ and each row $(b_{h1}, b_{h2},\cdots , b_{hk})$ of $B$, include the row $((a_{i1}, b_{h1}), (a_{i2}, b_{h2}),\\ \cdots, (a_{ik}, b_{hk}))$ as a row of $\overline{A}$, where $1\leq i\leq N_1, 1\leq h\leq N_2$.
From the typical weighting method in design theory, the resultant array $\overline{A}$ is an MCA$(N_1N_2;t, k,(v_1s_1,\\v_2s_2,\dots, v_ks_k))$, as both $A$ and $B$ are MCAs. By Lemma \[(1,t)-LA and MCA\], we only need to prove that $\overline{A}$ is a $(1,t)$-LA. Suppose that $\rho(\overline{A},T_1)= \rho(\overline{A},T_2)$, where $T_1=\{(i,(a_{hi},b_{ci})):i\in I, |I|=t, I\subset \{1,2,\cdots,k\}, h\in I_{N_1},c\in I_{N_2}\}$ and $T_2=\{(j,(a_{h'j},b_{c'j})):j\in I', |I'|=t, I'\subset \{1,2,\cdots,k\}, h'\in I_{N_1},c'\in I_{N_2}\}$ with $T_1\not =T_2$. It is noteworthy that the projection on the first component of $T_1$ and $T_2$ is the corresponding $t$-way interaction of $A$, while the projection on the second component is the corresponding $t$-way interaction of $B$. Therefore, $A$ is not a $(1,t)$-LA. The first assertion is then proved because a $(\bar{1},t)$-LA$(N_2;k,(s_1,s_2,\dots, s_k))$ is an MCA$(N_2;t,k,(s_1,s_2,\dots, s_k))$. The second assertion can be proven by the first assertion.
The following construction can be used to increase the number of levels for a certain factor.
\[twice construction\] If a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$ exists, then a $(\bar{1},t)$-LA$(2N;k,(v_1,v_2,\cdots, v_{i-1}, a, \\ v_{i+1}, \cdots,v_k))$ exists, where $i\in \{1,2,3,\cdots,k\}$ and $v_i<a\leq 2v_i$.
[[**Proof.**]{} ]{}Let $A=(a_{ij}), (i\in I_N,j\in I_k)$ be the given $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$ with entries in the $i$th column from a set $V_i$ of size $v_i$. For a certain $i\in I_k$, we replace the symbols $0,1,\cdots,a-v_i-1$ in the $i$th column of $A$ by $v_i,v_{i}+1,\cdots, a-1$, respectively. We denote the resultant array by $A'$. Clearly, permuting the symbols in a certain column does not affect the property of $(\bar{1},t)$-LAs. Thus, $A'$ is also a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_k))$, where entries in the $i$th column of $A'$ from the set $\{a-v_i, a-v_i+1,\cdots, v_i-1,v_i,v_i+1,\cdots, a-1\}$. Subsequently, write $M=(A^T|(A')^T)^T$. It is easy to prove that $M$ is a $(1,t)$-LA$(2N;k,(v_1,v_2,\cdots, v_{i-1}, a,v_{i+1}, \cdots,v_k))$ and an MCA$(2N;t,k,(v_1,v_2,\cdots, v_{i-1}, a,v_{i+1}, \cdots,v_k))$. By Lemma \[(1,t)-LA and MCA\], $M$ is the desired array.
The following example illustrates the idea in Construction \[twice construction\].
\[Ex3-1\]The transpose of the following array is a $(\bar{1},2)$-LA$(12;5,(2,2,2,2,3))$
1.8pt
--- --- --- --- --- --- --- --- --- --- --- ---
0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 1 1 1 0 0 1 1 1
0 0 0 1 0 1 1 1 1 0 0 1
0 0 1 0 1 0 1 1 1 0 1 0
0 2 1 0 1 1 2 0 1 0 2 2
--- --- --- --- --- --- --- --- --- --- --- ---
Replace the symbols $0,1$ by $2,3$ in the $3$th column, respectively. Juxtapose two such arrays from top to bottom to obtain the following array $M$; we list it as its transpose to conserve space.
1.8pt
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1
0 0 0 1 0 1 1 1 1 0 0 1 2 2 2 3 2 3 3 3 3 2 2 3
0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0
0 2 1 0 1 1 2 0 1 0 2 2 0 2 1 0 1 1 2 0 1 0 2 2
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
It is easy to verify that $M$ is a $(\bar{1},2)$-LA$(24;5,(2,2,4,2,3))$.
Replace the symbol $0$ by $2$ in the $3$th column. Juxtapose two such arrays from top to bottom to obtain the following array $M'$; we list it as its transpose to conserve space.
1.8pt
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1
0 0 0 1 0 1 1 1 1 0 0 1 2 2 2 1 2 1 1 1 1 2 2 1
0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0
0 2 1 0 1 1 2 0 1 0 2 2 0 2 1 0 1 1 2 0 1 0 2 2
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
It is easy to verify that $M'$ is a $(\bar{1},2)$-LA$(24;5,(2,2,3,2,3))$.
[**Remark:**]{} Construction \[twice construction\] may produce an optimal $(\bar{1},t)$-LA. For example, a $(\bar{1},2)$-LA$(16;(2,2,3,4))$ is shown in Table 1. By Construction \[twice construction\], we can obtain a $(\bar{1},2)$-LA$(32;(2,2,3,8))$, which is optimal by Lemma \[case 3\].
Fusion is an effective construction for MCAs from CAs. It causes any $d\ge 2$ levels to be identical; for example, see [@CGRS2010]. As with CAs, fusion for $(\bar{1},t)$-LAs guarantees the extension of uniform constructions to mixed cases. However, fusion for a $(\bar{1},t)$-LA$(N;k,v)$ may not produce mixed-level $(\bar{1},t)$-LAs. This problem can be circumvented by introducing the notion of detecting arrays (DAs). If, for any ${\cal T}\subseteq \mathcal {I}_t$ with $|{\cal T}| = d$ and any $T\in \mathcal {I}_t$, we have $\rho(A, T)\subseteq \rho(A, {\cal T}) \Leftrightarrow T\in {\cal T},$ then the array $A$ is called a $(d,t)$-DA or a $(d,t)$-DA$(N; k,v)$.
[(Fusion)]{}\[fusion\] Suppose that $A$ is a $(1,t)$-DA$(N; k,v)$ with $t\geq 2$. If $A$ is also a $(\lceil \frac{v}{v_i}\rceil,t)$-LA$(N; k,v)$, then a $(\bar{1},t)$-LA$(N;k,(v,\cdots,v,v_i,v,\cdots, v))$ exists, where $ 2\leq v_i< v$.
[[**Proof.**]{} ]{}Let $A$ be a $(1,t)$-DA$(N; k,v)$ over the symbol set $V$ of size $v$. Let $a_1+a_2+\cdots+a_{v_i}=v$, where $a_i(i=1,2,\cdots,v_i)\geq 1$. We can select one $a_i$ such that $a_i=\lceil \frac{v}{v_i}\rceil$ and $a_i\geq a_j$, where $1\leq i\not =j\leq v_i$. We select $a_1, a_2,\cdots, a_{v_i}$ elements from $V$ in the $i$th column of $A$ to form the element sets $A_i(1\leq i\leq v_i)$, respectively. The elements in $A_i(1\leq i\leq v_i)$ are identical with $1,2,\cdots, v_i$, respectively. Then, we obtain an $N\times k$ array $A'$. Clearly, $A'$ is an MCA. We only need to prove that $A'$ is a $(1,t)$-LA by Lemma \[(1,t)-LA and MCA\], i.e., for any two distinct $t$-way interactions $T_1=\{(a_1,u_{a_1}),\cdots,(a_t,u_{a_t})\}$ and $T_2=\{(b_1,s_{b_1}), \cdots,(b_t,s_{b_t})\}$, we have $\rho(A',T_1)\not =\rho(A',T_2)$. It is clear that $\rho(A,T_1)=\rho(A',T_1)$ and $\rho(A',T_2)=\rho(A,T_2)$ when $i\not \in \{a_1,\cdots,a_t\}$ and $i\not \in \{b_1,\cdots,b_t\}$. Hence, $\rho(A',T_1)\not=\rho(A',T_2)$.
When $i\in \{a_1,\cdots,a_t\}$ and $i\not \in \{b_1,\cdots,b_t\}$, we can obtain a $t$-way interaction $T_1'=\{(a_1,u_{a_1},\cdots,(i,a),\\\cdots,(a_t,u_{a_t})\}$ of $A$, where $a\in A_{u_i}$. If $\rho(A',T_1) =\rho(A',T_2)$, then $\rho(A,T_1')\subset \rho(A',T_1)=\rho(A',T_2)=\rho(A,T_2)$. However, $T_1'\not =T_2$; as such, it is a contradiction that $A$ is a $(1,t)$-DA$(N;k,v)$. If $i\not \in \{a_1,\cdots,a_t\}$ and $i \in \{b_1,\cdots,b_t\}$, then the similar argument can prove the conclusion.
When $i\in \{a_1,\cdots,a_t\}$ and $i\in \{b_1,\cdots,b_t\}$, it is clear that $\rho(A',T_1)\not =\rho(A',T_2)$ if $u_i\not =s_i$. The case $u_i=s_i$ remains to be considered. Without loss of generality, suppose that $a_j$ elements are identical with $u_i$. It is clear that $T_1$ and $T_2$ can be obtained from ${\cal T}_1$ and ${\cal T}_2$ by fusion, respectively, where ${\cal T}_1$ and ${\cal T}_2$ are sets of $t$-way interactions with $|{\cal T}_1|=|{\cal T}_2|=a_j$. If $\rho(A',T_1) =\rho(A',T_2)$, then $\rho(A',T_1)=\rho(A,{\cal T}_1) =\rho(A',T_2)=\rho(A,{\cal T}_2)$. It is a contradiction that $A$ is a $(\lceil \frac{v}{v_i}\rceil,t)$-LA$(N;k,v)$ because the existence of $(\lceil \frac{v}{v_i}\rceil,t)$-LA$(N;k,v)$ implies the existence of $(a_j,t)$-LA$(N;k,v)$ [@CM2008].
Constructions \[twice construction\] and \[fusion\] provide an effective and efficient method to construct a mixed-level $(\bar{1},t)$-LA from a $(1,t)$-LA$(N;k,v)$. The existence of $(d,t)$-DA$(N;k,v)$ with $d\geq 1$ implies the existence of $(d,t)$-LA$(N;k,v)$ [@CM2008]. Hence, the array $A$ in Construction \[fusion\] can be obtained by a $(d,t)$-DA$(N;k,v)$, which is characterized in terms of super-simple OAs. The existence of super-simple OAs can be found in [@Chen2011; @H2000; @STY2012; @SW2016; @SY2014; @TY2011]. It is noteworthy that the derived array is not optimal. In the remainder of this section, we present two “Roux-type” recursive constructions[@R1987].
\[increasing one group\] If both a $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\cdots,v_k))$ and a $(\bar{1},t-1)$-LA$(N_2;k-1,(v_1,v_2,\cdots,\\v_{i-1},v_{i+1},\cdots,v_k))$ exist, then a $(\bar{1},t)$-LA$(N_1+eN_2;k,(v_1,v_2,\cdots, v_{i-1}, v_i+e, v_{i+1}, v_{i+2}\cdots,v_k))$ exists, where $e\geq 0$.
[[**Proof.**]{} ]{}Let $A$ and $B$ be the given $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\cdots,v_k))$ and $(\bar{1},t-1)$-LA$(N_2;k-1,(v_1,v_2,\cdots,\\v_{i-1},v_{i+1},\cdots,v_k))$, respectively. Clearly, if $e=0$, then $A$ is the required array. Now, suppose that $e\geq 1$. Insert a column vector $(j,j,\cdots,j)$ of length $N_2$ to the front of the $i$th column of $B$ to form an $N_2\times k$ array $B_j$, where $j\in \{v_i,v_i+1,v_i+2,\cdots,v_i+e-1\}$. Let $M=(A^T|B_{v_i}^T|B_{v_i+1}^T|\cdots|B_{v_i+e-1}^T)^T$. Clearly, $M$ is an MCA$(N_1+eN_2;t,k,(v_1,v_2,\cdots, v_{i-1}, v_i+e, v_{i+1}, v_{i+2}\cdots,v_k))$ [@CSWY2011]. By Lemma \[(1,t)-LA and MCA\], we only need to prove that $M$ is a $(1,t)$-LA, i.e., $\rho(M,T_1)\not =\rho(M,T_2)$ for any two distinct $t$-way interactions $T_1$ and $T_2$, where $T_1=\{(a_1,u_{a_1}),\cdots,(a_t,u_{a_t})\}$ and $T_2=\{(b_1,s_{b_1}), \cdots,(b_t,s_{b_t})\}$. Next, we distinguish the following cases.
[Case 1. ]{} $i\not \in \{a_1,\cdots, a_t\}$ and $i\not \in \{b_1,\cdots, b_t\}$
In this case, because $A$ is a $(\bar{1},t)$-LA, $\rho(A,T_1)\not= \rho(A,T_2)$, $\rho(M,T_1)\not= \rho(M,T_2)$ as $A$ is part of $M$.
[Case 2. ]{} $i\not \in \{a_1,\cdots, a_t\}$ and $i\in \{b_1,\cdots, b_t\}$ or $i \in \{a_1,\cdots, a_t\}$ and $i \not \in \{b_1,\cdots, b_t\}$
When $i\not \in \{a_1,\cdots, a_t\}$ and $i\in \{b_1,\cdots, b_t\}$, if $s_i\not \in \{v_i,v_i+1,\cdots, v_i+e-1\}$, then $\rho(A,T_1)\not= \rho(A,T_2)$. Thus, $\rho(M,T_1)\not= \rho(M,T_2)$. If $s_i \in \{v_i,v_i+1,\cdots, v_i+e-1\}$, then $T_2$ must be included by rows of $B_i$, where $i\in \{v_i,v_i+1,\cdots, v_i+e-1\}$; however, it must not be included by any row of $A$. Clearly, $T_1$ must be included by some rows of $A$. Consequently, $\rho(M,T_1)\not= \rho(M,T_2)$. When $i \in \{a_1,\cdots, a_t\}$ and $i \not \in \{b_1,\cdots, b_t\}$, the same argument can prove the conclusion.
[Case 3. ]{} $i \in \{a_1,\cdots, a_t\}$ and $i\in \{b_1,\cdots, b_t\}$
Clearly, $\rho(M,T_1)\not= \rho(M,T_2)$ holds whenever $u_i\not =s_i$. If $u_i=s_i\not \in \{v_i,v_i+1,\cdots,v_i+e-1\}$, then $\rho(A,T_1)\not= \rho(A,T_2)$, which implies that $\rho(M,T_1)\not= \rho(M,T_2)$. If $u_i=s_i\in \{v_i,v_i+1,\cdots,v_i+e-1\}$, then $T_1$ and $T_2$ must be included by some rows for a certain $B_i$, where $i\in \{v_i,v_i+1,\cdots,v_i+e-1\}$. Because $B$ is a $(\bar{1},t-1)$-LA, $\rho(B_i, T_1)\not =\rho(B_i,T_2)$, which implies $\rho(M,T_1)\not= \rho(M,T_2)$.
More generally, we have the following construction.
\[increasing two groups\] Let $p\geq 0, q\geq 0$ and $1\leq i< j\leq k$. If a $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\cdots,v_{i-1},v_{i},v_{i+1},\cdots,\\ v_{j-1},v_{j},v_{j+1},\cdots,v_k))$, $(\bar{1},t-1)$-LA$(N_2;k-1,(v_1,v_2,\cdots ,v_{i-1},v_{i+1},\cdots,v_k))$, a $(\bar{1},t-1)$-LA$(N_3;k-1,(v_1,v_2,\cdots ,v_{j-1},v_{j+1},\cdots,v_k))$ and $(\bar{1},t-2)$-LA$(N_4;k-2,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_{j-1},v_{j+1},\cdots,\\v_k))$ exist, then a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_{i-1},v_{i}+p,v_{i+1},\cdots, v_{j-1},v_{j}+q,v_{j+1},\cdots,v_k))$ exists, where $N=N_1+pN_2+qN_3+pqN_4$.
[[**Proof.**]{} ]{}We begin with a $(\bar{1},t)$-LA$(N_1;k,(v_1,v_2,\cdots,v_{i-1},v_{i},v_{i+1},\cdots, v_{j-1},v_{j},v_{j+1},\cdots,v_k))$, an $N_1\times k$ array $A$ that is on $V_1\times\cdots \times V_{i-1}\times V_{i}'\times V_{i+1}\times\cdots \times V_{j-1}\times V_{j}'\times V_{j+1}\times\cdots \times
V_k$. Let $H_1$ and $H_2$ be two sets with $|H_1|=p$ and $|H_2|=q$ such that $H_1\bigcap V_i'=\emptyset$ and $H_2\bigcap V_j'=\emptyset$, respectively. Suppose that $B'$, an $N_2\times (k-1)$ array, is a $(\bar{1},t-1)$-LA$(N_2;k-1,(v_1,v_2,\cdots ,v_{i-1},v_{i+1},\cdots,v_k))$, which is on $V_1\times\cdots \times V_{i-1}\times V_{i+1}\times\cdots \times V_k$. For each row $(a_1$, $a_2,\cdots, a_{i-1},a_{i+1},\cdots, a_k)$ of $B'$, add $x\in H_1$ to obtain a $k$-tuple $(a_1$, $a_2,\cdots, a_{i-1},x,a_{i+1},\cdots,a_k)$. Then, we obtain a $pN_2\times k$ array from $B'$, denoted by $B$. Similarly, from a $(\bar{1},t-1)$-LA$(N_3;k-1,(v_1,v_2,\cdots ,v_{j-1},v_{j+1},\cdots,v_k))$, we obtain a $qN_3\times k$ array, denoted by $C$. For each pair $(x,y)\in H_1\times H_2$, we construct $k$-tuple $(a_1, a_2,\cdots, a_{i-1},x,a_{i+1},\cdots,a_{j-1},y,a_{j+1},\cdots,a_k)$ for each row of the given $(\bar{1},t-2)$-LA$(N_4;k-2,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_{j-1},v_{j+1},\cdots,v_k))$. These tuples result in a $pqN_4\times k$ array, denoted by $D$.
Denote $V_i'\cup H_1=V_i$, $V_j'\cup H_2=V_j$ and $
F=\left(
\begin{array}{l}
A\\
B\\
C\\
D\\
\end{array}
\right)
$. We claim that $F$, an $(N_1+pN_2+qN_3+pqN_4)\times k$ array, is a $(\bar{1},t)$-LA$(N;k,(v_1,v_2,\cdots,v_{i-1},v_{i}+p,v_{i+1},\cdots, v_{j-1},v_{j}+q,v_{j+1},\cdots,v_k)$ which is on $V_1\times\cdots \times V_{i-1}\times V_{i}\times V_{i+1}\times\cdots \times V_{j-1}\times V_{j}\times V_{j+1}\times\cdots \times
V_k$.
Clearly, $F$ is an MCA$(N;t,k,(v_1,v_2,\cdots,v_{i-1},v_{i}+p,v_{i+1},\cdots, v_{j-1},v_{j}+q,v_{j+1},\cdots,v_k)$. To prove this assertion, we only need to demonstrate that $\rho(F,T_a)\not =\rho(F,T_b)$ for any two distinct $t$-way interactions $T_a=\{(a_1,u_{a_1}),\cdots,(a_t,u_{a_t})\}$ and $T_b=\{(b_1,v_{b_1}),\cdots,(b_t,v_{b_t})\}$. By similar argument as the proof of Construction \[increasing one group\], we can prove the conclusion except for the case where $i,j\in \{a_1,a_2,\cdots,a_t\}$ and $i,j\in \{b_1,b_2,\cdots,b_t\}$, $u_i=v_i\in H_1$, and $ u_j=v_j\in H_2$. In this case, $T_a$ and $T_b$ are only included by some rows of $D$. If $\rho(F,T_a)=\rho(F,T_b)$, then $\rho(D,T_a) =\rho(D,T_b)=\rho(F,T_a)=\rho(F,T_b)$. Consequently, $\rho(D,T_a\setminus \{(i,u_i),(j,u_j)\}) =\rho(D,T_b\setminus \{(i,u_i),(j,u_j)\})$, which implies that $\rho(D',T_a\setminus \{(i,u_i),(j,u_j)\}) =\rho(D',T_b\setminus \{(i,u_i),(j,u_j)\})$ by the construction of $D$. It is a contradiction with $D'$ being a $(\bar{1},t-2)$-LA$(N_4;k-2,(v_1,v_2,\cdots,v_{i-1},v_{i+1},\cdots,v_{j-1},v_{j+1},\cdots,v_k))$. The proof is completed.
Optimal $(\bar{1},t)$-LA$(\prod_{i=k-t+1}^k;k, (v_1,v_2,\cdots, v_k))$
----------------------------------------------------------------------
In this subsection, some series of optimal mixed-level $(\bar{1},t)$-LAs are presented. First, we list some known results for later use.
\[OA(t,t+1,v)\] [@HSS1999] An OA$(v^t;t,t+1,v)$ exists for any integer $v\geq 2, t\geq 2$.
The existence of PDIMOA$^*(t,t+1,(v_1,v_2,\cdots, v_t))'$s is determined completely by the following theorem.
\[PDIMOA(t,t+1)\] Let $v_1<v_2<\cdots <v_{t+1}$. A PDIMOA$^*(\prod_{i=2}^{t+1} v_i;t,t+1,(v_1,v_2,\cdots, v_t,v_{t+1}))$ exists if and only if $v_1|v_i$ for $2\leq i\leq t+1$.
[[**Proof.**]{} ]{}The necessity can be easily obtained by Lemma \[NC1\]. For sufficiency, we write $v_i=v_1r_i$ for $i=2,3,\cdots, t+1$. Clearly, $r_i\geq 2$ and $r_i\not= r_j$ for $2\leq i\not=j\leq t+1$. We list all $t$-tuples from $Z_{r_2}\times Z_{r_3}\times \cdots \times Z_{r_{t+1}}$ to form an MOA$(\prod_{i=2}^{t+1}
r_i;t,t,(r_2,r_3,\cdots, r_t, r_{t+1})$, which is also a PDIMOA$^*(\prod_{i=2}^{t+1}r_i;t,t+1,(1,r_2,r_3,\cdots, r_t, r_{t+1})$. Apply Construction \[pc2\] with an OA$(v_1^t;t,t+1,v_1)$ given by Lemma \[OA(t,t+1,v)\] to obtain the required PDIMOA$^*$.
More generally, we have the following results.
Let $v_1<v_2<\cdots<v_k$ and $v_i=k_iv_1v_2\cdots v_{k-t}$, where $k_i\geq 2$, $i=k-t+1,k-t+2,\cdots, k$. Then, a PDIMOA$^*(\prod _{i=k-t+1}^ k v_i; t,k,(v_1,v_2,\cdots, v_k))$ exists.
[[**Proof.**]{} ]{}Let $M=v_1v_2\cdots v_{k-t}$. Then, $v_i=Mk_i$, where $i=k-t+1,\cdots, k$. By Theorem \[PDIMOA(t,t+1)\], a PDIMOA$^*(N;t,t+1,(M,v_{k-t+1},\cdots,v_k))$ with $N=\prod _{i=k-t+1}^ k v_i$ exists. Apply Construction \[f\] to obtain a PDIMOA$^*(\prod _{i=k-t+1}^ k v_i, t,k,(v_1,v_2,\cdots, v_k))$ as desired.
\[LA(2,3,v)\] Let $v_1\leq v_2\leq v_3$ with $v_2\geq 2v_1$. Then, an optimal $(\bar{1},2)$-LA$(v_2v_3;3,(v_1,v_2,v_3))$ exists.
[[**Proof.**]{} ]{}First, we construct a $v_2v_3\times 3$ array $A=(a_{ij})$ : $a_{i+rv_3,1}=(i-1+r)\%v_1$, where $i=1,2,\cdots, v_3$ and $r=0,1,\cdots,v_2-1$; $a_{i,2}=\left \lfloor\frac{i-1} {v_3} \right\rfloor $ and $a_{i,3}=(i-1)\%v_3$ for $i=1,2,\cdots, v_2v_3$.
We will prove that $A$ is an optimal $(\bar{1},2)$-LA. Optimality is guaranteed by Theorem \[L-bound\]. It is clear that $A$ is MCA$^*_2(v_2v_3, (v_1,v_2,v_3))$. Consequently, $|\rho(A,\{(1,a),(2,b)\})|\geq 2,
|\rho(A,\{(1,c),(3,d)\})|\geq 2$ and $|\rho(A,\{(2,e),(3,f)\})|=1$, where $a,c\in V_1,b,e\in V_2, d,f\in V_3$. It is clear that $\rho(A,\{(1,a),(2,b)\})\not =\rho(A,\{(2,e),(3,f)\})$ and $\rho(A,\{(1,c),(3,d)\})\not =\rho(A,\{(2,e),(3,f)\})$. We only need to prove $\rho(A,\{(1,a),(2,b)\}) \\\not=\rho(A,\{(1,c),(3,d)\})$. In fact, by construction, $\rho(A,\{(1,a),(2,b)\}) \subset \{rv_3+1,rv_3+2,\cdots, (r+1)v_3\}$ for a certain $r\in \{0,1,2,\cdots, v_2-1\}$ but $\{i,i+v_1v_3\}\subset \rho(A,\{(1,c),(3,d)\})$, where $i\in \{1,2,\cdots, v_1v_3\}$, which implies $\rho(A,\{(1,a),(2,b)\})\not =\rho(A,\{(1,c),(3,d)\})$. Thus, $A$ is a $(\bar{1},t)$-LA by Lemma \[(1,t)-LA and MCA\].
The following example illustrates the idea in Theorem \[LA(2,3,v)\].
The transpose of the following array is an optimal $(\bar{1},2)$-LA$(42;3,(3,6,7))$
2.0pt
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
\[LA(1,K,v)\] Let $2\leq w<v$ with $v\geq 2w$. Then, an optimal $(\bar{1},1)$-LA$(v;w+1,(w,w,\cdots,w,v))$ exists.
[[**Proof.**]{} ]{}First, we construct a $2w\times (w+1)$ array $A=(a_{ij})$ as follows: $$A=\left[
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & 0 \\
1 & 1 & \cdots & 1 & 1 \\
\vdots & \vdots &\vdots & \vdots &\vdots \\
w-1 & w-1 &\cdots & w-1 &w-1 \\
0 & 1 & \cdots & w-1 & w\\
1 & 2 & \cdots & 0 & w+1\\
\vdots & \vdots &\vdots & \vdots \\
w-1 & 0 &\cdots & w-2 & 2w-1 \\
\end{array}
\right]$$
When $v>2w$, let $C=(c_{ij})$ be a $(v-2w)\times (w+1)$ array with $c_{i,(w+1)}=i-1$ for $i=2w+1,2w+2,\cdots, v$ and $c_{i,j}$ be an arbitrary element for $\{0,1,\cdots, w-1\}$ with $i=2w+1,2w+2,\cdots,v, j=1,2,\cdots,w$. Let $M=A$ and $N=(A^T|C^T)^T$. It is easy to prove that $M$ and $N$ are the required arrays if $v=2w$ and $v>2w$, respectively.
Concluding Remarks
==================
LAs can be used to generate test suites for combinatorial testing and identify interaction faults in component-based systems. In this study, a lower bound on the size of $(\bar{1},t)$-LAs with mixed levels was determined. In addition, some constructions of $(\bar{1},t)$-LAs were proposed. Some of these constructions produce optimal locating arrays. Based on the constructions, some infinite series of optimal locating arrays satisfying the lower bound in Lemma \[case 1\] were presented. Obtaining new constructions for mixed-level $(\bar{1},t)$-LAs and providing more existence results are potential future directions.
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[^1]: Correspondence to: Ce Shi (shice060@lixin.edu.cn). The second author’s work was supported by NSFC grant 11301342 and Natural Science Foundation of Shanghai No. 17ZR1419900
|
---
abstract: 'We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expressions are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff matrix. These expressions involve the matrix of maximum out forests of the digraph. Every matrix of out forests with a fixed number of arcs and the normalized matrix of out forests are represented as polynomials of the Kirchhoff matrix; with the help of these identities, new proofs are given for the matrix-forest theorem and some other statements. A connection is specified between the forest dimension of a digraph and the degree of an annihilating polynomial for the Kirchhoff matrix. Some accessibility measures for digraph vertices are considered. These are based on the enumeration of spanning forests.'
author:
- 'R. P. Agaev'
- 'P. Yu. Chebotarev'
title: |
Spanning Forests of a Digraph\
and Their Applications[^1]
---
\#1\#2
[*\#1* ]{}[*\#2*]{}
\#1\#2
[*\#1* ]{}[*\#2*]{}
ł å n[1,…,n]{} \_\#1
Introduction
============
Directed graphs provide a simple and universal tool to model connection structures. It is not accidental that the first systematic monograph in the theory of digraphs [@HarNoCa] was titled “Structural Models: An Introduction to the Theory of Directed Graphs.” Digraphs frequently serve to model processes that can proceed in the direction of arcs. Physical transference, service, control, transmission of influences, ideas, innovations, and diseases are examples of such processes. If a process can start from a number of vertices and ends with the inclusion of all vertices, then the process can be modelled by the family of out forests (i.e., spanning diverging forests) of the digraph. The enumeration of all out forests allows one to determine the typical roles of the vertices in the process: one vertex is a typical starting point, another vertex is a typical intermediate point, some vertex is a typical terminating point of the process, etc. If an initial (weighted) digraph imposes some measure on the said processes, then the “role profile” of each vertex can be expressed numerically. Moreover, an exact answer can be given to the following important question: how likely is it that the process initiated at vertex $j$ arrives at vertex $i$. It is not surprising that out forests of a digraph turn out to be closely related with Markov chains realizable on the digraph.
The study of out forests has been started in [@Fiedler]. Generally, they were given less attention in the literature, than that given to spanning diverging trees (out arborescences), which exist only for a narrow class of digraphs. We mention in this connection [@Kelm; @LiuChow; @Cha; @MyrvoldA; @BapatConstantine; @Takacs; @Erdos; @Merr97; @Merr98], where still undirected forests were considered in most cases. The maximum out forests (i.e., out forests with the greatest possible number of arcs) of a digraph were studied in [@Che.rat; @Che.ra1]. It was established that the normalized matrix of such forests coincides with the matrix of limiting probabilities of every Markov chain [*related*]{} to the given digraph. Some results on spanning forests of directed and undirected multigraphs were given in [@CheSha97; @CheSha981].
In this paper, we study the normalized matrix of out forests (which has been also termed the matrix of relative forest accessibilities and the matrix of forest proximities) and the matrices of forests with fixed numbers of arcs.
Notation and some earlier results {#sec2}
=================================
In the terminology, we mainly follow [@HarNoCa; @Harary]. Suppose that $\G$ is a weighted digraph without loops, $V(\G)=\{\1n\}$ $(n>1)$ is its set of vertices, and $E(\G)$ its set of arcs. The weights of all arcs are supposed to be strictly positive. [*A subgraph*]{} of a digraph $\G$ is a digraph whose vertices and arcs belong to the sets of vertices and arcs of $\G$; the weights of subgraph’s arcs are the same as in $\G$. [*A restriction*]{} of $\G$ to $V'\subset V(\G)$ is a digraph whose arc set contains all the arcs in $E(\G)$ that have both incident vertices in $V'$. [*A spanning subgraph*]{} of $\G$ is a subgraph with vertex set $V(\G)$. The [*indegree*]{} id($w$) of vertex $w$ is the number of arcs that come to $w$, [*outdegree*]{} od($w$) of vertex $w$ is the number of arcs that come from $w$. A vertex $w$ will be called [*undominated*]{} if id($w$)=0 and [*dominated*]{} if id$(w)\ge 1$. A vertex $w$ is [*isolated*]{} if $\G$ contains no arcs incident to $w$.
A [*route*]{} in a digraph is an alternating sequence of vertices and arcs $w_0,e_1,$ $w_1\cdc e_k, w_k$ with every arc $e_i$ being $(w_{i-1},w_i)$. If every arc $e_i$ is either $(w_{i-1},w_i)$ or $(w_i,w_{i-1}),$ then the sequence is called a [*semiroute*]{}. A [*path*]{} in a digraph is a route all whose vertices are different. A [*circuit*]{} is a route with $w\_0=w_k$, the other vertices being distinct and different from $w_0$. A vertex $w$ [*is reachable*]{} from a vertex $z$ in $\G$ if $w=z$ or $\G$ contains a path from $z$ to $w$. A [*semicircuit*]{} is an alternating sequence of distinct vertices and arcs, $w_0,e_1,w_1\cdc
e_k,w_0,$ where every arc $e_i$ is either $(w_{i-1},w_i)$ or $(w_i,w_{i-1}$) and all vertices $w_0\cdc
w_{k-1}$ are different. The restriction of $\G$ to any maximal subset of vertices connected by semiroutes is called a [*weak component*]{} of $\G$. Let $E=(\e_{ij})$ be the matrix of arc weights. Its entry $\e_{ij}$ is zero if and only if there is no arc from vertex $i$ to vertex $j$ in $\G$. If $\G'$ is a subgraph of $\G$, then the weight of $\G'$, $\e(\G')$, is the product of the weights of all its arcs; if $\G'$ does not contain arcs, then $\e(\G')=1$. The weight of a nonempty set of digraphs $\GG$ is defined as follows: $$\e(\GG)=\suml_{H\in\GG}\e(H);$$ the weight of the empty set is 0.
The [*Kirchhoff matrix*]{} [@Tutte] of a weighted digraph $\G$ is the $n\times n$-matrix $L=L(\G)=(\l\_{ij})$ with elements $\l\_{ij}=-\e_{ji}$ when $j\ne i$ and $\l\_{ii}=-\suml_{k\ne i}\l\_{ik}$, $i,j=\1n$.
A [*diverging tree*]{} is a digraph without semicircuits that has a vertex (called the [*root*]{}) from which every vertex is reachable. The indegree of every non-root vertex of a diverging tree is 1. If $w$ is the root, then id$(w)=0$. A [*converging tree*]{} is a digraph without semicircuits that has a vertex (called the [*sink*]{}) reachable from every vertex.
A [*diverging forest*]{} ([*converging forest*]{}) is a digraph without circuits such that id$(w)\le1$ (respectively, od$(w)\le1$) for every vertex $w$. An [*out forest*]{} ([*in forest*]{}) of a digraph $\G$ is any its spanning diverging (respectively, converging) forest.
The weak components of diverging forests (converging forests) are diverging trees (respectively, converging trees).
\[De2\] An out forest $F$ of a digraph $\G$ is called a [*maximum out forest*]{} of $\G$ if $\G$ has no out forest with a greater number of arcs than in $F$. An in forest $F$ of a digraph $\G$ is a [*maximum in forest*]{} of $\G$ if $\G$ has no in forest with a greater number of arcs than in $F$.
Obviously, every maximum out forest of $\G$ has the minimum possible number of weak components (out trees); this number will be called the [*out forest dimension*]{} of the digraph and denoted by $v$. The number of arcs in any maximum out forest is obviously $n-v$. The number of weak components of every maximum in forest will be called the [*in forest dimension*]{} of the digraph and denoted by $v'$. Obviously, for every digraph, $v,v'\in\{1\cdc n\}$.
If a digraph $\G_1$ is obtained from $\G$ by the reversal of all arcs, then the out forests in $\G$ naturally correspond to the in forests in $\G_1$ and vice versa. Therefore, the out forest dimension and in forest dimension of $\G$ are respectively equal to the in forest dimension and out forest dimension of $\G_1$.
The following proposition states that the dimensions $v$ and $v'$ of a digraph are not connected, except for the case where $v=n$ and $v'=n$.
\[propvv’\] $1.$ Let $k,k'\in\{1\cdc n-1\}$. Then there exists a digraph on $n$ vertices such that $v=k$ and $v'=k'$.
$2.$ For every digraph $\G$ on $n$ vertices$,$ $v=n\Leftrightarrow v'=n\Leftrightarrow
E(\G)=\emptyset$.
The proofs are given in the Appendix.
Throughout the paper, we mainly deal with diverging forests. However, all the results have counterparts formulated in terms of converging forests. Simple properties of out forests have been studied in [@Che.ra1] (Section 3). We do not cite them here and only confine ourselves to the following
\[prop2\] If $i$ and $j$ belong to different trees in a maximum out forest $F$ of a digraph $\G,$ and $j$ is a root in $F,$ then $\G$ contains no paths from $i$ to $j$.
Let us adduce some definitions and results from [@Che.ra1] which are frequently used below.
\[De1\] A nonempty subset of vertices $K\subseteq V(\G)$ of digraph $\G$ is an [*undominated knot*]{}[^2] in $\G$ iff all the vertices that belong to $K$ are mutually reachable and there are no arcs $(w_j,w_i)$ such that $w_j\in V(\G)\setminus K$ and $w_i\in K$.
Suppose that $\ktil=\cupo^{u}_{i=1}K_i$, where $K_1\cdc K_u$ are all the undominated knots of $\G$, and $K_i^{+}$ is the set of all vertices reachable from $K_i$ and unreachable from the other undominated knots. For any undominated knot $K$ of $\G,$ denote by $\G_K$ the restriction of $\G$ to $K$ and by $\G_{-K}$ the subgraph with vertex set $V(\G)$ and arc set $E(\G)\setminus E(\G_K)$. For a fixed $K$, $\TT$ will designate the set of all spanning diverging trees of $\G_K$ and $\PP$ will be the set of all maximum out forests of $\G_{-K}$. By $\TT^k$, $k\in K$, we denote the subset of $\TT$ consisting of all trees that diverge from $k$, and by $\PP^{K \rightarrow i}$, $i\in V(\G)$, the set of all maximum out forests of $\G_{-K}$ such that $i$ is reachable from some vertex that belongs to $K$ in these forests.
By $\FF(\G)=\FF$ and $\FF_k(\G)=\FF_k$ we denote the set of all out forests of $\G$ and the set of all out forests of $\G$ with $k$ arcs, respectively; $\FF^{i\rightarrow j}_k$ will designate the set of all out forests with $k$ arcs where $j$ belongs to a tree diverging from $i$.
\[De3\] The matrix $\vj=(\q_{ij})=\si^{-1}Q_{n-v},$ where $\si=\e(\FF_{n-v}),$ $Q_{n-v}=(q\_{ij})=(\e(\FF^{j \to i}_{n-v}))$, will be called the [*normalized matrix of maximum out forests*]{} of a digraph.
\[t1.che.ra1\]
Suppose that $\G$ is an arbitrary digraph and $K$ is an undominated knot in $\G$. Then the following statements are true.
[1.]{} $\vj$ is a stochastic matrix$:$ $\q_{ij}\ge0,$ $\;\suml^n_{k=1}\q_{ik}=1,\;$ $i,j=\1n.$
[2.]{} $\q_{ij}\ne 0\;\Leftrightarrow\; (j\in \ktil$ and $i$ is reachable from $j$ in $\G).$
[3.]{} Suppose that $j\in K.$ For any $i\in V(\G),$ $\q_{ij}=\e(\TT^j)\e(\PP^{K\rightarrow i})\slash
\e(\FF_{n-v})$. Furthermore$,$ if $i\in K^{+},$ then $\q_{ij}=\q_{jj}=\e(\TT^j)\slash \e(\TT)$[.]{}
[4.]{} $\suml_{j\in K}\q_{jj}=1.$ In particular$,$ if $j$ is an undominated vertex$,$ then $\q_{jj}=1.$
[5.]{} If $j_1,j_2\in K$, then $\q_{\cdot j\_2}=(\e(\TT^{j_2}) \slash\e(\TT^{j_1}))\q_{\cdot j_1},$ i.e.$,$ the $j_1$ and $j_2$ columns of $\q$ are proportional[.]{}
\[t2.che.ra1\] For every weighted digraph$,$ $\vj$ is idempotent$:$ $\;\q^{2}=\q.$
\[t3.che.ra1\] For every weighted digraph$,$ $L\q=\q L=0$.
$($a parametric version of the matrix-forest theorem$).$ [*For any weighted multidigraph $\G$ with positive weights of arcs and any $\tau>0,$ there exists the matrix $Q(\tau)=(I+{\tau}L(\G))^{-1}$ and $$Q(\tau)=\frac{1}{s(\tau)} \suml^{n-v}_{k=0}{\tau}^k Q_k, \label{razlo}$$ where $$\label{raz2} s(\tau)=\suml^{n-v}_{k=0}{\tau}^k\e(\FF_k), \;\; Q_k=(q^k_{ij}), \;\; q^k_{ij}=\e(\FF^{j\to
i}_k), \;\; k=0\cdc n-v, \;\; i,j=\1n.$$* ]{}
\[DeQk\] The matrix $Q_k,\,$ $k=0\cdc n-v,$ will be called the [*matrix of out forests of $\G$ with $k$ arcs*]{}.
Theorem 4 represents $(I+\tau L)^{-1}$ via the matrices of out forests with various numbers of arcs.
\[DeQ(tau)\] The matrices $Q(\tau)=(I+\tau L)^{-1},\,$ $\tau>0,$ will be called the [*normalized matrices of out forests*]{} of a digraph.
In [@CheSha97], the matrices $Q(\tau)=(I+\tau L)^{-1}$ were referred to as the matrices of relative forest accessibilities of a digraph. In Section \[Sec\_Poli\], $Q(\tau)$ are expressed as polynomials of $L$ (Corollary from Theorem \[sumsum7\]).
\[t5.che.le2\] For every weighted digraph $\G,$ $\liml_{\tau\to\infty}Q(\tau)=\liml_{\tau\to\infty}(I+\tau\,L)^{-1}=\vj.$
Matrices of out forests and transition probabilities of Markov chains {#forest}
=====================================================================
It has been shown in [@Che.ra1] that the matrix of Cesàro limiting probabilities of a Markov chain coincides with the normalized matrix $\vj$ of maximum out forests of any digraph related to this Markov chain. Now we give a Markov chain interpretation for the normalized matrices of out forests $Q(\tau)$ with any $\tau>0.$
A homogeneous Markov chain with set of states $\{\1n\}$ and transition probability matrix $P$ is [*related to a weighted digraph*]{} $\G$ iff there exists $\alpha\ne0$ such that
$$\label{7.1} P=I-\alpha\,L(\G).$$
Let $\G$ be a weighted digraph. Consider an arbitrary Markov chain [*related to*]{} $\G$ and the following observation model.
[**The geometric model of random observation.**]{} [Suppose that a Bernoulli trial is performed at the point of time $t=0$ with success probability $q$ $(0<q<1)$. In case of success$,$ $t=0$ becomes the epoch of observation. Otherwise, Bernoulli trials are performed at $t=1,2,\ldots$—to the point of the first success. This point becomes the epoch of observation.]{}
This model determines a discrete probability distribution $p(k)$ of the epoch of observation on the set $\{0,1,2,\ldots\}$. This is obviously the geometric distribution (which gives the name of the model) with parameter $q$: $$\label{Geo} p(k)=q(1-q)^k,\quad k=1,2,\ldots$$
Consider Markov chain multistep transitions in a [*random number of steps*]{}: from the initial state at $t=0$ to the state at the random epoch of observation distributed geometrically with parameter $q$.
Suppose that $\Ptil(\aa,q)=\left(\ptil\_{ij}(\aa,q)\right)$ is the matrix of unconditional probabilities for such multistep transitions: from the initial state to the state at the epoch of observation.
\[prop1\] For any weighted digraph$,$ any $\tau>0$ and any Markov chains related to the weighted digraph$,$ $$Q(\tau)=\Ptil(\aa,q)$$ holds$,$ where $$\label{qtaual} q=({\tau/\aa}+1)^{-1}.$$
Theorem \[prop1\] provides an interpretation for the normalized matrix $Q(\tau)$ of out forests in terms of Markov chain transition probabilities. Conversely, for any Markov chain, the transition probabilities in the geometric observation model can be interpreted in terms of diverging forests of the corresponding digraphs.
The following corollary stresses the arbitrariness of Markov chains in Theorem \[prop1\].
[**1 from Theorem \[prop1\]**]{} [For every Markov chain$,$ every success probability $q\in\,]0,1[$ in the geometric observation model$,$ and every digraph related to the Markov chain$,$ $$\Ptil(\aa,q)=Q(\tau)$$ holds$,$ where $\tau=(q^{-1}-1)\aa.$ ]{}
[**2 from Theorem \[prop1\]**]{}[ $$\label{Cor1} \lim_{q\to+0}\Ptil(\aa,q)=\vj=\lim_{k\to\infty} \frac{1}{k} \suml_{p=0}^{k-1} P^{k}.$$ ]{}
By Corollary 2 from Theorem \[prop1\], at a vanishingly small success probability $q$, the transition probabilities in the geometric observation model are given by the matrix $\vj$ of maximum out forests of any weighted digraph to which this chain is related.
Representations of forest matrices via the Kirchhoff matrix and their consequences {#Sec_Poli}
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In this section, we represent the matrices $Q_k$ of out forests with $k$ arcs as polynomials of the Kirchhoff matrix $L$ (Theorem \[teo.allk\]). This allows one to obtain alternative proofs of Theorems 2–4 and to represent the matrix $Q(\tau)=(I+\tau L)^{-1}$ as a polynomial of $L$ (Theorem \[sumsum7\]). Proposition \[recur\] gives an easy way to calculate $Q_k,\,$ $k=1\cdc n-v,$ and $\q$.
By $\si\_{k}$ we denote the total weight of all out forests of $\G$ with $k$ arcs: $\si\_{k}=\e(\FF_{k}),\,$ $k=0\cdc n-v.$
\[pro.allk\] For any weighted digraph and any $k=0\cdc n-v,$ $$\label{QQQ} Q_{k+1}=\si\_{k+1}\!I-L Q_{k}.$$
Observe that since the weight of the empty set is 0, we have $Q_{n-v+1}=0$ and $\si\_{n-v+1}=0$.
Taking the traces on the left-hand side and the right-hand side of (\[QQQ\]) and using the fact that $$\label{TrQ} \tr{\x}(Q_{k})=(n-k)\si\_{k},\quad k=0\cdc n-v+1$$ (because every out forest with $k$ arcs has $n-k$ roots), we deduce $$\label{si_k+1} \si_{k+1}=\frac{\tr{\x}(LQ_k)}{k+1},\quad k=0\cdc n-v.$$
Substituting (\[si\_k+1\]) in (\[QQQ\]) provides
\[recur\] For every weighted digraph$,$ $$\label{QQ+} Q_{k+1}=\frac{\tr{\x}(LQ_k)}{k+1}I-LQ_{k},\quad k=0\cdc n-v.$$
Identity (\[QQ+\]) enables one to recursively determine the matrices $Q_k,\,$ $k=0\cdc n-v,$ and $\q$, starting with $Q_0=I.$ Note that this procedure essentially coincides with Faddeev’s algorithm [@Faddeev] for the computation of the characteristic polynomial as applied to $L$. Thus, the matrices involved in Faddeev’s method are precisely $Q_k$.
From Proposition \[pro.allk\], it follows
\[teo.allk\] For any weighted digraph and any $k=0\cdc n-v,$ $$\label{psibek} Q_{k}=\suml_{i=0}^{k} \si\_{k-i} {(-L)}^{i}.$$
[**1 from Theorem \[teo.allk\]**]{} [For every weighted digraph$,$ matrices $Q_k,$ $k=0\cdc n-v,$ commute with all matrices with which $L$ commutes$,$ in particular$,$ with $L,$ $\vj,$ $Q(\tau),$ and each other. ]{}
\[sumstr\] For any $k=0\cdc n-v,$ every row sum of $L Q_{k}$ is $0$.
From Proposition \[pro.allk\] and Lemma \[sumstr\], it follows
\[allmatrbek\] The matrices $L Q_{k},$ $k=0\cdc n-v,$ are the Kirchhoff matrices of some weighted digraphs.
[**2 from Theorem \[teo.allk\]**]{} [For any weighted digraph$,$ $LQ_{n-v}=Q_{n-v}L=0.$]{}
In view of Definition \[De3\], this corollary is equivalent to Theorem \[t3.che.ra1\]. Thus, we get a new proof of this theorem.
Consider the matrices $$\label{*} \q_k=\si_{k}^{-1}Q_k,\quad k=0{\x}\cdc n-v.$$ In particular, $\q_0=I$ and $\q_{n-v}=\q$.
Making use of the last corollary, we obtain
[**3 from Theorem \[teo.allk\]**]{} [For any $k\in\{\1n-v\},\,$ $\q_k\q=\q\q_k=\q.$ In particular$,$ $\q_{n-v}\vj=\vj^2=\vj.$ Moreover$,$ $Q(\tau)\q=\q Q(\tau)=\q$ for every $\tau>0.$ ]{}
This corollary provides a new proof of Theorem \[t2.che.ra1\].
By virtue of Proposition \[pro.allk\] and Corollary 1 from Theorem \[teo.allk\], the matrices $\q_{k}$ are connected as follows: $$\label{j_k+1} \q_{k+1}=I-\frac{\si\_k}{\si\_{k+1}}\q_{k} L, \quad k=0\cdc n-v-1,$$ and, by Lemma \[sumstr\], each their row is unity. The entries of $\q_k$ are nonnegative by definition, thus, we obtain
\[propQk1\] For every weighted digraph $\G,$ matrices $\q_k,\,$ $k=0\cdc n-v,$ are stochastic.
Completing Proposition \[pro.allk\] with the obvious equality $Q_{0}=I=\si\_0 I$ gives $$\label{system1} \cases{
Q_{0}=\si\_0 I, \cr
Q_{1}+L Q_{0}=\si\_1 I, \cr
\ldots\ldots\ldots\ldots\ldots\ldots \cr
Q_{n-v}+L Q_{n-v-1}=\si\_{n-v} I. \cr
}$$
Add up these equations and, using Corollary 2 from Theorem \[teo.allk\], substitute $(I+L)Q_{n-v}$ for $Q_{n-v}$: $$(I+L)Q_{0}+(I+L)Q_{1}+\ldots+(I+L)Q_{n-v} =\Bigl(\suml_{k=0}^{n-v}\si\_k\Bigr)I.$$
Making use of the nonsingularity of $I+L$ (Theorem 4) and the notation $s=\e(\FF)=\suml_{k=0}^{n-v}\si\_k$, we obtain $$\label{iden1} \suml_{k=0}^{n-v}Q_k = s(I+L)^{-1},$$ which provides
[**from Proposition \[pro.allk\]**]{} [For any weighted digraph$,$ $$Q(1)=(I+L)^{-1}=s^{-1}\suml_{k=0}^{n-v}Q_k.$$ ]{}
This statement coincides with the matrix-forest theorem for digraphs [@CheSha97] and with Theorem 4 in the case of $\tau=1$. Accordingly, we obtain a new proof of the matrix-forest theorem.
By means of Theorem \[teo.allk\], the matrices $Q(\tau)={(I+\tau L)}^{-1}=s^{-1}\suml_{k=0}^{n-v}\tau^k
Q_k$ including $Q(1)={(I+L)}^{-1}$ can be represented as polynomials of $L$.
\[sumsum7\] For any weighted digraph$,$ $$\label{eqth7} Q(1)={(I+L)}^{-1}=s^{-1}\suml_{i=0}^{n-v}s\_{n-v-i}(-L)^i,$$ where $s\_k=\suml_{j=0}^k\si\_j$ is the total weight of out forests of $\G$ with at most $k$ arcs$,$ $k=0\cdc
n-v$.
[**from Theorem \[sumsum7\]**]{} [For any weighted digraph and any $\tau>0,$ $$\label{eqco7} Q(\tau)={(I+\tau L)}^{-1}= s^{-1}(\tau)\suml_{i=0}^{n-v}s\_{n-v-i}(\tau)(-\tau L)^i,$$ where $s\_k(\tau)=\suml_{j=0}^{k}\tau^j\si\_j,\:$ $k=0\cdc n-v$. ]{}
Note that $s(I+L)^{-1}$ is the adjugate (the transposed matrix of cofactors) of $I+L$; $s(\tau)\,(I+\tau
L)^{-1}$ is the same for $I+\tau L$. Theorem \[sumsum7\] and the above corollary provide representations for these matrices as polynomials of $L$: $$\begin{aligned}
\label{iden2} s\,(I+L)^{-1} &=&\suml_{i=0}^{n-v}s\_{n-v-i}(-L)^i,\cr \label{iden3} s(\tau)\,(I+\tau L)^{-1}
&=&\suml_{i=0}^{n-v}s\_{n-v-i}(\tau)\,(-\tau L)^i.\end{aligned}$$
Since $L Q_k$ the is Kirchhoff matrix of some weighted digraph (Proposition \[allmatrbek\]), all its principal minors are nonnegative (by Theorem 6 in [@Fiedler]). Therefore, all $L Q_k$ are singular $M$-matrices (see, e.g., item (A1) of Theorem 4.6 in [@Abraham.Robert]). Alternatively, this can be concluded from the nonnegativity of the real parts of the eigenvalues (see Proposition \[Gosha\] below) and the nonpositivity of off-diagonal elements of $L$ (item (F12) of Theorem 4.6 in [@Abraham.Robert]). It follows from the representation $\si\_{k+1}I-Q_{k+1}=LQ_k$ (Proposition \[pro.allk\]) of the singular $M$-matrix $L Q_k$ that $\si\_{k+1}=\rho(Q_{k+1})$, i.e., $\si\_{k+1}$ is the spectral radius of $Q_{k+1}$, $k=0\cdc n-v-1$. This also follows from Proposition \[propQk1\] (see (\[\*\])).
On some linear transformations related to digraphs {#line}
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For a matrix $A\in\R^{n\times n}$, by ${\bf A}$ we denote the linear transformation ${\bf A}:\R^n\to\R^n$ induced by $A$ with respect to the standard basis of $\R^n$: ${\bf A}({\bf x})=A{\bf x}$. $\RR({\bf A})$ and $\NN({\bf A})$ will designate the range and the null space of ${\bf A}$, respectively.
As has been seen in [@Che.ra1], the dimensions of $\RJ$ and $\RLT$ are $v$ and $n-v$, respectively. Furthermore, $\RLT\cap\RJ=\{{\bf 0}\}$ and, since the dimensions of $\RLT$ and $\RJ$ sum to $n$, $\R^n$ decomposes to the direct sum of $\RLT$ and $\RJ$: $$\label{pr.summa} \R^n=\RLT\dot+\RJ.{\x}$$
Since $L\!\vj=0$ (Theorem \[t3.che.ra1\]), we get $\NL=\RJ$ and $\NJT=\RLT$, thus, the sum (\[pr.summa\]) is orthogonal.
Similarly, in view of $\vj\! L=0$, the orthogonal decomposition $$\R^n=\RL\dot+\RJT{\x}
$$ holds along with $\RL\cap\RJT=\{{\bf 0}\}$, $\NJ=\RL$, and $\NLT=\RJT$.
In accordance with (\[pr.summa\]), every vector ${\bf u}\in\R^n$ is uniquely represented as ${\bf u}={\bf
u}\_1+{\bf u}\_2,$ where ${\bf u}\_1\in \RLT=\NJT$ and ${\bf u}\_2\in\RJ=\NL$. For every ${\bf u}\ne{\bf
0}$, we have ${(L+\vj}^{\intercal}) {\bf u}={(L+\vj}^{\intercal}){\bf u}\_1+$ ${(L+\vj}^{\intercal}) {\bf
u}\_2$ $=L{\bf u}\_1+{\vj}^{\intercal}{\bf u}\_2.$ If $L{\bf u}\_1+{\vj}^{\intercal}{\bf u}\_2=0$, then, since $\RL\cap\RJT=\{{\bf 0}\}$, we have $L{\bf u}\_1={\vj}^{\intercal}{\bf u}\_2={\bf 0}$, whence, by $\NL\cap\NJT=\{{\bf 0}\}$, ${\bf u}\_1={\bf u}\_2={\bf 0}$ results. Therefore, the dimension of the range (rank) of ${\bf Z=L+\vj}^{\intercal}$ is $n$. Thus, we obtain
\[invert.Ltj\] For any weighted digraph $\G,$ the matrix $Z=L+\vj^{\intercal}$ is nonsingular.
We will also need the nonsingularity of $L+\vj$.
\[fulllj\] For any weighted digraph $\G,$ the matrix $L+\q$ is nonsingular.
[**from Theorem \[fulllj\]**]{} [For any weighted digraph and any $\aa\ne 0,$ the matrix $L+\aa\vj$ is nonsingular.]{}
It follows from $\vj^2=\vj$ (Theorem \[t2.che.ra1\]) that every nonzero columns of $\vj$ is an eigenvector of $\vj$ associated with the eigenvalue 1. Hence, for any ${\bf u}\in\RJ$, ${\bf\vj}({\bf
u})={\bf u}$ holds, therefore, $\RJ$ is exactly the subspace of fixed vectors of ${\bf\vj}$.
The Moore-Penrose and group inverses of the Kirchhoff matrix {#pseudo2}
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In this section, we obtain some expressions for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff matrix $L$. The [*Moore-Penrose generalized inverse*]{} of a rectangular complex matrix $A$ is the unique matrix $X$ such that
(1) $AXA=A,$
(2) $XAX=X,$
(3) $(AX)^*=AX,$ (4) $(XA)^*=XA,$ where $(AX)^*$ and $(XA)^*$ are the conjugate transposes (Hermitian adjoints) of $AX$ and $XA$, respectively.
For any matrix $A,$ the Moore-Penrose generalized inverse, $A^+$, does exist and is unique. If $A$ is nonsingular, then $A^+$ coincides with $A^{-1}$.
The Moore-Penrose inverses are of theoretical and practical interest. The latter is because $A^+$ provides the normal pseudosolution of the inconsistent equation $A{\bf x}={\bf b}$: it is ${\bf x}=A^+{\bf b}$. The normal pseudosolution is a vector of the minimum length that minimizes the length of $A{\bf x}-{\bf b}$ (the minimum norm least-squares solution). As applied to Laplacian matrices, such solutions, among others, were considered for some preference aggregation problems (more specifically, estimation from paired comparisons) [@CheSha99], in constructing geometrical representations for systems modelled by graphs [@Hall], in the analysis of social networks, and cluster analysis.
The group inverses are no less important (see, e.g., [@CampMey]). A matrix $X$ is the [*group inverse*]{} of a square matrix $A$, if $X$ satisfies the conditions (1) and (2) in the definition of Moore-Penrose generalized inverse and also
\(5) $AX=XA$.
The group inverse of $A$ is denoted by $\Apr$. Generally, group inverses need not exist, but if such a matrix exists, then it is unique, but $A^+=\Apr$ is not necessary.
If $L$ is symmetric (in particular, this is the case for symmetric digraphs, which can be identified with undirected graphs), then the matrix $(L+\aa\vj)^{-1}-\aa^{-1}\vj$ (with any $\aa>0$) is [@CheSha981] the Moore-Penrose generalized inverse and the group inverse of $L$. Moreover, the latter is true for every digraph.
\[groupinv\] For every weighted digraph and any $\aa\ne0,$ $$\label{qtil} \Lpr=(L+\aa\vj)^{-1}-\aa^{-1}\vj$$ and $$\Lpr L=L\Lpr=I-\q.$$
As well as in the case of undirected graphs, $\Lpr=(\lpr_{ij})$ can be obtained via a passage to the limit.
\[qtiLim\] For every weighted digraph$,$ $$\Lpr=\liml_{\tau\to\infty}\tau\Big(Q(\tau)-\vj\Big).$$
We now express $\Lpr$ in terms of the normalized matrices of out forests $\q_{n-v-1}$ and $\q_{n-v}=\q$ (see (\[\*\])). The following proposition is an analogue of Theorem 3 in [@CheSha981].
\[topol\] For every weighted digraph$,$ $$\Lpr=\frac{\si_{n-v-1}}{\si_{n-v}}\left(\q_{n-v-1}-\q\right).$$
Because of the nonsymmetry of $I-\q=\Lpr L=L\Lpr$, $\Lpr$ is not generally the Moore-Penrose generalized inverse of $L$ for digraphs. To obtain an explicit formula for $L^+,$ consider the matrix $Z=L+\vj^{\intercal}$ which, by Theorem \[invert.Ltj\], is nonsingular. Using the identity $L\vj=0$ (Theorem \[t3.che.ra1\]), we obtain $${(Z^{\intercal})}^{-1}Z^{-1} =(ZZ^{\intercal})^{-1} =(\vj^{\intercal}\!\vj+LL^{\intercal})^{-1}.$$
\[commut\] For every weighted digraph$,$ ${(ZZ^{\intercal})}^{-1}$ commutes with $LL^{\intercal}$ and $\vj^{\intercal}\!\vj$.
Matrices $L L^{\intercal}$, $\vj^{\intercal}\!\vj$, and ${(ZZ^{\intercal})}^{-1}$ are symmetric. The product of two symmetric matrices is symmetric iff they are commuting [@HoJo]. This implies the following corollary.
[**from Lemma \[commut\]**]{} [For every weighted digraph$,$ the matrices $LL^{\intercal}{(ZZ^{\intercal})}^{-1}$ and $\vj^{\intercal}\!\vj{(ZZ^{\intercal})}^{-1}$ are symmetric. ]{}
These facts are useful for the proof of the following theorem.
\[pseudoor\] For every weighted digraph$,$ the matrix $L^{\intercal}{(ZZ^{\intercal})}^{-1}=L^{\intercal}
(\vj^{\intercal}\!\vj+L L^{\intercal})^{-1}$ is the Moore-Penrose generalized inverse of $L$.
On the Geršgorin region and annihilating polynomials for the Kirchhoff matrix
=============================================================================
By the Geršgorin theorem (see, e.g., [@HoJo]), the eigenvalues of a matrix $A$ belong to the union $G(A)$ of $n$ discs: $$\label{Gershgorn} G(A)=\cupo_{i=1}^{n}\Bigl\{z\in\C\:\Big\vert\;|z-a_{ii}|\le R'_i(A)\Bigr\},$$ where $\C$ is the complex field and $R'_i(A)=\suml_{j \ne i}|a_{ij}|,$ $i=\1n,$ are the deleted absolute row sums of $A$.
Since $R'_i (L)=\l_{ii}$ holds, (\[Gershgorn\]) can be represented as follows: $$\label{GershgornL} G(L)=\cupo_{i=1}^{n}\Bigl\{z\in\C\;\Big\vert\:|z-\l_{ii}| \le\l_{ii}\Bigr\}.$$
Hence, we have
\[Gosha\] [(1)]{} The real part of every eigenvalue of $L$ is nonnegative$:$ every Geršgorin disc belongs to the right coordinate half-plane$;$
[(2)]{} the intersection of all Geršgorin discs contains zero$;$
[(3)]{} $G(L)=\Bigl\{(z+1) \maxl_{1\le i\le n} \l_{ii}\:\Big\vert\; |z| \le 1\Bigr\}$.
Obviously, the intersection of all Geršgorin discs consists of zero iff the digraph contain an undominated vertex.
Consider the characteristic polynomial of $L$: $$p\_L(\la)=\suml_{i=0}^{n}{(-1)}^i E_{i}(L)\la^{n-i},$$ where $E_{i}(L)$ is the sum of all principal minors of order $i$. By Theorem 6 in [@Fiedler], $E_{i}(L)=\si\_i$ for every $i=\1n$. Since every principal minor of order greater than $n-v$ is zero, we have $p\_L(\la)={(-1)}^{n-v}\la^v \suml_{i=0}^{n-v}
\si\_{i}{(-\la)}^{n-v-i}=\la^v\suml_{i=0}^{n-v}(-1)^{i} \si\_{i}{\la}^{n-v-i}$.
\[annul\] $p'_L(\la)=\la \suml_{i=0}^{n-v}\si\_{n-v-i}{(-\la)}^{i}$ is an annihilating polynomial for $L$.
Accessibility via forests and dense forests in digraphs {#dostup}
=======================================================
Forest accessibility {#dostup1}
--------------------
The entries of $Q(\tau)$ measure the proximity of the vertices of an undirected multigraph [@CheSha97; @CheSha981]. The matrix $\vj^{\intercal}=\lim_{\tau\to \infty}Q^{\intercal}(\tau)$ was analyzed in [@Che.ra1] as the matrix of limiting accessibilities of a multidigraph. Here, we study the matrix $P\_1(\tau)=Q^{\intercal}(\tau)$ with $\tau>0$ as an accessibility measure for digraph vertices. By Theorem 4, the $(i,j)$-entry of this matrix is the total weight of out forests that “connect” $i$ with $j$ in the digraph where the weights of all arcs are multiplied by $\tau$. Along with $P\_1(\tau)$, we consider the matrix of in forests $P\_2(\tau)$. Its $(i,j)$-entry is the total weight of in forests (of the modified digraph) where $j$ is a sink and $i$ belongs to a tree converging to $j$.
The following definition is formulated for an arbitrary vertex accessibility measure (formally, every square matrix of order $n$ or, more precisely, the corresponding matrix-valued function of a digraph can be considered as such a measure). A measure $P\_2$ is called to be [*dual*]{} to a measure $P\_1$ if under the reversal of all arcs in an arbitrary digraph (provided that the weights of the arcs are preserved), the matrix of $P\_2$ for the modified digraph coincides with $P_1^{\intercal}$ calculated for the initial digraph. It follows from this definition that $P\_2$ is dual to $P\_1$ if and only if $P\_1$ is dual to $P\_2.$ In [@CheSha981], three self-dual accessibility measures were studied.
Let us check the satisfaction of the characteristic conditions listed below for $P\_1(\tau)$ and $P\_2(\tau)$. Triangle inequality for accessibility measures requires the symmetry of the corresponding matrix (see, e.g., [@CheSha982]). For that reason, we will check this condition for $P\_3(\tau)=(P\_{1}(\tau)+P\_{2}(\tau)+ P_{1}^{\intercal}(\tau)+P_{2}^{\intercal}(\tau))/4$.
[**Nonnegativity. **]{} [For any digraph $\G,$ $\;p_{ij}\ge 0,\;\:i,j\in V(\G)$.]{}
[**Diagonal maximality. **]{}
For any digraph $\G$ and any distinct $i,j\in V(\G),$
[(1)]{} $p\_{ii}>p\_{ij}$ and
[(2)]{} $p\_{ii}>p\_{ji}$ hold.
[**Disconnection condition. **]{} [For any digraph $\G$ and any $i,j\in V(\G),\;$ $p\_{ij}=0$ if and only if $j$ is unreachable from $i$.]{}
[**Triangle inequality for accessibility measures. **]{} [For any digraph $\G$ and any $i,j,k\in V(\G),$ $p\_{ij}+p\_{ik}-p\_{jk}$ $\le p\_{ii}$ holds. If, in addition, $j=k$ and $i\ne j$, then the inequality is strict.]{}
[**Transit property. **]{} [For any digraph $\G$ and any $i,k,t\in V(\G),$ if $\G$ includes a path from $i$ to $k,$ $i\ne k\ne t,$ and every path from $i$ to $t$ contains $k,$ then [(1)]{} $p\_{ik}>p\_{it};$ [(2)]{} $p\_{kt}>p\_{it}$.]{}
[**Monotonicity. **]{}
Suppose that the weight of some arc $\e_{kt}^p$ in a digraph $\G$ increases. Then[:]{} $\D p\_{kt}>0$ and for any $i,j\in V(\G),$ $(i,j)\ne (k,t)$ implies $\D p\_{kt}>\D p\_{ij};$ For any $i\in V(\G),$ if there is a path from $k$ to $t,$ and each path from $k$ to $i$ includes $t,$ then $(a)$ $\D p\_{kt}>\D p\_{ki}$ and $(b)$ $\D p\_{ki}>\D p\_{ti};$
[(3)]{} For any $i\in V(\G),$ if there is a path from $i$ to $k$ and every path from $i$ to $t$ includes $k,$ then $(a)$ $\D p\_{kt}>\D p\_{it}$ and $(b)$ $\D p\_{it}>\D p\_{ik}.$
The results of testing $P\_1(\tau),$ $P\_2(\tau),$ and $P\_3(\tau)$ are collected in the following proposition.
\[otledostup\] The measures $P\_1(\tau)$ and $P\_2(\tau)$ are dual to each other for every $\tau>0$. They satisfy nonnegativity$,$ reversal property$,$ disconnection condition$,$ the first part of item $1,$ and item $2$ of monotonicity. Moreover$,$ $P\_1(\tau)$ satisfies items [1]{} of diagonal maximality and transit property$;$ $P\_2(\tau)$ satisfies items [2]{} of these conditions. With respect to the remaining statements of monotonicity$,$ $P\_1(\tau)$ satisfies items $2$ and $3b$, whereas $P\_2(\tau)$ satisfies items $3$ and $2b,$ and they both violate the second part of item $1.$ Furthermore$,$ $P\_1(\tau)$ breaks item $3a$ of monotonicity and item $2$ of transit property$,$ whereas $P\_2(\tau)$ breaks item $2a$ of monotonicity and item $1$ of transit property. Triangle inequality for $P\_3(\tau)$ is not satisfied.
As was noted in [@Che.ra1], the limiting accessibility $P=\vj^{\intercal}$ of a digraph does not completely correspond to the general concept of proximity. Notice that [*disconnection condition*]{}, which is satisfied for the limiting accessibility in one side only, is completely fulfilled for $P\_1(\tau)$ and $P\_2(\tau)$. Moreover, $P\_1(\tau)$ and $P\_2(\tau)$ obey a number of conditions which are satisfied by the limiting accessibility in the nonstrict form only.[^3]
Accessibility via dense forests {#dostup2}
-------------------------------
Now we consider a measure which is intermediate between the limiting accessibility (which depends on $Q_{n-v}$ only) and the forest accessibility $Q(\tau)$ (which is a weighted sum of all matrices $Q_k$). This new measure is determined by the matrices $Q_{n-v-1}$ and $Q_{n-v}$ (or, equivalently, by the matrices $\q_{n-v-1}$ and $\q_{n-v}=\q$, which also determine $\Lpr$ as stated in Proposition \[topol\]). This measure can be also obtained by the inversion of $L+\aa\vj$ with some values of $\aa$.
Thus, consider the matrices $R(\aa)=(r_{ij})=(L+\aa\vj)^{-1}$ with $\aa>0$. Using Theorem \[groupinv\] and Proposition \[topol\], we have $$\label{f21} (L+\aa\vj)^{-1}=\Lpr+\aa^{-1}\vj =\frac{\si_{n-v-1}}{\si_{n-v}}\q_{n-v-1}
+\left(\aa^{-1}-\frac{\si_{n-v-1}}{\si_{n-v}}\right)\q.$$
If $0<\aa<\frac{\si_{n-v}}{\si_{n-v-1}}$, then, by (\[f21\]), $(L+\aa\oj)^{-1}$ is the sum of $Q_{n-v-1}$ and $Q_{n-v}$ with positive coefficients. Spanning rooted forests (of an undirected multigraph) with $n-v$ or $n-v-1$ arcs are called in [@CheSha981] [*dense forests*]{}, and the undirected counterpart of the accessibility measure (\[f21\]) with $0<\aa<\frac{\si_{n-v}}{\si_{n-v-1}}$ is called [*accessibility via dense forests*]{}.
Consider two accessibility measures for digraphs: $P\_1(\aa)=R^{\intercal}(\aa)$, [*accessibility via dense diverging forests*]{} and $P\_2(\aa)$, [*accessibility via dense converging forests*]{}.
An important property of the set of dense diverging forests is as follows.
\[p1.2\] $1.$ For any vertex $i\in V(\G),$ there exists an out forest in $\FF_{n-v-1}$ where $i$ is a root. $2.$ For any path $($chain subgraph$)$ in $\G,$ there exists an out forest in $\FF_{n-v-1}\cup\FF_{n-v}$ that contains this path.
A similar proposition is true for converging forests. At the same time, the set of maximum out forests $\FF_{n-v}$ and the set of maximum in forests do not have this property. For example, on Fig. 1 in [@Che.ra1], no maximum out forest contains arc $(4,2)$.
We now test $P\_1(\aa)$ and $P\_2(\aa).$ Similar to the previous consideration, triangle inequality for accessibility measures will be checked for the index $P\_3(\aa)=(P\_1(\aa)+P_1^{\intercal}(\aa)+P\_2(\aa)+P_2^{\intercal}(\aa))/4$, since this inequality requires the symmetry of the corresponding matrix.
\[pogushe\] For any $\aa\in\:]\,0,\,\si_{n-v}/\si_{n-v-1}\/[,$ the measures $P\_1(\aa)$ and $P\_2(\aa)$ are dual to each other. They satisfy nonnegativity and disconnection condition. Moreover$,$ the nonstrict versions of items [1]{} of diagonal maximality and transit property are satisfied by $P\_1(\aa),$ and items [2]{} of these conditions by $P\_2(\aa)$. Both measures violate monotonicity. Triangle inequality for accessibility measures is not true for $P\_3(\aa)$.
Conclusion {#conclusion .unnumbered}
==========
The normalized matrices of out forests are stochastic and determine the transition probabilities in the geometric observation model applied to the Markov chains related to the digraph under consideration. Various matrices of forests can be represented by simple polynomials of the Kirchhoff matrix. The Moore-Penrose generalized inverse $L^+$ and the group inverse $\Lpr$ of the Kirchhoff matrix $L$ can be explicitly represented via $L$ and the normalized matrix $\q$ of digraph’s maximum out forests. The matrices of diverging and converging forests characterize the pairwise accessibility of vertices. These and other results enable one to consider the matrices of spanning forests as a useful tool for the analysis of digraph’s structure.
*Appendix* {#appendix .unnumbered}
==========
[[**Proof of Lemma \[sumstr\].**]{}]{}[ The $i$th row sum of $LQ_k=(a_{ij}^k)$ is $$\sum_{j=1}^n a^{k}_{ij}= \sum_{j=1}^n \sum_{s=1}^n\l_{is}q^{k}_{sj}= \sum_{s=1}^n\l_{is}\sum_{j=1}^n
q^{k}_{sj}= \sum_{s=1}^n\l_{is}\si\_{k}=0.$$ ]{}
[**Proof of Corollary 1 from Theorem \[teo.allk\].**]{} Multiplying both sides of (\[psibek\]) by any matrix that commutes with $L$ and using distributivity and associativity of matrix operations, we get the required statement. [**Proof of Corollary 2 from Theorem \[teo.allk\].**]{} Consider (\[aii.lj\]) at $k=n-v$. By virtue of Proposition \[prop2\], for every $(s,i)\in E(\G)$, $\,\e_{si} q^{n-v}_{\bar si}=0$ holds, i.e., $a^{n-v}_{ii}=0$ for all $i\in\{\1n\}$. In this way, Corollary 2 is derived from $a^{n-v}_{ii}=0$, inequality (\[aij2.lj\]), Lemma \[sumstr\], and Corollary 1 from Theorem \[teo.allk\]. [**Proof of Corollary 3 from Theorem \[teo.allk\].**]{} Postmultiplying both sides of (\[psibek\]) by $\q$ and using $L\q=0$ (Corollary 2 from Theorem \[teo.allk\]) provides $Q_k\q=\si\_k\q$. By commutativity, $\q\q_k=\q_k\q=\q$ holds. Using Theorem 4, we also get $Q(\tau)\q=\q Q(\tau)=\q$ for any $\tau>0.$
[**Proof of Corollary from Theorem \[sumsum7\].**]{} Observe that the digraph resulting from $\G$ by multiplying the weights of all arcs by $\tau$ has the Kirchhoff matrix $\tau L$, and its total weight of out forests with $j$ arcs is $\tau^j\si\_j.$ Hence, the required statement follows from Theorem \[sumsum7\].
[**Proof of Corollary from Theorem \[fulllj\].**]{} First, we prove the following lemma.
\[invert.Laj\] For every weighted digraph $\G$ of out forest dimension $1$ and any $\aa\neq 0,$ the matrix $L+\aa\vj$ is nonsingular.
[[**Proof of Lemma \[invert.Laj\].**]{}]{}[ This lemma is proved by the same argument as Lemma \[invert.Lj\] with the only difference that the analogue of (\[lj1\]) takes here a more general form: $$(L+\aa\vj)^{\intercal}{\bf b} =\left\|\matrix{ b_1\l_{11}+\ldots+b_n\l_{n1}+\aa\q_{11}(b_1+\ldots+b_n)\cr
b_1\l_{12}+\ldots+b_n\l_{n2}+\aa\q_{12}(b_1+\ldots+b_n)\cr
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\cr
b_1\l_{1n}+\ldots+b_n\l_{nn}+\aa\q_{1n}(b_1+\ldots+b_n)\cr }\right\|=\left\|\matrix {0\cr 0\cr\vdots\cr 0\cr
}\right\|.$$ ]{}
To complete the proof of Corollary from Theorem \[fulllj\], note that for any $\aa\ne0$, the matrix $L+\aa\vj$, as well as $L+\vj$, is a block lower triangular matrix with $v+1$ blocks. By item 2 of Theorem \[t1.che.ra1\], its $(v+1)$st diagonal block coincides with the corresponding block of $L+\vj$. Using Lemma \[invert.Laj\], we conclude that the other diagonal blocks are also nonsingular.
[[**Proof of Lemma \[commut\].**]{}]{}[ By virtue of the identity $\vj L=0$ (Theorem \[t3.che.ra1\]), matrices $LL^{\intercal}$ and $ZZ^{\intercal}=\vj^{\intercal}\!\vj+L L^{\intercal}$ commute, i.e., $L L^{\intercal}
(\vj^{\intercal}\!\vj+L L^{\intercal})=(\vj^{\intercal}\!\vj+L L^{\intercal})L
L^{\intercal}={(LL^{\intercal})}^2$. Premultiplying and postmultiplying both sides of the first equality by ${(ZZ^{\intercal})}^{-1}=(\vj^{\intercal}\!\vj+LL^{\intercal})^{-1}$, we obtain the desired $(\vj^{\intercal}\!\vj +LL^{\intercal})^{-1}LL^{\intercal}=LL^{\intercal}(\vj^{\intercal}\!\vj
+LL^{\intercal})^{-1}$. The second statement is proved similarly. ]{}
[99]{}
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[^1]: This work was supported by the Russian Foundation for Basic Research and INTAS.
[^2]: In [@Fiedler], undominated knots are called W-bases.
[^3]: By the nonstrict form of a condition we mean the result of substituting nonstrict inequalities ($\ge$ and $\le$) for the strict ones ($>$ and $<$) in it.
|
---
abstract: 'We consider heterotic string theories compactified on a K3 surface which lead to an unbroken perturbative gauge group of $\spnh$. All solutions obtained are combinations of two types of point-like instanton — one “simple type” as discovered by Witten and a new type associated to the “generalized second Stiefel-Whitney class” as introduced by Berkooz et al. The new type of instanton is associated to an enhancement of the gauge symmetry by $\Sp(4)$ and the addition of a massless tensor supermultiplet. It is shown that if four simple instantons coalesce at an orbifold point in the K3 surface then a massless tensor field appears which may be used to interpolate between the two types of instanton. By allowing various combinations of point-like instantons to coalesce, large gauge groups (e.g., rank 128) with many massless tensor supermultiplets result. The analysis is done in terms of F-theory.'
author:
- |
Paul S. Aspinwall\
Dept. of Physics and Astronomy,\
Rutgers University,\
Piscataway, NJ 08855\
title: |
Point-like Instantons and the\
$\spnh$ Heterotic String\
---
0
Introduction
============
There has recently been considerable progress in the understanding of the nonperturbative physics of string compactification. A fairly realistic model which would be very nice to understand would be the heterotic string compactified on a threefold as this leads to an $N=1$ theory in four dimensions. Here we will deal with the more modest model of a heterotic string compactified on a K3 surface to yield an $N=1$ theory in six dimensions.
Starting with the work of [@KV:N=2; @FHSV:N=2] it was realized that the structure of the heterotic string on a K3 surface could be related to the geometry of a threefold. In particular that the type IIA string on this space was dual to the heterotic string compactified on a product of a K3 surface and a 2-torus. Recall that the heterotic string requires a bundle structure for its compactification and so this product of K3 and a torus also comes equipped with a bundle.
By a process known as F-theory [@Vafa:F; @MV:F; @MV:F2] one can analyze only the parts of the threefold, $X$, that are relevent to the K3 part of the compactification and ignore the 2-torus part. In order to do this, $X$ must be in the form of an elliptic fibration with section $p:X\to\Theta$, where $\Theta$ is a complex surface. For a precise statement of this see [@me:lK3]. This may be viewed in two ways. Firstly one may take the area of the $T^2$ to be large and switch off any Wilson lines around it and watch how $X$ degenerates. Alternatively one may perform a fibre-wise mirror map and replace the type IIA theory with a type IIB string compactified on $\Theta$, where points in $\Theta$ corresponding to “bad fibres” are identified with D-branes embedded in the space. Either way, F-theory promises to yield a fairly complete understanding of the entire moduli space of heterotic strings on a K3 surface.[^1]
Since we are able to probe the moduli space so well, we should ask where the interesting points might be. An obvious place to look is where the underlying K3 surface itself degenerates to an orbifold. This, afterall, has been where the interesting physics lives when one compactifies a type IIA or IIB string on a K3 surface [@W:dyn; @me:enhg; @W:dyn2]. It turns out that a simpler question to answer concerns when the bundle data on the K3 surface degenerates. This has no analogue for the type II string on the K3 surface.
The most obvious type of degeneration of a bundle is that of the “point-like instanton”. That is, where the curvature of the bundle is concentrated in an infinitesimal region of the base space. The study of such objects in heterotic string theory began with Witten’s paper [@W:small-i]. Here it was argued that for the $\spnh$ heterotic string on a smooth K3 surface a point-like instanton induces massless vector multiplets enhancing the gauge symmetry by $\sp(1)$.[^2] This was suggested on general grounds, from the quaternionic nature of the moduli space of hypermultiplets and pictured in terms of Dirichlet 5-branes from the dual type I string theory. When $k$ point-like instantons coalesce at the same point in the K3 surface it was argued that the gauge symmetry is enhanced by $\sp(k)$.
In the context of point-like instantons, the $E_8\times E_8$ heterotic string appears, at first sight, to be a quite different animal. From duality to M-theory [@HW:E8M], it was argued in [@SW:6d] that point-like instantons induce peculiar “tensionless strings” and new moduli in tensor supermultiplets which allow one to move off in a new direction in moduli space corresponding to a new massless tensor multiplet in the theory.
This is far from the end of the story for point-like instantons however. What if the underlying K3 surface is singular and a point-like instanton sits right at the singularity? This may well provide new behaviour. The purpose of this paper is to give an example of such an instanton in the context of the $\spnh$ heterotic string and explore some of its rich properties.
There are two approaches to nonperturbative analysis of the heterotic string by duality. One method, which we use here, is F-theory, which may be viewed as finding a type II dual. The rival method is that of using duality to the type I string, in particular by using orientifold methods [@GP:open]. One should note that one may directly relate these two approaches to each other of course [@Sen:ort]. The orientifold approach has proven very powerful in its ability to find explicit spectra for given models — see, for example, [@DP:ort; @GJ:ort]. Indeed, the subject of massless tensors associated to the $\spnh$ heterotic string was analyzed in this context in [@Pol:ortt].
We wish to attempt to completely classify heterotic string theories on a K3 surface which contain the original ten-dimensional $\spnh$ as part of their unbroken gauge symmetry. This will lead us to the new instanton. In general one should expect the F-theory approach to give a much better coverage of the moduli space of theories than the orientifold approach. This is because the orientifold approach necessarily focuses on points in the moduli space corresponding the theories which are the global quotient of some other theory. One may also probe an infinitesimal region around this point by “twisted marginal operators”. F-theory on the other hand phrases questions in terms of elliptic fibrations. Since any smooth deformation of an elliptic threefold is also an elliptic threefold one might at first think one can probe the entire moduli space of a given theory. While this is almost true, current understanding of F-theory only treats enhanced gauge symmetry from the point of view of degenerate fibres. There is another potential contribution from the “Mordell-Weil group”. This arises when the fibration has an infinite number of sections. We will ignore this latter possibility.
It is not clear whether or not orientifold techniques could reproduce the results in this paper but it would be an interesting question to answer.[^3]
We will present the classical geometry of this new instanton in section \[s:bun\] and relate it to the “generalized second Stiefel-Whitney class” as introduced by Berkooz, Leigh, Polchinski, Schwarz, Seiberg, and Witten [@BLPSSW:so32]. This will allow us to build our new “hidden obstructer” point-like instanton in section \[s:ghol\] which has nonzero generalized second Stiefel-Whitney class but manages to not break any of the primordeal $\spnh$ gauge group.
In section \[s:F\] we build the F-theory picture of the new instanton which allows us to determine its nonperturabtive physics. The main result is the appearance of an $\sp(4)$ enhanced gauge symmetry and a massless tensor. In section \[s:eq\] the F-theory picture is tied to known results about the $E_8\times E_8$ heterotic string and to the Gimon-Polchinski models.
In section \[s:ph\] we show how to transform our new instantons into Witten’s simple instantons and [*vice versa*]{}. This will also show that four simple instantons coaleced at an orbifold point in the K3 surface induce a massless tensor.
In section \[s:co\] we tackle the question of what happens when the two types of instantons hit each other. Large spectra of gauge symmetries and hypermultiplets appear. Finally we include an appendix which reviews some properties of elliptic threefolds that we require.
Bundles on K3 and the Kummer Lattice {#s:bun}
====================================
Let us review the notion of a [*generalized second Stiefel-Whitney class*]{} following the work of Berkooz et al [@BLPSSW:so32]. Consider a smooth $G$-bundle, $E$, on a smooth K3 surface, $S$. How can we express the topology of this bundle? Consider a 2-sphere, $C$, within $S$ with a curve, $\gamma$, around its equator. An element, $g_\gamma\in G$, of the holonomy of $E$ may be found by parallel transport around this curve.
View $C$ as the union of its northern hemisphere, $C_N$, with its southern hemisphere, $C_S$. From the curvature, $F$, of $E$ we may then determine $$\begin{split}
g_\gamma &= \exp\left(\int_{C_N} iF\right)\\
&= \exp\left(-\int_{C_S} iF\right).
\end{split}$$ Thus $$\int_C F = 2\pi n, \label{eq:c1}$$ for some integer, $n$. Thus $\ff1{2\pi}F$ appears as an element of $H^2(S,\Z)$. This quantity will depend on the topology of $E$ but it may be that different values of $n$ specify the same topological class. To see how this works, consider the transition functions around $\gamma$ from the northern hemisphere to the southern hemisphere as a map from $\gamma$ into $G$. In order that $E$ be homotopically nontrivial we require that the image of $\gamma$ lie in a nontrivial element of $\pi_1(G)$. We may apply this construction to every homology 2-cycle within $S$.
We arrive at the result that a natural topological invariant of a $G$-bundle on $S$ is given by a homomorphism from $H_2(S,\Z)$ to $\pi_1(G)$. If $\pi_1(S)$ is trivial then the universal coefficients theorem [@BT:] says that this group of homomorphisms is isomorphic to $H^2(S,\pi_1(G))$.
There are two very familiar examples of this invariant. First if $E$ is the principle bundle of a holomorphic vector bundle then $G\cong\GU(r)$, for some $r$. Since $\pi_1(\GU(r))\cong\Z$ we have our invariant is simply an element of $H^2(S,\Z)$. This is the [*first Chern class*]{}, $c_1(E)$. If $E$ is the principle bundle of a real vector bundle then $G\cong\SO(r)$. Since $\pi_1(SO(r))\cong\Z_2$ we have an object in $H^2(S,\Z_2)$. This is the [*second Stiefel-Whitney class*]{}, $w_2(E)$.
We are interested in the case $G\cong\spnh$. Clearly $\pi_1(\spnh)\cong\Z_2$ and so we are in a situation analogous to the second Stiefel-Whitney class. Following [@BLPSSW:so32] we denote this $\tilde w_2\in H^2(S,\Z_2)$ and consider it to be a “generalized second Stiefel-Whitney class”.
It will be convenient to represent $\tilde w_2$ as a 2-cycle rather than a 2-cocycle. Dual to $H^2(S,\Z)$ is $H_2(S,\Z)$ in the usual way. We may then take $H_2(S,\Z)$ to be dual to itself by Poincaré duality. Thus we may take the dual of the dual of an element of $H^2(S,\Z_2)$ as an element of $H_2(S,\Z_2)$. When seen this way, $\tilde w_2$, as an element of $\Hom(H_2(S),\Z_2)$, may be viewed as $$\tilde w_2:C\to\#(\tilde w_2\cap C)\pmod 2,$$ where “$\#$” represents the intersection number. We will simply use a dot to represent this natural inner product in $H_2(S,\Z)$ from now on.
We will be particularly interested in the case where the K3 surface, $S$, is a [*Kummer Surface*]{}. That is, when it has been obtained as the blow-up of the orbifold $T^4/\Z_2$ in the usual way. The Kummer surface gives a natural set of elements in $H_2(S,\Z)$. These are
1. The image of the six 2-cycles in the $T^4$ under the quotient map.
2. The sixteen 2-spheres that appear as the exceptional divisors under blowing up.
Although these 22 elements may be used as a basis for $H_2(S,\Q)$, they are not correctly normalized to form a basis for $H_2(S,\Z)$. That is, they generate only a finite-index sublattice of $H_2(S,\Z)$. This sublattice is called the [*Kummer Lattice*]{}. $H_2(S,\Z)$ is even self-dual, whereas the matrix of inner products on the generators of the Kummer Lattice has determinant not equal to one.
Let us use $C_i, i=1\ldots16$, to denote the sixteen exceptional divisors. Since $C_i.C_i=-2$ by the usual arguments (see, for example, [@me:lK3]), then $C_i/n$ cannot be an element of $H_2(S,\Z)$ for any integer, $n>1$. Having said that, certain sums of $C_i's$ will be multiples of elements in $H_2(S,\Z)$. This partially accounts for why $\{C_i\}$ are not good generators for $H_2(S,\Z)$. Let $$D = \sum_{i=1}^{16}\xi_iC_i,$$ where $\xi_i$ is either 0 or 1. One may then show that $D$ will be [*twice*]{} an element of $H^2(S,\Z)$ if the following is true. The 16 exceptional divisors come from the 16 fixed points of the $\Z_2$ action on $T^4$. The latter sixteen points naturally form the vertices of a hypercube. Consider every two-dimensional face of the hypercube. Each such face will contain four $C_i$’s and thus can be associated with four $\xi_i$’s. The sum of these four $\xi_i$’s must be an even number.
A simple solution is to set all $\xi_i$ to zero, which is trivial, or to set all $\xi_i$ to one. The other possibilities correspond to having eight $\xi_i$’s equal to zero and eight $\xi_i$’s equal to one in suitable combinations.
In section 4.1 of [@BLPSSW:so32] a picture of an instanton with $\tilde w_2\neq0$ was given locally for an open neighbourhood of one of the exceptional divisors. This was given in terms of the curvature of the bundle which could be given compact support near the exceptional divisor. One may try to treat a $\spnh$-bundle as if it were a $\Spin(32)$-bundle simply by viewing the transition functions as elements of $\Spin(32)$ rather than $\spnh$. As such we may try to build a bundle in the vector representation. Let the curvature of this resulting bundle be $F_{\mathbf{32}}$. One may then show that $$\int_C \ff1{2\pi}F_{\mathbf{32}} = \ff12(\tilde w_2.C) + n,$$ for some integer, $n$. Thus $\tilde w_2$ can violate the quantization condition (\[eq:c1\]) and obstruct the existence of a vector representation — just as $w_2$ obstructs a spin structure. If $C_i$ is the exceptional divisor in question then the instanton of [@BLPSSW:so32] satisfies $$\int_{C_i} \ff1{2\pi}F_{\mathbf{32}} = \ff12, \label{eq:s32}$$ and thus obstructs a vector structure over $C_i$. We will call this instanton a “$C_i$-obstructer”. A $C_i$-obstructer can be seen to satisfy $\tilde w_2.C_i=1$.
Since the curvature of a $C_i$-obstructer is meant to arise from local geometry, $\tilde w_2$ should be proportional to $C_i$. Since $C_i.C_i=-2$, we have $$\tilde w_2 = \ff12C_i.$$
Now let us fit the local $C_i$-obstructer picture into the global geometry of the K3 surface, $S$. It is clear that a single obstructer is not a valid configuration since $\tilde w_2$ does not lie in integral homology. We may consider a situation where we place a single obstructer over more than one exceptional divisor. Now, if our set of exceptional divisors satisfies the Kummer lattice condition above then we are in business. The solution considered in [@BLPSSW:so32] was to put an obstructer at all sixteen sites and so $$\tilde w_2 = \ff12\sum_{i=1}^{16}C_i,$$ which is in integral homology.
Unbroken Gauge Symmetry and Global Holonomy {#s:ghol}
===========================================
When considering a compactification of a heterotic string on a bundle, $E\to B$, an important piece of information about $E$ is its [*global holonomy*]{}, $H$. Let $G_0$ be the “primordial” gauge group, i.e., $E_8\times E_8$ or $\spnh$, of the heterotic string in ten dimensions. When compactified on $E$, this will be broken to the [*centralizer*]{} of $H\subset G_0$. That is, any element of $G_0$ which commutes with all of $H$ will remain a symmetry after compactification.
There are two contributions to the global holonomy group, $H$. Firstly there is the local holonomy generated by the curvature of $E$. Secondly there is the contribution from non-contractable loops from $\pi_1(B)$ of the base space of $E$.
We will be most interested in the case where the global holonomy group is trivial and thus all of the primordial gauge symmetry remains in the lower-dimensional compactified theory. We need to make both the local holonomy and the contribution from $\pi_1$ trivial.
We know from the work of [@W:small-i] how to make the local holonomy trivial. Since this comes from the curvature of the bundle, we need to squeeze all of the region of the nonzero curvature into points over the base space. This limit is called “point-like instantons”. Of course, we haven’t really justified that the paths which happen to exactly pass through the point where a small instanton lives don’t pick up holonomy but the evidence is considerable [@W:small-i; @AG:sp32] that string theory really does allow one to ignore such paths.
Once we have shrunken all instantons down to zero size, we need only worry about non-contractable loops breaking the gauge group. Actually we should consider loops that are non-contractable [*after*]{} the points within $B$ where the point-like instantons live have been removed since we are required to ignore paths which pass through such points.
Let us consider the case where $B$ is a K3 surface. Since a K3 surface is simply connected, we need only worry about non-contractable loops produced by removing the locations of point-like instantons. If the instanton happens to sit at a smooth point inside the K3 surface then the open neighbourhood of the instanton, minus the point where it sits, may be retracted onto $S^3$ — which is simply connected. Thus, a point-like instanton at a smooth point in a K3 surface breaks non of the primordial gauge group. For the $\spnh$ heterotic string, such point-like instantons are precisely the ones discovered by Witten [@W:small-i]. We will denote such point-like instantons “simple”.
In the case that the K3 surface is a Kummer surface at an orbifold limit, we have a singularity locally of the form $\C^2/\Z_2$. If the instanton happens to be sat right on this singular point, then the neighbourhood retracts onto the lens space $S^3/\Z_2$. Since $\pi_1(S^3/\Z_2)$ equals $\Z_2$ we now have the possibility that the point-like instanton breaks part of the primordial gauge symmetry.
As shown in [@BLPSSW:so32] this breaking of the gauge symmetry by $\pi_1$ effects is intimately connected to $\tilde w_2$ of the instanton. Let us review this fact. Consider blowing up the orbifold slightly so that we have an exception 2-sphere, $C_i$, in a small open neighbourhood of the K3 surface. We may also put the lens space $S^3/\Z_2$ in this open neighbourhood, surrounding the 2-sphere. We show this in figure \[fig:Lens\].
Now, the lens space, $L$, may be viewed as an $S^1$-bundle over the 2-sphere, $C_i$. We may then use the Leray spectral sequence (see example 15.15 in [@BT:]) to write the cohomology of $L$ in terms of that of $C_i$. The important point is that there is an isomorphism $$\phi:H^2(C_i,\Z_2) \cong H^2(L,\Z_2)\cong\Z_2.$$ This maps the topological class of bundles over $C_i$ as measured by $\tilde w_2$ into the class of bundles over $L$ given by $\phi(\tilde
w_2)$. The generator of $H^2(L,\Z_2)$ may be associated to the generator of $\pi_1(L)$. This follows from the universal coefficients theorem [@BT:] and the fact that $H^2(L,\Z)$ is pure torsion. Let us call this latter generator, $\gamma$. We show $\gamma$ as a non-contractable loop in figure \[fig:Lens\].
If $\tilde w_2.C_i=1$ then the bundle is nontrivial. The only way the bundle on $L$ may be nontrivial is if the holonomy element generated by $\gamma$ is nontrivial. Thus the global holonomy of the instanton is precisely measured by $\tilde w_2$.
As discussed in [@BLPSSW:so32], the $\Z_2$ subgroup of $\spnh$ generated by $\gamma$ is unique, up to endomorphisms. It is not the central $\Z_2$ in $\spnh$ and actually breaks the primordial gauge group down to $\GU(16)/\Z_2$. That is, [*a point-like instanton in the form of a $C_i$-obstructer breaks $\spnh$ to $\GU(16)/\Z_2$.*]{}
This isn’t what we want however. We want to see if we can leave the entire $\spnh$ unbroken. There is a very simple way of producing a bundle with $\tilde w_2\neq0$ and yet keeping the primordial $\spnh$ intact. Consider the case where $$\tilde w_2 = C_i.$$ Now we have $\tilde w_2(C_i)=C_i.C_i\pmod2=0$. That is, the bundle over $C_i$, and hence $L$, is now topologically trivial. The holonomy around $\gamma$ will be trivial and so $\spnh$ remains unbroken when $C_i$ is blown down to a point. Let us call this the hidden-$C_i$-obstructer.
At first sight it looks like we have constructed something rather trivial but consider the case where we have precisely one curve, $C_i$, in the K3 surface over which we put the hidden obstructer. Then as $C_i$ is not twice an element of $H^2(S,\Z_2)$ we really do have a nontrivial value for $\tilde w_2$. Following our discussion of the Kummer lattice in the previous section, the situation for hidden obstructers is somewhat the opposite as for the previously discussed non-hidden obstructers:
- Non-hidden obstructers must appear in multiples of eight so that the associated curves add up to twice an element of the Picard lattice.
- Hidden obstructers must [*not*]{} appear in such multiples of eight since they would then form a trivial bundle.
Let us recap the trick we have used here to find an instanton with nontrivial $\tilde w_2$ which manages to keep the entire $\spnh$ unbroken. Take, say, one exceptional $S^2$, call it $C_i$, and set $\tilde w_2=C_i$. This value of $\tilde w_2$ is trivial is far as $C_i$ is concerned since the self-intersection of $C_i$ is even. There will be another curve $C_i^*$ dual to $C_i$, for which $C_i^*.C_i=
C_i^*.\tilde w_2=1$ and so $\tilde w_2$ is nontrivial over this curve. Now go to the limit where we blow down $C_i$. We may put the support of the curvature near $C_i$ as shown in [@BLPSSW:so32] and so the curvature becomes zero everywhere except inside the point-like instanton. Thus, all that matters for global holonomy are the non-contractable loops around the lens space surrounding $C_i$. As we have shown, this is trivial. It is important to notice that $C_i^*$ is [*not*]{} blown down during this process and so does not build a lens space which would pick up global holonomy.
In [@BLPSSW:so32] it was shown that a single obstructing instanton locally contributes one to the second Chern class of the bundle. To build our hidden obstructer we essentially double the value of $F_{\mathbf{32}}$ in (\[eq:s32\]). Since $c_2$ goes as the square of the curvature, we see that we multiply the second Chern class by four. That is, [*the hidden obstructer contributes four to the second Chern class*]{}.
We thus know the two possibilities for producing compactifications of the $\spnh$ heterotic string which preserve the $\spnh$ gauge symmetry:
1. Point-like instantons at smooth points in the K3 surface which have instanton number (i.e., contribution to $c_2$) one and $\tilde w_2=0$.
2. Point-like instantons stuck at orbifold points in the K3 surface which have instanton number four and $\tilde w_2\neq0$.
The Dual Picture {#s:F}
================
To understand nonperturbatively how string theory behaves on our new instantons we need to find a dual picture. This is provided by F-theory. Recall that F-theory associates a threefold, $X$, to a heterotic string on a K3 surface [@Vafa:F; @MV:F]. For F-theory to work, $X$ must be in the form of an elliptic fibration $p:X\to\Theta$. One may regard the heterotic string on the K3 surface as dual to either the type IIB string on $\Theta$ with some D-brane insertions or, alternatively, to some special large radius limit of the type IIA string on $X$. Either way, the non-perturbative physics of the heterotic string becomes encoded in the elliptic fibration, $p:X\to\Theta$. See [@MV:F2; @me:lK3] for more details. In particular, we will use the notation from [@me:lK3] and assume a knowledge of many of the results in section 6 of that paper.
We would like to completely classify all heterotic string compactifications on a K3 surface which lead to a gauge symmetry containing $\spnh$ in the perturbatively-understood part of the gauge symmetry. The only assumption we will make (subject to a few caveats outlined in [@me:lK3]) is that none of this gauge symmetry arises from the Mordell-Weil group.
\#1[[**F**]{}\_[\#1]{}]{} \#1[\^\*\_[\#1]{}]{} We wish to understand theories which at least begin as a perturbatively-understood heterotic string theory. We thus want to begin with one tensor multiplet and, as such, we assume $\Theta$ is of the form of a Hirzebruch surface $\HS n$. The Hirzebruch surface is a $\P^1$-bundle over $\P_1$ with a natural zero section, $C_0$. We will denote the class of the fibre, $f$. We will call such fibres, “$f$-curves”, to avoid any confusion with the fibres of $X$ as an elliptic fibration. To obtain an $\so(32)$ term in the gauge algebra we require a line of $\Ist{12}$ fibres in $\Theta$. To make this $\so(32)$ part of the perturbatively-understood symmetry we put it along a section of $\HS n$. Let us assume it is the zero section, $C_0$. We may do this without loss of generality so long as we do [*not*]{} impose $n\geq0$.
To make the group precisely $\spnh$, it was shown in [@AG:sp32] that one required $X$ to have precisely two global sections, as an elliptic fibration. This forces a factorization of the Weierstrass form of the elliptic fibration: $$\begin{split}
y^2&=x^3+ax+b\\
a &= q - p^2\\
b &= -pq\\
\delta &= (q+2p^2)^2(4q-p^2),
\end{split}$$ where $p$ and $q$ are functions over $\HS n$ and $\delta$ is the discriminant. The $\Ist{12}$ condition forces $(a,b,\delta)$ to vanish to order $(2,3,18)$ along $C_0$. Denote $$\begin{split}
m_1 &= q+2p^2\\
m_2 &= 4q-p^2.
\end{split} \label{eq:m1m2}$$ One may then show that $m_1$ must vanish to order 8 along $C_0$ and $m_2$ vanishes to order 2. Let us use upper case letters to denote the divisors in $\HS n$ associated to the various functions above. The condition imposes $$\begin{split}
\Delta &= 2M_1+M_2\\
M_1 = M_2 &= 8C_0 + (8+4n)f.
\end{split}$$ Let us split off from $M_1$ and $M_2$ the parts giving the $\Ist{12}$ along $C_0$: $$\begin{split}
M_1 &= M_1' + 8C_0\\
M_2 &= M_2' + 2C_0,
\end{split}$$ with $\Delta'$ defined similarly. Now, to make sure that the fibres along $C_0$ are generically nothing worse than $\Ist{12}$, neither $M_1'$ nor $M_2'$ should contain any more of $C_0$. This means that the intersection numbers $$\begin{split}
M_1'.C_0 &= 8+4n\\
M_2'.C_0 &= 2(4-n),
\end{split}$$ must be nonnegative. Thus $-2\leq n\leq 4$.
Let us treat the remainder of the discriminant, given by $M_1'$ and $M_2'$ in turn. $M_1'$ is simply $8+4n$ copies of $f$. Generically this will mean it is $8+4n$ parallel lines along the $f$ direction. As $\Delta$ contains $2M_1'$, this will produce lines of $\mathrm{I}_2$ fibres. Thus, the gauge symmetry is enhanced nonperturbatively by $8+4n$ $\sp(1)$ terms. These are precisely Witten’s simple point-like instantons of [@W:small-i].
If $k$ of these instantons are brought together, $k$ lines of $\mathrm{I}_2$ will merge to form a line of $\mathrm{I}_{2k}$. As explained in [@AG:sp32], monodromy turns the $\su(2k)$ gauge algebra one might first associate to this into an $\sp(k)$ gauge algebra. One can potentially have monodromy whenever a curve in the discriminant, whose associated gauge algebra may admit nontrivial outer automorphisms, collides with another component of the discriminant. Whether or not there is monodromy can be determined purely in terms of the local geometry of the collision, and with what type of curve it collided. In our case we have a transverse collision of a line of $\Ist{12}$ fibres with a line of $\mathrm{I}_{2k}$ fibres. One may show that such a collision induces no monodromy in the $\Ist{12}$ fibre but has a $\Z_2$ action in the $\mathrm{I}_{2k}$ fibre. We show how to determine how the monodromy acts in the appendix.
As well as the gauge algebra, we may also determine the spectrum of hypermultiplets as discussed in [@AG:sp32; @BKV:enhg; @KV:hyp]. The transverse collision of the $\Ist{12}$ and $\mathrm{I}_{2k}$ produce a half hypermultiplet in the $(\mbf{32},\mbf{2k})$ representation of $\so(32)\oplus\sp(k)$.
As discussed in [@W:small-i] we should also expect a hypermultiplet in the $\mbf{k(2k-1)-1}$ (i.e., antisymmetric tensor) representation of $\sp(k)$. Call this the $\mbf{A_2}$ representation for brevity. Let us use $\Delta''$ to denote the discriminant after the contribution from $C_0$ [*and*]{} all the $f$-curves has been subtracted. To see how the hypermultiplets arise note that an $f$-curve is topologically a sphere. Thus, if we are to have a nontrivial action of monodromy on the fibre of the elliptic fibration around this sphere, we must have more than one branch point. At present we have only found one collision — that of $f$ with $C_0$. There must be further collisions of $\Delta''$ with the $\mathrm{I}_{2k}$ line to produce more monodromy. As explained by Morrison [@Mor:TASIF], these collisions will produce the $\mbf{A_2}$ representation required.
To see this we use the results of [@KMP:enhg] which say that if a curve of bad fibres is of genus $g$, then we expect $g$ hypermultiplets in the adjoint of the associated gauge algebra, in addition to the usual adjoint of vectors. When monodromy acts within the curve, the algebra is split between the part invariant under the monodromy and the rest which varies. The vectors are only associated with the monodromy-invariant part (see, for example, [@me:lK3]) but we may pick up hypermultiplets in the part that varies depending on the genus of the base curve after we have taken the monodromy into account. Thus, suppose we have a $\Z_2$ monodromy acting on a rational curve in $\Theta$ associated to a gauge algebra (before monodromy is taken into account) $\mathfrak{g}$. The outer automorphism induced by the monodromy leaves $\mathfrak{g}_0$ invariant. The adjoint of $\mathfrak{g}$ may then be decomposed into the adjoint of $\mathfrak{g}_0$ plus a representation $R'$. Suppose the monodromy is branched over $n_p$ points within the rational curve. Then as far as the representation $R'$ is concerned, the base curve is actually a double cover of the rational curve branched at $n_p$ points. This has genus $\ff12n_p-1$. We therefore expect $\ff12n_p-1$ hypermultiplets in the $R'$ representation.
In our case, we are reducing $\su(2k)$ to $\sp(k)$. It is easy to show that $R'$ is indeed the $\mbf{A_2}$ representation. Now we need to know how many points there are within each $f$-curve over which the $\Z_2$ monodromy is branched. This will allow us to count the $\mbf{A_2}$’s.
Let us introduce affine coordinates $(s,t)$ to parameterize $\HS n$ locally. Let $C_0$ be given by $s=0$ and let us fix a particular $f$-curve to be given by $t=0$. To associate an $\sp(k)$ gauge symmetry with this $f$-curve we require $m_1$ to be of order $k$ in $t$. Let us put $m_1=t^k$ for the simplest case. Then $$\delta = t^{2k}(4t^k-9p^2). \label{eq:simplek}$$ Thus, $\Delta$ will collide with this $f$-curve whenever $p(s,t)$ has a zero. $P$ is in the class $4C_0+(4+2n)f$ and thus collides with $f$ a total of $P.f=4$ times. One of these collisions is the transverse collision with the line of $\Ist{12}$ fibres along $C_0$. The other three collision are generically non-transverse collisions with a curve of $\mathrm{I}_1$ fibres along $\Delta''$. As we will see in the appendix, all four collisions induce monodromy — we have $n_p=4$ and thus one hypermultiplet in the $\mbf{A_2}$ representation as desired.
Thus far we have recovered the simple $\sp(1)$ point-like instantons. Now we discover something new when we look at collisions between $\Delta''$ and $C_0$. Since $M_1$ is order 8 along $C_0$ as explained above, let us put $m_1=s^8$. It follows that $$\delta = s^{16}(4s^8-9p^2).$$ We know that $m_2$ vanishes to order 2 along $s=0$ so we must be able to factorize $p=sp_1$. Therefore $$\delta = s^{18}(4s^6-9p_1^2). \label{eq:p1C}$$ Thus there will be collisions between $\Delta''$ and $C_0$ whenever $p_1$ has extra zeros. This happens at $C_0.P_1=4-n$ points.
Put $p_1=t$ to get a local form of the collision. (Note that now $t=0$ is [*not*]{} the equation of an $f$-curve for simple instantons.) Adding the degrees in $s$ and $t$ together, we see that $(a,b,\delta)$ have degrees $(4,6,20)$ respectively. As explained in [@me:lK3], whenever these degrees are greater than, or equal to, $(4,6,12)$, one must blow-up the base to resolve $X$. Therefore, these collision of $\Delta''$ with $C_0$ induce new massless [*tensor*]{} degrees of freedom.
Let $E_1$ be the resulting exceptional $\P^1$ in the blown-up $\Theta$. The order to which $(a,b,\delta)$ vanish over generic points in $E_1$ is given by subtracting $(4,6,12)$ from the orders at the point which was blown-up. That is, the orders are $(0,0,8)$. Thus we have $\mathrm{I}_8$ fibres along $E_1$. The collision between $E_1$ and $C_0$ produces monodromy. This results in a gauge symmetry of $\sp(4)$.
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As shown in figure \[fig:bup1\], $\Delta''$ collides with $E_1$ just once to produce another monodromy branch point. This means that $n_p=2$ for this $\sp(4)$ gauge symmetry and so no hypermultiplets in the $\mbf{A_2}$ representation appear. There will be hypermultiplets in the $(\mbf{32},\mbf{8})$ representation from the collision of $E_1$ with $C_0$.
Let us review the spectrum we have obtained. Let the simple instantons clump together in $\mu$ groups of $k_i$ (so that $\sum_{i=1}^{\mu}
k_i=8+4n$) but leave the collisions of $C_0$ and $\Delta''$ isolated. We have
1. A gauge algebra $$\so(32)\oplus\left(\bigoplus_{i=1}^\mu \sp(k_i)\right)
\oplus\sp(4)^{\oplus(4-n)}.$$
2. $5-n$ massless tensor supermultiplets (including the dilaton).
3. Hypermultiplets (or half-hypermultiplets if the representation is not complex) in the following representations: $$\begin{split}
(\mbf{32},\mbf{2k_i})\quad&\mbox{of}\quad \so(32)\oplus \sp(k_i)\\
\mbf{k_i(2k_i-1)-1}\quad&\mbox{of}\quad \sp(k_i)\\
(\mbf{32},\mbf{8})\quad&\mbox{of}\quad \so(32)\oplus \sp(4)
\quad\quad\mbox{($4-n$ times)}
\end{split}$$ as well as some chargeless hypermultiplets.
At this point the interpretation of this model in terms of point-like instantons discussed at the end of section \[s:ghol\] should be fairly evident. As mentioned above, the simple instantons are associated to the $8+4n$ zeros of $m_1$. Each of these have instanton number one. If we assign instanton number four to each of the $4-n$ collisions between $C_0$ and $\Delta''$ then $$\sum_{i=1}^{\mu} k_i + 4(4-n) = 24,$$ for the total instanton number as expected for the bundle, $E$, on a K3 surface. [*We therefore identify these $4-n$ collisions as point-like hidden-obstructer instantons in the dual heterotic string.*]{}
We see that each hidden obstructer instanton is associated to a massless tensor multiplet and an $\sp(4)$ gauge symmetry.
Let us be precise about what we mean exactly by this statement. As discussed in [@SW:6d], phase transitions between tensor moduli and hypermultiplet moduli are fairly exotic in nature. This is exactly what we have here — when the size of the new instanton is shrunk down to zero size (by hypermultiplets) a new modulus appears as the scalar component of a tensor supermultiplet which allows us to move off into a new component of the moduli space. Rather than speak of the theory right at the phase transition point, which has “tensionless strings” roughly speaking, we will assume that we switch on the new tensor modulus slightly to move away from this peculiar theory. As a result, we have a more conventional six-dimensional theory (although it has no covariant action) and we may ask sensible questions about anomalies etc. We shall not attempt to say anything in this paper about the theory which sits right on the phase transition point.
To complete the spectrum we should count the number of chargeless hypermultiplets. Roughly speaking, this is given by the number of deformations of complex structure of $X$ plus one. One needs to be a little careful however. It may be that F-theory counts some of the linear combinations of charged hypermultiplets which can also act as deformations. This latter effect is due to the appearance of “elliptic scrolls” in $X$ and is tied to Wilson’s work on the phenomenon of the Kähler cone jumping for special values of complex structure [@Wil:Kc] (see also [@me:lK3] for a brief account of this). This indeed happens when simple instantons coalesce. To avoid this issue let us assume all $8+4n$ simple instantons are isolated.
We know $h^{1,1}(X)$ from the blow-ups in both the base (the tensor multiplets) and the fibre (the rank of the gauge group). We have $h^{1,1}=3+(4-n)+24+16=47-n$. To calculate $h^{2,1}(X)$ we need the Euler characteristic of $X$. This is done by adding the contributions from all of the bad fibres in $X$ as an elliptic fibration. For an example see [@me:lK3]. In our case we need some Euler characteristics of some of the fibres appearing over collisions within $\Delta$. We calculate those required in the appendix. The result is[^4] $$\begin{split}
\chi(X) &= \sum_{i=1}^{\mu}\left\{2k_i.(2-4)\right\}+18.(2-\mu-4+n)
+8.(4-n).(2-2) \\
&\qquad+ (-24-3n-3\mu-3\mu-(4-n))+3\sum_{i=1}^{\mu}(2+k_i)
+\sum_{i=1}^{\mu}(18+k_i)\\
&\qquad\qquad+(4-n).22+(4-n).6\\
&= 48-12n.
\end{split}$$ This gives $h^{2,1}=h^{1,1}(X)-\ff12\chi(X)=23+5n$. Therefore there are $24+5n$ chargeless hypermultiplets.
As always, one may check this F-theory calculation to ensure that anomalies cancel (as they must). The gravitational anomaly yields $$\begin{split}
273-29n_T-n_H+n_V&=273-29.(5-n)-\ff12.2.32.24-(24+5n)\\
&\qquad+3.(8+4n)+36.(4-n)+496\\
&=0.
\end{split}$$ Similarly one may check the gauge anomalies.[^5]
This counting of chargeless hypermultiplets fits nicely with the heterotic interpretation. The underlying K3 surface has 20 (quaternionic) deformations and the simple instantons may be placed anywhere giving $8+4n$ more deformations. Each hidden obstructer requires an orbifold point locally of the form $\C^2/\Z_2$, which reduces the number of deformations of the K3 by one. The location of each hidden obstructer is then fixed at this orbifold point. Thus, the total number of deformations is $20+(8+4n)-(4-n)=24+5n$ as expected.
One may also check the above calculations in the case that some of the simple instantons coalesce. In this case the topology of $X$ is actually unchanged but the interpretation of some of the hypermultiplets is modified.
Some Equivalences {#s:eq}
=================
Recall the behaviour of the $E_8\times E_8$ heterotic string on a K3 surface as regards F-theory [@SW:6d; @MV:F; @MV:F2]. The topology of the required $E_8\times E_8$-bundle on the K3 surface is specified by how the total second Chern class is split between the two $E_8$’s. In particular, F-theory on the Hirzebruch surface $\HS n$ is dual to a split of $12+n$ and $12-n$.
This shows the T-duality between the $E_8\times E_8$ heterotic string on a K3 surface and the $\spnh$ heterotic string on another K3 surface. For example, as has been known for some time [@MV:F], the $\spnh$ heterotic string with $\tilde w_2=0$ must be dual to the $E_8\times
E_8$ string with the second Chern class split $(8,16)$ between the two $E_8$’s. This follows since $\tilde w_2=0$ implies that there can be no obstructers, hidden or non-hidden, which implies that $4-n=0$.
This raises a point which, at least at first sight, looks puzzling. The global diffeomorphisms of the underlying K3 surface, on which the heterotic string lives, can transform one value of $\tilde
w_2$ into another. In particular there are only three equivalence classes once this is taken into account [@BLPSSW:so32]:
1. $\tilde w_2=0$,
2. $\tilde w_2\neq0$ and $\tilde w_2.\tilde w_2=0\pmod4$,
3. $\tilde w_2\neq0$ and $\tilde w_2.\tilde w_2=2\pmod4$.
Since $(4-n)$ hidden obstructers over disjoint $(-2)$-curves yields $\tilde w_2.\tilde w_2=2(n-4)$ we are implying equivalences between certain $E_8\times E_8$ string vacua. Actually these equivalences do exist.
To see this we need to look at the strange properties of the Hirzebruch surface.[^6] The topology of $\HS n$ is actually only specified by whether $n$ is even or odd. Indeed one may build a family of surfaces $\pi:Z\to D$, where $D$ is a complex disc with coordinate $z$ such that the fibre at $z\neq0$ is $\HS n$ but at $z=0$ it becomes $\HS{n+2}$. At $z=0$ a new algebraic curve within the fibre jumps into existence with self-intersection $-n-2$. This causes the Kähler cone to contract, relative to that of $\HS n$, but nothing has changed topologically.
This equivalence between Hirzebruch surfaces is used to show the equivalence of the $n=0$ model and the $n=2$ model as in [@AG:mulK3; @MV:F]. The elliptic threefold fibred over $\HS2$ is a codimension one subset (which can be realized as a hypersurface in a weighted projective space) of the more general member of the family which is fibred over $\HS0$. The jumping Kähler cone of the Hirzebruch surface is transfered to the threefold whose Kähler cone also shrinks over this special sub-family.
It was shown in [@Wil:Kc] that jumping Kähler cones could only happen in smooth threefolds if the algebraic class that jumped into existence for special values of the complex structure was an “elliptic scroll”. That is, an elliptic curve times a rational curve. Thus, this rational curve is a $(0,-2)$-curve within the threefold. Within the base of an elliptic fibration therefore, the only curve which is allowed to jump into existence is a $(-2)$-curve, which would come from the Hirzebruch surface $\HS2$. We appear to have shown that the only equivalence allowed between models is the $n=0$ to $n=2$ equivalence.
We may obtain the rest of the equivalences by relaxing the constraint that the elliptic threefold be smooth. Now any smooth transition between $\HS n$ and $\HS{n+2}$ may be turned into a “smooth” transition between singular elliptic threefolds. In the case we are studying the threefold has a curve of $D_{16}$-type singularities inducing the $\spnh$ gauge symmetry. It is certainly singular.
Given this equivalence between threefolds, there is no contradiction between $\tilde w_2$ equivalence classes and F-theory equivalence classes.
Now let us turn our attention to the connection between the hidden obstructer theories we have described into terms of F-theory and other models in the same $\tilde w_2$ class which break at least part of $\spnh$. We focus on the Gimon-Polchinski models of [@GP:open]. As explained in [@BLPSSW:so32], we expect these models to all be in the F-theory class with $n=0$.
To see this simply deform $m_2$ of (\[eq:m1m2\]) so that it no longer vanishes along $C_0$. This will turn the line of $\Ist{12}$ line of fibres along $C_0$ into a line of $\mathrm{I}_{16}$ fibres. This changes the class of $\Delta''$ but it will still collide with $C_0$ at four points (doubly at each point). These collisions will induce monodromy and so the $C_0$ line now generates a $\sp(8)$ gauge symmetry.
This breaking of $\spnh$ may be seen by the maximal subgroup $$\frac{\Spin(32)}{\Z_2} \supset
\SO(3)\times\frac{\Sp(8)}{\Z_2}\times\Z_2.$$ Giving the hidden obstructer nonzero size can turn it into a smooth $\SO(3)$-bundle. (The group must be non-simply-connected since $\tilde
w_2\neq0$.) Thus the global holonomy breaks the primordial gauge symmetry to $\Sp(8)/\Z_2$ consistent with what we saw from F-theory.
Further deformations can be used to bunch the four points of collision between $\Delta''$ and $C_0$ into two coalesced pairs. This will remove the monodromy and so the gauge symmetry $\sp(8)$ will turn into $\su(16)$.[^7] To fit in with the work of [@GP:open] (see also the earlier work of [@BSag:u16]) we may then identify the two points of collision of $\Delta''$ with $C_0$ as each yielding a hypermultiplet in the $\mbf{120}$ of $\su(16)$.
The line of fibres along $C_0$ can be broken up into a parallel set of lines of $\mathrm{I}_{2l_j}$ fibres so that $\sum_jl_j=8$. This makes the class $C_0$ analogous to the class $f$ in which we have a set of parallel lines of $\mathrm{I}_{2k_i}$ fibres satisfying $\sum_ik_i=8$. This is as it should be since $\HS0$ has an obvious symmetry between the classes $C_0$ and $f$.
This allows us to reproduce all of the Gimon-Polchinski models in terms of F-theory. We show an example in figure \[fig:GP\]. Note that the collisions within the discriminant will produce massless hypermultiplets in various representations in the usual way.
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Phase Transitions {#s:ph}
=================
Let us return to the $E_8\times E_8$ heterotic string with its second Chern class split as $12+n$ and $12-n$ between the two $E_8$’s. When any of the instantons become point-like in the $E_8\times E_8$ a new massless tensor supermultiplet appears. One may then use this new direction in the moduli space to move to another theory with a point-like instanton with the second Chern class split $(11+n,13-n)$. This instanton can then be given size to remove the massless tensor. Thus, by a process that involves two phase transitions, we may effectively change the topology of the $E_8\times
E_8$-bundle.
In terms of M-theory [@SW:6d] this was understood by a 5-brane peeling itself off one “end of the universe” and moving over (by varying the tensor degree of freedom) to the other end of the universe.
In terms of F-theory [@MV:F2], which is the approach we use here, this is achieved by first blowing up a point in $\HS n$. The proper transform of the fibre that passed through this point then has self-intersection $-1$ allowing it to be blown down. This blow down results in the Hirzebruch surface $\HS {n\pm 1}$ (depending on whether the original point blown up was on $C_0$ or not).
Our new point-like hidden obstructer instanton is very similar is the point-like $E_8$ instanton in that a new massless tensor results. We may therefore follow a phase transition to another Hirzebruch surface and see what happens. We will discover that we may transform hidden obstructer instantons into simple instantons and [*vice versa*]{}.
Begin with the collision of $\Delta''$ with $C_0$ in the Hirzebruch surface, $\HS n$, as in the previous section. As we discussed above, to resolve $X$, such a collision must be blown-up within $\HS n$. The exceptional divisor, $E_1$, results. Let $\tilde f$ be the proper transform of the $f$-curve that passed through the collision. As blow-ups decrease self-intersections by one and $f.f=0$, we see that $\tilde f.\tilde f=-1$.
We know that $\Delta''.f=6$ and that the collision with $C_0$ accounts for two of these intersections. Thus, away from $C_0$, $\Delta''$ hits our particular $f$-curve four times. Assuming everything else is generic, these will be at four distinct points. Thus the proper transform of $\Delta''$, which we also denote $\Delta''$, hits $\tilde f$ at four distinct points.
We may now blow down $\tilde f$. This gives the proper transform of $E_1$ a self-intersection of 0 and it becomes a fibre, $f$, of the Hirzebruch surface $\HS{n+1}$ which we have now made. Now $\Delta''$ will hit this new $f$-curve four times at the same point (where $\tilde f$ used to hit $E_1$).
Now deform $X$ so that this quadruple collision of $\Delta''$ with $f$ divides into two double collisions. What we have done is to produce exactly the F-theory picture of four coalesed simple instantons giving a gauge group $\sp(4)$.
Let us repeat what we have done in the language of the heterotic string. Begin with a point-like hidden obstructer instanton. Then move along in moduli space from one phase to another using the massless tensor degree of freedom. Then deform using hypermultiplets to get rid of the massless tensor. The hidden obstructer has disappeared ($n$ has increased by one) but four new simple instantons have appeared.
We see therefore that our two types of point-like instantons may be transformed into each other by using massless tensors. This also gives a way of changing the topology (as given by $\tilde w_2$) of the associated vector bundle. Thus we see that the picture is very analogous to the $E_8\times E_8$ heterotic string.
Note that the geometry of the K3 surface is given by hypermultiplet deformations and so is fixed while we vary the tensor. As we knew that the hidden obstructer lived on an orbifold point, the orbifold point must still be there after moving along the tensor direction. What’s more we know that the location of the simple instantons are also given by hypermultiplets. This means that, before we get rid of the massless tensor by moving the simple instantons, the four simple instantons must have been sat right on the orbifold point. This implies that [*four simple instantons on a $\C^2/\Z_2$ quotient singularity in the K3 surface produce a massless tensor supermultiplet.*]{}
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In figure \[fig:phase\] we show the phase transition we described above backwards. Start with a heterotic string with, say $\tilde
w_2=0$ (and therefore $\Theta\cong\HS4$). Then bring four simple point-like instantons together to form a theory containing a gauge symmetry of $\sp(4)$. Now give the K3 surface a $\C^2/\Z_2$ quotient singularity and put this coalesced instanton at that point. Now a massless tensor appears. Use this massless tensor to turn $\HS 4$ into $\HS 3$. Now we have a hidden obstructer instanton at the orbifold point. If we wish, the massless tensor may be given mass by giving the new point-like instanton size (which will break $\so(32)$).
A natural question to ask is what happens if fewer than four simple instantons coalesce at an orbifold point. Let us consider $k$ simple instantons. The collision of the associated $f$-curve with $\Delta''$ was given in (\[eq:simplek\]). Consider the zeros of $p$ as $s$ is varied to move along $f$. Generically the zeros are isolated. It is evident from the above discussion that the orbifold condition amounts to $p$ having a zero of order two. We are therefore interested in a collision roughly of the form $$\begin{split}
a &= t^k-3s^4\\
b &= -s^2(t^k-2s^4)\\
\delta &= t^{2k}(4t^k-9s^4).
\end{split}$$ Adding the degrees of $s$ and $t$ together we see that $(a,b,\delta)$ have degrees $(\min(k,4),2+\min(k,4),2k+\min(k,4))$ respectively. Thus we hit the required $(4,6,12)$ for a massless tensor precisely when $k\geq4$. That is, fewer than four simple instantons at an orbifold point are not enough to produce the massless tensor.
Coalesced Instantons {#s:co}
====================
Now we know that four simple instantons at an orbifold point produce a massless tensor which connects the theory to a hidden obstructer, the natural question to ask is what happens when more than four simple instantons coalesce at an orbifold point. This is equivalent to asking what happens when a simple instanton hits a hidden obstructer. It is then natural to ask what happens when two hidden obstructers coalesce.
A simple instanton meets a hidden obstructer {#ss:sh}
--------------------------------------------
Let $k$ simple instantons hit a hidden obstructer. Recall that a hidden obstructer corresponds to a collision of $\Delta''$ with $C_0$. Consider the $f$-curve passing through this collision point. From our discussion of simple instantons above and their relationship to $M_1$, it is clear that we require $M_1$ to contain $k$ times this $f$-curve. That is, $\Delta$ includes $2k$ times this $f$-curve.
Following (\[eq:p1C\]), the form of the discriminant is $$\delta = s^{18}t^{2k}(4s^6-9p_1^2),$$ where $p_1$, where $s=0$, has a single zero at $t=0$. Adding the degrees of $s$ and $t$ together gives the degrees of $(a,b,\delta)$ equal to $(4,6,20+2k)$. Thus we have a blow-up in the base. Now the degrees along the exceptional divisor, $E_1$, are $(0,0,8+2k)$. This gives a gauge symmetry $\sp(4+k)$. The proper transform, $\tilde f$, of the $f$-curve that passed through the collision is still a line of $\mathrm{I}_{2k}$ fibres and so we also have an $\sp(k)$ gauge symmetry. Since this curve hits $E_1$, we expect a hypermultiplet in the $(\mbf{8+2k},\mbf{2k})$ representation of the $\sp(4+k)\oplus\sp(k)$ part of the gauge algebra. This can be seen by applying monodromy to the results of [@BSV:D-man]. We also have hypermultiplets from the $C_0$ collision with $E_1$ but there is no collision between $C_0$ and $\tilde f$.
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As shown in figure \[fig:sh\] and the appendix, there are only two points of monodromy in the curves generating both the $\sp(4+k)$ and the $\sp(k)$ gauge algebras. Thus we have no hypermultiplets in the $\mbf{A_2}$ representations of either of these algebras.
As an example, suppose $k_1$ of the simple point-like instantons collide with one of the $4-n$ hidden obstructers and let the remaining $8+4n-k_1$ clump into groups of $k_i$, $i=2\ldots\mu$. The spectrum is
1. A gauge algebra $$\so(32)\oplus\sp(k_1)\oplus\sp(4+k_1)
\oplus\left(\bigoplus_{i=2}^\mu \sp(k_i)\right)
\oplus\sp(4)^{\oplus(3-n)}.$$
2. $5-n$ massless tensor supermultiplets (including the dilaton).
3. Hypermultiplets (or half-hypermultiplets if the representation is not complex) in the following representations: $$\begin{split}
(\mbf{32},\mbf{8+2k_1})\quad&\mbox{of}\quad \so(32)\oplus\sp(4+k_1)\\
(\mbf{8+2k_1},\mbf{2k_1})\quad&\mbox{of}\quad\sp(4+k_1)\oplus\sp(k_1)\\
(\mbf{32},\mbf{2k_i})\quad&\mbox{of}\quad \so(32)\oplus \sp(k_i)\\
\mbf{k_i(2k_i-1)-1}\quad&\mbox{of}\quad \sp(k_i)\\
(\mbf{32},\mbf{8})\quad&\mbox{of}\quad \so(32)\oplus \sp(4)
\quad\quad\mbox{($3-n$ times)},
\end{split}$$ where $i=2\ldots\mu$, as well as $20+(\mu-1)-(4-n)$ chargeless hypermultiplets.
The reader may check that anomalies cancel.
Two hidden obstructers meet {#ss:hh}
---------------------------
The natural thing to identify with two coalesced hidden obstructers is when two of the zeroes of $p_1$ in (\[eq:p1C\]) coalesce. This may be achieved by putting $p_1=t^2+\alpha st+\beta s^2$ for some generic $\alpha,\beta$. We obtain a total degree for $(a,b,\delta)$ at $s=t=0$ equal to $(6,9,22)$ respectively. When we blow up this point, we obtain the exceptional divisor, $E_1$, with degrees $(2,3,10)$. Thus $E_1$ is a curve of $\Ist{4}$. There is no monodromy within this and so a gauge algebra $\so(16)$ results.
We are not done however. The collision of the curve of $\Ist{12}$ fibres along $C_0$ and $\Ist{4}$ fibres along $E_1$ has total degree $(4,6,28)$. Therefore we are required to blow-up this point too. This introduces an exceptional divisor $E_2$. Along this curve, the degrees are $(0,0,16)$. In this case there is monodromy and so the gauge algebra is $\sp(8)$. Finally there are two collision of $E_1$ with the proper transform of $\Delta''$ which also require blowing up. The collisions each have total degree $(4,6,12)$ and so the resulting two exceptional divisors, $E_3$ and $E_4$, carry smooth fibres and hence no further gauge algebra. See figure \[fig:hh\] for this process.
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It is not much harder to go directly to the case of $k$ more simple instantons joining the two coalesced hidden obstructers. In this case the $f$-curve passing through the complicated collision of $\Delta''$ with $C_0$ will now carry $\mathrm{I}_{2k}$. Now the blow-up process is similar to the above case except that more singular fibres appear. This process is shown in figure \[fig:hhs\].
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Let us give the spectrum that results in this case. Let the remaining $8+4n-k$ simple instantons that have not joined the hidden obstructers be disjoint. It is an easy matter to generalize to the case where these coalesce amongst themselves but it will clutter the notation. The result is
1. A gauge algebra $$\sp(1)^{\oplus(8+4n-k)}\oplus\so(32)\oplus\sp(8+k)\oplus
\so(16+4k)\oplus\sp(k)^{\oplus3}\oplus\sp(4)^{\oplus(2-n)}.$$
2. $7-n$ massless tensor supermultiplets (including the dilaton).
3. Hypermultiplets (or half-hypermultiplets if the representation is not complex) in the following representations: $$\begin{split}
(\mbf{32},\mbf{2})\quad&\mbox{of}\quad \so(32)\oplus \sp(1)
\quad\quad\mbox{($8+4n-k$ times)}\\
(\mbf{32},\mbf{16+2k})\quad&\mbox{of}\quad \so(32)\oplus \sp(8+k)\\
(\mbf{16+4k},\mbf{16+2k})\quad&\mbox{of}\quad \so(16+4k)\oplus \sp(8+k)\\
(\mbf{16+4k},\mbf{2k})\quad&\mbox{of}\quad \so(16+4k)\oplus \sp(k)
\quad\quad\mbox{($3$ times)}\\
(\mbf{32},\mbf{8})\quad&\mbox{of}\quad \so(32)\oplus \sp(4)
\quad\quad\mbox{($2-n$ times)},
\end{split}$$ as well as $22+5n-k$ chargeless hypermultiplets.
As usual the anomalies miraculously cancel.
A couple of points are worth noting. Firstly the gauge group is getting pretty large. For example, putting $n=2$ and $k=16$ in the above yields a rank 128 gauge group. It also contains an $\so(80)$ factor in this case. This is interesting as we know that a rank 40 gauge symmetry can never be understood perturbatively. Therefore, there is no heterotic string theory dual to our model whose conformal field theory knows about this gauge symmetry factor.
Secondly the counting of moduli, i.e., chargeless hypermultiplets is curious. This should be equal to the number of deformations of the K3, plus the number of deformations of the simple instantons still free, minus the number of hidden obstructers, minus the number of moduli required to force the two obstructers to meet. The fact that there are $22+5n-k$ moduli shows that this latter number of moduli, required to be tuned to make to two obstructers meet, is equal to two.
This tuning must correspond to bending the K3 around as to bring two orbifold points together in the right way. This should result in a more complicated orbifold singularity. It looks as if the number of blow-ups required to smooth this orbifold singularity is equal to two, from the blow-up modes we already had, plus two more from the tuning required. This suggests that the resulting quotient singularity is either of the type $A_4$ (i.e., $\C^2/\Z_5$) or $D_4$ (i.e., $\C^2$ divided by the discrete quaternion group). It would be interesting to study this further.
If we continue further and attempt to bring three hidden obstructers together by giving $p_1$ a triple zero, we obtain a total degree at the collision equal to $(8,12,24)$. After blowing up this point, the degrees along the exceptional divisor are $(4,6,12)$. While degrees greater than, or equal to, $(4,6,12)$ are admissible at points within the discriminant, they are not acceptable along curves. The condition is violated if we attempt to blow up. We therefore have no further extremal transitions associated to three colliding hidden obstructers.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank M. Gross for explaining to me much of the technology of elliptic fibrations used in this paper. It is also a pleasure to thank O. Aharony, C. Johnson, S. Kachru, D. Morrison, J. Polchinski, N. Seiberg and E. Silverstein for useful conversations. The author is supported by DOE grant DE-FG02-96ER40959.
Appendix {#s:app .unnumbered}
========
Let $X$ be an algebraic threefold which admits an elliptic fibration, $p:X\to\Theta$, for some complex surface, $\Theta$. The elliptic fibres degenerate over the discriminant, $\Delta\subset\Theta$. At a smooth point in $\Delta$, the bad fibres are classified by the Weierstrass classification (see, for example, [@me:lK3]). In general however $\Delta$ has singularities, usually formed by intersections of irreducible components of $\Delta$. In this appendix we discuss what happens to the bad fibre over such singularities in $\Delta$.
This problem was studied by Miranda in [@Mir:fibr]. It has also been analyzed in [@BKV:enhg] in terms of Tate’s algorithm. We will adopt Miranda’s method as it is slightly better suited to our approach and yields some Euler characteristics which are required for some points in the main text. Part of Miranda’s approach was to blow up $\Delta$ until it had only double points. In other words, he needed only to consider [*transverse*]{} collisions of two curves within $\Delta$. Such collisions are classified by the generic fibre type over each of the two curves. In addition some collisions could be reduced to other types by blowing up the double point. As such he needed only to consider a subset of all possible collisions.
Our problem is not quite the same as Miranda’s. Blowing up the base, $\Theta$, will affect the canonical class of $X$, which we want to be trivial. Sometimes one [*must*]{} blow up the base (as in many example in the main text) in order to achieve $K_X=0$. In many other cases blowing up the base would destroy $K_X=0$. We find therefore that Miranda’s classification is not sufficient for us. We must often deal with collisions within $\Delta$ without blowing them up. As such there are considerably many more possibilities than Miranda considered. See [@BJ:collide] for a discussion of some aspects of F-theory which do fall into Miranda’s classification.
Fortunately Miranda’s methods did not rely on the assumption that $\Delta$ contained only double points. Let us review the construction. Begin with the case of a complex [*surface*]{}, $S$, which is an elliptic fibration, $\pi:S\to B$, where $B$ is an algebraic curve. Let $z$ be an affine coordinate in $B$. If this fibration has a global section then we may write the fibration in Weierstrass form $$y^2 = x^3 + a(z)x + b(z). \label{eq:Wei}$$ The discriminant is then given by $\delta=4a^3+27b^2$.
An elliptic curve may be written as a double cover of $\P^1$ branched at four points. Indeed, the Weierstrass form exhibits this property — $y$ has two solutions for any $x$ except at the roots of the right hand side of (\[eq:Wei\]). There are three roots of this cubic plus one solution “at infinity”. As such $S$ may be considered as a double cover of a $\P^1$-bundle over $B$ branched over the curve $x^3 + a(z)x + b(z)$ and the global section at infinity.
We may draw a typical model for $S$ as $$\setlength{\unitlength}{0.008750in}%
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\put(125,670){\makebox(0,0)[lb]{$\mathrm{I}_0$}}
\put(185,670){\makebox(0,0)[lb]{$\mathrm{I}_1$}}
\end{picture}$$ In this graph, the solid lines represent the branch locus (with the section at infinity at the top) and the two dotted lines represent $\P^1$ fibres for fixed values of $z$. The generic fibre on the left intersects the branch locus 4 times. The double cover of this is a smooth elliptic. This is an $\mathrm{I}_0$ fibre. On the fibre on the right, two of the branch points have coalesced. This amounts to shrinking a cycle in the elliptic down to a point and, as such, is a curve with a double point. This is an $\mathrm{I}_1$ fibre. $\delta$ will have a single zero at this point in $B$. Even though the $\mathrm{I}_1$ fibre is itself singular, $S$ is smooth.
It is possible for the branch locus to degenerate further to produce higher zeros in $\delta$. As an example let $$\begin{split}
y^2 = x^3 - 3x + 2+z^N.
\end{split}$$ If $N$ is even this looks like $$\setlength{\unitlength}{0.006250in}%
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\end{picture}$$ Now $S$ is singular at $(x,y,z)=(1,0,0)$. We may resolve $S$ by blowing this point up. We may follow the blow up in our picture. For example in the case $N=4$ we have $$\setlength{\unitlength}{0.006250in}%
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\label{eq:resI4}$$ The new curves may, or may not, be in the branch locus. The rule is that they are in the branch locus if and only if the total degree of branch divisor at the point blown up is odd. In the above case this degree is always two. We denote the fact that the new curves are not in the branch locus by drawing them as dotted lines. $S$ will be smooth when the branch locus is smooth and so after these two blow-ups we are done. The curve $f_0$ is the proper transform of the original bad fibre. Note that it only intersects the branch locus twice. Thus, the double cover of this is a rational curve, rather than an elliptic. The new curve $f_2$ is also branched twice and so will map to a rational curve in the double cover. $f_1$ is not branched at all and so must map to [*two*]{} rational curves in the double cover. The resulting configuration of curves in the double cover is $$\setlength{\unitlength}{0.006250in}%
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\put(140,805){\makebox(0,0)[lb]{$f_0$}}
\end{picture}$$ This is Kodaira’s $\mathrm{I}_4$ fibre. Subtracting the curve $f_0$, which was already in $S$, we see that the exceptional divisor within $S$ produced by the blow-up is a chain of three $\P^1$’s. This is the resolution of the surface singularity $A_3$ in the usual $A$-$D$-$E$ classification. This method of using a double cover is probably the best for finding the blow-ups of surface singularities and may be applied to all of the $A$-$D$-$E$ series. The type of bad fibres can be classified according to the degree of vanishing of $a$, $b$, and $\delta$.
Now let us return to our elliptic threefold, $X$. Let $s$ and $t$ be affine coordinates in the base, $\Theta$. Over a generic point in the discriminant we may put $z$ equal to a generic linear combination of $s$ and $t$ and reduce to the elliptic surface case. There is nothing to stop us putting such a generic slice through a bad point in the discriminant. The degrees of $(a,b,\delta)$ will jump at such a point.
Consider a transverse intersection of two curves, $D_1$ and $D_2$, within $\Delta$. The degrees of $(a,b,\delta)$ in our generic slice given by $z$ will then simply be the sum of the corresponding degrees along $D_1$ and $D_2$. One may expect them to be higher for non-transverse intersections however.
For example, let us consider the case given by (\[eq:simplek\]) of a curve of $\mathrm{I}_1$ fibres colliding with a curve of $\mathrm{I}_{2k}$ fibres given by $$\begin{split}
a &= t^k-3s^2\\
b &= -s(t^k-2s^2)\\
\delta &= t^{2k}(4t^k-9s^2).
\end{split}$$ The degrees along $4t^k-9s^2=0$ are $(0,0,1)$ (for an $\mathrm{I}_1$ fibre) and along $t=0$ are $(0,0,2k)$ (for an $\mathrm{I}_{2k}$ fibre). At $s=t=0$ these curves collide and the total degrees are $(2,3,2k+2)$, assuming $k\geq2$, (which is an $\Ist{2k-4}$ fibre).
To resolve $X$ we certainly need to begin by blowing up the fibres along the generic parts of $\Delta$ as in the surface case. Each time we do a blow-up of the generic points, the fibres at the collisions will also be partially resolved. In some simple cases the fibres at the collisions will automatically be fully resolved as usual by this process but, more usually, we will only end up with a partial resolution. At this point in the resolution process, $X$ may already be smooth or it may require a “small resolution” at the collision. Occasionally it cannot be resolved but this will not happen in any examples here.
Let us follow our example for this process. The $\mathrm{I}_1$ fibres require no blow-ups so we just have to consider the $\mathrm{I}_{2k}$ blow-up. Let us assume $k=2$ so that the resolution follows the sequence given in (\[eq:resI4\]). In this case the partial resolution of the $\Ist{0}$ fibre at the collision proceeds as $$\setlength{\unitlength}{0.006250in}%
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\end{picture}$$ Note that the first blow-up occurs at a degree 3 point in the branch locus. The exceptional curve is therefore in the branch locus. Note that at the end the branch locus is still colliding with itself. If this were a generic point in the discriminant we would have to continue the blow-up. In this case however, we are done. $X$ is now smooth. The fibre over the collision is the double cover of this which is given as follows: $$\setlength{\unitlength}{0.006250in}%
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\put(200,805){\makebox(0,0)[lb]{$f_2$}}
\put(140,805){\makebox(0,0)[lb]{$f_0$}}
\end{picture}$$ This has Euler characteristic 4. For general $k$ the Euler characteristic is $2+k$.
We are now in a position to read off the monodromy. Note that $f_1$ appears within the branch locus at the collision point and yet there are two $f_1$ curves in the $\mathrm{I}_4$ fibre away from the collision. So long as the Weierstrass form gives $y^2$ as a generic function of $s$ then an orbit around $s=0$ within the $t=0$ line will exchange the two $f_1$ curves in the fibre. Thus, this particular collision will induce monodromy. In general this collision will produce the expected monodromy in $\mathrm{I}_{2k}$ fibres to produce $\sp(k)$.
One should contrast this with to a [*transverse*]{} collision of an $\mathrm{I}_{m}$-curve and an $\mathrm{I}_{n}$-curve in which case there is no monodromy (unless, of course, it’s induced by a collision elsewhere). Analysis in [@Mir:fibr] shows that the Euler characteristic of the fibre over such a collision is $m+n$. Miranda also considered the case of an $\mathrm{I}_{2k}$-curve collision with a $\Ist{2m}$-curve which is relevant for our purposes. In this case the resulting fibre at the collision point has Euler characteristic $2m+k+6$. Monodromy is induced on the $\mathrm{I}_{2k}$ fibre but not the $\Ist{2m}$ fibre.
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[^1]: Although the moduli associated with the R-R sector in the type IIA string may be troublesome.
[^2]: Unless we really need to, we will speak in terms of the algebra, rather than the group, of the gauge symmetry. This will allow us to ignore a lot of awkward $\Z_2$ factors.
[^3]: I thank C. Johnson for a correspondence on this question.
[^4]: The only awkward step is calculating the Euler characteristic of $\Delta''$. This is done by using the adjunction formula and then compensating for the various high-order tacnodes which appear in $\Delta''$.
[^5]: I thank N. Seiberg for explaining this to me and showing that they do indeed cancel.
[^6]: I thank D. Morrison for conversations about this.
[^7]: Actually there are good reasons to expect the Mordell-Weil group to enhance this further to $\gu(16)$. This is because $X$ can be written as a K3 fibration whose generic fibre can be written as a double cover of a rational elliptic surface. The rational elliptic surface thus obtained is known from the classification of [@Pers:RES] to have a Mordell-Weil group of rank one. I thank M. Gross for conversations on this point.
|
---
abstract: 'Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Lévy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.'
author:
- 'Anastasia Borovykh[^1]'
- 'Andrea Pascucci[^2]'
- 'Cornelis W. Oosterlee[^3]'
bibliography:
- 'Biblio.bib'
title: Efficient Computation of Various Valuation Adjustments Under Local Lévy Models
---
Fast Fourier Transform, CVA, XVA, BSDE, characteristic function
Introduction {#sec1}
============
After the financial crisis in 2007, it was recognized that Counterparty Credit Risk (CCR) poses a substantial risk for financial institutions. In 2010 in the Basel III framework an additional capital charge requirement, called Credit Valuation Adjustment (CVA), was introduced to cover the risk of losses on a counterparty default event for over-the-counter (OTC) uncollateralized derivatives. The CVA is the expected loss arising from a default by the counterparty and can be defined as the difference between the risky value and the current risk-free value of a derivatives contract. CVA is calculated and hedged in the same way as derivatives by many banks, therefore having efficient ways of calculating the value and the Greeks of these adjustments is important.
One common way of pricing CVA is to use the concept of expected exposure, defined as the mean of the exposure distribution at a future date. Calculating these exposures typically involve computationally time-consuming Monte Carlo procedures, like nested Monte Carlo schemes or the more efficient least squares Monte Carlo method (LSM)[@LongstaffSchwartz]. Recently the Stochastic Grid Bundling method (SGBM)[@shashi2013] was introduced as an improvement of the standard LSM. This method was extended to pricing CVA for Bermudan options in [@feng14]. Another recently introduced alternative is the so-called finite-differences Monte Carlo method (FDMC) [@degraaf]. The FDMC method uses the scenario generation from the Monte Carlo method combined with finite-difference option valuation.
Besides CVA, many other valuation adjustments, collectively called XVA, have been introduced in option pricing in the recent years, causing a change in the way derivatives contracts are priced. For instance, a companies own credit risk is taken into account with a debt value adjustment (DVA). The DVA is the expected gain that will be experienced by the bank in the event that the bank defaults on its portfolio of derivatives with a counterparty. To reduce the credit risk in a derivatives contract, the parties can include a credit support annex (CSA), requiring one or both of the parties to post collateral. Valuation of derivatives under CSA was first done in [@piterbarg10]. A margin valuation adjustment (MVA) arises when the parties are required to post an initial margin. In this case the cost of posting the initial margin to the counterparty over the length of the contract is known as MVA. Funding value adjustments (FVA) can be interpreted as a funding cost or benefit associated to the hedge of market risk of an uncollateralized transaction through a collateralized market. While there is still a debate going on about whether to include or exclude this adjustment, see [@hullwhite2] and [@fvadebate] for an in-depth overview of the arguments, most dealers now seem to indeed take into account the FVA. The capital value adjustment (KVA) refers to the cost of funding the additional capital that is required for derivative trades. This capital acts as a buffer against unexpected losses and thus, as argued in [@green], has to be included in derivative pricing.
For pricing in the presence of XVA, one needs to redefine the pricing partial differential equation (PDE) by constructing a hedging portfolio with cashflows that are consistent with the additional funding requirements. This has been done for unilateral CCR in [@piterbarg10], bilateral CCR and XVA in [@burgard11] and extended to stochastic rates in [@lesniewski16]. This results in a non-linear option valuation PDE.
Non-linear PDEs can be solved by e.g. finite-difference methods or the LSM for solving the corresponding backward stochastic differential equation (BSDE). In [@piterbarg15] an efficient forward simulation algorithm that gives the solution of the non-linear PDE as the optimum over solutions of related but linear PDEs is introduced, with the computational cost being of the same order as one forward Monte Carlo simulation. The downside of these numerical methods is the computational time that is required to reach an accurate solution. An efficient alternative might be to use Fourier methods for solving the (non-)linear PDE or related BSDE, such as the COS method, as was introduced in [@FangO08], extended to Bermudan options in [@FangO09] and to BSDEs in [@ruijter15]. In certain cases the efficiency of these methods is further increased due to the ability to additionally use the fast Fourier transform (FFT).
In this paper we consider an exponential Lévy-type model with a state-dependent jump measure and propose an efficient Fourier-based method to solve for Bermudan derivatives, including options and swaptions, with XVA. We derive, in the presence of state-dependent jumps, a non-linear partial integro-differential equation (PIDE) and its corresponding BSDE for an OTC derivative between a bank $B$ and its counterparty $C$ in the presence of CCR, bilateral collateralization, MVA, FVA and KVA, by setting up a hedging portfolio in which we focus on hedging the default risks and take into account the different rates associated with different types of lending. We extend the Fourier-based method known as the BCOS method, developed in [@ruijter15], to solve the BSDE under Lévy models with non-constant coefficients. As this method requires the knowledge of the characteristic function of the forward process, which, in the case of the Lévy process with variable coefficients, is not known, we will use an approximation of the characteristic function obtained by the adjoint expansion method developed in [@pascucci-riga], [@LorigPP2015] and extended to the defaultable Lévy process with a state-dependent jump measure in [@borovykh]. Compared to other state-of-the-art methods for calculating XVAs, like Monte Carlo methods and PDE solvers, our method is more efficient and/or flexible. The efficiency is both due to the availability of the characteristic function in closed form through the adjoint expansion method and the fast convergence of the COS method. Furthermore we propose an alternative Fourier-based method for explicitly pricing the CVA term in case of unilateral CCR for Bermudan derivatives under the local Lévy model. The advantage of this method is that is allows us to use the FFT, resulting in a fast and efficient calculation. The Greeks, used for hedging CVA, can be computed at almost no additional cost.
The rest of the paper is structured as follows. In Section \[sec2\] we introduce the Lévy models with non-constant coefficients. In Section \[sec3\] we derive the non-linear PIDE and corresponding BSDE for pricing contracts under XVA. In Section \[sec4\] we propose the Fourier-based method for solving this BSDE and in Section \[sec51\] this method is extended to pricing Bermudan contracts. In Section \[sec52\] an alternative FFT-based method for pricing and hedging the CVA term is proposed and Section \[sec6\] presents numerical examples validating the accuracy and efficiency of the proposed methods.
The model {#sec2}
=========
We consider a defaultable asset $S_t$ whose risk-neutral dynamics are given by $$\begin{aligned}
S_t &= \caratt_{\{t<\zeta\}}e^{X_t},\\
dX_t &= \mu (t,X_t)dt+\sigma (t,X_t)dW_t+\int_\mathbb{R}q d\tilde N_t(t,X_{t-},dq),\\\label{eq:hetmodel}
d\tilde N_t(t,X_{t-},dq) &= dN_t(t,X_{t-},dq)-a(t, X_{t-})\nu (dq)dt,\\
\zeta &= \inf\{t\geq 0 :\int_0^t\gamma(s,X_{s})ds\geq \varepsilon\},\end{aligned}$$ where $d\tilde N_t(t,X_{t-},dq)$ is a compensated random measure with state-dependent Lévy measure $$\nu(t,X_{t-},dq) = a(t, X_{t-})\nu(dq).$$ The default time $\zeta$ of $S_t$ is defined in a canonical way as the first arrival time of a doubly stochastic Poisson process with local intensity function $\gamma(t,x)\geq 0$, and $\varepsilon
\sim \mathrm{Exp}(1)$ and is independent of $X_t$. This way of modeling default is also considered in a diffusive setting in [@JDCEV] and for exponential Lévy models in [@capponi]. Thus, our model includes a local volatility function, a local jump measure, and a default probability which is dependent on the underlying. We define the filtration at time $t$ of the market observer to be $\mathcal{G}_t=\mathcal{F}^X_t\vee \mathcal{F}^D_t$, where $\mathcal{F}^X_t$ is the filtration generated by $X$ upto time $t$ and $\mathcal{F}_t^D:=\sigma(\{\zeta\leq u\},u\leq t)$, for $t\ge0$, is the filtration of the default. Using this definition of default, the probability of default is $$\begin{aligned}
\label{eq:probdef}
\text{PD}(t) := \mathbb{P}(\zeta\leq t)= 1-\mathbb{E}\left[e^{-\int_0^t\gamma(s,X_s)ds}\right].
\end{aligned}$$ We assume furthermore $$\label{nusomm}
\int_\mathbb{R}e^{|q|}a(t,x)\nu(dq)<\infty.$$ Imposing that the discounted asset price $\tilde S_t := e^{-rt}S_{t}$ is a $\mathcal{G}$-martingale under the risk-neutral measure, we get the following restriction on the drift coefficient: $$\begin{aligned}
\label{eq:martdrift}
\mu(t,x) = \gamma(t,x)+r-\frac{\sigma^2(t,x)}{2}-a(t,x)\int_\mathbb{R}\nu(dq)(e^q-1-q),\end{aligned}$$ with $r$ being the risk-free (collateralized) rate. In the whole of the paper we assume deterministic, constant interest rates, while the derivations can easily be extended to time-dependent rates. The integro-differential operator of the process is given by (see e.g. [@Pascucci2011]) $$\begin{aligned}
L u(t,x) =&\partial_t u(t,x)+\mu(t,x)\partial_xu(t,x) -\gamma(t,x)u(t,x)+\frac{\sigma^2(t,x)}{2}\partial_{xx}u(t,x)\nonumber\\
&+a(t,x)\int_\mathbb{R}\nu(dq)(u(t,x+q)-u(t,x)-q\partial_x u(t,x)).\label{eq:opL}
\end{aligned}$$
XVA computation {#sec3}
===============
Consider a bank $B$ and its counterparty $C$, both of them might default. Assume they enter into a contract paying $\Phi(S_t)$ at maturity. Let $\phi(x) = \Phi(e^x)$, and assume the risk-neutral dynamics of the underlying as in with the drift given by . Define $\hat u(t,x)$ to be the value to the bank of the (default risky) portfolio with valuation adjustments referred to as XVA and $u(t,x)$ to be the risk-free value. Note that the difference between these two values is called the *total valuation adjustment* and in our setting this consists of $$\begin{aligned}
\label{eq:xva}
\textnormal{TVA}:=\hat u(t,x)-u(t,x)=\textnormal{CVA}+\textnormal{DVA}+\textnormal{KVA}+\textnormal{MVA}+\textnormal{FVA}.\end{aligned}$$ The risk-free value $u(t,x)$ solves a linear PIDE: $$\begin{aligned}
\label{eq:linearpdenoxva}
Lu(t,x) &= ru(t,x),\\
u(T,x) &= \phi(x),\end{aligned}$$ where $L$ is given in . Assuming the dynamics in , this linear PIDE can be solved with the methods presented in [@borovykh].
Derivative pricing under CCR and bilateral CSA agreements {#sec31}
---------------------------------------------------------
In [@burgard11], the authors derive an extension to the Black-Scholes PDE in the presence of a bilateral counterparty risk in a jump-to-default model with the underlying being a diffusion, using replication arguments that include the funding costs. In [@lesniewski16] this derivation is extended to a multivariate diffusion setting with stochastic rates in the presence of CCR, assuming that both parties $B$ and $C$ are subject to default. To mitigate the CCR, both parties exchange collateral consisting of the initial margin and the variation margin. The parties are obliged to hold regulatory capital, the cost of which is the KVA and face the costs of funding uncollateralized positions through collateralized markets, known as FVA. Both [@burgard11] and [@lesniewski16] extend the approach of [@piterbarg10], in which unilateral collateralization was considered. We extend their approach to derive the value of $\hat u(t,x)$ when the underlying follows the jump-diffusion defined in . We assume a one-dimensional underlying diffusion and consider all rates to be deterministic and, for ease of notation, constant. We specify different rates, defined in Table \[tab000\], for different types of lending.
\[tab000\]
Rate Definition Rate Definition
------------- ---------------------- ------------- ----------------------
$r$ $r_R$
$r_D$ $r_F$
$r_B$ $r_C$
$\lambda_B$ $\lambda_B := r_B-r$ $\lambda_C$ $\lambda_C := r_C-r$
$\lambda_F$ $\lambda_F := r_F-r$ $R_B$
$R_C$
: Definitions of the rates used throughout the paper.
Assume that the parties $B$ and $C$ enter into a derivative contract on the spot asset that pays the bank $B$ the amount $\phi(X_T)$ at maturity $T$. The value of this derivative to the bank at time t is denoted by $\hat u(t,x,\mathcal{J}^B,\mathcal{J}^C)$ and depends on the value of the underlying $X$ and the default states $\mathcal{J}^B$ and $\mathcal{J}^C$ of the bank $B$ and counterparty $C$, respectively. Define $I^{TC}$ to be the initial margin posted by the bank to the counterparty, $I^{FC}$ the initial margin posted by the counterparty to the bank and $I^V(t)$ to be the variation margin on which a rate $r_I$ is paid or received. The initial margin is constant throughout the duration of the contract. Let $K(t)$ be the regulatory capital on which a rate of $r_K$ is paid/received.
The cashflows are viewed from the perspective of the bank $B$. At the default time of either the counterparty or the bank, the value of the derivative to the bank $\hat u(t,x)$ is determined with a mark-to-market rule $M$, which may be equal to either the derivative value $\hat u(t,x,0,0)$ prior to default or the risk-free derivative value $u(t,x)$, depending on the specifications in the ISDA master agreement. Denote by $\tau^B$ and $\tau^C$ the random default times of the bank and the counterparty respectively. We will use the notation $x^+=\max(x,0)$ and $x^-=\min(x,0)$. In a situation in which the counterparty defaults, the bank is already in the possession of $I^V+I^{FC}$. If the outstanding value $M-(I^V+I^{FC})$ is negative, the bank has to pay the full amount $(M-I^V-I^{FC})^-$, while if the contract has a positive value to the bank, it will recover only $R_C(M-I^V-I^{FC})^+$. Using a similar argument in case the bank defaults, we find the following boundary conditions: $$\begin{aligned}
\theta^B_t &:=\hat u(t,x,1,0)= I^V(t)-I^{TC}+(M-I^V(t)+I^{TC})^++R^B(M-I^V(t)+I^{TC})^-,\\
\theta^C_t &:=\hat u(t,x,0,1)= I^V(t)+I^{FC}+R^C(M-I^V(t)-I^{FC})^++(M-I^V(t)-I^{FC})^-,\end{aligned}$$ so that the portfolio value at default is given by $$\theta_\tau = 1_{\tau^C<\tau^B}\theta_\tau^C+1_{\tau^B<\tau^C}\theta_\tau^B,$$ with $\tau = \min(\tau^B,\tau^C)$. Further we introduce the default risky, zero-recovery, zero-coupon bonds (ZCBs) $P^B$ and $P^C$ with respective maturities $T^B$ and $T^C$ with face value one if the issuer has not defaulted, and zero otherwise. Assume the dynamics for $P^B_t$ and $P_t^C$ to be given by $P_t^B=\caratt_{\{\tau^B>t\}}e^{r_Bt}$ and $P_t^C=\caratt_{\{\tau^C>t\}}e^{r_Ct}$, so that $$\begin{aligned}
dP_t^B &= r_BP^B_tdt-P_{t-}^Bd\mathcal{J}_t^B,\\
dP_t^C &= r_CP^C_tdt-P_{t-}^Cd\mathcal{J}_t^C,\end{aligned}$$ with $\mathcal{J}_t^B=\caratt_{\tau^B\leq t}$ and $\mathcal{J}_t^C=\caratt_{\tau^C\leq t}$, where the default times $\tau^B$ and $\tau^C$ are defined in a canonical way as the first arrival time of a doubly stochastic Poisson process with intensity functions $\gamma^B$ and $\gamma^C$, respectively (see also the definition of the defaultable asset in ). We define the market interest rates for $B$ and $C$ to be $r_B=r+\gamma^B$ and $r_C=r+\gamma^C$, so that by the usual arguments (see, for instance, [@linetsky2006bankruptcy Section 2.2]) the discounted bonds $e^{-rt}P_t^B$ and $e^{-rt}P_t^C$ are martingales under the risk-neutral measure.
We construct a hedging portfolio consisting of the shorted derivative, $\alpha_C$ units of $P^C$, $\alpha_B$ units of $P^B$ and $g$ units of cash: $$\Pi(t) = -\hat u(t,x) + \alpha_B(t)P^B_t+\alpha_C(t)P^C_t+g(t).$$ In other words, since we assume both the underlying asset process and the tradeable bonds $P_B$ and $P_C$ to be risk-neutral, we focus on hedging the risk arising from the defaults of both $B$ and $C$ by means of the default-risky bonds.
If the value of the derivative is positive to $B$, it will incur a cost at the counterparties’ default. To hedge this, $B$ shorts $P^C$, i.e. $\alpha_C\leq 0$. If we assume $B$ can borrow the bond close to the risk-free rate $r$ (i.e. no haircut) through a repurchase agreement, it will incur financing costs of $r\alpha_C(t)P_t^Cdt$. The cashflows from the collateralization follow from the rate $r_{TC}$ received and $r_{FC}$ paid on the initial margin and the rate $r_I$ paid or received on the collateral, depending on whether $I^V>0$, and the bank receives collateral, or $I^V<0$, and the bank pays collateral respectively. From holding the regulatory capital we incur a cost of $r_KK(t)$. Finally, the rates $r$ and $r_F$ are respectively received or paid on the surplus cash in the account. This cash consists of the gap between the shorted derivative value and the collateral and the cost of buying $\alpha_B$ bonds $P^B$ in order for $B$ to hedge its own default, i.e. $-\hat u(t,x)-I^V(t)+I^{TC}-\alpha_B(t)P_t^B$. Thus, the total change in the cash account is given by $$\begin{aligned}
dg(t) =& [-r\alpha_C(t)P^C_t+r_{TC}I_{TC}-r_{FC}I_{FC}-r_II^V(t)-r_KK(t)\\
&+r(-\hat u(t,x)-I^V(t)+I_{TC}-\alpha_B(t)P^B_t)+\lambda_F(-\hat u(t,x)-I^V(t)+I_{TC}-\alpha_B(t)P^B_t)^-]dt.\end{aligned}$$ Note that this is in contrast with the change in cash in a portfolio without the XVA arising from the different types of funding, i.e. where we assume the cash in the portfolio simply earns the risk-free rate $$\begin{aligned}
dg(t) = -r \hat u(t,x)dt.\end{aligned}$$ Assuming the portfolio is self-financing we have $$\begin{aligned}
d\Pi(t)=& -d\hat u(t,x)+\alpha_B(t)dP^B_t+\alpha_C(t)dP^C_t+dg(t).
$$ Applying Itô’s Lemma to $\hat u(t,x)$ gives us: $$\begin{aligned}
d\hat u(t,x) =& L\hat u(t,x)dt +\sigma(t,x)\partial_x\hat u(t,x)dW_t+\int_\mathbb{R}(\hat u(t,x+q)-\hat u(t,x))d\tilde N(t,x,dq)\\
&-(\theta^B-\hat u(t,x))d\mathcal{J}^B_t-(\theta^C-\hat u(t,x))d\mathcal{J}^C_t,\end{aligned}$$ with the operator $L$ as in . Thus, we find, $$\begin{aligned}
d\Pi =& -L\hat u(t,x)dt -\sigma(t,x)\partial_x\hat u(t,x)dW_t-\int_\mathbb{R}(\hat u(t,x+q)-\hat u(t,x))d\tilde N(t,x,dq)\\
&+(\theta^B-\hat u(t,x))d\mathcal{J}^B_t+(\theta^C-\hat u(t,x))d\mathcal{J}^C_t-\alpha^B(t)P_{t-}^Bd\mathcal{J}_t^B-\alpha^C(t)P_{t-}^Cd\mathcal{J}_t^C\\
&+[\alpha^B(t)\lambda_BP_t^B+\alpha^C(t)\lambda_CP^C_t+(r_{TC}+r)I^{TC}-r_{FC}I^{FC}-(r_I+r)I^V(t)\\
&-r_KK(t)+r\hat u(t,x)+\lambda_F(-\hat u(t,x)-I^V(t)+I^{TC}-\alpha^B(t)P^B_t)^-]dt.\end{aligned}$$ By choosing $$\begin{aligned}
\alpha_B = -\frac{\theta^B - \hat u(t,x)}{P_B}, \;\;\; \alpha_C = -\frac{\theta^C - \hat u(t,x)}{P_C},\end{aligned}$$ we hedge the jump-to-default risk in the hedging portfolio, i.e., $$\begin{aligned}
d\Pi =& -L\hat u(t,x)dt +\sigma(t,x)\partial_x\hat u(t,x)dW_t-\int_\mathbb{R}(\hat u(t,x+q)-\hat u(t,x))d\tilde N(t,X_{t-},dq)\\
&+[-(\theta^B-\hat u(t,x))\lambda_B-(\theta^C-\hat u(t,x))\lambda_C+(r_{TC}+r)I^{TC}-r_{FC}I^{FC}-(r_I+r)I^V(t)\\
&-r_KK(t)+r\hat u(t,x)+\lambda_F(\theta^B-I^V(t)+I^{TC})^-]dt.\end{aligned}$$ Then, using the fact that the portfolio has to satisfy the martingale condition in the risk-neutral world, i.e. $\mathbb{E}[d\Pi] = 0$, we find the non-linear pricing PIDE to be $$\begin{aligned}
\label{eq:thepde}
L\hat u(t,x) =& f(t,x,\hat u(t,x)),\end{aligned}$$ where we have defined $$\begin{aligned}
f(t,x,\hat u(t,x))=&-(\theta^B(t)-\hat u(t,x))\lambda_B-(\theta^C(t)-\hat u(t,x))\lambda_C+(r_{TC}+r)I^{TC}-r_{FC}I^{FC}\\
&-(r_I+r)I^V(t)-r_KK(t)+r\hat u(t,x)+\lambda_F(\theta^B-I^V(t)+I^{TC})^-.\end{aligned}$$
BSDE representation {#sec32}
-------------------
In this section we will cast the PIDE in in the form of a Backward Stochastic Differential Equation. In the methods where we make use of BSDEs we assume $\gamma(t,x)=0$. We begin by recalling the non-linear Feynman-Kac theorem in the presence of jumps, see Theorem 4.2.1 in [@delong13].
\[theorem1\] Consider $X_t$ as in . We assume $\mu$, $\sigma$ and $a$ to be Lipschitz continuous in $x$ and additionally $|a(t,x)|\leq K$. Consider the BSDE $$\begin{aligned}
Y_t =\ & \phi(X_T) + \int_t^T
f\left(s,X_s,Y_s,Z_s,a(s,X_{s-})\int_\mathbb{R}V_s(q)\delta(q)\nu(dq)\right)ds-\int_t^TZ_sdW_s\\\label{eq:setfbsdes1}
&-\int_t^T\int_\mathbb{R}V_s (q)d\tilde N_s(s,X_s,q),\end{aligned}$$ where the generator $f$ is continuous and satisfies the Lipschitz condition in the space variables, $\delta$ is a measurable, bounded function and the terminal condition $\phi(x)$ is measurable and Lipschitz continuous. Consider the non-linear PIDE $$\begin{aligned}
\label{eq:bsdepde}
\begin{cases}
Lu(t,x) = f(t,x,u(t,x),\partial_xu(t,x)\sigma(t,x),a(t,x)\int_\mathbb{R}(u(t,x+q)-u(t,x))\delta(q)\nu(dq)),\\
u(T,x) = \psi(x).
\end{cases}\end{aligned}$$ If the PIDE in has a solution $u(t,x)\in C^{1,2}$, the FBSDE in has a unique solution $(Y_t,Z_t,V_t(q))$ that can be represented as $$\begin{aligned}
&Y_s^{t,x} = u(s,X_s^{t,x}),\\
&Z_s^{t,x} = \partial_xu(s,X_s^{t,x})\sigma(s,X_s^{t,x}),\\
&V_s^{t,x}(q) = u(s,X_s^{t,x}+q)-u(s,X_s^{t,x}),\qquad q\in\mathbb{R},\end{aligned}$$ for all $s\in[t,T]$, where $Y$ is a continuous, real-valued and adapted process and where the control processes $Z$ and $V$ are continuous, real-valued and predictable.
In our case, the BSDE corresponding to the PIDE in reads $$\begin{aligned}
\label{eq:fundbsde}
Y_t = \phi(X_T) + \int_t^T f(s,X_s,Y_s)ds-\int_t^TZ_sdW_s-\int_t^T\int_\mathbb{R}V_s(q)d\tilde N(s,X_s,dq),\end{aligned}$$ where we have defined the driver function to be $$\begin{aligned}
f(t,x,y) =& -\lambda_B(\theta^B-y)-\lambda_C(\theta^C-y)+(r_{TC}+r)I^{TC}-r_{FC}I^{FC}-(r_I+r)I^V(t)\\
&-r_KK(t)+ry+\lambda_F(\theta^B-I^V(t)+I^{TC})^-.\end{aligned}$$
A simplified driver function {#sec323}
----------------------------
Following [@green], one can derive that the KVA is a function of trade properties (i.e. maturity, strike) and/or the exposure at default, which in turn is a function of the portfolio value, so that the cost of holding the capital can be rewritten as $r_KK(t)=r_Kc_1\hat u(t,x),$ with $c_1$ being a function of the trade properties. The collateral is paid when the portfolio has a negative value, and received when the collateral has a positive value. Assuming the collateral is a multiple of the portfolio value we have $I^V(t)= c_2\hat u(t,x)$, where $c_2$ is some constant. Then, the driver function is simply a function of the portfolio value.
Note that in the case of ‘no collateralization’ or ‘perfect collateralization’, the driver function reduces to $f(t,\hat u(t,x)) = r_u(t)\max(\hat u(t,x),0)$, for a function $r_u$ here left unspecified. In this case the BSDE is similar to the one considered in [@piterbarg15].
Solving FBSDEs {#sec4}
==============
In this section we extend the BCOS method from [@ruijter15] to solving FBSDEs under local Lévy models with variable coefficients and jumps (without default, i.e. $\gamma(t,x)=0$). The conditional expectations resulting from the discretization of the FBSDE are approximated using the COS method. This requires the characteristic function, which we approximate using the [Adjoint Expansion Method]{} of [@pascucci-riga] and [@borovykh].
Discretization of the BSDE {#sec41}
--------------------------
Consider the forward process $X_t$ as in and the BSDE $Y_t$ as in with a more general driver function $f(t,x,y,z)$. Define a partition $0=t_0<t_1<...<t_N=T$ of $[0,T]$ with a fixed time step $\Delta t = t_{n+1}-t_n$, for $n=N-1,...0$. Rewriting the set of FBSDEs we find, $$\begin{aligned}
X_{n+1}=&X_n+\int_{t_n}^{t_{n+1}}\mu(s,X_s)ds+\int_{t_n}^{t_{n+1}}\sigma(s,X_s)dW_s+\int_{t_n}^{t_{n+1}}\int_\mathbb{R}qd\tilde N_s(s,X_{s-},dq),\\ \label{eq:discry}
Y_{n}=&Y_{n+1}+\int_{t_n}^{t_{n+1}}f\left(s,X_s,Y_s,Z_s\right)ds-\int_{t_n}^{t_{n+1}}Z_sdW_s-\int_{t_n}^{t_{n+1}}\int_\mathbb{R}V_s(q)d\tilde N_s(s,X_{s-},dq).\end{aligned}$$ One can obtain an approximation of the process $Y_t$ by taking conditional expectations with respect to the underlying filtration $\mathcal{G}_n$, using the independence of $W_t$ and $\tilde N_t(t,X_{t-},dq)$ and by approximating the integrals that appear with a theta-method, as first done in [@zhao12] and extended to BSDEs with jumps in [@ruijter15]: $$\begin{aligned}
Y_n &\approx \mathbb{E}_n[Y_{n+1}]+\Delta t \theta_1f\left(t_n,X_n,Y_n,Z_n\right)+\Delta t (1-\theta_1)\mathbb{E}_n\left[f\left(t_{n+1},X_{n+1},Y_{n+1},Z_{n+1}\right)\right].\end{aligned}$$ Let $\Delta W_s:=W_s-W_n$ for $t_n\leq s\leq t_{n+1}$. Multiplying both sides of equation by $\Delta W_{n+1}$, taking conditional expectations and applying the theta-method gives $$\begin{aligned}
Z_n &\approx -\theta_2^{-1}(1-\theta_2)\mathbb{E}_n[Z_{n+1}]+\frac{1}{\Delta t}\theta_2^{-1}\mathbb{E}_n[Y_{n+1}\Delta W_{n+1}]\\
&+\theta_2^{-1}(1-\theta_2)\mathbb{E}_n\left[f\left(t_{n+1},X_{n+1},Y_{n+1},Z_{n+1}\right)\Delta W_{n+1}\right].\end{aligned}$$ Since in our scheme the terminal values are functions of time $t$ and the Markov process $X$, it is easily seen that there exist deterministic functions $y(t_n,x)$ and $z(t_n,x)$ so that $$\begin{aligned}
Y_n=y(t_n,X_n), \;\;\; Z_n=z(t_n,X_n).\end{aligned}$$ The functions $y(t_n,x)$ and $z(t_n,x)$ are obtained in a backward manner using the following scheme $$\begin{aligned}
\label{eq:scheme1}
y(t_N,x)=&\phi(x), \;\;\; z(t_N,x) = \partial_x\phi(x)\sigma(t_N,x),\\
&\textnormal{for $n=N-1,...,0$:}\\\label{eq:scheme2}
y(t_n,x) =& \mathbb{E}_n[y(t_{n+1},X_{n+1})]+\Delta t \theta_1f\left(t_n,x\right)+\Delta t (1-\theta_1)\mathbb{E}_n\left[f(t_{n+1},X_{n+1})\right],\\\label{eq:scheme3}
z(t_n,x) =& -\frac{1-\theta_2}{\theta_2}\mathbb{E}_n[z(t_{n+1},X_{n+1})]+\frac{1}{\Delta t}\theta_2^{-1}\mathbb{E}_n[y(t_{n+1},X_{n+1})\Delta W_{n+1}]\\
&+\frac{1-\theta_2}{\theta_2}\mathbb{E}_n\left[f(t_{n+1},X_{n+1})\Delta W_{n+1}\right],\end{aligned}$$ where we have simplified notations with $$\begin{aligned}
f(t,X_t) := f\left(t,X_t,y(t,X_t),z(t,X_t)\right).\end{aligned}$$ In the case $\theta_1>0$ we obtain an implicit dependence on $y(t_n,x)$ in and we use $P$ Picard iterations starting with initial guess $\mathbb{E}_n[y(t_{n+1},X_{n+1})]$ to determine $y(t_n,x)$.
The characteristic function {#sec42}
---------------------------
Is it well-known (see, for instance, [@linetsky2006bankruptcy Section 2.2]) that the risk-free pre-default price $u(t,x)$ of a European option on the defaultable asset $S_t$ with maturity $T$ and payoff $\phi(X_{T})$ is given by $$\begin{aligned}
\label{e1}
u(t,x) = \caratt_{\{\zeta>t\}} e^{-r(T-t)}\mathbb{E} \left[e^{-\int_t^T \gamma(s,X_s) ds}\phi(X_T) | X_t \right],\;\;\; t\leq T,\end{aligned}$$ in the measure corresponding to the dynamics in . Thus, in order to compute the price of an option, we must evaluate functions of the form $$\begin{aligned}
\label{expectation}
v(t,x):= \mathbb{E} \left[e^{-\int_t^T \gamma(s,X_s) ds}\phi(X_T)| X_t = x \right] .\end{aligned}$$ Under standard assumptions, by the Feynman-Kac theorem, $v$ can be expressed as the classical solution of the following Cauchy problem $$\begin{aligned}
\label{eq:v.pide}
&\begin{cases}
L v(t,x)=0,\qquad & t\in[0,T[,\ x\in\mathbb{R}, \\
v(T,x) = \phi(x),& x \in\mathbb{R},
\end{cases}\end{aligned}$$ with $L$ as in .
The function $v$ in can be represented as an integral with respect to the transition distribution of the defaultable log-price process $\log S_t$: $$\begin{aligned}
\label{eq:v.def1}
v(t,x) = \int_\mathbb{R} \phi(y)\Gamma(t,x;T,dy),\end{aligned}$$ where $\Gamma(t,x;T,dy)$ is the Green’s function of the PIDE in and we say that its Fourier transform $$\hat\Gamma(t,x;T,\xi):=\mathcal{F}(\Gamma(t,x;T,\cdot))(\xi):= \int_\mathbb{R}e^{i\xi y}\Gamma(t,x;T,dy),\qquad \xi\in\mathbb{R},$$ is the characteristic function of $\log S$. Following [@pascucci-riga] and [@borovykh] we expand the state-dependent coefficients $$s(t,x):=\frac{\sigma^2(t,x)}{2},\qquad \mu(t,x), \qquad \gamma(t,x),\qquad a(t,x),$$ around some point $\bar{x}$. The coefficients $s(t,x)$, $\gamma(t,x)$ and $a(t,x)$ are assumed to be continuously differentiable with respect to $x$ up to order $n\in\mathbb{N}$.
Introduce the $n$th-order approximation of $L$ in : $$\begin{aligned}
L_n =&\ L_0 + \sum_{k=1}^n\Big( (x-\bar x)^k\mu_k(t)+(x-\bar x)^k s_k(t) \partial_{xx}-(x-\bar x)^k\gamma_k(t)\\
&+\int_\mathbb{R}(x-\bar x)^ka_k(t)\nu(dq)(e^{q\partial_x}-1-q\partial_x)\Big),
\end{aligned}$$ where $$\begin{aligned}
L_0 &= \partial_t +\mu_0(t)\partial_x+ s_0(t) \partial_{xx}-\gamma_0(t)
+\int_\mathbb{R}a_0(t)\nu(dq)(e^{q\partial_x}-1-q\partial_x),
\end{aligned}$$ and $$\begin{aligned}
s_k= \frac{\partial_x^k s(\cdot,\bar x)}{k!},\qquad
\gamma_k = \frac{\partial_x^k \gamma(\cdot,\bar x)}{k!},\qquad
\mu_k(dq) = \frac{\partial_x^k \mu (\cdot,\bar x)}{k!},\qquad a_k= \frac{\partial_x^k a(\cdot,\bar x)}{k!}\qquad\ k\ge 0.
\end{aligned}$$ The basepoint $\bar x$ is a constant parameter which can be chosen freely. In general the simplest choice is $\bar x = x$ (the value of the underlying at initial time $t$).
Assume for a moment that $L_{0}$ has a fundamental solution $G^{0}(t,x;T,y)$ that is defined as the solution of the Cauchy problem $$\begin{cases}
L_0 G^{0}(t,x;T,y) =0\qquad & t\in[0,T[,\ x\in\mathbb{R}, \\
G^{0}(T,\cdot;T,y) =\delta_{y}.
\end{cases}$$ In this case we define the $n$th-order approximation of $\Gamma$ as $$\Gamma^{(n)}(t,x;T,y) = \sum_{k=0}^n G^{k}(t,x;T,y),$$ where, for any $k\ge 1$ and $(T,y)$, $G^{k}(\cdot,\cdot;T,y)$ is defined recursively through the following Cauchy problem $$\begin{cases}
L_0 G^{k}(t,x;T,y) = -\sum\limits_{h=1}^k(L_h-L_{h-1})G^{k-h}(t,x;T,y)\qquad & t\in[0,T[,\ x\in\mathbb{R}, \\
G^{k}(T,x;T,y) =0,& x \in\mathbb{R}.
\end{cases}$$ Notice that $$\begin{aligned}
L_k-L_{k-1}=& (x-\bar x)^k \mu_h(t) \partial_{x} +(x-\bar x)^k s_k(t) \partial_{xx}-(x-\bar x)^k\gamma_k(t)\\
&+\int_\mathbb{R}(x-\bar x)^ka_k(t)\nu(dq)(e^{q\partial_x}-1-q\partial_x).
\end{aligned}$$ Correspondingly, the $n$th-order approximation of the characteristic function $\hat \Gamma$ is defined to be $$\label{adapprox}
\hat \Gamma^{(n)}(t,x;T,\xi)=\sum_{k=0}^n \mathcal{F}\left(G^{k}(t,x;T,\cdot)\right)(\xi):=\sum_{k=0}^n \hat G^{k}(t,x;T,\xi),\qquad \xi\in\mathbb{R}.$$ Now, by transforming the simplified Cauchy problems into adjoint problems and solving these in the Fourier space we find $$\begin{aligned}
\label{eq:Ghat}
\hat G^{0}(t,x;T,\xi) &= e^{i\xi x}e^{\int_t^T\psi(s,\xi)ds},\\
\hat G^{k}(t,x;T,\xi) &= -\int_t^Te^{\int_s^T\psi(\tau,\xi)d\tau}\mathcal{F}\left(\sum_{h=1}^k\left(\tilde
L_h^{(s,\cdot)}(s)-\tilde L_{h-1}^{(s,\cdot)}(s)\right)G^{k-h}(t,x;s,\cdot)\right)(\xi)ds,\end{aligned}$$ with $$\begin{aligned}
\psi(t,\xi) = i\xi\mu_0(t)
+s_0(t)\xi^2+\int_\mathbb{R}a_0\nu(t,dq)(e^{iz\xi}-1-iz\xi),
\end{aligned}$$ $$\begin{aligned}
\tilde L_h^{(t,y)}(t)-\tilde L_{h-1}^{(t,y)}(t) &= \mu_h(t)h(y-\bar x)^{h-1}+\mu_h(t)(y-\bar x)^h\partial_y-\gamma_h(t)(y-\bar x)^h\\
& +s_h(t) h(h-1)(y-\bar x)^{h-2}+s_h(t) (y-\bar x)^{h-1} \left(2h\partial_y+(y-\bar x)\partial_{yy}\right)\\
&+\int_\mathbb{R}a_h(t)\bar\nu(dq)\left((y+q-\bar x)^he^{q\partial_y}-(y-\bar x)^h-q\left(h(y-\bar x)^{h-1}-(y-\bar
x)^h\partial_y\right)\right),\end{aligned}$$ where $\bar \nu(dq) = \nu(-dq)$.
\[r3\] After some algebraic manipulations it can be shown, see [@borovykh], that the characteristic function approximation of order $n$ is a function of the form $$\label{eq:struc3}
\hat\Gamma^{(n)}(t,x;T,\xi):= e^{i\xi x} \sum_{k=0}^n (x-\bar x)^k g_{n,k}(t,T,\xi),$$ where the coefficients $g_{n,k}$, with $0\leq k\leq n$, depend only on $t,T$ and $\xi$, but not on $x$. The approximation formula can thus always be split into a sum of products of functions depending only on $\xi$ and functions that are linear combinations of $(x-\bar x)^m e^{i\xi x}$, $m\in\mathbb{N}_{0}$.
Similar to the derivation in [@borovykh], one can derive the error bounds for the characteristic function approximation. Let $n=0,1$ and assume the coefficients $s(t,x)$, $\gamma(t,x)$ and $a(t,x)$ are continuously differentiable with bounded derivatives up to order $n$. For the $n$th-order approximation $\Gamma^{(n)}(t,x;T,\xi)$, for any $\bar x\in \mathbb{R}$, $$\begin{aligned}
\left|\Gamma(t,x;T,\xi)-\Gamma^{(n)}(t,x;T,\xi)\right|\leq C(T,\xi)((T-t)^2+(T-t)(x-\bar x))^{\frac{n+1}{2}}.\end{aligned}$$ Note that if $\bar x = x$, the bound reduces to $C(T,\xi)(T-t)^{n+1}$.
The COS formulae {#sec43}
----------------
The conditional expectations are approximated using the COS method, which was developed in [@FangO09] and applied to FBSDEs with jumps in [@ruijter15]. The conditional expectations arising in the equations - are all of the form $\mathbb{E}_n[h(t_{n+1},X_{n+1})]$ or $\mathbb{E}_n[h(t_{n+1},X_{n+1})\Delta W_{n+1}]$. The COS formula for the first type of conditional expectation reads $$\begin{aligned}
&\mathbb{E}_n^x[h(t_{n+1},X_{n+1})]\approx \sideset{}{'}\sum_{j=0}^{J-1}H_j(t_{n+1})\textnormal{Re}\left(\hat\Gamma\left(t_n,x;t_{n+1},\frac{j\pi}{b-a}\right)\exp\left(ij\pi\frac{-a}{b-a}\right)\right),\end{aligned}$$ where $\sideset{}{'}\sum$ denotes an ordinary summation with the first term weighted by one-half, $J>0$ is the number of Fourier-cosine coefficients we use, $H_j(t_{n+1})$ denotes the $j$th Fourier-cosine coefficients of the function $h(t_{n+1},x)$ and $\hat\Gamma\left(t_n,x;t_{n+1},\xi \right)$ is the conditional characteristic function of the process $X_{n+1}$ given $X_n=x$. For the second type of conditional expectation, using integration by parts, we obtain $$\begin{aligned}
\mathbb{E}_n^x&[h(t_{n+1},X_{n+1})\Delta W_n]\\
&\approx \Delta t\sigma(t_n,x)\sideset{}{'}\sum_{j=0}^{J-1}H_j(t_{n+1})\textnormal{Re}\left(i\frac{j\pi}{b-a}\hat\Gamma\left(t_n,x;t_{n+1},\frac{j\pi}{b-a}\right)\exp\left(ij\pi\frac{-a}{b-a}\right)\right).\end{aligned}$$ See [@ruijter15] for the full derivations.
Note that these formulas are obtained by using an Euler approximation of the forward process and using the 2nd-order approximation of the characteristic function of the actual process. We have found this to be more exact than using the characteristic function of the Euler process, which is equivalent to using just the 0th-order approximation of the characteristic function.
Finally we need to approximate the Fourier-cosine coefficients $H_j(t_{n+1})$ of $h(t_{n+1},x)$ at time points $t_n$, where $n=0,...,N$. The Fourier-cosine coefficient of $h$ at time $t_{n+1}$ is defined by $$\begin{aligned}
\label{eq:fouriercos}
&H_j(t_{n+1})=\frac{2}{b-a}\int_a^bh(t_{n+1},x)\cos\left(j\pi\frac{x-a}{b-a}\right)dx.\end{aligned}$$ Due to the structure of the approximated characteristic function of the local Lévy process, see , the coefficients of the functions $z(t_{n+1},x)$ and the explicit part of $y(t_{n+1},x)$ can be computed using the FFT algorithm, as we do in Appendix \[app1\], because of the matrix in being of a certain form with constant diagonals. In order to determine $F_j(t_{n+1})$, the Fourier-Cosine coefficient of the function $$f\left(t_{n+1},x,y(t_{n+1},x),z(t_{n+1},x)\right),$$ due to the intricate dependence on the functions $z$ and $y$ we choose to approximate the integral in $F_j$ by a discrete Fourier-Cosine transform (DCT). For the DCT we compute the integrand, and thus the functions $z(t_{n+1},x)$ and $y(t_{n+1},x)$, on an equidistant $x$-grid. Note that in this case we can easily approximate *all* Fourier-Cosine coefficients with a DCT (instead of the FFT). If we take $J$ grid points defined by $x_i:=a+(i+\frac{1}{2})\frac{b-a}{J}$ and $\Delta x = \frac{b-a}{J}$ we find, using the mid-point integration rule, the approximation $$\begin{aligned}
H_j(t_{n+1})\approx \frac{2}{J}\sideset{}{'}\sum_{i=0}^{J-1}h(t_{n+1},x_i)\cos\left(j\pi \frac{2i+1}{2J}\right),\end{aligned}$$ which can be calculated using the DCT algorithm, with a computational complexity of $O(J\log J)$.
We define the truncation range $[a,b]$ as follows: $$\begin{aligned}
\label{eq:truncrange}
[a,b]:=\left[c_1-L\sqrt{c_2+\sqrt{c_4}},c_1+L\sqrt{c_2+\sqrt{c_4}}\right],\end{aligned}$$ where $c_n$ is the $n$th cumulant of log-price process $\log S$, as proposed in [@FangO08]. The cumulants are calculated using the 0th-order approximation of the characteristic function.
XVA computation for Bermudan derivatives {#sec5}
========================================
The method in Section \[sec4\] allows us to compute the XVA as in , consisting of CVA, DVA, MVA, KVA and FVA. In this section, we apply this method to computing Bermudan derivative values with XVA. The resulting method – the solution of the non-linear XVA PDE through a BSDE-type method – is an efficient alternative to finite-difference methods as well as to the Monte-Carlo based method developed in [@piterbarg15]. The efficiency is both due to the availability of the characteristic function in closed form through the adjoint expansion method and the fast convergence of the COS method. Furthermore, in finite difference methods complications may arise in the implementation of the scheme for jump diffusions. Since our proposed method works in the Fourier space, the jump component is easily handled by means of an additional term in the characteristic function and does not cause any further difficulties.
For the CVA component in the XVA we develop an alternative method, which due to the ability of the FFT, results in a particularly efficient computation.
XVA computation {#sec51}
---------------
Consider an OTC derivative contract between the bank $B$ and the counterparty $C$ on the underlying asset $S_t$ given by with $\gamma(t,x)=0$ with a Bermudan-type exercise possibility: there is a finite set of so-called exercise moments $\{t_1,...,t_{M}\}$ prior to the maturity, with $0\le t_{1}<t_2<\cdots<t_{M}=T$. The payoff from the point-of-view of bank $B$ is given by $\phi(t_m, X_{t_m})$. Denote $\hat u(t,x)$ to be the risky Bermudan option value and $c(t,x)$ the continuation value. By the dynamic programming approach, the value for a Bermudan derivative with XVA and $M$ exercise dates $t_1,...,t_M$ can be expressed by a backward recursion as $$\begin{aligned}
\hat u(t_{M},x)=\phi(t_M,x),\end{aligned}$$ and the continuation value solves the non-linear PIDE defined in $$\begin{aligned}
\label{eq:bermud2}
\begin{cases}
\begin{cases}
Lc(t,x) =f(t,x,c(t,x)),\qquad \;\; t\in[t_{m-1},t_{m}[\\
c(t_m,x) = \hat u(t_m,x)
\end{cases}\\
\hat u(t_{m-1},x)=\max\{\Phi(t_{m-1},x),c(t_{m-1},x)\}, \;\;m\in\{2,\dots,M\}.
\end{cases}\end{aligned}$$ The derivative value is set to be $\hat u(t,x)=c(t,x)$ for $t\in]t_{m-1},t_m[$, and, if $t_1>0$, also for $t\in[0,t_1[$. The payoff function might take on various forms:
1. (Portfolio) Following [@piterbarg15], we can consider $X_t$ to be the process of a portfolio which can take on both positive and negative values. Then, when exercised at time $t_m$, bank $B$ receives the portfolio so that $\phi(t_m, x) =e^x$.
2. (Bermudan option) In case the Bermudan contract is an option, the option value to the bank can not have a negative value for the bank. At the same time, in case of default of the bank itself, the counterparty loses nothing. In this case the framework simplifies to one with unilateral collateralization and default risk and the payoff at time $t_m$, if exercised, is given by $\phi(t_m,x)=(K-e^x)^+$ for a put and $\phi(t_m,x)=(e^x-K)^+$ for a call with $K$ being the strike price.
3. (Bermudan swaptions) A Bermudan swaption is an option in which the holder, bank $B$, has the right to exercise and enter into an underlying swap with fixed end date $t_{M+1}$. If the swaption is exercised at time $t_m$ the underlying swap starts with payment dates $\mathcal{T}_m=\{t_{m+1},...,t_{M+1}\}$. Working under the forward measure corresponding to the last reset date $t_M$, the payoff function is given by $$\begin{aligned}
\phi(t_m,x) = N^S\left(\sum_{k=m}^M\frac{P(t_m,t_{k+1},x)}{P(t_m,t_M)}\Delta t\right)\max(c_p(S(t_m,\mathcal{T}_m,x)-K),0),\end{aligned}$$ where $N^S$ is the notional, $c_p=1$ for a payer swaption and $c_p=-1$ for a receiver swaption, $P(t_m,t_k,x)$ is the price of a ZCB conditional on $X_{t_m}=x$ and $S(t_m,\mathcal{T}_m,x)$ is the forward swap rate given by $$\begin{aligned}
S(t_m,\mathcal{T}_m,x)=\left(1-\frac{P(t_m,t_{m+1},x)}{P(t_m,t_M,x)}\right)\big /\left(\sum_{k=m}^M\frac{P(t_m,t_{k+1},x)}{P(t_m,t_M,x)}\Delta t\right ).\end{aligned}$$
To solve for the continuation value we define a partition with $N$ steps $t_{m-1}=t_{0,m}<t_{1,m}<t_{2,m}<...<t_{n,m}<...<t_{N,m}=t_m$ between two exercise dates $t_{m-1}$ and $t_m$, with fixed time step $\Delta t_n :=t_{n+1,m}-t_{n,m}$. Applying the method developed in Section \[sec4\], we find the following time iteration for the continuation value: $$\begin{aligned}
\label{eq:formulaym}
\textnormal{At time $t_{N,m}$ set:}&\\
c(t_{N,m},x)&=\hat u(t_m,x)\\
\textnormal{for }n=N-1,...,&0 \textnormal{ compute:}\\
c(t_{n,m},x)&\approx \Delta t_{n} \theta_1f(t_{n,m},x,c(t_{n,m},x))+\sideset{}{'}\sum_{j=0}^{J-1}\Psi_j(x)(C_j(t_{n+1,m})+\Delta t_n(1-\theta_1)F_j(t_{n+1,m})),\label{eq:formulay}\end{aligned}$$ where we have defined $$\begin{aligned}
&\Psi_j(x) =\textnormal{Re}\left(\hat\Gamma\left(t_{n,m},x;t_{n+1,m},\frac{j\pi}{b-a}\right)\exp\left(ij\pi\frac{-a}{b-a}\right)\right),\end{aligned}$$ and the Fourier-cosine coefficients are given by $$\begin{aligned}
&C_j(t_{n+1,m})=\frac{2}{b-a}\int_a^bc(t_{n+1,m},x)\cos\left(j\pi\frac{x-a}{b-a}\right)dx,\\
&F_j(t_{n+1,m})=\frac{2}{b-a}\int_a^bf(t_{n+1,m},x,c(t_{n+1,m},x))\cos\left(j\pi\frac{x-a}{b-a}\right)dx.\end{aligned}$$ In order to determine the function $c(t_n,x)$, we will perform $P$ Picard iterations. To evaluate the coefficients with a DCT we need to compute the integrands $c(t_{n+1,m},x)$ and $f(t_{n+1,m},x,c(t_{n+1,m},x))$ on the equidistant $x$-grid with $x_i$, for $i=0,...,J-1$. In order to compute this at each time step $t_{n,m}$ we thus need to evaluate $c(t_{n,m},x)$ on the $x$-grid with $J$ equidistant points using formula . The matrix-vector product in the formula results in a computational time of order $O(J^2)$.
A Picard iteration is used to find the fixed-point $c$ of $c = \Delta t\theta_1f(t_{n,m},x,c)+h(t_{n,m},x),$ where $f(t,x,c)$ and $h(t,x)$ are respectively the implicit and explicit parts of the equation. Due to the computational domain of $c(t,x)$ being bounded by $[a,b]$, we can thus say that $f(t,x,c(t,x))$ is also bounded. If the driver function $f(t,x,c)$ is Lipschitz continuous in $c$, i.e. $\exists$ $L^{Lipz}$ such that $|f(t,x,c_1)-f(t,x,c_2)|\leq L^{Lipz}|c_1-c_2|$, and $\Delta t_n$ is small enough such that $\Delta t \theta_1 L^{Lipz}<1$, a unique fixed-point exists and the Picard iterations converge towards that point for any initial guess. In particular, for the XVA case the non-linearity is of the form $f(t,x,c) = -r\max(c,0)$, and this is Lipschitz continuous with $L^{Lipz}=1$. Thus for $\Delta t$ sufficiently small, the Picard iteration converges to a unique fixed-point.
The total algorithm for computing the value of a Bermudan contract with XVA can be summarised as in Algorithm 1 in Figure \[fig0\]. The total computational time for the algorithm is of order $$\begin{aligned}
\label{eq:complex1}
O(M\cdot N(J + J^2+PJ+J\log_2 J)),\end{aligned}$$ consisting of the computation for $M\cdot N$ times the computation of the characteristic function on the $x$-grid (due to the availability of the analytical approximation) of $O(J)$, computation of the matrix-vector multiplications in the formulas for $c(t_{n,m},x)$ and $z(t_{n,m},x)$ of $O(J^2)$, initialization of the Picard method with $\mathbb{E}_n[c(t_{n+1},X_{n+1}]$ in $O(J^2)$ operations, computation of the $P$ Picard approximations for $c(t_{n,m},x)$ in $O(PJ)$ and computing the Fourier coefficients $F_j(t_n)$ and $C_j(t_n)$ with the DCT in $O(J\log_2 J)$ operations.
\[fig0\]
1. Define the $x$-grid with $J$ grid points given by $x_i=a+(i+\frac{1}{2})\frac{b-a}{J}$ for $i=0,...,J-1$.
2. Calculate the final exercise date values $c(t_{N,M},x)=\hat u(t_M,x)$ on the $x$-grid and compute the terminal coefficients $C_j(t_M)$ and $F_j(t_M)$ using the DCT.
3. Recursively for the exercise dates $m = M-1,...,0$ do:
1. For time steps $n=N-1,...,0$ do:
1. Compute $c(t_{n,m},x)$ using formula and use this to determine $f(t_{n,m},x,c(t_{n,m},x))$ on the $x$-grid.
2. Subsequently, use these to determine $F_j(t_{n,m})$ and $C_j(t_{n,m})$ using the DCT.
2. Compute the new terminal condition $c(t_{N,m-1},x)=\max\{\phi(t_{0,m},x),c(t_{0,m},x)\}$ (either analytically or numerically) and the corresponding Fourier-cosine coefficient.
4. Finally $\hat u(t_0,x_0) = c(t_{0,0},x_0)$.
An alternative for CVA computation {#sec52}
----------------------------------
In this section we present an efficient alternative way of calculating the CVA term in in the case of unilateral CCR using a Fourier-based method. Due to the ability of using the FFT this method is considerably faster for computing the CVA than the method presented in Section \[sec51\]. We use the definition of CVA at time $t$ given by $$\textnormal{CVA}(t) =
\hat u(t,X_t)- u(t,X_t),$$ where $u(t,X_t)$ is as usual the default-free value of the Bermudan option ($\gamma(t,x)=0$), while $\hat u(t,X_t)$ is the value including default ($\gamma(t,x)\neq 0$). We consider the model as defined in . We will compute $u(t,X_t)$ and $\hat u(t,X_t)$ using the COS method and the approximation of the characteristic function (as derived in Section \[sec43\]), without default and with default, respectively. In case of a default the payoff becomes zero. Note that the risky option value $\hat u(t,x)$ computed with the characteristic function for a defaultable underlying corresponds exactly to the option value in which the counterparty might default, with the probablity of default, $PD(t)$, defined as in . Thus, in this case we have unilateral CCR and $\zeta = \tau_C$, the default time of the counterparty.
Using the definition of the defaultable $S_t$, it is well-known (see, for instance, [@linetsky2006bankruptcy Section 2.2]) that the risky no-arbitrage value of the Bermudan option on the defaultable asset $S_t$ at time $t$ is given by $$\begin{aligned}
\hat u\left(t,X_{t}\right)=\caratt_{\{\zeta>t\}}\sup_{\tau \in \{t_1,...,t_M\}}\mathbb{E}\left[e^{-\int_{t}^{\tau}
\left(r+\gamma(s,X_s)\right) ds}\phi(\tau,X_{\tau})|X_{t}\right].\end{aligned}$$
By allowing the dependence of the default intensity on the underlying, a simplified form of wrong-way risk is already incorporated into the CVA valuation.
For a Bermudan put option with strike price $K$, we simply have $\phi(t,x)=\left(K-x\right)^{+}$. By the dynamic programming approach, the option value can be expressed by a backward recursion as $$\begin{aligned}
\hat u(t_{M},x)=\caratt_{\{\zeta>t_{M}\}}\max(\phi(t_{M},x),0),
\end{aligned}$$ and $$\begin{aligned}
c(t,x)=
\mathbb{E}\left[
e^{\int_{t}^{t_m}\left(r+\gamma(s,X_s)\right)ds}\hat u(t_{m},X_{t_{m}})|X_{t}=x\right],\qquad &t\in[t_{m-1},t_{m}[\\
\hat u(t_{m-1},x)=\caratt_{\{\zeta>t_{m-1}\}}\max\{\phi(t_{m-1},x),c(t_{m-1},x)\},\qquad
&m\in\{2,\dots,M\}.\label{eq:bermud}\end{aligned}$$ Thus to find the risky option price $\hat u(t,X_t)$ one uses the defaultable asset with $\gamma(t,x)$ representing the default intensity of the counterparty and in order to get the default-free value $u(t,X_t)$ one uses the default-free asset by setting $\gamma(t,x)=0$. The CVA adjustment is calculated as the difference between the two. Both $\hat u(t,x)$ and $u(t,x)$ are calculated using the approximated characteristic function and the COS method applied to the continuation value [@borovykh]. Due to the characteristic function being of the form , we are able to use the FFT in the matrix-vector multiplication when computing the continuation values of the Bermudan option with and without default, reducing this operation from $O(J^2)$ to $O(J\log_2 J)$. For more details, we refer to Appendix \[app1\]. The total complexity of the calculation of the CVA value for a Bermudan option with $M$ exercise dates is then $O(M J \log_2 J)$. Comparing this to , in which the most time-consuming operations were indeed the matrix-vector products of order $O(J^2)$ that resulted from the computation of the functions on the $x$-grid of size $J$, we conclude that the method for CVA computation is indeed significantly faster due to the ability of using the FFT.
### Hedging CVA {#sec521}
In practice CVA is hedged and thus practitioners require efficient ways to compute the sensitivity of the CVA with respect to the underlying. The widely used bump- and revalue- method, while resulting in precise calculations, might be slow to compute. Using the Fourier-based approach we find explicit formulas allowing for an easy computation of the first- and second-order derivatives of the CVA with respect to the underlying. For the first-order and second-order Greeks we have $$\begin{aligned}
\Delta &=\ e^{-r(t_{1}-t_0)}\sideset{}{'}\sum_{j=0}^{J-1}\textnormal{Re}\left(e^{ij\pi\frac{x-a}{b-a}}\left(\frac{ij\pi}{b-a}g_{n,0}^d
\left(t_0,t_{1},\frac{j\pi}{b-a}\right)+g_{n,1}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\right)\right) V_j^d(t_1)\\
&-\ e^{-r(t_{1}-t_0)}\sideset{}{'}\sum_{j=0}^{J-1}\textnormal{Re}\left(e^{ij\pi\frac{x-a}{b-a}}\left(\frac{ij\pi}{b-a}g_{n,0}^r
\left(t_0,t_{1},\frac{j\pi}{b-a}\right)+g_{n,1}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\right)\right) V_j^r(t_1),
\end{aligned}$$ $$\begin{aligned}
\frac{\partial \Delta}{\partial X} &=\ e^{-r(t_{1}-t_0)}\sideset{}{'}\sum_{j=0}^{J-1}\textnormal{Re}\bigg(e^{ij\pi\frac{x-a}{b-a}}\bigg(-\frac{ij\pi}{b-a}
g_{n,0}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)-g_{n,1}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\\
&+\ 2\frac{ij\pi}{b-a}g_{n,1}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)
+\left(\frac{ij\pi}{b-a}\right)^2g_{n,0}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)+2g_{n,2}^d\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\bigg)\bigg) V_j^d(t_1)\\
&-\ e^{-r(t_{1}-t_0)}\sideset{}{'}\sum_{j=0}^{J-1}\textnormal{Re}\bigg(e^{ij\pi\frac{x-a}{b-a}}\bigg(-\frac{ij\pi}{b-a}
g_{n,0}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)-g_{n,1}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\\
&-\ 2\frac{ij\pi}{b-a}g_{n,1}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)
+\left(\frac{ij\pi}{b-a}\right)^2g_{n,0}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)+2g_{n,2}^r\left(t_0,t_{1},\frac{j\pi}{b-a}\right)\bigg)\bigg) V_j(t_1)^r,\end{aligned}$$ where $V_k^d$ and $V_k^r$ are the Fourier-cosine coefficients with the defaultable and default-free characteristic function terms, $g_{n,h}^d$ and $g_{n,h}^r$, respectively.
Numerical experiments {#sec6}
=====================
In this section we present numerical examples to justify the accuracy of the methods in practice. We compute the XVA with the method presented in Section \[sec51\] and the CVA in the case of unilateral CCR with the method from Section \[sec52\], which we show is more efficient for cases in which one only needs to compute the CVA. We compare the results of solving the BSDE with the COS method and the adjoint expansion of the characteristic function to the values obtained by using a least-squares Monte-Carlo method for computing the conditional expected values in the BSDE as done in e.g. [@bender12].
The computer used in the experiments has an Intel Core i7 CPU with a 2.2 GHz processor. We use the second-order approximation of the characteristic function. We have found this to be sufficiently accurate by numerical experiments and theoretical error estimates. The formulas for the second-order approximation are simple, making the methods easy to implement.
A numerical example for XVA {#sec61}
---------------------------
Here, we check the accuracy of the method from Section \[sec51\]. We will compute the Bermudan option value with XVA using a simplified driver function given by $f(t,\hat u(t,x)) = -r\max(\hat u(t,x),0)$. Our method is easily extendible to the driver function in Section \[sec32\]. Consider $X_t$ to be a portfolio process and the payoff, if exercised at time $t_m$, to be given by $\Phi(t_m,x) =
x$. In this case the value we can receive at every exercise date is the value of the portfolio. Consider the model in Section \[sec2\] without default, with a local jump measure and a local volatility function with CEV-like dynamics and Gaussian jumps defined by $$\begin{aligned}
\label{eq:parameters1}
&\sigma(x) = be^{\beta x},\\ \label{eq:parameters2}
& \nu(x,dq) = \lambda e^{\beta x}
\frac{1}{\sqrt{2\pi\delta^2}}\exp\left(\frac{-(q-m)^2}{2\delta^2}\right)dq.\end{aligned}$$ We assume the following parameters in equations -, unless otherwise mentioned: $b=0.15$, $\beta = -2$, $\lambda = 0.2$, $\delta = 0.2$, $m=-0.2$, $r=0.1$, $K=1$ and $X_0=0$ (so that $S_0 = 1$). In the LSM the number of time steps is taken to be 100 and we simulate $10^5$ paths. In the COS method we take $J=256$, $\theta_1=0.5$ and $N=10$, $M=10$, making the total number of time steps $N\cdot M = 100$. The truncation range is determined as in with $L=10$. Due to the state-dependent coefficients in the underlying dynamics in - we use the approximated characteristic function as derived in Section \[sec42\] with the second-order approximation, i.e. $\hat \Gamma^{(2)}(t,x;T,\xi)$ and take $\bar x = x$, where $x=\{x_i\}_{i=0}^{J-1}$. Note that we thus compute the values, including those of the characteristic function, on the complete $x$-grid. In the final iteration when computing $\hat u(t_0,X_0)$ we use $\bar x = X_0$.
In Table \[taberr\] we analyse the error in the approximation of $\hat u(t_0,X_0)$ with $S_0=0.4$ for different values of the discretization parameter $N$ and the number of grid points (and Fourier-cosine coefficients) $J$. We compare the approximated COS value to the 95% confidence interval obtained by a LSM. Accurate results are quickly obtained for small values of both $J$ and $N$. In Figure \[figconv\] we plot the upper bound of the 95% confidence interval of the absolute error in the approximation for varying $J$ and $N$. We observe approximately a linear convergence and note that the error stops decreasing at some point for increasing values of $J$ and $N$. This can be due to the error being dominated by the approximated characteristic function. In particular we observe that $J=32$ and $N=10$ seem to be sufficient parameters to achieve a satisfactory accuracy in the approximation.
The results for $\hat u(t_0,X_0)$ of the COS approximation method compared to a 95% confidence interval of the value obtained through a LSM are presented in Table \[tab1\]. These results show that our method is able to solve non-linear PIDEs accurately. The CPU time of the approximating method depends on the number of time steps $M\cdot N$ and is approximately $5\cdot (N\cdot M)$ ms.
$N=1$ $N=10$ $N=20$ $N=30$
--------- ------------------- ------------------- ------------------- -------------------
$J=8$ 6.4E-03$-$6.9E-03 4.3E-03$-$4.8E-03 4.9E-03$-$5.3E-03 5.3E-03$-$5.8E-03
$J=16$ 2.3E-03$-$2.7E-03 8.8E-04$-$1.3E-03 6.2E-04$-$1.1E-03 5.4E-04$-$9.2E-04
$J=32$ 1.7E-03$-$2.0E-03 4.2E-04$-$8.3E-04 2.4E-04$-$6.3E-04 1.6E04$-$5.8E-04
$J=64$ 1.4E-03$-$1.9E-03 2.2E-04$-$6.5E-04 1.6E-04$-$2.3E-04 1.2E-04$-$2.9E-04
$J=128$ 1.7E-04$-$6.0E-04 2.1E-04$-$6.6E-04 2.3E-04$-$6.5E-04 1.9E-04$-$6.1E-04
$J=256$ 2.1E-04$-$6.6E-04 3.7E-04$-$7.7E-04 1.5E-04$-$5.7E-04 1.2E-04$-$3.1E-04
: The 95% confidence interval of the absolute error in the COS approximation of $\hat u(0,X_0)$ with $S_0 = 0.4$ compared to a LSM for varying parameters $J$ and $N$.
\[taberr\]
maturity $T$ $S_0$ MC value with XVA COS value with XVA
-------------- ------- ------------------- --------------------
0.5 0 0.03770$-$0.03838 0.03809
0.2 0.2326$-$0.2330 0.2320
0.4 0.4251$-$0.4254 0.4243
0.6 0.6169$-$0.6171 0.6158
0.8 0.8077$-$0.8079 0.8069
1 1.000$-$1.000 1.0000
1 0 0.07374$-$0.07453 0.07228
0.2 0.2611$-$0.2617 0.2606
0.4 0.4461$-$0.4465 0.4454
0.6 0.6288$-$0.6291 0.6288
0.8 0.8126$-$0.8129 0.8113
1 1.001$-$1.001 1.000
: A Bermudan put option with XVA (10 exercise dates, expiry $T=0.5,1$) in the CEV-like model for the 2nd-order approximation of the characteristic function, and an LSM comparison.
\[tab1\]
A numerical example for CVA {#sec62}
---------------------------
In this section we validate the accuracy of the method presented in Section \[sec52\] and compute the CVA in the case of unilateral CCR under the model dynamics given in Section \[sec2\] with a local jump measure and a local volatility function with CEV-like dynamics, Gaussian jumps defined by defined as in and a local default function $\gamma(x)=ce^{\beta x}$. We assume the same parameters as in Section \[sec62\], except $r=0.05$ and we take $c = 0.1$ in the default function. In the LSM the number of time steps is taken to be 100 and we simulate $10^5$ paths. In the COS method we take $L = 10$ and $J=100$. Again, due to the state-dependent coefficients in the underlying dynamics we use the approximated characteristic function as derived in Section \[sec42\] with the second-order approximation, i.e. $\hat \Gamma^{(2)}(t,x;T,\xi)$ and take $\bar x = X_0$.
The results for the CVA valuation with the FFT-based method and with LSM are presented in Table \[tab0\]. The CPU time of the LSM is at least 5 times the CPU time of the approximating method, which for $M$ exercise dates is approximately $3\cdot M$ ms, thus more efficient than the computation of the XVA with the method in Section \[sec51\]. The optimal exercise boundary in Figure \[fig2\] shows that the exercise region becomes larger when the probability of default increases; this is to be expected: in case of the default probability being greater, the option of exercising early is more valuable and used more often.
\[tab0\]
maturity $T$ strike $K$ MC CVA COS CVA
-------------- ------------ ----------------------------------------- ----------------------
0.5 0.6 $4.200\cdot 10^{-4}-4.807\cdot 10^{-4}$ $1.113\cdot 10^{-4}$
0.8 0.001525$-$0.001609 9.869$\cdot 10^{-4}$
1 0.01254$-$0.01273 0.01138
1.2 0.005908$-$0.005931 0.005937
1.4 0.006657$-$0.06758 0.006898
1.6 0.007795$-$0.008008 0.007883
1 0.6 8.673E-04$-$9.574E-04 4.463E-04
0.8 0.005817$-$0.006040 0.003535
1 0.02023$-$0.02054 0.01882
1.2 0.01221$-$0.01222 0.1272
1.4 0.01378$-$0.01391 0.01360
1.6 0.01532$-$0.01502 0.01554
: CVA for a Bermudan put option (10 exercise dates, expiry $T=0.5,1$) in the CEV-like model for the 2nd-order approximation of the characteristic function, and an LSM comparison.
Conclusion
==========
In this paper we considered pricing Bermudan derivatives under the presence of XVA, consisting of CVA, DVA, MVA, FVA and KVA. We derived the replicating portfolio with cashflows corresponding to the different rates for different types of lending. This resulted in the PIDE in and its corresponding BSDE . We propose to solve the BSDE using a Fourier-cosine method for the resulting conditional expectations and an adjoint expansion method for determining an approximation of the characteristic function of the local Lévy model in . This approach is extended to Bermudan option pricing in Section \[sec51\]. In Section \[sec52\] we presented an alternative for computing the CVA term in the case of unilateral collateralization (as is the case when the derivative is an option) without the use of BSDEs. This results in an even more efficient method due to the ability to use the FFT. We verify the accuracy of both methods in Sections \[sec61\] and \[sec62\] by comparing it to a LSM and conclude that the method from Section \[sec51\] is able to achieve a rapid convergence and gives, already for small values of the discretization parameters an accurate result. The alternative method for CVA computation from Section \[sec52\] is indeed more efficient than the BSDE method for computing just the CVA term.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank two anonymous referees for the comments and suggestions that have improved the quality of this paper. This research is supported by the European Union in the the context of the H2020 EU Marie Curie Initial Training Network project named WAKEUPCALL.
The COS formulae {#app1}
================
Let, as usual, $J$ denote the number of Fourier-cosine coefficients. Remembering that the expected value $c(t,x)$ in can be rewritten in integral form, we have $$\begin{aligned}
c(t,x) = e^{-r(t_{m}-t)}\int_\mathbb{R} v(t_m,y)\Gamma(t,x;t_{m},dy),\qquad t\in[t_{m-1},t_{m}[,\end{aligned}$$ where, $v(t_m,y)$ can be either $u(t_m,y)$ or $\hat u(t_m,y)$. Then we use the Fourier-cosine expansion to get the approximation: $$\begin{aligned}
\label{eq:conti}
&\hat c(t,x)= e^{-r(t_{m}-t)}\sideset{}{'}\sum_{j=0}^{J-1}
\textnormal{Re}\left(
e^{-ij\pi\frac{a}{b-a}}\hat\Gamma\left(t,x;t_{m},\frac{j\pi}{b-a}\right)\right)V_j(t_{m}),\qquad t\in[t_{m-1},t_{m}[\\
&V_j(t_m)=\frac{2}{b-a}\int_a^b \cos\left(j\pi \frac{y-a}{b-a}\right)\max\{\phi(t_{m},y),c(t_{m},y)\}dy,\end{aligned}$$ with $\phi(t,x)=\left(K-e^{x}\right)^{+}$.
We can recover the coefficients $\left(V_j(t_m)\right)_{j=0,1,...,J-1}$ from $\left(V_j(t_{m+1})\right)_{j=0,1,...,J-1}$. To this end, we split the integral in the definition of $V_j(t_m)$ into two parts using the early-exercise point $x_m^*$, which is the point where the continuation value is equal to the payoff, i.e. $c(t_m,x_m^*)=\phi(t_m,x_m^*)$; this point can easily be found by using the Newton method. Thus, we have $$V_j(t_m)=F_j(t_{m},x_m^*)+C_j(t_{m},x_m^*),\qquad m=M-1,M-2,...,1,$$ where $$\label{eq:vcoef}
\begin{split}
F_j(t_{m},x_m^*)&:=\frac{2}{b-a}\int_a^{x_m^*}\phi(t_m,y)\cos\left(j\pi\frac{y-a}{b-a}\right)dy,\\
C_j(t_{m},x_m^*)&:=\frac{2}{b-a}\int_{x_m^*}^b c(t_m,y)\cos\left(j\pi\frac{y-a}{b-a}\right)dy,
\end{split}$$ and $V_j(t_M) =F_j(t_{M},\log K).$
The coefficients $F_j(t_m,x_m^*)$ can be computed analytically using $x_m^*\leq \log
K$, and by inserting the approximation for the continuation value into the formula for $C_j(t_{m},x_{m}^*)$ have the following coefficients $\hat C_j$ for $m =M-1,M-2,...,1$: $$\begin{aligned}
\hat C_j(t_{m},x_m^*) =& \frac{2e^{-r(t_{m+1}-t_{m})}}{b-a}\nonumber\\\label{eq:contin}
&\cdot\sideset{}{'}\sum_{k=0}^{J-1}V_k(t_{m+1})\int_{x_m^*}^{b}
\mathrm{Re}\left(e^{-ik\pi\frac{a}{b-a}}\hat\Gamma\left(t_{m},x;t_{m+1},\frac{k\pi}{b-a}\right)\right)
\cos\left(j\pi\frac{x-a}{b-a}\right)dx.\end{aligned}$$ From we know that the $n$th-order approximation of the characteristic function is of the form: $$\begin{aligned}
\hat\Gamma^{(n)}(t_m,x;t_{m+1},\xi)= e^{i\xi x} \sum_{h=0}^n (x-\bar x)^h g_{n,h}(t_m,t_{m+1},\xi),\end{aligned}$$ where the coefficients $g_{n,h}(t,T,\xi)$, with $0\leq k\leq n$, depend only on $t,T$ and $\xi$, but not on $x$.
To find $u(t,x)$ we use $$\hat\Gamma^r(t_m,x;t_{m+1},\xi):=e^{i\xi x} \sum_{h=0}^n (x-\bar x)^h g_{n,h}^r(t_m,t_{m+1},\xi),$$ the characteristic function with $\gamma(t,x)=0$. For $\hat u (t,x)$ we use $$\hat\Gamma^d(t_m,x;t_{m+1},\xi):=e^{i\xi x} \sum_{h=0}^n (x-\bar x)^h g_{n,h}^d(t_m,t_{m+1},\xi),$$ where $\gamma(t,x)$ is chosen to be some specified function.
Using we can write the Fourier coefficients of the continuation value in vectorized form as: $$\begin{aligned}
\bold{\hat C}(t_{m},x_m^*) =\sum_{h=0}^n e^{-r(t_{m+1}-t_m)}
\mathrm{Re}\left(\bold V(t_{m+1})\mathcal{M}^h(x_m^*,b)\Lambda^h\right),\end{aligned}$$ where $\bold V(t_{m+1})$ is the vector $[V_0(t_{m+1}),...,V_{J-1}(t_{m+1})]^T$ and $\mathcal{M}^h(x_m^*,b)\Lambda^h$ is a matrix-matrix product with $\mathcal{M}^h$ a matrix with elements $\{M_{k,j}^h\}_{k,j=0}^{J-1}$ defined as $$\begin{aligned}
M_{k,j}^h(x_m^*,b) := \frac{2}{b-a}\int_{x_m^*}^{b} e^{ij\pi\frac{x-a}{b-a}}(x-\bar x)^h\cos\left(k\pi\frac{x-a}{b-a}\right)dx\label{eq:integraal},\end{aligned}$$ and $\Lambda^h$ is a diagonal matrix with elements $$g_{n,h}\Big(t_m,t_{m+1},\frac{j\pi}{b-a}\Big),\qquad j=0,\dots,J-1.$$ One can show, see [@borovykh], that the resulting matrix $\mathcal{M}^h$ is a sum of a Hankel and Toeplitz matrix and thus the resulting matrix vector product can be calculated using a FFT.
[^1]: Dipartimento di Matematica, Università di Bologna, Bologna, Italy. (**e-mail**:anastasia.borovykh2@unibo.it)
[^2]: Dipartimento di Matematica, Università di Bologna, Bologna, Italy. (**e-mail**:andrea.pascucci@unibo.it)
[^3]: Centrum Wiskunde & Informatica, Amsterdam, The Netherlands and Delft University of Technology, Delft, The Netherlands. (**e-mail**:c.w.oosterlee@cwi.nl)
|
---
abstract: |
In this paper we prove that the square of an essentially 2-edge connected graph with an additional property has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge connected graph does not contain a connected even factor with bounded maximum degree.
[**Keywords**]{}: connected even factors; (essentially) 2-edge connected graphs; square of graphs
author:
- 'Jan Ekstein[^1]'
- 'Baoyindureng Wu[^2]'
- 'Liming Xiong[^3]'
title: 'Connected even factors in the square of essentially 2-edge connected graphs'
---
Introduction
=============
We consider only finite undirected simple graphs. For terminology and notation not defined in this paper we refer to [@WES]. Let $G$ be a connected graph. For vertices $x, y$ of $G$, let $N_G(x)$ denote the *neighborhood* of $x$ in $G$, $d_G(x)=|N_G(x)|$ the *degree* of $x$ in $G$, and $\mbox{dist}_G(x,y)$ the *distance* between $x, y$ in $G$. The *square* of a graph $G$, denoted by $G^2$, is the graph with same vertex set as $G$ in which two vertices are adjacent if their distance in $G$ is at most 2. Thus $G\subseteq G^2$. There are several papers (e.g. see [@EL], [@EKS], [@FAU], [@FLE], [@GOU], [@HEN], [@CHA], and [@CHI]) about hamiltonian properties in the square of a graph. This paper deals with connected even factors which generalize some previous known results.
A *factor* in a graph $G$ is a spanning subgraph of $G$. A *connected even factor* in $G$ is a connected factor in $G$ in which every vertex has positive even degree. A *$[2, 2s]$-factor* is a connected even factor in $G$ in which every vertex has degree at most $2s$. Hence a hamiltonian cycle is a $[2, 2s]$-factor with $s=1$. It is well known the following result by Fleischner in [@FLE] concerning the existence of a hamiltonian cycle (a $[2, 2]$-factor) in the square of a 2-connected graph. Recently, Müttel and Rautenbach in [@MUT] gave a shorter proof of this result.
***[@FLE]*** \[Fleischner\] If $G$ is a 2-connected graph and $v_1$ and $v_2$ are two distinct vertices of $G$, then $G^2$ contains a hamiltonian cycle $C$ such that both edges of $C$ incident with $v_1$ and one edge of $C$ incident with $v_2$ belong to $G$. Furthermore, if $v_1$ and $v_2$ are neighbors in $C$, then these are three distinct edges.
Theorem \[Fleischner\] was a base for proving the following theorem by Abderrezzak et al. in [@EL] using forbidden subgraphs.
***[@EL]*** \[Abderrezzak\] If $G$ is a connected graph such that every induced $S(K_{1,3})$ has at least three edges in a block of degree at most 2, then $G^{2}$ is hamiltonian.
Theorem \[Abderrezzak\] was generalized by Ekstein et al. in [@EKS] for $[2, 2s]$-factors.
***[@EKS]*** \[Ekstein\] Let $s$ be a positive integer and $G$ be a connected graph such that every induced $S(K_{1, 2s+1})$ has at least three edges in a block of degree at most two. Then $G^2$ has a $[2, 2s]$-factor.
Recall that a graph $G$ is *essentially $k$-edge connected* if deleting less than $k$ edges from $G$ cannot result in two nontrivial components. In this paper, we shall answer the question how it is for the existence of a $[2, 2s]$-factor in the square of a graph with 2-edge (or essentially 2-edge) connectivity instead of (vertex) connectivity of a graph.
Let $G$ be a connected graph. A vertex of degree 1 is called [*a leaf*]{}. A cut vertex $y$ is [*trivial*]{} in $G$, if $y$ is not a cut vertex in $G-M$, where $M$ is a set of all leaves adjacent to $y$, otherwise is [*non-trivial*]{}. If $M=\{x\}$ and the neighbor of $x$ is a trivial cut vertex of $G$, then $x$ is called [*a bad leaf*]{}. [*A trivial bridge*]{} is a cut-edge of $G$ containing a leaf, otherwise is [*non-trivial*]{}. [*A bad bridge*]{} is a trivial bridge of $G$ adjacent to a bad leaf. For illustration see Fig. 1.
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The following theorems are the main results of this paper.
\[Main-\] For any fixed positive integer $s$, there exists an infinite class of essentially 2-edge connected graphs $G$ such that $G^{2}$ has no $[2, 2s]$-factor, even if the resulting graph obtained from $G$ by deleting its all leaves is 2-connected.
\[Main++\] Let $G$ be a connected graph without non-trivial bridges and without any two bad leaves at distance exactly 4. Then $G^{2}$ has a $[2,4]$-factor.
A *Useful lemma*
================
Before presenting this lemma, we need some additional notation. [*Block graph*]{} of a graph $G$, denoted by $BC(G)$, is the graph whose vertex set consists of all blocks and cut vertices of $G$, and two vertices are adjacent in $BC(G)$ if one of them is a block of $G$ and the second one is its vertex. It is easy to see that $BC(G)$ is a tree for a connected graph $G$. Note that for any tree, we may choose any vertex as its root. Hence without loss of generality, we may assume that $B_1, \ldots, B_t$ be all blocks of $G$ such that $B_1$ corresponds to the root of $BC(G)$. For a cut-vertex $v$ of $G$, [*the parent block*]{} of $v$ is the block containing $v$ and its corresponding vertex in $BC(G)$ has the smallest distance to the root of $BC(G)$. The remaining blocks containing $v$ are called [*children blocks*]{} of $v$ with respect to the root of $BC(G)$.
The following lemma, we call it a *Useful lemma*, is a key for proofs of main results (Theorem \[Main-\] and Theorem \[Main++\]).
(Useful lemma) \[Main\] Let $G$ be a connected graph without non-trivial bridges and without bad leaves (except $K_{1,2}, K_{1,3}$) and $u$ be a vertex of $G$ that is neither a cut vertex nor a leaf (if any).
Then $G^{2}$ has a $[2,4]$-factor $F$ such that
- $d_F(x)=2$, where $x$ is not a cut vertex of $G$;
- both edges of $F$ incident with $u$ belong to $G$;
- for each cut vertex $y$ of $G$ it holds that $d_F(y)=4$ and at least two edges of $F$ incident with $y$ belong to $G$, moreover if $y$ is a trivial cut vertex, then these two edges are trivial bridges;
- for any cut vertex $y$ of $G$, the two edges incident with $u$ in $F$ are distinct from the two edges incident with $y$ in $F$ as specified in (c);
- for any two cut vertices $y_1$ and $y_2$ of $G$, the two edges of $F$ incident with $y_1$ as specified in $(c$) are distinct from those with $y_2$.
If $G$ is $K_{1,s}$, for $s\geq4$, then the result is obvious. Now we assume that $G$ contains at least one cyclic block and $G'=G-M$, where $M$ is a set of all leaves adjacent with all trivial cut vertices of $G$.
Let $\mathbb{O}=B_{1}, B_{2}, ..., B_{k}$ be an ordering of all blocks of $G'$ such that either $u\in V(B_{1})$, if any, or we choose arbitrary cyclic block as $B_{1}$, satisfying the following properties:
- for any cut vertex $v$ of $G'$, all children blocks of $v$ with respect to the root $r$ of $BC(G')$ corresponding to $B_{1}$ appear consecutively in $\mathbb{O}$ such that bridges containing $v$ are in $\mathbb{O}$ before cyclic blocks containing $v$;
- $\mbox{dist}_{BC(G')}(r, v_{i})<\mbox{dist}_{BC(G')}(r, v_{j})$ implies $i<j$, where $v_{i}, v_{j}$ are vertices of $BC(G')$ corresponding to $B_{i}, B_{j}$, respectively.
Then $G'$ is a connected graph without non-trivial bridges and without bad leaves and we prove by induction on $k$ that $(G')^{2}$ contains a $[2,4]$-factor $F'$ such that
- $d_{F'}(x)=2$, where $x$ is not a cut vertex of $G'$;
- both edges of $F'$ incident with $u$, if any, belong to $B_{1}$;
- for each cut-vertex $y$ of $G'$, it holds that $d_{F'}(y)=4$ and at least two edges of $F'$ incident with $y$ belong to $G'$. Moreover:
- if $y$ belongs to exactly two blocks of $G'$, then at least two edges of $F'$ incident with $y$ are edges from the children block of $y$ with respect to $r$ (the root of $BC(G')$ corresponding to $B_{1}$);
- if $y$ belongs to more than two blocks of $G'$, then at least two edges of $F'$ incident with $y$ are edges from two different children blocks of $y$ with respect to $r$.
For $k=1$, $G'=B_{1}$ and $(G')^{2}$ has even a hamiltonian cycle $C$ such that both edges of $F'$ incident with $u$, if any, belong to $B_{1}$ by Theorem \[Fleischner\].
Let $k>1$ and assume that lemma is true for all integers less than $k$. By the definition of $G'$ and $\mathbb{O}$, $B_{k}$ is an end cyclic block of $G'$ and let $v_{0}$ be the cut vertex of $G'$ with $v_{0}\in V(B_{k})$.
If $B_{k-1}=v_{0}l$ (i.e. $B_{k-1}$ is a bridge) and $B_{k-1}, B_{k}$ are only children blocks of $v_{0}$ with respect to $r$, then we set $G_{1}=G' - \{V(B_{k})\cup\{l\}\setminus\{v_{0}\}\}$, otherwise we set $G_{2}=G' - \{V(B_{k})\setminus\{v_{0}\}\}$. Hence $G_{1}, G_{2}$ are connected graphs without non-trivial bridges and without bad leaves and have $k-2, k-1$ blocks, respectively. Hence by the induction hypothesis, $(G_{1})^{2}, (G_{2})^{2}$ have a $[2,4]$-factor $F_{1}, F_{2}$ with properties 1), 2), and 3), respectively.
By Theorem \[Fleischner\], there is a Hamiltonian cycle $C$ in $(B_{k})^{2}$ such that two edges $f_{1}, f_{2}$ of $C$ incident with $v_{0}$ belong to $B_{k}$ and thus belong to $G'$.
*Case 1:* $G_{1}$ exists.
Let $f_{1}=v_{0}v_{k}$. Then $F'=(F_{1}\cup C)+\{v_{0}l,v_{k}l\}-\{f_{1}\}$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).
*Case 2:* $G_{1}$ does not exist and $v_{0}$ is not a cut vertex in $G_{2}$.
Hence $v_{0}$ belongs to exactly two blocks of $G'$ and $F'=F_{2}\cup C$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).
*Case 3:* $G_{1}$ does not exist and $v_{0}$ is a cut vertex in $G_{2}$.
Let $f_{1}=v_{0}v_{k}$. We consider two possibilities depending on the property 3).
If exactly two blocks of $G_{2}$ contain $v_{0}$, then by the induction hypothesis $d_{G_{2}}(v_{0})=4$ and there are two edges of $F_{2}$ incident with $v_{0}$ from a children block $B_{k-1}$ of $v_{0}$. (Note that $B_{k-1}$ is a cyclic block, since $G_{1}$ does not exist.) Let $e_{k-1}=v_{0}v_{k-1}$ be such an edge of $F_{2}$. Since $\mbox{dist}_{G'}(v_{k-1},v_{k})=2$, the edge $v_{k-1}v_{k}$ is an edge of $(G_{2})^{2}$. Thus $F'=(F_{2} \cup C)+\{v_{k-1}v_{k}\}-\{e_{k-1},f_{1}\}$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).
If there are more than two blocks of $G_{2}$ containing $v_{0}$, then by the induction hypothesis $d_{G_{2}}(v_{0})=4$ and there are two edges $e_{k-2},e_{k-1}$ of $F_{2}$ incident with $v_{0}$ in $B_{k-2},B_{k-1}$, respectively. Let $e_{k-2}=v_{0}v_{k-2}$. Since $\mbox{dist}_{G'}(v_{k-2},v_{k})=2$, the edge $v_{k-2}v_{k}$ is an edge of $(G_{2})^{2}$. Thus $F'=(F_{2}\cup C)+\{v_{k-2}v_{k}\}-\{e_{k-2},f_{1}\}$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).
Now we extend $F'$ to a $[2,4]$-factor $F$ in $G^{2}$ with required properties. Note that the properties 1), 2), and 3) imply the properties a)-e) in Lemma \[Main\].
Let $u_{1},u_{2},...,u_{t}$ be all trivial cut vertices of $G$ and $l_{i}^{1},l_{i}^{2},...,l_{i}^{s_{i}}$ be all leaves incident with $u_{i}$, for $i=1,2,...,t$. Note that $s_{i}\geq2$, otherwise we have a bad bridge in $G$, a contradiction. For $i=1,2,...,t$, let $C_{i}=u_{i}l_{i}^{1}l_{i}^{2}...l_{i}^{s_{i}}u_{i}$ be cycles in $G^{2}$ and $C'=\cup_{j=1}^{t}C_{j}$. Since $d_{F'}(u_{i})=2$ and $u_{i}l_{i}^{1}, l_{i}^{s_{i}}u_{i}$ are edges from $G$, $F=F'\cup C'$ is the $[2,4]$-factor of $G^{2}$ with properties a)-e).
Note that clearly the square of $K_{1,2}$, $K_{1,3}$ is hamiltonian but there is no $[2,4]$-factor with a vertex of degree 4 in the square of $K_{1,2}$, $K_{1,3}$, respectively.
Proofs of main results
======================
In this section we prove Theorem \[Main-\] and Theorem \[Main++\]. The proof of Theorem \[Main-\] is made to be convenient for finding an infinite class of graphs.
\[Figure2\] $$\beginpicture
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It is easy to see that the square of all essentially 2-edge connected graphs on Fig. 2 does not contain $[2,2s]$-factor for any fixed positive integer $s$.
Finally we prove Theorem \[Main++\].
Firstly if $G$ is $K_{1,2}$ or $K_{1,3}$, then clearly $G^{2}$ is even hamiltonian.
Now let $X$ be a set of all bad leaves of $G$ and $G'=G-X$. For $x_{i}\in X$, we denote $y_{i}$ its unique neighbor in $G$. By Lemma \[Main\], there is a \[2,4\]-factor $F'$ of $(G')^2$ with properties a)-e). Note that $d_{F'}(y_{i})=2$ for each $y_{i}$.
By the definition, any two bad leaves have a distance at least 3. Let $X_{0}\subseteq X$ be the set of all bad leaves that has a bad leaf at the distance exactly 3 in $G$. Then, for all $x_{i}\in X_{0}$, corresponding $y_{i}$’s induce a subgraph of $G'$ in which all components (denoted by $H_{1},H_{2},...,H_{s}$) are complete graphs, otherwise we have in $G$ two bad leaves at distance 4, a contradiction.
Let $V(H_{i})=\{y_{i,1},y_{i,2},...,y_{i,t_{i}}\}$, $t_{i}\geq 2$ for $i=1,2,...,s$. Then we set
$$M_{i}=\bigcup_{j=1}^{t_{i-1}}\{x_{i,j}y_{i,j+1}, x_{i,j+1}y_{i,j}\}~\bigcup~\{x_{i,1}y_{i,1}, x_{i,t_i}y_{i,t_i}\}.$$
All bad leaves of $X\setminus X_{0}$ are pairwise at distance at least 5 and we order them in a sequence $x_{1},x_{2}, ...,x_{k_{1}},x_{k_{1}+1},...,x_{k_{1}+k_{2}},x_{k_{1}+k_{2}+1}, ...,x_{k_{1}+k_{2}+k_{3}}$ in the following way (see Fig. \[Figure3\] for illustration):
\[Figure3\] $$\beginpicture
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\plot 40 0 45 0 45 -5 /
\endpicture$$
- for $i=1,2, ..., k_{1}$, there exists $y_{i}z_{i}\in E(F)\cap E(G')$ for some $z_{i}\in V(G')$;
- for $i=k_{1}+1, k_{1}+2, ..., k_{1}+k_{2}$, it does not hold 1) and there exists no cut vertex $z_{i}$ of $G'$ such that $y_{i}z_{i}\in E(G')$;
- for $i=k_{1}+k_{2}+1, k_{1}+k_{2}+2, ..., k_{1}+k_{2}+k_{3}$, it does not hold 1) and 2). It means that there exists only a cut vertex $z_{i}$ of $G'$ such that $y_{i}z_{i}\in E(G')$.
Note that by Lemma \[Main\] we have $d_{F'}(z_{i})=2$, for $i=k_{1}+1, k_{1}+2, ..., k_{1}+k_{2}$, and $d_{F'}(z_{i})=4$ and at least two edges (namely $z_{i}z'_{i}, z_{i}z''_{i}$) of $F'$ incident with $z_{i}$ belong to $G'$, for $i=k_{1}+k_{2}+1, k_{1}+k_{2}+2, ..., k_{1}+k_{2}+k_{3}$.
We set $$E_0=\bigcup _{i=1}^{s}M_{i},~~~ E_{1}=\bigcup_{i=1}^{k_{1}}\{x_{i}y_{i}, x_{i}z_{i}\},~~~ E'_{1}=\bigcup_{i=1}^{k_{1}}\{y_{i}z_{i}\},$$ $$E_{2}=\bigcup_{i=k_{1}+1}^{k_{1}+k_{2}}\{x_{i}y_{i}, x_{i}z_{i}, y_{i}z_{i}\},$$ $$E_{3}=\bigcup_{i=k_{1}+k_{2}+1}^{k_{1}+k_{2}+k_{3}}\{x_{i}y_{i}, x_{i}z_{i}, y_{i}z'_{i}\},~~~
E'_{3}=\bigcup_{i=k_{1}+k_{2}+1}^{k_{1}+k_{2}+k_{3}}\{z_{i}z'_{i}\}.$$
For $i=1, 2, ..., k_{1}+k_{2}+k_{3}$, all $z_{i}$’s are different, otherwise if $z_{i}=z_{j}$, for $i\neq j$, then $x_{i}y_{i}z_{i}(=z_{j})y_{j}x_{j}$ is a path of length 4 in $G$ joining two bad leaves, a contradiction. Similarly, none of $z_{i}$’s is a neighbor of a bad leaf in $G$.
Possibly, $z_{i_{1}}z_{i_{2}}...z_{i_{k}}$ is a path in $F'$ for $i_{1},i_{2},...,i_{k}\in\{k_{1}+k_{2}+1, k_{1}+k_{2}+2, ..., k_{1}+k_{2}+k_{3}\}$. In order to have different edges in $E_{3}$ and $E'_{3}$ we set $z'_{j}=z_{j+1}$, for $j=i_{1},i_{2},...,i_{k-1}$, and $z'_{i_{k}}$ as arbitrary neighbor of $z_{i_{k}}$ in $F'$ and in $G$ different from $z_{i_{k-1}}$. Note that by 3) and Lemma \[Main\] such a vertex exists and could be some $z_{j}$, for $j\in \{i_{1},i_{2},...,i_{k-2}\}$.
Hence we conclude that $F=F'+(E_{0}\cup E_{1}\cup E_{2}\cup E_{3})-(E'_{1}\cup E'_{3})$ is a \[2,4\]-factor of $G^2$.
Conclusion
==========
The following corollaries are immediate consequences of Theorem \[Main++\].
\[Cor1\] If $G$ is a 2-edge connected graph, then $G^2$ contains a $[2,4]$-factor.
\[Cor2\] If $G$ is an essentially 2-edge connected graph without bad leaves, then $G^2$ contains a $[2,4]$-factor.
\[Main+\] Let $G$ be a connected graph without non-trivial bridges. If any two bad leaves have distance at least 5 in $G$, then $G^{2}$ has a $[2,4]$-factor.
The graph in Fig. 2 also shows that the distance 5 in Corollary \[Main+\] can not be replaced by distance 4.
Now we could answer the question from Introduction. By Theorem \[Fleischner\] we know that the square of 2-connected graph has a $[2,2s]$-factor for $s=1$. In this paper we prove that the square of 2-edge connected graph has a $[2,2s]$-factor for $s=2$ (Corollary \[Cor1\]) and that the square of essentially 2-edge connected graph without bad leaves has a $[2,2s]$-factor also for $s=2$ (Corollary \[Cor2\]). In general, there exist essentially 2-edge connected graphs whose square have no $[2,2s]$-factor for every $s$. Such an example $G$ even exists under an additional condition that the graph obtained from $G$ by deleting all leaves is 2-connected (Theorem \[Main-\]).
[**Acknowledgements**]{}.
This work was supported by the European Regional Development Fund (ERDF), project NTIS - New Technologies for Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.
The first author was supported by project GA14-19503S of the Grant Agency of the Czech Republic.
The second author was supported by NSFC (No.11161046) and by Xinjiang Talent Youth Project (No.2013721012).
The third author was supported by NSFC (No.11471037 and No.11171129) and by Specialized Research Fund for the Doctoral Program of Higher Education (No.20131101110048).
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J. Ekstein, P. Holub, T. Kaiser, L. Xiong, and S. Zhang, Star subdivisions and connected even factors in the square of a graph, Discrete Mathematics 312 (2012), 2574-2578.
R. J. Faudree and R. H. Schelp, The square of a block is strongly path connected, Journal of Combinatorial Theory, Series B 20 (1976), 47-61.
H. Fleischner, In the square of graphs, Hamiltonicity and pancyclicity, hamiltonian connectedness and panconnectedness are equivalent concepts, Monatshefte für [0]{}Mathematik 82 (1976), 125-149.
R. J. Gould and M. S. Jacobson, Forbidden Subgraphs and Hamiltonian Properties in the Square of a Connected Graph, Journal of Graph Theory 8 (1984), 147-154.
G. Hendry and W. Vogler, The square of a $S(K_{1,3})$ - free graph is vertex pancyclic, Journal of Graph Theory 9 (1985), 535-537.
G. Chartrand, A. M. Hobbs, H. A. Jung, S. F. Kapoor, and C. St. J. A. Nash-Williams, The square of a block is Hamiltonian connected, Journal of Combinatorial Theory, Series B 16 (1974), 290-292.
G. L. Chia, S. Ong, and L. Y. Tan: On graphs whose square have strong hamiltonian properties, Discrete Mathematics 309 (2009), 4608-4613.
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[^1]: University of West Bohemia, Pilsen, Czech Republic,e-mail: .
[^2]: Xinjiang University, Urumgi, Xinjiang, P.R.China,e-mail: .
[^3]: Beijing Institute of Technology, Beijing, P.R.China,e-mail: .
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---
abstract: 'Hermes has measured longitudinal double spin asymmetries as a function of transverse momentum $p_T$ using charged inclusive hadrons from electroproduction off a deuterium target. At $p_T>1$ GeV, the asymmetries are sensitive to the spin dependent gluon distribution $\Delta g$. To extract the gluon polarization $\Delta g/g$, information on the background asymmetry and the subprocess kinematics has been obtained from a Leading Order Monte Carlo model. Values for $\Delta g/g$ have been calculated both as a function of the measured $p_T$ and $x$, using two different methods, in the region $p_T>1.05$ GeV.'
author:
- 'P. Liebing, on behalf of the Hermes Collaboration'
bibliography:
- 'proceedings.bib'
title: 'Extraction Of $\Delta g/g$ From Hermes Data On Inclusive Charged Hadrons'
---
[ address=[Riken-BNL research Center, Upton, NY, 11973, USA]{} ]{}
Introduction
============
A direct, model dependent extraction of $\Delta g/g$ has been performed by Hermes [@hermes1],SMC [@smc] and Compass [@compass1; @compass2] for different channels and data sets. This report presents a refined extraction method, using the high statistics data sample of antitagged (vetoed by electrons or positrons), inclusive charged hadrons. A detailed study was performed to estimate the systematic error arising within the model which uses Pythia 6.2 [@pythia], and parametrizations of spin dependent parton distributions of the nucleon and of the photon.
Experimental Data
=================
The data sample used for this analysis was collected using the Hermes spectrometer. Charged inclusive hadrons were selected from events where neither a positron nor an electron were detected. The asymmetries were calculated as $$A_{\parallel}(p_T)=\frac{N^-L^+-N^+L^-}{N^-P^++N^+P^-},$$ where $N^{+(-)}$ are the number of hadrons detected with beam and target spins parallel (antiparallel), $L^{+(-)}$ are the corresponding integrated luminosities and $P^{+(-)}$ the integrated luminosities weighted with the product of beam and target polarizations. The transverse momentum was calculated with respect to the beam axis. The results for positive and negative hadrons from proton and deuteron targets are shown in figure \[asym\_measured\]. The asymmetries have not been corrected for acceptance and trigger efficiency. It was confirmed that the trigger efficiency does neither introduce a significant bias to the asymmetries nor to the final results.
![\[asym\_measured\] Measured asymmetries for antitagged inclusive charged hadrons. The top row shows the asymmetries for proton- , the bottom row for deuteron target. The curves show the asymmetries calculated using Monte Carlo and spin dependent quark distributions and the assumptions $\Delta g/g(x)=-1,\,0$ and $+1$ (from top to bottom).](asym_photo_prelim.eps){height=".3\textheight"}
Also shown in the figure are the asymmetries expected from the model, using the assumptions $\Delta g/g(x)=-1,\,0$ and $+1$ (lines from top to bottom) over the full covered $x$-range. More detailed information on how the model asymmetries were calculated will be given below. The differences between the measured asymmetries are due to the quarks, and are well described by the curves at low $p_T$, where the contribution from gluons is negligible.
Interpretation in Terms of the Gluon Polarization
=================================================
The measured asymmetries arise from a superposition of different subprocesses contributing to the production of hadrons at a given measured $p_T$. In order to decompose the asymmetries and extract the signal from processes initiated by a hard gluon, the asymmetries and relative contributions of the background processes have to be known as well as the hard subprocess kinematics of the signal processes. This information was obtained from a simulation of the data using the Pythia 6.2 Monte Carlo program and a model of the Hermes detector.
Data – Monte Carlo Comparison
-----------------------------
The Vector Meson Dominance (VMD) Model in Pythia was adapted to reproduce the observed exclusive $\rho^0$ cross section [@patty]. The fragmentation process simulated in Jetset was tuned to multiplicities of identified hadrons at $Q^2>1$ GeV$^2$ measured at Hermes [@achim]. For $Q^2>0.1$ GeV$^2$, the observed semiinclusive cross sections agree typically within 15% in variables [*integrated*]{} over $p_T$. In contrast to this generally good agreement, the observed cross sections do not agree vs. $p_T$. For $p_T>0.7$ GeV, the Monte Carlo underestimates the data by factors 2 to 4. The disagreement is most likely due to large NLO corrections as calculated in Ref. [@nlopt]. The LO result for $\Delta g/g$ might therefore also be subject to large NLO corrections.
Subprocess Fractions
--------------------
The soft background processes from exclusive and diffractive VMD as well as nondiffractive VMD (“low-$p_T$”) are suppressed at high $p_T$, although the “low-$p_T$” process still contributes significantly for $p_T<1.5$ GeV. The quark initiated hard QCD processes together contribute less than 20% at $p_T>1$ GeV. At $p_T>1.5$ GeV, the LO DIS process dominates the cross section. Hadrons from this process originate predominantly from events with a large lepton scattering angle, where the transverse momentum calculated with respect to the beam axis overestimates the transverse momentum in the center of mass frame. The signal processes are Photon-Gluon-Fusion (PGF) and the gluon initiated $2\rightarrow2$ (resolved photon) processes, each contributing 10-20% in the relevant $p_T$ range.
Extraction Methods and Results
==============================
The background asymmetries for hard subprocesses were estimated using the MC information on the particle types and subprocess kinematics with the nucleon PDFs from [@grsv] and the photon PDFs from [@grs], where the average of the maximal and minimal scenarios was taken. The asymmetry for exclusive VMD was set to 0, that of the “low-$p_T$” process was set to $A^{low-p_T}=g_1/F_1$, using an extrapolation of a fit to world data to lower $x\approx10^{-4}$. By subtracting the background asymmetry weighted with the background fraction from the measured asymmetry, $A^{meas}_{\parallel}-R_{BG}A^{BG}=R_{sig}A^{sig}$ the signal asymmetry can be obtained. The signal asymmetry contains a convolution of $\Delta g(x)/g(x)$ with the (polarized) hard subprocess cross section over the $x$-range covered by the data. Two methods have been applied to extract the average $\langle\Delta g/g\rangle(p_T)$ from this asymmetry using different assumptions on the shape of $\Delta g(x)/g(x)$. Method I assumes that $\Delta g(x)/g(x)$ is essentially constant in the relevant $x$-range. Then, $\langle\Delta g/g\rangle(p_T)$ can be found by solving the equation $$A^{meas}_{\parallel}-R_{BG}A^{BG}=R_{sig}\left\langle \hat a
\frac{\Delta f^\gamma}{f^\gamma}\right\rangle\left\langle \frac{\Delta g}{g}\right\rangle$$ for each bin in $p_T$. Here, $\hat a$ is the hard subprocess asymmetry, and $\Delta f^\gamma/f^\gamma$ is the polarization of partons in the resolved photon.
In Method II, a functional form is assumed for $\Delta g(x)/g(x)$ which is used to calculate the integral $A^{sig}$ for each $p_T$ bin. The functional parameter(s) are determined by minimizing $\chi^2$ for the difference $A^{meas}_{\parallel}-R_{BG}A^{BG}-R_{sig}A^{sig}$ using all bins in $p_T$.
Values for $\langle\Delta g/g\rangle(p_T)$ have been obtained from the deuterium data on charge combined hadrons, for 4 bins in $p_T$ between 1.05 and 2.5 GeV. The results are shown in figure \[deltag\] for both Methods. Note that, in Method II, the results and errors are correlated through the fit function.
![\[deltag\] Results for $\langle\Delta g/g\rangle(p_T)$ from Method I (left) and Method II (right) for the antitagged deuterium data. The inner error bars (band) correspond to the statistical errors, the outer error bars (band) to the total errors.](deltagpoints_total_pt_prelim_nohorizontal.eps){height=".3\textheight"}
Method I was used to confirm the overall consistency between different independent data sets from proton and deuteron targets and positive and negative hadrons. The experimental systematic error is approximately 14% and arises from the uncertainties in the beam and target polarization measurements. It is small compared to the model uncertainty which was estimated by varying the Pythia model parameters, the unpolarized PDFs in the MC generation, the polarized PDFs in the asymmetry calculation and the assumption used for the asymmetry of the “low-$p_T$” process. For Method II, the functional form was also varied. No error was assigned on the Pythia model itself, and, because this is a leading order approach, also no error was assigned to account for NLO corrections. Method II allows to determine the average $x$ of the measurement, and by integrating over $1.05<p_T<2.5$ GeV a value of $\Delta g/g=0.071\pm 0.034 (stat)\pm 0.010
(sys-exp)^{+0.127}_{-0.105}(sys-Models)$ has been obtained at $\langle x\rangle = 0.22$ and $\langle\mu^2\rangle=1.35~{\rm GeV}^2$.
|
---
abstract: 'The literature in social network analysis has largely focused on methods and models which require complete network data; however there exist many networks which can only be studied via sampling methods due to the scale or complexity of the network, access limitations, or the population of interest is hard to reach. In such cases, the application of random walk-based Markov chain Monte Carlo (MCMC) methods to estimate multiple network features is common. However, the reliability of these estimates has been largely ignored. We consider and further develop multivariate MCMC output analysis methods in the context of network sampling to directly address the reliability of the multivariate estimation. This approach yields principled, computationally efficient, and broadly applicable methods for assessing the Monte Carlo estimation procedure. In particular, with respect to two random-walk algorithms, a simple random walk and a Metropolis-Hastings random walk, we construct and compare network parameter estimates, effective sample sizes, coverage probabilities, and stopping rules, all of which speaks to the estimation reliability.'
address:
- 'School of Statistics, University of Minnesota, Twin Cities, 313 Ford Hall 224 Church St SE, Minneapolis, MN 55455'
- 'Department of Sociology and eScience, University of Washington'
author:
- Haema Nilakanta
- 'Zack W. Almquist'
- 'Galin L. Jones'
bibliography:
- 'mybibfile.bib'
title: Ensuring Reliable Monte Carlo Estimates of Network Properties
---
Monte Carlo ,Markov chains ,output analysis ,sampling ,estimation ,estimation reliability
Introduction and Background {#sec:intro}
===========================
Much of the network literature has focused on complete network data [@scott2017social; @wasserman1994social; @Kolaczyk2009]; but in many practically relevant settings, the full network is difficult to study due to its scale or complexity (e.g., geospatial social networks) or the network represents a hidden population (e.g., homeless friendship networks in United States). In such cases, traditional survey sampling methods, e.g., simple random sampling (SRS), are not practical due to the absence of a sampling frame. Alternatively, one can collect an approximately uniform sample from a network by traversing the structure in a nondeterministic manner. Features of interest can then be estimated using sample statistics. A particular focus within the network sampling literature is on traversing networks with random walk-based algorithms, a Markov chain Monte Carlo (MCMC) method.
Overall, there are three core approaches to sample networks: (1) SRS of nodes, also known as egocentric sampling [@wasserman1994social], (2) sampling edges at random, or (3) MCMC sampling, commonly referred to as traversal sampling or link trace sampling. Practical advantages to each of these methods exist, although SRS and link trace sampling are more common than edge sampling. There exists an extensive literature which looks at SRS and other more complex sampling designs (e.g., cluster or stratified designs) in the social network literature [@marsden2011survey].
Link trace sampling methods are particularly popular in the social sciences. With these traversal approaches, one can leverage the underlying network structure without a sampling frame to obtain population level measures. One of the most popular versions is Respondent Driven Sampling (RDS); first introduced in [@heckathorn1997respondent]. Since its introduction, there have been several extensions of RDS that further underline the appeal of link trace methods to study social interactions [see e.g., @salganik2004sampling; @gile2010respondent; @handcock2010modeling]. More recently, the growth of large Online Social Networks (OSNs) has also brought rising attention to traversal methods. For example, [@gjoka2010walking] and others [e.g., @kurant2012coarse; @gjoka2011multigraph] have used these methods to obtain asymptotically unbiased estimates of core network features (e.g., the degree distribution or clique census) or subgroup populations.
Among large OSNs, random walk-based algorithms have been regularly used to estimate key features such as average connectedness or clustering coefficients [@gjoka2011practical]. These random walk algorithms have also been employed to obtain information about hard to reach populations, such as estimating disease prevalence among individuals at high risk for HIV [@thompson:2017]. While the use of these MCMC methods to estimate network features is common, the quality of estimation with these Monte Carlo samples has not been directly addressed in a computationally efficient way. We contribute to this area by considering and further developing MCMC output analysis methods in the context of network sampling that directly address the reliability of estimation.
Constructing MCMC sampling algorithms to efficiently traverse a network can be challenging and is an active area of research. As a result, there has been substantial work on comparing various MCMC sampling methods for networks, but the comparisons usually only consider the properties of univariate point estimates, computation speed (i.e., clock time or percent of network sampled), or the difference in empirical distributions using the Kullback-Leibler divergence, Kolmogorov-Smirnov D-statistic, or the total variation distance [see, among others, @lee2006statistical; @avrachenkov2018revisiting; @gjoka2011practical; @gile2010respondent; @blagus2017empirical; @ahmed2014network; @zhou2016faster; @li2015random; @leskovec2006sampling; @wang2011understanding; @salamanos2017deterministic; @lee2012beyond; @joyce2011kullback]. Typically the goal is to estimate many network features based on one Monte Carlo sample, while comparisons typically focus on univariate summaries. That is, the multivariate nature of the estimation problem has been broadly ignored.
Moreover, separate from the natural variability in the data, the estimates produced by these Monte Carlo methods are also subject to Monte Carlo error in that different runs of the sampling algorithm will result in different estimates. Thus, the algorithm used will impact the quality of the estimation. Of course, if the Monte Carlo sample sizes are large enough, then the differences in run estimates will be negligible. This then raises the question, how large is large enough? That is, how large does the Monte Carlo sample need to be so that the estimates are trustworthy?
The current tools used in the network sampling literature to determine when to terminate the sampling process are insufficient. Popular methods rely on the use of so-called convergence diagnostics [@cowl:carl:1996; @gelm:rubi:1992; @gewe:1992; @heid:welc:1983], but none of these methods make any attempt to assess the quality of estimation [@fleg:etal:2008; @jone:etal:2006]. Moreover, these diagnostics have been shown to stop the sampling process prematurely [@jone:etal:2006; @vats:2018revisiting]. Another common approach is to study the running mean plot and determine the point at which it stabilizes to find approximately when the estimates have settled [@gjoka2011practical; @lu2012sampling; @lee2006statistical; @ribeiro:2010]. This approach is inadequate since its interpretation is subject to how much one zooms in on a section of the plot.
Although the network sampling literature on Monte Carlo estimation reliability is relatively sparse, [@Avrachenkov2016; @lee2006statistical; @chiericetti2016sampling; @salamanos2017deterministic], and [@wang2011understanding] considered the relative error or normalized root mean squared error of sample estimates from various sampling methods. However, neither approach takes into account the multivariate nature of the problem nor tries to calculate the sample variance from the correlated sampling procedure. In addition, [@mohaisen:2010] and [@zhou2016faster] discuss the theoretical mixing time of the sampling algorithms they propose, although theoretically valid, are impractical to implement. We are unaware of any other work that directly address the reliability of the multivariate estimation with these MCMC samples.
We consider and further develop multivariate MCMC output analysis methods [see e.g. @vats2015multivariate; @vats2018strong; @flegal2015mcmcse] in the context of network sampling with respect to two MCMC algorithms: a simple random walk and a random walk-based version of the Metropolis-Hastings algorithm. This approach yields principled, computationally efficient, and broadly applicable methods for assessing the reliability of the Monte Carlo estimation procedure. In particular, we construct and compare network parameter estimates, effective sample sizes, coverage probabilities, and stopping rules.
The rest of the paper is organized as follows. In Section 2 we introduce some basic network notation and MCMC methods on networks. We also introduce output analysis tools to determine multivariate MCMC estimation reliability. In Sections 3 and 4 we further develop these output analysis tools in the context of network sampling, providing three examples of their use on a simple simulated high school social network to illustrate the concepts and progressively move to more complicated, larger networks. Finally, we conclude with a discussion in Section 5.
Methods
=======
Markov Chain Monte Carlo Methods on Networks {#Monte Carlo Methods on Networks}
--------------------------------------------
We represent the network of interest in terms of a graph [see @wasserman1994social], which is a relational structure comprised of two elements: a set of nodes or vertices (used interchangeably), and a set of vertex pairs representing edges or ties (i.e., a relationship between two nodes). Formally, let $V$ denote a non-empty countable set of nodes, $E \subseteq V \times V$ denote the set of edges between the vertices, and $G=(V,E)$ denote the network. We only consider simple networks that are binary, undirected, well-connected, and without self loops. Define the network size, $n$, to be the set cardinality of $V$. Similarly, $n_e$ is the number of edges in the graph. The network features of interest can be expressed as the mean of a function over the entire network. More formally, suppose $h: V \rightarrow {\mathbb{R}}^p$ where $p$ is the number of features of interest and let $\lambda$ be the uniform distribution on $V$. Then, if $X \sim \lambda$, we want to calculate the $p$-dimensional mean vector $$\label{eq:network means}
E_{\lambda}[h(X)] = \frac{1}{n} \sum\limits_{ v \in V} h(v),$$ where the subscript indicates that the expectation is calculated with respect to $\lambda$. It will be notationally convenient to denote $E_{\lambda}[h(X)] = \mu_h$ and we will use both interchangeably. Specific network features of interest might include: mean degree, degree distribution, mean clustering coefficient, and proportion of nodes with specific nodal attributes, e.g., proportion of female users in an OSN.
Computing $\mu_h$ is often difficult in practically relevant applications and hence we turn to MCMC methods. Let $\{V_0, V_1, V_2, \ldots \}$ be an irreducible, aperiodic Markov chain with invariant distribution $\lambda$ [for definitions see @bremaud_2010; @levin2009markov]. Then by Birkhoff’s ergodic theorem we have that, if $E_{\lambda}|h(X)| < \infty$, with probability 1, $$\label{eq:slln}
\mu_m = \frac{1}{m} \sum_{t=0}^{m-1} h(V_t) \to \mu_h, ~\text{as } ~ m \to \infty.$$ Thus estimation of $\mu_h$ is straightforward; simulate $m$ steps of the Markov chain and use the sample mean. However, the quality of estimation depends on the Monte Carlo sample size, $m$, since for a finite $m$ there will be an unknown *Monte Carlo error*, $\mu_m - \mu_h$. We can begin to assess this error through a central limit theorem [see e.g. @aldous1997mixing; @jones2004markov; @vats2015multivariate]. That is, for any initial distribution of the Markov chain, as $m \to \infty$, $$\label{eq:clt}
\sqrt{m}(\mu_m - \mu_h) \stackrel{d}{\to} N_p(0, \Sigma),$$ where $$\label{eq:sigma}
\Sigma = \text{Var}_\lambda(h(V_0)) + \sum\limits_{t=1}^\infty \left[\text{Cov}_\lambda (h(V_0), h(V_t)) + \text{Cov}_\lambda (h(V_0), h(V_t))^T \right].$$ If $\|\cdot\|$ denotes the standard Euclidean norm, then, given our assumptions on the Markov chain, the main requirement for is that $E_{\lambda}[\|h\|^2] < \infty$, which typically will hold. Also, since the chain is on the finite state space $V$, it is uniformly ergodic. [@aldous1997mixing].
The matrices $\Sigma$ and $\Lambda := \text{Var}_\lambda(h(V_0))$ will be fundamental to the remainder. Estimating $\Lambda$ is straightforward using the sample covariance, denoted $\Lambda_m$, but estimating $\Sigma$ is a nontrivial matter which has attracted a significant research interest [@andr:1991; @chen1987multivariate; @dai2017multivariate; @liu:fleg:2018; @liu:fleg:2018spec; @hobe:etal:2002; @jones2006fixed; @kosorok2000monte; @seil:1982; @vats2018strong; @vats2015multivariate; @vats:fleg:2018]. There are several approaches to estimate $\Sigma$ that use spectral variance estimators, but these are computationally demanding especially with large Monte Carlo sample sizes [@liu:fleg:2018] . Therefore due to computational feasibility, we will only consider the method of batch means, which we present now.
Let $\{X_t, \, t \ge 0\} = \{h(V_t), \, t \ge 0\}$ and set $m = a_m b_m$ where $a_m$ is the number of batches and $b_m$ is the batch size. For $k = 0, \ldots, a_m-1$ set
$$\bar{X}_k := b_m^{-1} \sum_{t=0}^{b_m-1} X_{k b_m +t}.$$ Then $\bar{X}_k$ is the mean vector for batch $k$ and the estimator of $\Sigma$ is $$\Sigma_m = \frac{b_m}{a_m-1} \sum\limits_{k=0}^{a_m-1} (\bar{X}_k -\mu_m)(\bar{X}_k - \mu_m)^T.$$ For $\Sigma_m$ to be positive definite, $a_m > p$. It is common to choose $a_m = \lfloor m^{1/2} \rfloor$ or $a_m = \lfloor m^{1/3} \rfloor$ where $a_m > p$ is met. Batch means produces a strongly consistent estimator of $\Sigma$ [@vats2015multivariate] under conditions similar to those required for and is implemented in the `mcmcse` R package [@flegal2015mcmcse].
MCMC Output Analysis {#sec:output}
--------------------
It would be natural to use the CLT and $\Sigma_m$ to form asymptotically valid confidence regions for $\mu_h$. The volume of the confidence region could then be used to describe the precision in the estimation and, indeed, this sort of procedure has been advocated [@jone:etal:2006]. More specifically, if $T^2_{1-\alpha, p, q}$ denotes the $1-\alpha$ quantile of a Hotelling’s $T$-squared distribution where $q = a_m - p$, then a $100(1-\alpha)$% confidence ellipsoid for $\mu_h$ is the set $$C_\alpha(m) = \{\mu_h \in {\mathbb{R}}^p: m(\mu_m -\mu_h)^T \Sigma_m^{-1}(\mu_m - \mu_h) < T^2_{1-\alpha, p, q}\} .$$ The volume of the ellipsoid is given by $$\text{Vol}(C_\alpha(m)) = \frac{2\pi^{p/2}}{p\Gamma(p/2)} \left(\frac{T_{1-\alpha, p, q}}{m}\right)^{p/2} |\Sigma_m|^{1/2}.$$ One could then terminate a simulation when the volume is sufficiently small, indicating that our Monte Carlo error is sufficiently low. However, the fixed-volume approach is difficult to implement even when $p$ is small [@flegal2015mcmcse; @vats2015multivariate; @glynnwhitt:1992].
An alternative is to terminate the simulation when the volume is small compared to the generalized variance [@wilk:1932] of the target distribution, that is, if $|\cdot|$ denotes determinant, small compared to $|\Lambda|$. The intuition is that when the Monte Carlo error is small compared to the variation in the target distribution, then it is safe to stop. More formally, letting $m^* > 0$ and $\epsilon >0$ be given, then we terminate the simulation at the random time $T_{SD}(\epsilon)$ defined as, $$T_{SD}(\epsilon) = \inf\left\{m \ge 0: \text{Vol}(C_\alpha(m))^{1/p} + \epsilon |\Lambda_m|^{1/2p}I(m < m^*) + m^{-1} \le \epsilon |\Lambda_m|^{1/2p}\right\}.$$ The role of $m^*$ is to require some minimum simulation effort. It should be large enough so that both $\Lambda_{m^*}$ and $\Sigma_{m^*}$ are positive definite and the lower bound on the ESS is achievable.
We can connect $T_{SD}(\epsilon)$ to effective sample size, the equivalent number of independent and identically distributed (*iid*) samples that would give the same standard error as the correlated sample, $$\label{eq:ESS}
\text{ESS} = m \left[\frac{| \Lambda |}{ |\Sigma| }\right]^{1/p}$$ and naturally estimated with $$\widehat{\text{ESS}} = m \left[\frac{| \Lambda_m |}{ |\Sigma_m| }\right]^{1/p}. \label{estimatedessequation}$$ By rearranging the defining inequality of $T_{SD}(\epsilon)$ we see that terminating at $T_{SD}(\epsilon)$ is essentially equivalent, for large $m$, to terminating when the estimated effective sample size satisfies $$\widehat{\text{ESS}} \ge \frac{2^{2/p}\pi}{(p\Gamma(p/2))^{2/p}}
\frac{\chi^2_{1-\alpha, p}}{\epsilon^2}.$$ Notice that the right-hand side of the inequality can be calculated prior to running the simulation and hence yielding a minimum simulation effort based on a desired confidence level $1-\alpha$ and relative precision $\epsilon$.
Later, we will require the delta method [see e.g. @sen:sing:1993 Ch. 3]. This substantially broadens the application of the methodology so far described. We are often interested in estimating $g(\mu_h)$ where $g: {\mathbb{R}}^p \rightarrow {\mathbb{R}}^p$. If $g$ is such that it has a non-null derivative $\nabla g(\mu_h)$ at $\mu_h \in {\mathbb{R}}^p$ and is continuous in a neighborhood of $\mu_h$, then, as $m \to \infty$, the strong law at ensures $g(\mu_m) \to g(\mu_h)$, with probability 1, and the CLT at ensures that $$\label{eq:dm.clt}
\sqrt{m} (g(\mu_m) - g(\mu_h)) \stackrel{d}{\to} \text{N}\left(0,
[\nabla g(\mu_h)]^T \Sigma [\nabla g(\mu_h)] \right) .$$ It is straightforward to estimate the asymptotic covariance with $$[\nabla g(\mu_m)]^T \Sigma_m [\nabla g(\mu_m)].$$ Thus we can proceed with the output analysis as described above. Notice that $$\label{prop:dESS}
\text{ESS}_{g} := m \left[\frac{|[\nabla g(\mu_h)]^T \Lambda
[\nabla g(\mu_h)]|}{|[\nabla g(\mu_h)]^T \Sigma [\nabla
g(\mu_h)] |}\right]^{1/p} = m \left[\frac{| [ \nabla
g(\mu_h)]^T| |\Lambda| |[\nabla g(\mu_h)]|}{|[\nabla
g(\mu_h)]^T| |\Sigma|| [\nabla
g(\mu_h)] |}\right]^{1/p} = \text{ESS}$$ and hence ESS is unaffected by the delta method transformation.
Two MCMC Sampling Methods {#RW sampling methods}
-------------------------
We will consider two random walk-based MCMC methods, a simple random walk (SRW) and a Metropolis-Hastings (MH) algorithm with a simple random walk proposal. MH is constructed to have its invariant distribution as $\lambda$, the uniform distribution over nodes. SRW has a different invariant distribution, necessitating the use of importance sampling in estimation. The details are considered below.
First, we require some notation. If there is an edge from node $i$ to node $j$ we say $i$ and $j$ are neighbors. The number of neighbors of node $i$ is its degree, $d_i$.
Then the SRW works as follows, if the current state is $i$, then the transition probability of moving to node $j$ is $$P(i,j)^{SRW} = \begin{cases}
\frac{1}{d_i} &\quad \text{if $j$ is a neighbor of $i$}\\
0 &\quad \text{otherwise.}
\end{cases}$$ The stationary density of the SRW is $\lambda^*(i) = d_i / 2 n_e$, which is not the uniform.
[@gjoka2011practical] suggested using a Metropolis-Hastings algorithm with SRW as the proposal distribution (for a summary of the Metropolis-Hastings algorithm refer to [@bremaud_2010]). This gives rise to MH transition probabilities of the form $$P(i,j)^{MH} = \begin{cases}
\frac{1}{d_i} \min\left(1, \frac{d_i}{d_j}\right) \quad &\text{if $j$ is a neighbor of $i$}\\
1 - \sum_{k\ne i} \frac{1}{d_i} \min\left(1, \frac{d_i}{d_k}\right) \quad &\text{if $j =i$}\\
0 \quad &\text{otherwise.}
\end{cases}$$ In this case, the stationary density is the uniform over $V$, $\lambda(i) = 1/n$.
Monte Carlo Methods for Network Descriptive Statistics and Inference {#Simulations}
====================================================================
We focus on estimating popular network features, these include: mean degree, degree distribution (e.g., proportion of nodes with $k$ neighbors), mean clustering coefficient, and mean of nodal attributes. For a given node $v$, let $d_v$ be the degree, $t_v$ be the number of triangles, and a categorical attribute, $x_v$, (e.g., race) having $c$ levels $x(1), x(2), \ldots, x(c)$. We keep these estimators general as one can easily see that the list can be expanded. In terms of the notation from the previous section where $\mathbb{I}$ denotes the indicator function, we want to estimate $\mu_h$ where $$\label{eq:newh}
h(v) = (d_v,\, \mathbb{I}(d_v = k), \, 2t_v \mathbb{I}(d_v \ge 2)/d_v (d_v - 1), \, \mathbb{I}(x_v = x(c)))^T .$$ When using MH, estimation proceeds by using $\mu_m$. When using SRW, estimation will proceed using importance sampling [@hest:1995; @mcbook; @robe:case:2013] with $$\mu_{m}^{SRW} = \frac{ \sum_{t=0}^{m-1} \left[ \frac{h(V_t)}{d_{V_t}} \right]}{\sum_{t=0}^{m-1} \frac{1}{d_{V_t}}}.$$ Other names for this approach include reweighted random walk or respondent driven sampling as MCMC [@goel2009respondent; @gjoka2011practical; @Avrachenkov2016; @salganik2004sampling]. To find the form of the CLT, we use a transformed version of $h$.
Namely, let $h^*(v) = (1/d_v,\, \mathbb{I}(d_v = k)/d_v, \, 2t_v \mathbb{I}(d_v \ge 2)/d_v^2 (d_v - 1), \, \mathbb{I}(x_v = x(c))/d_v)^T$ so that if $$\mu^*_m= \frac{1}{m} \sum_{t=0}^{m-1} h^*(V_t),$$
then, by the CLT, we have, as $m \to \infty$, $$\sqrt{m}(\mu^*_m - \mu_{h^*}) \to \text{N}(0, \Sigma^*).$$ We then apply the delta method with $g(a,b,c,d)= (1/a, b/a, c/a, d/a)^T$ so that $$\nabla g = \begin{pmatrix} -1/a^2 & -b/a^2 & -c/a^2 & -d/a^2\\
0 & 1/a & -0 & 0 \\
0 & 0 & 1/a & 0 \\
0 & 0 & 0 & 1/a
\end{pmatrix},$$ to obtain, via , that, as $m \to \infty$, $$\sqrt{m}(g(\mu^*_m) - g(\mu_{h^*})) \to \text{N}\left(0, [\nabla g(\mu_{h^*})]^T \Sigma^* [\nabla g(\mu_{h^*})]\right)$$ and we can estimate the asymptotic variance with $$[\nabla g(\mu^*_m)]^T \Sigma_m^* [\nabla g(\mu^*_m)] .$$
Again, the goal is to obtain estimates of these network properties and measures on the reliability of those estimates.
We now consider the algorithms and output analysis methods described above as applied to three social networks. We begin with a simple example to illustrate the concepts and progressively move to more complicated, larger networks.
Application to Social Networks
==============================
To demonstrate the applicability of this work we look into classic cases in the literature: (1) a simulated network based on Ad-Health data [@handcock2008statnet; @resnick1997protecting], (2) a college Facebook friendship network [@traud2008community], and (3) the Friendster network to showcase its use on large scale graphs. These three cases allow us to demonstrate the effectiveness of the output analysis methods.
High School Social Network Data {#Toy}
-------------------------------
The `faux.magnolia.high` social network is in the `ergm` R package [@handcock2008statnet; @resnick1997protecting]. It is a simulation of a within-school friendship network representative of those in the southern United States. All edges are undirected and we removed 1,022 nodes out of 1,461 to ensure a well-connected graph. This resulting social network has 439 nodes (students) and 573 edges (friendships). Other nodal attributes besides structural are grade, race, and sex. The population parameters are in Tables \[tab:magnoliaSummaryStats\] and \[tab:magnoliaSummaryStatsOther\].
Min 25% Median Mean 75% Max
------------------------ ------ ------ -------- ------ ------ -------
Degree 1.00 1.00 2.00 2.61 4.00 8.00
Triples 0.00 0.00 1.00 3.15 6.00 28.00
Triangles 0.00 0.00 0.00 0.90 1.00 10.00
Clustering Coefficient 0.00 0.00 0.00 0.13 0.20 1.00
: Population parameters of well-connected `faux.magnolia.high` social network.[]{data-label="tab:magnoliaSummaryStats"}
------- ------- -------- ------- ------ ------- -------
Grade Mean SD
9.42 1.62
Sex Male Female
% 42.82 57.18
Race White Black Asian Hisp NatAm Other
% 79.73 12.07 3.19 3.19 1.37 0.46
------- ------- -------- ------- ------ ------- -------
: Other population parameters of well-connected `faux.magnolia.high` social network.[]{data-label="tab:magnoliaSummaryStatsOther"}
We ran a single chain of both the SRW and MH walks on this network with random starting nodes repeating this 1000 times independently, constructing estimates for the mean degree, mean clustering coefficient, mean grade, proportion of females, and proportion of students who identified as white. The minimum ESS for $p=5, \epsilon=0.05$, and $\alpha = 0.05$ is 10363. We also constructed the 95% confidence region and used the corresponding volume to determine the termination time using the relative fixed-volume sequential stopping rule with multivariate batch means with the square root batch size, $\epsilon=0.05$, and $m^* = 10,000$. At this random terminating point we also noted the univariate mean estimates, multivariate effective sample size, and the number of unique nodes visited by the termination step.
### Results
The univariate estimates with standard errors from both the SRW and MH are in Figure \[fig:MagnoliaMeanDegree\] and Table \[tab:MagnoliaMeans\].
[.45]{} ![Mean estimates from SRW and MH on well-connected `faux.magnolia.high` network. Replications = 1000. Blue dashed line indicates population quantity.[]{data-label="fig:MagnoliaMeanDegree"}](Graphics/SRWmeanHistograms.pdf "fig:"){width=".9\linewidth"}
[.45]{} ![Mean estimates from SRW and MH on well-connected `faux.magnolia.high` network. Replications = 1000. Blue dashed line indicates population quantity.[]{data-label="fig:MagnoliaMeanDegree"}](Graphics/MHmeanHistogramsAtTermination.pdf "fig:"){width=".9\linewidth"}
Type Degree Clustering coeff Grade Prop female Prop white
------- ----------------- ------------------ ----------------- ----------------- -----------------
Truth 2.6105 0.1956 9.4146 0.5718 0.7973
SRW 2.6106 (0.0004) 0.1956 (0.0001) 9.4145 (0.0026) 0.5716 (0.0157) 0.7969 (0.0127)
MH 2.6103 (0.0004) 0.1956 (0.0001) 9.4158 (0.0024) 0.5719 (0.0157) 0.7973 (0.0127)
: Mean estimates from SRW and MH on the well-connected `faux.magnolia.high` network at termination time. Replications = 1000 and standard errors in parentheses.[]{data-label="tab:MagnoliaMeans"}
All SRW samples terminated on average around 341,000 steps (average computer run time 425 seconds) whereas the MH samples did not achieve the stopping criterion until around 689,115 steps on average (average computer run time 352 seconds). Results are shown in Table \[tab:magnoliaESS\]. Since the network is relatively small, all runs of the two sampling methods captured all the nodes in the network. The mean acceptance rate of the MH samples was 0.29. Auto correlation function (ACF) plots for the five estimates from one terminated chain of the SRW and MH are shown in Figure \[fig:MagnoliaACFandTrace\].
Termination Step ESS Unique Nodes $T(\epsilon=0.05)$
----- ------------------ ------------------ -------------- --------------------
SRW 341190 (452.481) 10639.67 (3.113) 439 (0) 0.0497 (0)
MH 689115 (698.090) 10550.03 (2.273) 439 (0) 0.0498 (0)
: Termination time, effective sample size, unique nodes sampled by termination for $\epsilon=0.05$, and $T(\epsilon=0.05)$ at termination step on the well-connected `faux.magnolia.high` network. Replications = 1000 and standard errors are in parentheses. []{data-label="tab:magnoliaESS"}
[.45]{} ![ACF plots from one terminated chain of SRW and MH on `faux.magnolia.high` network.[]{data-label="fig:MagnoliaACFandTrace"}](Graphics/acfRWTerminationMagnolia.png "fig:"){height="2in"}
[.45]{} ![ACF plots from one terminated chain of SRW and MH on `faux.magnolia.high` network.[]{data-label="fig:MagnoliaACFandTrace"}](Graphics/acfMHTerminationMagnolia.png "fig:"){height="2in"}
NYU Facebook Data
-----------------
The New York University (NYU) Facebook (FB) dataset is a snapshot of anonymized Facebook data from the NYU student population in 2005 [@traud2008community]. Nodes are NYU FB users and edges are online friendships. The data was obtained directly from FB and is a complete set of users at NYU at the time. Other nodal attributes in this data are: gender, class year, major, high school, and residence. Some nodes had missing attribute data, so we created a new category labeled “Not Reported” (NR). The full NYU FB dataset contains 21,679 nodes (users) and 715,715 undirected edges (online friendships). We only considered the largest well-connected component, NYU WC FB, which has 21,623 users and 715,673 undirected edges. The population parameters of this network are in Table \[tab:nyuSummaryStats\]. We estimated the mean degree, mean clustering coefficient, proportion of female users, and proportion of users with major = 209.
Min 25% Median Mean 75% Max
------------------------ -------- -------- --------- --------- --------- ------------
Degree 1.00 21.00 50.00 66.20 93.00 2315.00
Triples 0.00 210.00 1225.00 4666.47 4278.00 2678455.00
Triangles 0.00 39.00 197.00 502.24 598.00 39402.00
Clustering Coefficient 0.00 0.10 0.15 0.19 0.23 1.00
Gender Female Male NR
% 55.05 37.39 7.57
Major 209 Other NR
% 6.02 77.82 16.16
: Population parameters of well-connected NYU FB social network, NR = Not Reported. $n=21,623$, $n_e=715,673$.[]{data-label="tab:nyuSummaryStats"}
Again we ran a single chain of both the SRW and MH on this network with random starting nodes, repeating this 1000 times independently, constructing the 95% confidence region and determining the termination time with the square root batch size, $\epsilon=0.05$ and $m^* = 10,000$. The minimum ESS for $p=4, \epsilon=0.05$, and $\alpha = 0.05$ is 9992. We constructed coverage probabilities by noting if the confidence region was below the Hotellings $T$-squared quantile.
### Results
The univariate network mean estimates are noted in Figure \[fig:NYUMeanDegree\] and Table \[tab:NYUMeans\]. The mean degree estimate from the SRW and MH on average both slightly overestimate the true mean degree. Otherwise, the estimates from both the SRW and MH algorithms are close to the population means.
[.45]{} ![Mean estimates from SRW and MH on NYU WC FB at termination. Replications = 1000. Blue dashed line indicates population quantity.[]{data-label="fig:NYUMeanDegree"}](Graphics/SRWmeanHistogramsAtTerminationNYU.png "fig:"){width=".9\linewidth"}
[.45]{} ![Mean estimates from SRW and MH on NYU WC FB at termination. Replications = 1000. Blue dashed line indicates population quantity.[]{data-label="fig:NYUMeanDegree"}](Graphics/MHmeanHistogramsAtTerminationNYU.png "fig:"){width=".9\linewidth"}
Type Degree Clustering coeff Prop female Prop Major=209
------- ------------------- ------------------ ------------------ ------------------
Truth 66.1955 0.1939 0.5505 0.0602
SRW 66.2708 (0.04714) 0.1939 (0.0002) 0.5504 (0.01573) 0.0605 (0.00754)
MH 66.2803 (0.02853) 0.194 (0.00012) 0.5508 (0.0157) 0.0601 (0.0075)
: Mean estimates from SRW and MH on NYC WC FB at termination time. Replications = 1000 and standard errors in parentheses.[]{data-label="tab:NYUMeans"}
Termination Step ESS Coverage Prob Unique Nodes $T(\epsilon=0.05)$
----- ------------------- ------------------ --------------- ------------------ --------------------
SRW 14676.78 (51.02) 10558.7 (25.36) 0.938 (0.002) 8703.88 (17.55) 0.048 (0.00)
MH 85948.61 (416.40) 6824.317 (11.38) 0.91 (0.003) 16790.81 (19.96) 0.049 (0.00)
: Termination times, effective sample size, coverage probabilities, number of unique nodes sampled by termination time for $\epsilon = 0.05$, and $T(\epsilon=0.05)$ at termination for NYU WC FB. Replications = 1000 and standard errors in parentheses.[]{data-label="tab:nyuESS"}
All SRW samples terminated on average around 14,700 steps (average computer run time 8.1 seconds) whereas among the MH samples terminated on average by 86,000 steps (average computer run time 30.9 seconds), see Table \[tab:nyuESS\]. The mean acceptance rate of the MH walks was 0.5621. ACF plots for one chain of both the SRW and MH are shown in Figure \[fig:NYUacfAndTrace\].
[.5]{} ![ACF plots from one chain of SRW and MH on NYU WC FB network.[]{data-label="fig:NYUacfAndTrace"}](Graphics/acfRWnyu.png "fig:"){width=".95\linewidth"}
[.5]{} ![ACF plots from one chain of SRW and MH on NYU WC FB network.[]{data-label="fig:NYUacfAndTrace"}](Graphics/acfMHRWnyu.png "fig:"){width=".95\linewidth"}
Friendster Data
---------------
The Friendster dataset is hosted on the Stanford Large Network Dataset (SNAP) web site [@leskovec2016snap]. Friendster was an online social gaming and social networking site, where members had user profiles and could link to one another. Friendster also allowed users to form groups which other members could join. The SNAP-hosted Friendster dataset is the largest well-connected component of the induced subgraph of nodes that belonged to at least one group or were connected to other nodes that belonged to at least one group. This social network has 65,608,366 nodes (users) and 1,806,067,135 undirected edges (friendships). There are no other nodal attributes in this data. We estimated the mean degree and mean clustering coefficient.
### Implementation
We ran 100 chains of length 100,000 from random starting nodes. To find these random starting nodes we generated random numbers and searched if it existed in the network. If it existed, the sample began at this node, if not we generated another random number until it was accepted. During the sampling procedure we collected the visited node’s id, neighborhood, and calculated its degree. Running all 100 independent chains on five cores, took around 80 minutes for the SRW samples and 116 minutes for the MH samples. After completing the walks, we queried the file again to count the number of triangles for each visited node. Counting triangles is a computationally expensive step, so we only computed triangles on the chains up to length 10,000. Therefore, the multivariate results we present are on shorter chains of length 10,000, but we also present full 100,000 results on the univariate estimate of mean degree.
### Shorter chain results
Results are in Figure \[fig:friendstermean1e4\] and Tables \[tab:FriendsterMeans\] and \[tab:FriendsterESS1e4\]. The mean degree estimate from both the SRW and MH is around 55 with more variability in the MH samples and the mean clustering coefficient for both algorithms is around 0.16.
![Mean estimates from SRW and MH walks on the Friendster network for 10,000 length chains. Replications = 100.[]{data-label="fig:friendstermean1e4"}](Graphics/Mean1e4Friendster.png)
Type Degree Clustering coeff
------ --------------- ------------------
SRW 55.51 (0.414) 0.163 (0.002)
MH 54.97 (0.765) 0.159 (0.009)
: Mean estimates from the SRW and MH on Friendster network with chain length 10,000. Replications = 100 and standard errors in parentheses.[]{data-label="tab:FriendsterMeans"}
The striking difference between the SRW and MH is in the effective sample size and number of unique nodes captured. The MH walks on average collect only around 25% of the unique nodes that the SRW does. And in the multivariate ESS, the MH on average is less than 20% of the SRW. The mean acceptance rate in the MH walks was 0.2904. The minimum ESS for $p=2, \epsilon=0.05$, and $\alpha=0.05$ is 7530, where none of the simulations achieved the minimum ESS for reliable estimation by 10,000 steps. This implies more samples are needed. ACF plots for one chain are shown in Figure \[fig:Friendster1e4acfAndTrace\].
$T(\epsilon=0.05)$ ESS Unique Nodes
----- -------------------- ------------------- ---------------
SRW 0.058 (0.0004) 3865.95 (212.399) 9797 (2.096)
MH 0.0985 (0.0002) 462.918 (6.467) 2437 (27.023)
: Multivariate: $T_{SD}(\epsilon=0.05)$, effective sample size, and number of unique nodes sampled by 10,000 steps in Friendster network. Replications = 100 and standard errors in parentheses.[]{data-label="tab:FriendsterESS1e4"}
[.5]{} ![ACF plots from one 1e4 chain of SRW and MH on Friendster network.[]{data-label="fig:Friendster1e4acfAndTrace"}](Graphics/acfRWfriendster1e4.png "fig:"){width=".9\linewidth"}
[.5]{} ![ACF plots from one 1e4 chain of SRW and MH on Friendster network.[]{data-label="fig:Friendster1e4acfAndTrace"}](Graphics/acfMHRWfriendster1e4.png "fig:"){width=".9\linewidth"}
### Full chain results
If we consider estimating the mean degree of the 100,000 length chains, we see the mean degree estimates from the SRW and MH walks are again similar. Likewise, the ESS and number of unique nodes are on starkly different scales (Figure \[fig:friendstermeandegree1e5t\] and Table \[tab:FriendsterESS1e5\]). We use the result from Proposition \[prop:dESS\], with $p=1$, $g(x) = 1/x$ and the square root batch means estimation to calculate the univariate ESS. The mean acceptance rate of of the MH walks was 0.2905. ACF plots for one chain are shown in Figure \[fig:Friendster1e5acfAndTrace\].
![Mean estimates from SRW and MH walks on the Friendster network for 100,000 length chains. Replications = 100.[]{data-label="fig:friendstermeandegree1e5t"}](Graphics/1e5MeanDegreeFriendster.png)
Degree ESS Unique Nodes
----- --------------- ----------------- ----------------
SRW 55.15 (0.149) 36229 (1408.53) 97474 (14.124)
MH 55.07 (0.245) 6002 (53.507) 24477 (91.33)
: Univariate: mean degree, effective sample size, and number of unique nodes sample by 100,000 steps for $\epsilon=0.05$ for Friendster network. Replications = 100 and standard errors in parenthesis.[]{data-label="tab:FriendsterESS1e5"}
[.5]{} ![ACF plots from one 1e5 chain of SRW and MH on Friendster network.[]{data-label="fig:Friendster1e5acfAndTrace"}](Graphics/acfRWfriendster1e5.png "fig:"){width=".95\linewidth"}
[.5]{} ![ACF plots from one 1e5 chain of SRW and MH on Friendster network.[]{data-label="fig:Friendster1e5acfAndTrace"}](Graphics/acfMHRWfriendster1e5.png "fig:"){width=".95\linewidth"}
Summary of results
------------------
Consistently across all three networks, the SRW was more efficient than the MH, either with respect to the termination time to achieve the stopping criterion or with respect to the effective sample size. Our results confirm what other authors have found in univariate settings [@gjoka2011practical; @Avrachenkov2016]. In addition, as clearly indicated in the histograms, repeated runs of the algorithms obtained slightly different estimates. However, when the minimum effective sample size was reached, the variation in these estimates was small. This further emphasizes that prior to running the algorithms on any of these networks, a researcher can determine the simulation effort required via the minimum ESS. Once that minimum ESS has been reached, researchers will have an approximately $100(1-\alpha)$% confidence with precision $\epsilon$ for the $p$ many estimates (as shown in Table \[tab:MinESS\]).
$p$ Conf level $\epsilon$ Minimum ESS
----- ------------ ------------ -------------
5 95% 0.05 10363
4 95% 0.05 9992
2 95% 0.05 7530
: Minimum ESS required for $p$ estimated features at a $100(1-\alpha)$% confidence level and threshold level $\epsilon$.[]{data-label="tab:MinESS"}
Discussion
==========
The use of MCMC methods on networks without sampling frames to estimate multiple features is common. However, the error associated with the estimation in the multivariate setting has not been studied closely. We contribute to the literature by further developing multivariate MCMC output analysis methods in the context of network sampling that directly addresses the reliability of the multivariate estimation.
We support existing findings that the MH is less efficient than the SRW in univariate estimation and extend the results to a multivariate setting. We have also extended the MCMC output analysis framework more generally so that it can be applied to other MCMC algorithms. If a researcher plans to use an MCMC method to collect a sample, they can now find the minimum number of effective samples they should collect before they terminate the sampling procedure. Moreover, they have the tools to assess the reliability of the inference they make from that sample. By using such tools, researchers can have greater confidence in the consistency and reproducibility of their results. This reduces the chance of outlier results or non-reproducible estimates due to insufficient Monte Carlo sample sizes.
There are multiple extensions of this work that could benefit from further research. First, it would be interesting to extend this research to handle edge sampling algorithms to estimate network edge properties. In addition, we focused on binary networks, so generalizing the framework to work on weighted networks that convey relationship strength or weakness would be useful. Another extension is to develop these methods to work on directed networks. The most practically beneficial extension, though, may be to use these reliable estimation tools, such as minimum effective sample size, in the context of RDS. However, the assumptions required for the output analysis tools are not met in RDS, therefore further work is required to apply the methods we propose.
|
---
abstract: 'In this paper, we present an approach to exploit phrase tables generated by statistical machine translation in order to map French discourse connectives to discourse relations. Using this approach, we created , a lexicon of French discourse connectives and their PDTB relations. When evaluated against LEXCONN, achieves a recall of 0.81 and an Average Precision of 0.68 for the <span style="font-variant:small-caps;">Concession</span> and <span style="font-variant:small-caps;">Condition</span> relations.'
author:
- |
Majid Laali Leila Kosseim\
Department of Computer Science and Software Engineering\
Concordia University, Montreal, Quebec, Canada\
[{m\_laali, kosseim}@encs.concordia.ca]{}\
title: |
Automatic Mapping of French Discourse Connectives\
to PDTB Discourse Relations
---
Introduction
============
Discourse connectives (DCs) (e.g. *because*, *although*) are terms that explicitly signal discourse relations within a text. Building a lexicon of DCs, where each connective is mapped to the discourse relations it can signal, is not an easy task. To build such lexicons, it is necessary to have linguists manually analyse the usage of individual DCs through a corpus study, which is an expensive endeavour both in terms of time and expertise. For example, LEXCONN [@roze12], a manually built lexicon of French DCs, was initiated in 2010 and released its first edition in 2012. The latest version, LEXCONN V2.1 [@danlos15], contains 343 DCs mapped to an average of 1.3 discourse relations. This project is still ongoing as 37 DCs still have not been assigned to any discourse relation. Because of this, only a limited number of languages currently possess such lexicons (e.g. French [@roze12], Spanish [@alonsoalemany02], German [@stede98]).
In this paper, we propose an approach to automatically map French DCs to their associated PDTB discourse relations using parallel texts. Our approach can also automatically identify the usage of a DC where the DC signals a specific discourse relation. This can help linguists to study a DC in parallel texts and/or to find evidence for an association between discourse relations and DCs. Our approach is based on phrase tables generated by statistical machine translation and makes no assumption about the target language except the availability of a parallel corpus with another language for which a discourse parser exists; hence the approach is easy to expand to other languages. We applied our approach to the Europarl corpus [@koehn05] and generated [^1], a lexicon mapping French DCs to their associated Penn Discourse Treebank (PDTB) discourse relations [@prasad08]. To our knowledge, is the first lexicon of French discourse connectives mapped to the PDTB relation set. When compared to LEXCONN, achieves a recall of 0.81 and an Average Precision of 0.68 for the <span style="font-variant:small-caps;">Concession</span> and <span style="font-variant:small-caps;">Condition</span> discourse relations.
Related Work {#sec:related-work}
============
Lexicons of DCs have been developed for several languages: English [@knott96], Spanish [@alonsoalemany02], German [@stede98], Czech [@polakova13], and French [@roze12]. However, constructing such lexicons requires linguistic expertise and is a time-consuming task. Discourse connectives and their translations have been studied within parallel texts by many [@meyer11; @meyer11-a; @taboada12; @cartoni13; @zufferey14; @zufferey14-a; @zufferey15; @hoek15]. These works have either focused on the effect of the translation of discourse connectives on machine translation systems [@meyer11; @meyer11-a; @cartoni13] or on a small number of discourse connectives due to the cost of manual annotations [@taboada12; @zufferey14; @zufferey14-a; @zufferey15; @hoek15].
To our knowledge, very little research has addressed the automatic construction of lexicons of DCs. @hidey16 proposed an automatic approach to identify English expressions that signal the <span style="font-variant:small-caps;">Causal</span> discourse relation. On the other hand, @laali14 automatically extracted French DCs from parallel texts; however, they did not associate discourse relations to the extracted DCs. The proposed approach goes beyond this work by mapping DCs to their associated discourse relations.
Methodology {#sec:method}
===========
Corpus Preparation
------------------
For our experiments, we used the English-French part of Europarl [@koehn05] which contains 2 million[^2] parallel sentences. To prepare the dataset, we parsed the English sentences with the CLaC discourse parser [@laali16] to identify English DCs and the discourse relation that they signal. The CLaC parser has been learned on Section 02-20 of the PDTB and can disambiguate the usage of the 100 English DCs listed in the PDTB with an F1-score of 0.90 and label them with their PDTB discourse relation with an F1-score of 0.76 when tested on the blind test set of the CoNLL 2016 shared task [@xue16]. This parser was used because its performance is very close to that of the state of the art [@oepen16] (i.e. 0.91 and 0.77 respectively), but is more efficient at running time than @oepen16.
Note that since the CoNLL 2016 blind test set was extracted from Wikipedia and its domain and genre differ significantly from the PDTB, the 0.90 and 0.76 F1-scores of the CLaC parser can be considered as an estimation of its performance on texts with a different domain/genre such as Europarl.
Mapping Discourse Relations {#sec:build-dictionaries}
---------------------------
To label French DCs with a PDTB discourse relation, we assumed that if a French DC is aligned to an English DC tagged with a discourse relation *Rel*, then it should signal the same discourse relation *Rel*. For our experiment, we used the inventory of 100 English DCs from the PDTB [@prasad08] and the 371 French DCs from LEXCONN V2.1 [@danlos15]. For the mapping, we used the subset of 14 PDTB discourse relations that was used in the CoNLL shared task [@xue15]. This list is based on the second-level types and a selected number of third-level subtypes of the PDTB discourse relations.
To have statistically reliable results, we ignored French DCs that appeared less than 50 times in Europarl. Out of the 371 French DCs listed in LEXCONN, seven do not appear in Europarl and 55 have a frequency lower than 50. This means that 89% (309/371) of the French DCs have a frequency higher than 50 and were thus used in the analysis. A manual inspection of the infrequent DCs shows that they are either informal (e.g. *des fois que*) or rare expression (e.g. *en dépit que*). Table \[tbl:fr-dc-freq\] shows the distribution of the LEXCONN French DCs in Europarl.
**Freq.** $\mathbf{= 0}$ $\mathbf{\le 50}$ $\mathbf{> 50}$ **Total**
----------- ---------------- ------------------- ----------------- -----------
\# FR-DC 7 55 309 371
: Distribution of LEXCONN French DCs in the Europarl corpus.[]{data-label="tbl:fr-dc-freq"}
We used the Moses statistical machine translation system [@koehn07] to extract the number of alignments between French DCs and English DCs. As part of its translation model, Moses generates a phrase table (see Table \[tbl:phrase-table\]) which aligns phrases between the language pairs. The phrase table is constructed based on statistical word alignment models and contains the frequency of the alignments between phrase pairs. We used the @och03 heuristic and combined IBM Model 4 word alignments [@brown93] to construct the phrase table.
Because an English DC can signal different discourse relations, to ensure that Moses’s phrase table distinguishes the different usages of the same English DC, we modified its English tokenizer so that each English DC and its discourse relation make up a single token. For example, the token ‘*although*<span style="font-variant:small-caps;">-Concession</span>’ will be created for the DC *although* when it signals the discourse relation <span style="font-variant:small-caps;">Concession</span>. Table \[tbl:phrase-table\] shows a few entries of the phrase table for the French DC *même si*. As the table shows, *même si* was aligned to three English DCs: *although*, labeled by the CLaC parser as a <span style="font-variant:small-caps;">Contrast</span> or as a <span style="font-variant:small-caps;">Concession</span> and to *even if* and *even though* which were not tagged .
In total, 1,970 entries of the phrase table contained a French DC, an English DC and a discourse relation[^3]. From these, we computed the number of times a French DC was aligned to each discourse relation, then, created : tuples of <*FR-DC, Rel, Prob*>, where *FR-DC* and *Rel* indicate a French DC and a discourse relation and *Prob* indicates the probability that *FR-DC* signals *Rel*. To calculate *Prob*, we divided the number of times *FR-DC* is associated to *Rel* by the frequency of *FR-DC* in Europarl. In total, the approach generated a lexicon of 900 such tuples, a few of which are shown in Table \[tab:tuples\].
Evaluation {#sec:evaluation}
==========
To evaluate , because LEXCONN uses a different inventory of discourse relations than the PDTB, we only considered the discourse relations that are common across these inventories: <span style="font-variant:small-caps;">Concession</span> and <span style="font-variant:small-caps;">Condition</span>. According to LEXCONN, 61 French DCs can signal a <span style="font-variant:small-caps;">Concession</span> or a <span style="font-variant:small-caps;">Condition</span> discourse relation. Out of these, 44 have a frequency higher than 50 in Europarl.
Automatic Evaluation
--------------------
To measure the quality of , we ranked the <*FR-DC, Rel, Prob*> tuples based on their probability and measured the quality of the ranked list using 11-point interpolated average precision [@manning08]. This curve shows the highest precision at the 11 recall levels of 0.0, 0.1, 0.2, ..., 1.0. This method allows us to evaluate the ranked list without considering any arbitrary cut-off point. As Figure \[fig:curve-map-by-definition\] shows, the approach retrieved 50% of the French DCs in LEXCONN with a precision of 0.81.
table \[x=R, y=P, col sep=comma\] [by-def-en.csv]{};
In addition, we also computed Average Precision (AveP) [@manning08]; the average of the precision obtained after seeing a correct LEXCONN entry in . More specifically, given a list of ranked tuples: $$AveP=\frac{1}{N}\sum_{i=1}^{N}Precision(DC_i)$$ where $N$ is the number of LEXCONN French DCs that signals the <span style="font-variant:small-caps;">Concession</span> or <span style="font-variant:small-caps;">Condition</span> discourse relations (i.e. 44), $DC_i$ is the rank of the $i^{th}$ LEXCONN DC in , and $Precision(DC_i)$ is the precision at the rank $DC_i$ of the ranked tuples. It can be shown that $AveP$ approximates the area under the interpolated precision-recall curve [@manning08]. The proposed approach identified 36 (81%) of these 44 French DCs with an $AveP$ of 0.68.
Manual Evaluation
-----------------
In addition to the quantitative evaluation, we also performed a manual analysis of the false-positive errors to see if they really constituted errors. To do so, we looked at the tuples with a probability higher than 0.01 but which did not appear in LEXCONN. 14 such cases, shown in Table \[tab:true-positive\], were found.
For example, while the French connective *à défaut de* (\#1 in Table \[tab:true-positive\]) signals a <span style="font-variant:small-caps;">Condition</span> discourse relation in Sentence \[ex:condition\] below, only the <span style="font-variant:small-caps;">Explanation</span> and the <span style="font-variant:small-caps;">Concession</span> discourse relations were associated with this connective in LEXCONN.
1. \[ex:condition\] **FR:** se montrer très ambitieux, notre industrie, nos chercheurs et nos experts ne disposeront purement et simplement pas du brevet moderne dont ils ont besoin.\
**EN:** we are anything less than ambitious in this field, we shall simply not provide our industry, our research and development experts with the modern patent which they need.
To evaluate if these 14 cases were true mistakes, we randomly selected five English-French parallel sentences from Europarl that contained the French DC and one of its English DC translations signalling the discourse relation. Then, we showed the French DCs within their sentence to two native French speakers and asked them to confirm if the discourse relation identified was indeed signaled by the French DCs or not. The Kappa agreement between the two annotators was 0.72. For 9 French connectives, both annotators agreed that indicated that in at least one of the five sentences, the discourse relation was signalled by the connective. This indicates that 64% (9/14) are in fact true-positives, i.e. correct mappings that are not listed in LEXCONN. Table \[tab:true-positive\] shows the 14 pairs of <FR-DC/English translation, Discourse relation> used in the manual evaluation and indicates the newly discovered mappings by .
We also observed that if multiple explicit connectives occur in the same clause (e.g. *certes* and *mais*), one of them can affect the discourse relation signaled by the other. This is an interesting phenomenon as it seems to indicate that the connectives are not independent. For example, in Sentence \[ex:dc-dependent\], the combination of *certes* and *mais* signals a <span style="font-variant:small-caps;">Concession</span> discourse relation.
1. \[ex:dc-dependent\] **FR:** Cela coûte un peu plus cher, est sans conséquence pour l’environnement.\
**EN:** it is a little more expensive, it does not harm the environment.
Note that according to LEXCONN, neither *certes* nor *mais* can signal a <span style="font-variant:small-caps;">Concession</span> discourse relation. The same phenomenon was also reported in the PDTB corpus [@prasad08-a p. 5].
Conclusion and Future Work {#sec:conclusion}
==========================
In this paper, we proposed a novel approach to automatically map PDTB discourse relations to French DCs. Using this approach, we generated : a lexicon of French DCs and their PDTB discourse relations. When compared with LEXCONN, our approach achieved a recall of 0.81 and an Average Precision of 0.68 for the <span style="font-variant:small-caps;">Concession</span> and <span style="font-variant:small-caps;">Condition</span> discourse relations. A manual error analysis of the false-positives showed that the approach identified new discourse relations for 9 French DCs which are not included in LEXCONN. As future work, we plan to evaluate all the discourse relations in and apply the approach to other languages.
### Acknowledgement {#acknowledgement .unnumbered}
The authors would like to thank the anonymous referees for their insightful comments on an earlier version of the paper. Many thanks also to Andre Cianflone for his help on the evaluation of this work. This work was financially supported by an NSERC grant.
[^1]: is publicly available at <https://github.com/mjlaali/ConcoLeDisCo>.
[^2]: 2,007,723 to be exact.
[^3]: We only considered entries whose texts are an exact match of an English DC listed in the PDTB and a French DC listed in LEXCONN.
|
---
abstract: 'It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for $K_t$-free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.'
author:
- Marthe Bonamy
- Oscar Defrain
- Marc Heinrich
- |
\
Michał Pilipczuk
- 'Jean-Florent Raymond'
date: March 2019
title: |
Enumerating minimal dominating sets\
in $K_t$-free graphs and variants[^1]
---
Introduction {#sec:intro}
============
Countless algorithmic problems in graph theory require to detect a structure with prescribed properties in an input graph. Rather than finding one such object, it is sometimes more desirable to generate all of them. This is for instance useful in certain applications to database search [@yan2005substructure], network analysis [@grochow2007network], bioinformatics [@Marino2015; @10.1007/978-3-540-28639-4_1], and cheminformatics [@barnard1993substructure]. Enumeration algorithms for graph problems seem to have been first mentioned in the early 70’s with the pioneer works of Tiernen [@tiernan1970efficient] and Tarjan [@tarjan1973enumeration] on cycles in directed graphs and of Akkoyunlu [@akkoyunlu1973enumeration]. However, they already appeared in disguise in earlier works [@5222697; @5392200]. To this date, several intriguing questions on the topic remain unsolved. We refer the reader to [@Marino2015enum] for a more in-depth introduction to enumeration algorithms and to [@wasa2016enumeration] for a listing of enumeration algorithms and problems.
The objects we wish to enumerate in this paper are the (inclusion-wise) minimal dominating sets of a given graph. In general, the number of these objects may grow exponentially with the order $n$ of the input graph. Therefore, in stark contrast to decision or optimization problems, looking for a running time polynomially bounded by $n$ is not a reasonable, let alone meaningful, efficiency criterion. Rather, we aim here for so-called *output-polynomial* algorithms, whose running time is polynomially bounded by the size of both the input and output data.
Because dominating sets are among the most studied objects in graph theory and algorithms, their enumeration (and counting) have attracted an increasing attention over the past 10 years. The problem of enumerating minimal dominating sets (hereafter referred to as ) has a notable feature: it is equivalent to the extensively studied hypergraph problem [<span style="font-variant:small-caps;">Trans-Enum</span>]{}. In [<span style="font-variant:small-caps;">Trans-Enum</span>]{}, one is given a hypergraph $\mathcal{H}$ ( a collection of sets, called *hyperedges*) and is asked to enumerate all the minimal *transversals* of $\mathcal{H}$ ( the inclusion-minimal sets of elements that meet every hyperedge). It is not hard to see that is a particular case of [<span style="font-variant:small-caps;">Trans-Enum</span>]{}: the minimal dominating sets of a graph $G$ are exactly the minimal transversals of the hypergraph of closed neighborhoods of $G$. Conversely, Kanté, Limouzy, Mary, and Nourine proved that every instance of [<span style="font-variant:small-caps;">Trans-Enum</span>]{} can be reduced to a co-bipartite[^2] instance of [@kante2014enumeration]. Currently, the best output-sensitive algorithm for [<span style="font-variant:small-caps;">Trans-Enum</span>]{} is due to Fredman and Khachiyan and runs in quasi-polynomial time [@fredman1996complexity]. It is a long-standing open problem whether this complexity bound can be improved (see for instance the surveys [@eiter2002hypergraph; @eiter2008computational]). Therefore, the equivalence between the two problems is an additional motivation to study , with the hope that techniques from graph theory will be used to obtain new results on the [<span style="font-variant:small-caps;">Trans-Enum</span>]{} problem. So far, output-polynomial algorithms have been obtained for in several classes of graphs, including planar graphs and degenerate graphs [@eiter2003new], classes of graphs of bounded tree-width, clique-width [@courcelle2009linear], or LMIM-width [@Golovach2018], path graphs and line graphs [@kante2012neighbourhood], interval graphs and permutation graphs [@kante2013enumeration], split graphs [@kante2015polynomial], graphs of girth at least 7 [@Golovach2015], chordal graphs [@kante2015polynomial], and chordal bipartite graphs [@GOLOVACH201630]. A succinct survey of results on can be found in [@Kante2008].
In this paper, we investigate the complexity of in graph classes defined by forbidding an induced subgraph $H$, hereafter referred to as *$H$-free* graphs. For every $t\in {\mathbb{N}}$, we denote by $K_t$ the complete graph on $t$ vertices, by $K_t-e$ the graph obtained by removing any edge in $K_t$ and by $K_t+e$ the disjoint union of $K_t$ and $K_2$. Our main result is the following.
\[thm:op\] There is an algorithm enumerating, for every $t \in {\mathbb{N}}$, the minimal dominating sets in $(K_t+e)$-free graphs in output-polynomial time and polynomial space.
In particular, this yields an output-polynomial time algorithm for $K_t$-free graphs. A notable special case is that of bipartite graphs, where the question of the existence of an output-polynomial time algorithm for was explicitly stated in [@kante2015polynomial] and later papers [@Kante2008; @GOLOVACH201630]. We stress that we provide in the proof of Theorem \[thm:op\] a single algorithm that deals with all values of $t$ and that this algorithm does not require the knowledge of $t$. We discuss the complexity in greater details in Sections \[sec:triangle-free\] and \[sec:kt-free\].
In order to push our techniques to their limits, we investigate cases that are close to but not covered by Theorem \[thm:op\]. Namely, we consider two particular choices of the graph $H$: the *paw*, which is the graph obtained by adding a vertex of degree one to $K_3$ ( $H = \tikz[every node/.style = black node, scale = 0.2, baseline=-0.1cm]{
\draw (0:1) node (a) {} -- (120:1) node {} -- (-120:1) node {} -- (a) -- ++ (1.4759, 0) node {};
}$), and $K_4-e$, also known as *diamond* graph ( $H = \tikz[every node/.style = black node, scale = 0.2, baseline=-0.1cm]{
\draw (0:1) node (a) {} -- (120:1) node (b) {} -- (-120:1) node (c) {} -- (a) ++ (-2.95189, 0) node (d) {} (b) -- (d) -- (c);
}$). We combine our main tools to some ad hoc tricks so as to handle those two cases, and obtain the following.
\[th:pawdiam\] There is an algorithm enumerating minimal dominating sets in paw-free (resp. diamond-free) graphs in output-polynomial time and polynomial space.
Our algorithms first decompose the input graph by successively removing closed neighborhoods in the fashion of [@eiter2003new]. We then follow this decomposition to construct partial minimal dominating sets, adding the neighborhoods back one after the other. A crucial point of our approach is that we can relate the enumeration of potential extensions of a partial minimal dominating set to the problem in a simpler class.
The paper is organized as follows. In Section \[sec:prelim\] we give the necessary definitions. The graph decompositions that we use, called *peelings*, are introduced in Section \[sec:bt\] along with their main properties. In Section \[sec:triangle-free\], we give an algorithm for that runs in output-polynomial time in triangle-free graphs with better time bound than that coming from Theorem \[thm:op\]. A generalization of this algorithm for $K_t$-free graphs is given in Section \[sec:kt-free\] (Theorem \[thm:kt-free\]). This algorithm is then extended to $(K_t+e)$-free graphs in Section \[sec:variants\] (Theorem \[thm:ktme\]). In the same section, algorithms are given for diamond-free graphs (Theorem \[thm:diamond-free\]) and paw-free graphs (Theorem \[thm:paw-free\]), , the two cases of Theorem \[th:pawdiam\]. We discuss in Section \[sec:beyond\] the obstacles to stronger theorems using the same tools. Finally, we conclude with possible future research directions in Section \[sec:concl\].
Preliminaries {#sec:prelim}
=============
#### Graphs.
All graphs in this paper are finite, undirected, simple, and loopless. If $G$ is a graph, then $V(G)$ is its set of vertices and $E(G)\subseteq V(G)^2 $ is its set of edges. Edges are denoted by $xy$ (or $yx$) instead of $\{x,y\}$. We assume that vertices are assigned distinct indices; these will be used to choose vertices in a deterministic way, typically selecting the vertex of smallest index. A [*clique*]{} (respectively an *independent set*) in a graph $G$ is a set of pairwise adjacent (respectively non-adjacent) vertices. The subgraph of $G$ *induced* by $X\subseteq V(G)$, denoted by $G[X]$, is the graph $(X,E(G)\cap (X\times X))$; $G\setminus X$ is the graph $G[V(G)\setminus X]$. For every graph $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$. If a vertex $v \in V(G)$ is adjacent to every vertex of a set $S \subseteq V(G)$, we say that $v$ is *complete* to $S$.
If the vertex set of a graph $G$ can be partitioned into one part inducing a clique and one part inducing an independent set (respectively two independent sets, two cliques), we say that $G$ is a *split* (respectively *bipartite*, *co-bipartite*) graph. If $f$ is a function, we write $f(n) = \operatorname{poly}n$ when there is a constant $c\in \mathbb{N}$ such that $f(n) = O(n^c)$.
#### Neighbors and domination.
Let $G$ be a graph and $x \in V(G)$. We note $N(x)$ the set of [*neighbors*]{} of $x$ in $G$ defined by $N(x)=\{y\in V(G)\mid xy\in E(G)\}$; $N[x]$ is the set of [*closed neighbors*]{} defined by $N[x]= N(x)\cup\{x\}$. For a given $X\subseteq V(G)$, we respectively denote by $N[X]$ and $N(X)$ the sets defined by $\bigcup_{x\in X} N[x]$ and $N[X]\setminus X$. Let $D$ be a set of vertices of $G$. We say that $D$ is *dominating* a subset $S \subseteq V(G)$ if $S \subseteq N[D]$. It is *minimally dominating* $S$ if no proper subset of $D$ dominates $S$. The set $D$ is a (*minimal*) *dominating set* of $G$ if it (minimally) dominates $V(G)$. The set of all minimal dominating sets of $G$ is denoted by $\D(G)$ and the problem of enumerating $\D(G)$ given $G$ is denoted by . Let $S \subseteq V(G)$. A vertex $y \in V(G)$ is said to be a *private neighbor* of some $x \in S$ if $y\not\in N[S\setminus \{x\}]$. Intuitively, this means that $y$ is not dominated by any other vertex of $S$. Note that $x$ can be its own private neighbor. The set of private neighbors of $x\in S$ in $G$ is denoted by $\operatorname{Priv}_G(S,x)$ and we drop the subscript when it can be inferred from the context. Observe that $S$ is a minimal dominating set of $G$ if and only if $V(G) \subseteq N[S]$ and for every $x\in S$, $\operatorname{Priv}(S,x)\neq \emptyset$.
#### Enumeration.
The aim of graph enumeration algorithms is to generate a set of objects $\mathcal{X}(G)$ related to a graph $G$. We say that an algorithm enumerating $\mathcal{X}(G)$ with input an $n$-vertex graph $G$ is *output-polynomial* if its running time is polynomially bounded by the size of the input and output data, , $n + |\mathcal{X}(G)|$. If an algorithm enumerates $\mathcal{X}(G)$ by spending $\operatorname{poly}(n)$-time (respectively $O(n)$-time) before it outputs the first element, between two output elements, and after it outputs the last element, then we say that it runs with *polynomial delay* (respectively *linear delay*). It is easy to see that every polynomial delay algorithm is also output-polynomial. Note however that some problems have output-polynomial algorithms but no polynomial delay ones, unless P=NP [@strozecki2010enumeration]. When discussing the space used by an enumeration algorithm, we ignore the space where the solutions are output. If the existence of an output-polynomial algorithm for a problem implies the existence of one for , we say that this problem is -hard. As mentioned in the introduction, we have the following.
\[thm:cobip-hard\] restricted to co-bipartite graphs is -hard.
Ordered generation in bicolored graphs {#sec:bt}
======================================
In this section, we give a general procedure that will be used in the rest of this paper for enumerating minimal dominating sets in graphs and variants. The algorithm constructs minimal dominating sets one neighborhood at a time, in a variant of what is known as the [*backtrack search technique*]{} in [@read1975bounds; @fukuda1997analysis; @mary2017efficient], and referred to as [*ordered generation*]{} in [@eiter2003new].
In what follows, we find it more convenient to deal with the slightly more general setting of domination in bicolored graphs. A *bicolored graph* is a graph together with a subset of its vertex set. For a graph $G$ and a subset $A \subseteq V(G)$, we denote by $G(A)$ the bicolored graph $G$ with prescribed set $A$. We also say that $G$ has *bicoloring* $(A,V(G)\setminus A)$. Then, a *dominating set of $G(A)$* is a set $D \subseteq V(G)$ such that $A \subseteq N[D]$. It is called *minimal* if it does not contain any dominating set of $G(A)$ as a proper subset. Intuitively, the vertices of $G-A$ may be used in the dominating set, but do not need to be dominated. For every graph $G$ and subset $A\subseteq V(G)$, we denote by $\D(G, A)$ the set of minimal dominating sets of $G(A)$.
A *peeling* of a bicolored graph $G(A)$ is a sequence of vertex sets $(V_0, \dots, V_{p+1})$ such that $V_{p+1} = V(G)$, $V_p = A$, $V_0 = \emptyset$, and for every $i\in {\left \{ 1, \dots, p \right \}}$, there is a vertex $v_i \in V_i$ such that $$V_{i-1} = V_i \setminus N[v_i].$$ We call $(v_1, \dots, v_p)$ the *vertex sequence* of the peeling; note that the value of $p$ is only known after peeling the whole graph.
In the remaining of this section, we consider a bicolored graph $G$ with prescribed set $A\subseteq V(G)$, together with a fixed peeling $(V_0, \dots, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1,\dots,v_p)$. Observe that $\D(G,V_p)=\D(G,A)$ and $\D(G,V_{p+1})=\D(G)$. We now define the relation that will be used by our algorithm to enumerate minimal dominating sets without repetition. Recall that the sets of $\D(G, V_i)$ may contain vertices of $G-V_i$, which is a crucial point.
\[def:parent\] Let $i\in {\left \{ 0, \dots, p-1 \right \}}$ and $D\in \D(G, V_{i+1})$. We denote by $\operatorname{\sf Parent}(D,i+1)$ the pair $(D^*,i)$ where $D^*$ is obtained from $D$ by successively removing the vertex $x$ of smaller index in $D$ satisfying $\operatorname{Priv}(D,x)\cap V_i=\emptyset$, until no such vertex exists.
Clearly, there is a unique way to build $\operatorname{\sf Parent}(D,i+1)$ given $D$ and $i$. By construction, the obtained set $D^*$ is a minimal dominating set of $G(V_i)$. Hence every set in $\D(G,V_{i+1})$ can be obtained by completing some $D^*$ in $\D(G,V_i)$; we develop this point below.
\[prop:parent-vi\] Let $i\in {\left \{ 0, \dots, p-1 \right \}}$ and $D^*\in \D(G,V_i)$;
- if $D^*$ dominates $V_{i+1}$ then $D^* \in \D(G,V_{i+1})$ and $\operatorname{\sf Parent}(D^*,i+1)=(D^*,i)$;
- otherwise, $D^*\cup \{v_{i+1}\} \in \D(G,V_{i+1})$ and $\operatorname{\sf Parent}(D^*\cup \{v_{i+1}\},{i+1})=(D^*,i)$.
First note that since $D^*\in \D(G,V_i)$, $\operatorname{Priv}(D^*,x)\cap V_i\neq \emptyset$ for all $x\in D^*$. Hence $\operatorname{\sf Parent}(D^*,i+1)=(D^*,i)$ whenever $D^*$ dominates $V_{i+1}$. If $D^*$ does not dominate $V_{i+1}$ then $D=D^*\cup \{v_{i+1}\}$ does. Moreover, $\operatorname{Priv}(D,v_{i+1})\cap V_{i+1}\neq\emptyset$. Since $v_{i+1}$ is not adjacent to any vertex in $V_i$, it cannot steal any private neighbors from the elements of $D^*$. Hence $\operatorname{Priv}(D,x)\cap V_{i+1}\neq \emptyset$ for all $x\in D$. Now, remark that since $v_{i+1}$ does not steal private neighbors to the elements of $D^*$, it is indeed itself the only node with no privates in $V_i$ and is removed by the parent function. Hence $\operatorname{\sf Parent}(D^*\cup\{v_{i+1}\},i+1)=(D^*,i)$.
The $\operatorname{\sf Parent}$ relation as introduced in Definition \[def:parent\] defines a tree on vertex set $$\{(D,i) \mid i \in {\left \{ 1, \dots, p \right \}},\ D \in \D(G,V_i)\},$$ with leaves $\{(D,p) \mid D \in \D(G, A)\}$, and root $(\emptyset,0)$ (the empty set being the only dominating set of the empty vertex set $V_0$). Our algorithm will search this tree in order to enumerate every minimal dominating set of $G$. Proposition \[prop:parent-vi\] guarantees that for every $i<p$ and every $D^* \in \D(G,V_i)$, the pair $(D^*,i)$ is the parent of some $(D,i+1)$ with $D \in \D(G,V_{i+1})$ (possibly $D = D^*$). Consequently, every branch of the tree leads to a different minimal dominating set of $G$. In particular, for every $i \in {\left \{ 0, \dots, p-1 \right \}}$, we have $$\label{eq:depthibound}
|\D(G,V_i)| \leq |\D(G,V_{i+1})| \leq |\D(G)|.$$ Given a set $D^* \in \D(G, V_i)$, we now focus on the enumeration of every $D \in \D(G,V_{i+1})$ such that $(D, i+1)$ has $(D^*, i)$ for parent. We call [*candidate extension*]{} of $(D^*, i)$ any (inclusion-wise) minimal set $X\subseteq V(G)$ such that $V_{i+1}\subseteq N[D^*\cup X]$. In other words, $X$ is a candidate extension of $(D^*,i)$ if and only if it is a minimal dominating set of the bicolored graph $G$ with prescribed set $V_{i+1}\setminus N[D^*]$. Then, we denote by $\C(D^*,i)$ the set of all candidate extensions of $(D^*,i)$, , $$\C(D^*,i){\overset{\text{\tiny{def}}}{=}}\D(G,V_{i+1}\setminus N[D^*]).$$ From Proposition \[prop:parent-vi\], we know that one of $(D^*, i + 1)$ and $(D^*\cup\{v_{i+1}\}, i + 1)$ has $(D^*, i)$ for parent. Note that we have no guarantee that any other candidate extension forms a minimal dominating set of $V_{i+1}$, together with $D^*$. We show that it is still reasonable to test each of the candidate extensions even though $D^*$ might have a unique child.
\[lem:cand-ext-bound\] Let $H(B)$ be a bicolored graph and $D \subseteq V(H)$. Then $$|\D(H, B \setminus N[D])| \leq \D(H,B).$$
We argue that for every $X \in \D(H, B \setminus N[D])$ there is an element $D'$ of $\D(H, B)$ that contains $X$. For this, we consider a minimal dominating set $D'$ of $H[B]$ that is subset of $D \cup X$. Such a set exists as $D \cup X$ dominates $G$. By definition, every vertex of $X$ has a private neighbor in $B \setminus N[D]$ so we have $X \subseteq D'$. This implies the desired inequality.
As a consequence of Lemma \[lem:cand-ext-bound\] and Inequality , we have the following.
\[cor:cibound\] Let $i\in {\left \{ 0, \dots, p-1 \right \}}$ and $D^*\in \D(G,V_i)$. Then $|\C(D^*,i)|\leq|\D(G, A)|$.
We conclude the ordered generation procedure with the following theorem that reduces the existence of an output-polynomial algorithm enumerating $\D(G,A)$, to the existence of one enumerating $\C(D^*,i)$ for any $i\in {\left \{ 0, \dots, p-1 \right \}}$ and $D^*\in \D(G,V_i)$.
\[thm:ordered-generation\] Let $f,s\colon {\mathbb{N}}\to {\mathbb{N}}$ be two functions and let $c \geq 1$. Assume that there is an algorithm that, given a bicolored graph $G$ on $n$ vertices with prescribed set $A\subseteq V(G)$, a peeling $(V_0,\dots, V_{p+1})$ of $G(A)$, $i\in {\left \{ 0, \dots, p-1 \right \}}$, and $D^*\in \D(G,V_i)$, enumerates the candidate extensions of $(D^*, i)$ in time at most $f(n)\cdot |\D(G,A)|^c$ and space at most $s(n)$. Then there is an algorithm that, given a bicolored graph $G(A)$ on $n$ vertices, enumerates the set $\D(G,A)$ in time $$O(n^4\cdot |\D(G,A)|^2 + n\cdot f(n)\cdot |\D(G,A)|^{c+1})$$ and space $O(n \cdot s(n))$.
Let us assume that there exists an algorithm [`B`]{} that, given a bicolored graph $G(A)$, a peeling $(V_0,\dots, V_{p+1})$ of $G(A)$, $i\in {\left \{ 0, \dots, p \right \}}$ and $D^*\in \D(G,V_i)$, enumerates $\C(D^*,i)$ in time at most $f(n)\cdot |\D(G,A)|^c$ and space at most $s(n)$. We describe an algorithm [`A`]{} that enumerates $\D(G,A)$ in the specified time.
The algorithm first checks if $A = \emptyset$ and, if so, returns $\{\emptyset\}$. Otherwise, we compute a peeling $(V_0,\dots,V_{p+1})$ of $G(A)$ in time $O(n^2)$ and using $O(n)$ space. Recall that the $\operatorname{\sf Parent}$ relation defines a tree $T$ on vertex set $$\{(D,i) \mid i \in {\left \{ 0, \dots, p \right \}},\ D \in \D(G,V_i)\},$$ with leaves $\{(D,p) \mid D \in \D(G, A)\}$ and root $(\emptyset,0)$. Therefore, in order to enumerate $\D(G,A)$, it is enough for [`A`]{} to enumerate the leaves of $T$. To do so, the algorithm performs a depth-first search (DFS) of $T$ outputting each visited leaf. For each node $(D^*,i)$, $i\in {\left \{ 0, \dots, p-1 \right \}}$ of $T$, the algorithm runs [`B`]{} on input $(G(A), (V_0,\dots,V_{p+1}), i, D^*)$ to generate $\C(D^*,i)$ in time $f(n)\cdot |\D(G,A)|^c$ and space $s(n)$. For every $X\in \C(D^*,i)$ generated by [`B`]{}, the algorithm tests whether $D^*\cup X$ is a minimal dominating set of $V_{i+1}$, and whether it has $(D^*,i)$ for parent. This requires $O(n^3)$ steps per candidate extension, and a total space of $O(n)$. As by Corollary \[cor:cibound\], $|\C(D^*,i)|\leq |\D(G,A)|$, the total time spent by [`A`]{} at each node of $T$ is bounded by $O(n^3\cdot |\D(G,A)| + f(n)\cdot |\D(G,A)|^c)$. As by Inequality , $|V(T)|\leq p\cdot |\D(G,A)|$, and $p\leq n$, the total running time of [`A`]{} is bounded by $$O(n^4\cdot |\D(G,A)|^2 + n\cdot f(n)\cdot |\D(G,A)|^{c+1}).$$
Regarding the space, we observe that whenever we visit a node of $T$, we do not need to compute the whole set of its children. Instead, it is enough in order to continue the DFS to compute the next unvisited child only, which can be done using [`B`]{} and pausing it afterward. Therefore, when we visit some $(D, i) \in V(T)$, we only need to store the data of the $i-1$ (paused) executions of [`B`]{} enumerating the children of the ancestors of $(D, i)$, plus the data of the algorithm enumerating the children of $D$, , $i\cdot (O(n)+s(n))$ space. As $s(n) = \Omega(n)$ (because [`B`]{} already needs $\Omega(n)$ space for storing its input), the described algorithm uses $O(n\cdot s(n))$ space, as claimed.
Candidate extensions in triangle-free graphs {#sec:triangle-free}
============================================
We show that candidate extensions can be enumerated in output-polynomial time in triangle-free graphs, which by Theorem \[thm:ordered-generation\] leads to an output-polynomial algorithm enumerating minimal dominating sets in this class of graphs. In fact, our result holds in the more general context where only the graph induced by the color to dominate is required to be triangle-free, not necessarily the whole graph.
In the following, we consider a bicolored graph $G$ on $n$ vertices, with prescribed set $A\subseteq V(G)$ such that $G[A]$ is triangle-free, together with a fixed peeling $(V_0,\dots, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1,\dots,v_p)$. Then we consider $$\begin{aligned}
& i \in {\left \{ 0, \dots, p-1 \right \}}
& D^* \in \D(G,V_i)\end{aligned}$$ and define $\C(D^*,i)$ as in Section \[sec:bt\]. We will show how to enumerate $\C(D^*,i)$ in output-polynomial time based on the observation that $N(v_{i+1})$ is an independent set.
In the following and for any two subsets $D$ and $X$, we denote by $D_X$ the intersection of $D$ with $X$, , $D_X=D\cap X$. We need some properties on minimal domination in split graphs. We say that a family of set $\mathcal{S}$ is an independence system if $\emptyset\in \mathcal{S}$ and for all $\emptyset\neq S\in \mathcal{S}$ and $x\in S$, $S\setminus\{x\}\in \mathcal{S}$.
\[prop:split-properties\] Let $H$ be a split graph with independent set $S$ and clique $C$, where $S$ is taken to be maximal. Let $\D_C(H)$ be the set defined by $\D_C(H)=\{D_C \mid D\in \D(H)\}$. Then for all $D\in \D(H)$ we have $S\subseteq N[D]$ and $D_S= S\setminus N(D_C)$, i.e., every minimal dominating set of $H$ is characterized by its intersection with the clique. Also,
1. $\D_C(H)=\{B \subseteq C \mid \forall x \in B,\ \operatorname{Priv}(B,x)\cap S\neq\emptyset\}$, and\[it:split1\]
2. $\D_C(H)$ and $\D(H)$ are in bijection.
Furthermore, $\D_C(H)$ is an independence system that can be enumerated with delay $O(n^2)$ and using $O(n^2)$ space.
\[lem:triangle-free-candidates-characterization\] Let $S=V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$ and let us assume that $S$ is an independent set. Let $C=N(S)\setminus \{v_{i+1}\}$ and let $H$ be the split graph induced by $S$ and $C$, where $C$ has been completed into a clique;
- if $V_{i+1} \subseteq N[D^*]$ then $\C(D^*,i) = \{\emptyset\}$;
- otherwise, if $S=\emptyset$, then $\C(D^*,i)=\{\{x\} \mid x\in N[v_{i+1}]\}$;
- otherwise, if $D^*\cap N(v_{i+1})\neq \emptyset$, then $\C(D^*,i)=\D(H)\cup \{\{v_{i+1}\}\}$;
- otherwise $$\C(D^*,i)=
\left \{
D\cup \{u\} \left |
\begin{array}{l}
D \in \{D\in \D(H) \mid D_S=\emptyset\},\\
u\in N(v_{i+1}),\ \text{and}\ \forall x\in D,\\
\operatorname{Priv}(D \cup \{u\}, x) \cap V_{i+1}\neq \emptyset
\end{array}
\right .
\right \}
\begin{array}{l}
\cup\ \{D\in \D(H) \mid D_S\neq\emptyset\}\\
\cup\ \{\{v_{i+1}\}\}.
\end{array}$$
The first case is a consequence of Proposition \[prop:parent-vi\]. In the second case, only $v_{i+1}$ is to be dominated by candidate extensions of $(D^*,i)$. Hence $\C(D^*,i)=\{\{x\} \mid x\in N[v_{i+1}]\}$. Let $S=V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$ and $C=N(S)\setminus \{v_{i+1}\}$. Let $H$ be the split graph induced by $S$ and $C$, where $C$ has been completed into a clique. Note that $S$ is a maximal independent set of $H$, as $C\subseteq N(S)$. In the third case, observe that as $D^*$ dominates $v_{i+1}$, $S=V_{i+1} \setminus N[D^*]$. As $S\neq \emptyset$, $\{v_{i+1}\}$ is a candidate extension. Let $D\in \D(H)$. Clearly, $D$ dominates $S$ in $G$. By Proposition \[prop:split-properties\], $\operatorname{Priv}(D_C,x)\cap S\neq\emptyset$ for every $x \in D_C$. Hence $D\in \C(D^*,i)$. Let $X\in \C(D^*,i)$, $X\neq \{v_{i+1}\}$. Clearly, $X\subseteq C\cup S$ and $X$ dominates $S$ in $H$. If $X\subseteq S$ then $X=S$ as $S$ is an independent set. If $X\not\subseteq S$ then $X\cap C\neq\emptyset$. In both cases, $X$ dominates $C$ in $H$. By definition, $\operatorname{Priv}(X,x)\cap S\neq\emptyset$ for all $x\in X$. By Proposition \[prop:split-properties\], we conclude that $X\in \D(H)$, hence that $\C(D^*,i)=\D(H)\cup \{\{v_{i+1}\}\}$.
From now on and until the end of the proof we assume that $S\neq\emptyset$ and $D^*\cap N(v_{i+1})=\emptyset$. As $S\neq \emptyset$, $\{v_{i+1}\}$ is a candidate extension. Let $X\in \C(D^*,i)$, $X\neq \{v_{i+1}\}$. Then $X$ dominates $S$ in $H$. If $X\subseteq S$ then $X=S$ as $S$ is an independent set. If $X\not\subseteq S$ then $X\cap C\neq\emptyset$. In both cases, $X$ dominates $C$ in $H$, and we conclude that $X$ is a dominating set of $H$. Now, if $X$ is a minimal dominating set of $H$, then by Proposition \[prop:split-properties\] we know that $\operatorname{Priv}(X,x)\cap S\neq\emptyset$ for all $x\in X$, and we deduce that $X\subseteq C\cup S$. Since $X$ has to dominate $v_{i+1}$ as well, we conclude that $X \cap S \neq \emptyset$, hence that $X \in \{D\in \D(H) \mid D_S\neq\emptyset\}$. Otherwise, $X$ is not a *minimal* dominating set of $H$. This implies that it has a vertex $u$ with no private neighbor in $H$. By definition of $\C(D^*, i)$, this means that $\operatorname{Priv}(D^* \cup X, u) \cap V_{i+1} = \{v_{i+1}\}$. Therefore there is exactly one such vertex. Then, if we write $D = X \setminus \{u\}$, $D$ is a minimal dominating set of $S$, hence of $H$. Since $v_{i+1}$ is a private neighbor of $u$, we must have $D_S = \emptyset$, and consequently $D \in \{D\in \D(H) \mid D_S=\emptyset\}$. Finally, by definition of $C(D^*, i)$, for any $x \in D\subsetneq X$, we have $\operatorname{Priv}(D\cup \{u\},x) \cap V_{i + 1} \neq \emptyset$. This shows that we have $$X\in
\left \{
D\cup \{u\} \left |
\begin{array}{l}
D \in \{D\in \D(H) \mid D_S=\emptyset\},\\
u\in N(v_{i+1}),\ \text{and}\ \forall x\in D,\\
\operatorname{Priv}(D \cup \{u\}, x) \cap V_{i+1}\neq \emptyset
\end{array}
\right .
\right \}$$ and proves the first inclusion. To prove the reverse inclusion, first recall that as $S\neq \emptyset$, $\{v_{i+1}\}$ is a candidate extension. We now consider $X\in \{D\in \D(H) \mid D_S\neq\emptyset\}$. By Proposition \[prop:split-properties\], $S\subseteq N[X]$ and $\operatorname{Priv}(X,x)\cap S\neq\emptyset$ for all $x\in X$. Since by hypothesis $X\cap S\neq\emptyset$, $S\cup\{v_{i+1}\}\subseteq N[X]$. Thus $X\in \C(D^*,i)$. Now we consider a set $X$ of the form $D\cup \{u\}$, for some $D \in \{D\in \D(H) \mid D_S=\emptyset\}$ and $u\in N(v_{i+1})$ such that $\forall x \in D$, $\operatorname{Priv}(D \cup \{u\}, x) \cap V_{i+1} \neq \emptyset$. By Proposition \[prop:split-properties\], $\operatorname{Priv}(D,x)\cap S\neq\emptyset$ for all $x\in D$. Since $\operatorname{Priv}(D\cup\{u\}, x) \cap V_{i+1} \neq \emptyset$ for all $x\in D$ and $v_{i+1}\in \operatorname{Priv}(X,u)$, $\operatorname{Priv}(X,x)\cap V_{i+1}\neq\emptyset$ for all $x\in X$. Since $S\cup\{v_{i+1}\}\subseteq N[X]$, $X\in \C(D^*,i)$. This proves the reverse inclusion and concludes the proof.
Notice that given $i$ and $D^*$ as in the statement of Lemma \[lem:triangle-free-candidates-characterization\], it is easy to construct $S$ and check whether $S=\emptyset$ and $D^*$ dominates $V_{i+1}$ in polynomial time and, in the two first cases of the statement of the lemma, to output the solution. We will show that the third and fourth cases can be handled using the algorithm of Proposition \[prop:split-properties\].
\[lem:triangle-free-candidates-enumeration\] There is an algorithm enumerating $\C(D^*,i)$ in total time $O(n^4 \cdot|\D(G,A)|)$ and $O(n^2)$ space whenever $V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$ is an independent set.
Let $S=V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$. Lemma \[lem:triangle-free-candidates-characterization\] above allows us to consider four cases depending on whether $D^*$ dominates $V_{i+1}$, $V_{i+1}\setminus \{v_{i+1}\}$, $\{v_{i+1}\}$, or none of these three sets. Clearly, the first two cases can be handled within the specified time and space bounds.
We discuss the time complexity of enumerating $\C(D^*,i)$ in the other two cases. Let $C=N(S)\setminus \{v_{i+1}\}$ and $H$ be the split graph induced by $S$ and $C$, where $C$ has been completed into a clique. Note that $H$ can be constructed in time $O(n^2)$ and require $O(n^2)$ space to store. To enumerate $\C(D^*,i)$, we start by generating $\D_C(H)$. This can be done in time $O(n^2)$ per solution and $O(n^2)$ space using the algorithm of Proposition \[prop:split-properties\]. In the third case of Lemma \[lem:triangle-free-candidates-characterization\], for every $B\in \D_C(H)$ generated by the above algorithm, we extend $B$ into its unique corresponding minimal dominating set $D\in \D(H)$ such that $D_C=B$ using Proposition \[prop:split-properties\], and output $D$. This adds an additional time of $O(n)$ to the delay, for a total time of $O(n^2\cdot |\C(D^*,i)|)$, since $|\D_C(H)|=|\D(H)|=|\C(D^*,i)|$ in that case. In the fourth case of Lemma \[lem:triangle-free-candidates-characterization\], for every set $B\in \D_C(H)$ generated by the above algorithm, we check in time $O(n)$ whether $B$ dominates $H$ (that is, whether $B$ belongs to the set $\{D_C\in \D(H) \mid D_S=\emptyset\}$). If it does not, we extend $B$ into its unique corresponding minimal dominating set of $H$ in time $O(n)$ as before. Otherwise, for every $u\in N(v_{i+1})$ such that for all $x\in B$, $\operatorname{Priv}(B\cup\{u\}, x) \cap V_{i+1} \neq \emptyset$, we output $B \cup \{u\}$. These steps add $O(n^3)$ time to the delay and do not require more than $O(n^2)$ space. Note that the only elements $D\in \D(H)$ which do not lead to an element of $\C(D^*, i)$ are the $D \in \{D\in \D(H) \mid D_S=\emptyset\}$ for which no vertex $u \in N(v_{i+1})$ satisfies the desired condition. However, we will show that $$\begin{aligned}
|\{D\in \D(H) \mid D_S=\emptyset\}| \leq n |\{D\in \D(H) \mid D_S\neq \emptyset\}|.\label{eq:triangle-free-trashbound}\end{aligned}$$ Indeed, consider the map $f$ that, given $D \in \{D\in \D(H) \mid D_S=\emptyset\}$ removes one arbitrary vertex from $D$, and completes the dominating set by adding the vertices in the independent set which are no longer dominated. Then, $f$ maps elements of $\{D\in \D(H) \mid D_S=\emptyset\}$, to the set $\{D\in \D(H) \mid D_S\neq\emptyset\}$. Moreover, every element in this second set is the image of at most $|C| \leq n$ elements by $f$. This implies the desired bound. By Lemma \[lem:triangle-free-candidates-characterization\], $\C(D^*,i)\supseteq \{D\in \D(H) \mid D_S\neq\emptyset\}$. Consequently, this means that while enumerating $\D(H)$, we might throw out at most $n\cdot |\C(D^*,i)|$ of all the solutions we found which do not lead to elements in $\C(D^*, i)$. Hence, the described algorithm uses $O(n^2)$ space and has a running time of $O(n^4\cdot |\C(D^*,i)|)$. By Corollary \[cor:cibound\], we conclude that the algorithm has running time $O(n^4\cdot |\D(G,A)|)$.
We conclude with the following theorem that we state in a more general way than in Section \[sec:intro\], and which is a consequence of Theorem \[thm:ordered-generation\] and Lemma \[lem:triangle-free-candidates-enumeration\].
\[thm:triangle-free\] There is an algorithm that, given a bicolored graph $G$ on $n$ vertices with prescribed set $A\subseteq V(G)$ such that $G[A]$ is triangle-free, enumerates the set $\D(G,A)$ in time $$O(\operatorname{poly}(n) \cdot |\D(G,A)|^2)$$ and $O(n^3)$ space.
Minimal dominating sets in Kt-free graphs {#sec:kt-free}
=========================================
In this section, we generalize the characterization of Lemma \[lem:triangle-free-candidates-characterization\] and show how to use it to enumerate minimal dominating sets in $K_t$-free graphs, at the cost of an increased complexity (see Theorem \[thm:kt-free\]).
We start with a lemma that, roughly, implies that any output-polynomial time algorithm that may repeat outputs can be turned into an output-polynomial algorithm without repetition. The different uses we make of this lemma required such a generic statement.
\[lem:avoid-repetitions\] Let ${\Sigma_{\rm in}}{}, {\Sigma_{\rm out}}{}$ be two sets and $R$ be a relation of ${\Sigma_{\rm in}}{} \times {\Sigma_{\rm out}}{}$. Let $f,s \colon {\Sigma_{\rm in}}{} \to {\mathbb{N}}$ be two functions. Suppose that there is a deterministic algorithm enumerating, given any $x \in {\Sigma_{\rm in}}{}$, the set $\{y \in {\Sigma_{\rm out}}{} \mid xRy\}$ in time at most $f(x)$ and space at most $s(x)$, possibly with repetition. Then there is an algorithm that, on the same input, return the same output without repetition, in time $O(f(x)^2)$ and space $O(s(n))$.
Let us call [`B’`]{} the algorithm that on input $x\in {\Sigma_{\rm in}}{}$ outputs $\{y \in {\Sigma_{\rm out}}{} \mid xRy\}$, possibly with repetition. We now describe an algorithm [`B`]{}. We will run two executions of [`B’`]{} and use two counters $i$ and $j$. Given an input $x \in {\Sigma_{\rm in}}{}$, we proceed as follows. We first initialize $i$ to 0 and call [`B’`]{} on $x$. For each $y$ output by this call, we increment $i$, set $j$ to 0 and call [`B’`]{} a second time on $x$. For each $z$ output by this second call, we increment $j$. Note that $i$ and $j$ are then the indices (in the sequence of outputs of a call of [`B’`]{} in $x$) of $y$ and $z$, respectively. We then consider the following cases:
- if $j<i$ and $y = z$, then we terminate the second call, discard $y$, and consider the next output of the first call to [`B’`]{};
- if $j<i$ and $y \neq z$, we discard $z$ and consider the next output of the second call;
- if $j\geq i$, we then terminate the second call, output $y$ and consider the next output of the first call.
Notice that since [`B’`]{} is deterministic, it always return the same sequence of outputs when called on an input $x \in {\Sigma_{\rm in}}$. Suppose that some $y' \in {\Sigma_{\rm out}}{}$ is such that $xRy'$. By definition, it is output by [`B’`]{}: let $k$ be the index of the first occurrence of $y'$ in the output sequence of [`B’`]{} when called on $x$. Observe then that because of the definition of $k$, the algorithm [`B`]{} will reach the third case above for $i=k$. Hence every element of $\{y \in {\Sigma_{\rm out}}{} \mid xRy\}$ is output by [`B`]{} on input $x$. Let us now suppose that $y'$ appears in the output sequence of [`B’`]{} at some index $k'>k$. When this output is considered by the above algorithm ( $i = k'$), we end up in the first case (with $j = k$) and $y'$ is not output a second time. This shows that every element output by [`B`]{} is unique.
Notice that [`B`]{} simultaneously keeps two instances of [`B’`]{} running, so its worst-case space complexity is asymptotically the same. Besides, [`B`]{} starts a second instance of [`B’`]{} for each output of the first instance, hence its time complexity on input $x$ is at most $O(f(x)^2)$. Therefore [`B`]{} has the desired properties.
By combining Lemma \[lem:avoid-repetitions\] and Theorem \[thm:ordered-generation\], we get the following corollary that we will use later.
\[cor:candexrep-to-dom\] Let $f \colon {\mathbb{N}}^2 \to N$ and $s\colon {\mathbb{N}}\to {\mathbb{N}}$ be two functions. Suppose that there is an algorithm that, given a bicolored graph $G$ on $n$ vertices with prescribed set $A\subseteq V(G)$, a peeling $(V_0, \dots, V_{p+1})$ of $G(A)$, $i \in {\left \{ 0, \dots, p-1 \right \}}$ and $D^* \in \D(G, V_i)$, enumerates the set $\C(D^*,i)$ in time at most $f(n, |\D(G,A)|)$ and space at most $s(n)$, possibly with repetition. Then there is an algorithm that, given a bicolored graph $G(A)$ on $n$ vertices, enumerates the set $\D(G,A)$ in time $$O(n^4\cdot d^2 + f(n, d)^2\cdot n d)$$ and space $O(n\cdot s(n))$, where $d = |\D(G,A)|$.
The aforementioned generalization of Lemma \[lem:triangle-free-candidates-characterization\] is the following.
\[lem:candidates-characterization\] Let $G$ be a bicolored graph with prescribed set $A\subseteq V(G)$. Let $(V_0,\allowbreak{}\dots,V_{p+1})$ be a fixed peeling of $G(A)$ with vertex sequence $(v_1,\dots,v_p)$, and let $i\in {\left \{ 0, \dots, p-1 \right \}}$, $D^*\in \D(G,V_i)$ and $S=V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$;
- if $V_{i+1} \subseteq N[D^*]$ then $\C(D^*,i) = \{\emptyset\}$;
- otherwise, if $S=\emptyset$, then $\C(D^*,i)=\{\{x\} \mid x\in N[v_{i+1}]\}$;
- otherwise, if $D^*\cap N(v_{i+1})\neq \emptyset$, then $\C(D^*,i)=\D(G,S)$;
- otherwise $$\C(D^*,i)= \left \{
Q\cup \{w\} \left |
\begin{array}{l}
w \in N[v_{i+1}]\ \text{and}\ Q \in \D(G, S\setminus N[w])\\
\text{s.t.}\ N[Q] \cap (S \cup \{v_{i+1}\}) \nsupseteq N[w]\cap (S \cup \{v_{i+1}\})
\end{array}
\right .
\right \}.$$
In this case, for every $w \in N[v_{i+1}]$ and $Q \in \D(G,S\setminus N[w])$, one of $Q$ and $Q \cup \{w\}$ is a candidate extension of $(D^*, i)$.
The first case is a consequence of Proposition \[prop:parent-vi\]. In the second case, only $v_{i+1}$ is to be dominated by candidate extensions of $(D^*,i)$. Hence $\C(D^*,i)=\{\{x\} \mid x\in N[v_{i+1}]\}$. In the third case, observe that as $D^*$ dominates $v_{i+1}$, $S=V_{i+1} \setminus N[D^*]$. By definition, we have that $\C(D^*,i)=\D(G,S)$.
From now on and until the end of the proof we assume that $S\neq\emptyset$ and $D^*\cap N(v_{i+1})=\emptyset$. Hence $\C(D^*,i)\neq \emptyset$ and $N[v_{i+1}] \cap X\neq \emptyset$ for all $X\in \C(D^*,i)$. Let $X\in \C(D^*,i)$ and $w\in N[v_{i+1}]\cap X$. We show that $X \setminus \{w\} \in \D(G, S\setminus N[w])$. Two cases arise. Either $w=v_{i+1}$ or $w\neq v_{i+1}$. In the first case, $S\setminus N[w]=\emptyset$ and the inclusion holds. In the second case, observe that every $x\in X\setminus \{w\}$ has a private neighbor in $V_{i+1}\setminus N[D^*]\setminus N[w]=S\setminus N[w]$. Clearly $X \setminus \{w\}$ dominates $S \setminus N[w]$ or otherwise $X$ does not dominate $V_{i+1}\setminus N[D^*]$. We conclude that $X \setminus \{w\} \in \D(G, S\setminus N[w])$. By minimality of $X$, $w$ has a private neighbor in $V_{i+1}\setminus N[D^*]=S \cup \{v_{i+1}\}$, hence $N[X \setminus \{w\}] \cap (S \cup \{v_{i+1}\})\nsupseteq N[w] \cap (S\cup \{v_{i+1}\})$.
Conversely, let $w$ and $Q$ be as in the right hand side of the equality, and let us show that $Q \cup \{w\}$ is a candidate extension of $D^*$. The vertex $w$ dominates both $v_{i+1}$ and $N[w] \cap S$, and $Q$ dominates $S \setminus N[w]$, so $Q \cup \{w\}$ dominates $S \cup \{v_{i+1}\}=V_{i+1}\setminus N[D^*]$. Also, every $u\in Q$ has a private neighbor (with respect to $Q$) in $S \setminus N[w]$, so it has a private neighbor with respect to $Q \cup \{w\}$. Finally, because $N[Q] \cap (S \cup \{v_{i+1}\}) \nsupseteq N[w] \cap (S \cup \{v_{i+1}\})$, $w$ has a private neighbor in $S \cup \{v_{i+1}\}=V_{i+1}\setminus N[D^*]$ with respect to $Q \cup \{w\}$. Consequently $Q \cup \{w\}$ is a minimal dominating set of $S \cup \{v_{i+1}\} = V_{i+1} \setminus N[D^*]$, , a candidate extension of $(D^*, i)$.
Regarding the last remark, we just proved that if $N[Q] \cap (S \cup \{v_{i+1}\}) \nsupseteq N[w] \cap (S \cup \{v_{i+1}\})$ then $Q \cup \{w\} \in \C(D^*,i)$. If $N[Q] \cap (S \cup \{v_{i+1}\}) \supseteq N[w]\cap (S \cup \{v_{i+1}\})$, then $Q$ minimally dominates $S \cup \{v_{i+1}\}$ and it is a candidate extension of $(D^*, i)$.
We point out that the last case of Lemma \[lem:candidates-characterization\] differs from the one of Lemma \[lem:triangle-free-candidates-characterization\] as the bound obtained in Lemma \[lem:triangle-free-candidates-enumeration\], Inequality does not hold in general (when $S$ induces at least an edge). This is why we separately consider $\D(G,S\setminus N[w])$ for each $w\in N[v_{i+1}]$. Repetitions will be handled using Lemma \[lem:avoid-repetitions\] at the cost of an increased complexity.
Observe that the only obstacles that prevents us from directly using Lemma \[lem:candidates-characterization\] to enumerate the candidate extensions in general (and thus the minimal dominating sets, using Theorem \[thm:ordered-generation\]) in output-polynomial time are the two last cases where one has to enumerate the minimal dominating sets of $G(S)$ or $G(S \setminus N[w])$. For bicolored graphs $G(A)$ such that $G[A]$ is $K_t$-free, this can be done by exploiting the fact that $G[S]$ is $K_{t-1}$-free and running the same algorithm on $G(S)$, as we describe now.
\[thm:kt-free\] There is a function $p \colon {\mathbb{N}}\to {\mathbb{N}}$ and an algorithm that, given a bicolored graph $G$ on n vertices with prescribed set $A \subseteq V(G)$ such that $G[A]$ is $K_t$-free for some integer $t\geq 1$, enumerates the set $\D(G,A)$ in time at most $$p(t) \cdot n^{2^{t+1} - 3} \cdot |\D(G,A)|^{2^t-1}$$ and space at most $p(t) \cdot n^{\max(2, t-1)}$.
When $A = V(G)$, we have $\D(G) = \D(G,A)$. Hence, Theorem \[thm:kt-free\] implies the existence of an algorithm enumerating, for every integer $t\geq 1$, the minimal dominating sets in $K_t$-free graphs in output-polynomial time and polynomial space. We stress that we provide a single algorithm for all values of $t$ and not one per value.
In this proof we consider two algorithms [`A`]{} and [`B`]{} that recursively call each other in order to enumerate the minimal dominating sets of a bicolored graph. We first give their specifications, then describe them, and finally prove that they perform as specified. Let $f \colon {\mathbb{N}}^3\to {\mathbb{N}}$ be defined by $f(n,d,t) = n^{2^{t+1} - 3} \cdot d^{2^t-1}$, for every $n,d,t \in {\mathbb{N}}$.
#### Specifications of [`A`]{} and [`B`]{}.
We will show that the aforementioned algorithms have the following properties:
- there is a constant $p(t) \in {\mathbb{N}}$ such that given an $n$-vertex graph $G$ and a set $A \subseteq V(G)$ such that $G[A]$ is $K_t$-free, [`A`]{} outputs $\D(G,A)$ in time at most $p(t) \cdot f(n,|\D(G,A)|,t)$ and space at most $p(t) \cdot n^{\max(2,t)}$; and
- there is a constant $q(t)$ such that given an $n$-vertex graph $G$, a set $A \subseteq V(G)$ such that $G[A]$ is $K_t$-free, a peeling $(V_0, \dots, V_p, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1, \dots, v_p)$, $i \in {\left \{ 0, \dots, p-1 \right \}}$, and $D^* \in \D(G, V_i)$, [`B`]{} outputs $\C(D^*,i)$ in time at most $q(t) \cdot n^2 \cdot f(n, |\D(G,A)|, t-1)^2$ and space at most $q(t)
\cdot n^{\max(2, t-1)}$,
for every integer $t\geq 1$ for $P(t)$ and for every integer $t\geq 2$ for $Q(t)$.
Observe that that the statement of Theorem \[thm:kt-free\] is implied by $\forall t\in {\mathbb{N}}_{\geq 1}, P(t)$. In order to prove it, we will also show that $Q(t)$ holds for every integer $t\geq 2$. Let us first describe [`A`]{}.
#### Description of [`A`]{}.
The algorithm [`A`]{} is the algorithm given by Theorem \[thm:ordered-generation\] that takes as input a bicolored graph $G$ with prescribed set $A\subseteq V(G)$, using [`B`]{} as a routine to enumerate candidate extensions. We will show below that [`B`]{} indeed does so.
#### Description of [`B`]{}.
Recall that [`B`]{} takes as input a bicolored graph $G$ with prescribed set $A\subseteq V(G)$, a peeling $(V_0, \dots, V_p, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1, \dots, v_p)$, an integer $i \in {\left \{ 0, \dots, p-1 \right \}}$, and a set $D^* \in \D(G, V_i)$.
We first describe an auxiliary routine [`B’`]{}. Let $S = V_{i+1}\setminus \{v_{i+1}\}\setminus N[D^*]$. Lemma \[lem:candidates-characterization\] above allows us to consider four cases depending on whether $D^*$ dominates $V_{i+1}$, $V_{i+1}\setminus \{v_{i+1}\}$, $\{v_{i+1}\}$, or none of these three sets:
(i) \[it:dejadom\] if $V_{i + 1} \subseteq N[D^*]$, we output $\{\emptyset\}$;
(ii) \[it:singleton\] otherwise, if $S=\emptyset$, we output $\{\{x\} \mid x\in N(v_{i+1})\}$;
(iii) \[it:vidom\] otherwise, if $D^*\cap N(v_{i+1})\neq \emptyset$, we call algorithm [`A`]{} on $G(S)$ to enumerate $\D(G,S)$ and we give the same output;
(iv) \[it:repet\] in the remaining case, we iterate over $w \in N[v_{i+1}]$ and $Q \in \D(G, S\setminus N[w])$ (obtained via a call to [`A`]{}) and output $D \cup X$ if and only if the following holds $$\label{eq:discarde2}
N[Q] \cap (S \cup \{v_{i+1}\})
\nsupseteq N[w] \cap (S \cup \{v_{i+1}\}).$$
We are now done with [`B’`]{}. As we will show later, [`B’`]{} enumerates $\C(D^*, i)$, however each element may be repeated, up to $n$ times. Then [`B`]{} is obtained from [`B’`]{} using Lemma \[lem:avoid-repetitions\]. This concludes the description of [`B`]{}.
#### Correctness of [`A`]{} and [`B`]{}.
Now that we described the algorithms [`A`]{} and [`B`]{}, we show that they conform to their specifications, , we prove that $P(t)$ holds for every $t\geq 1$ and that $Q(t)$ holds for every $t \geq 2$. The proof by induction on $t$ is split in lemmas.
\[lem:base\] $P(1)$ holds.
The statement $P(1)$ deals with pairs $(G,A)$ such that $G[A]$ is $K_1$-free, so $A = \emptyset$. In these cases we clearly have $\D(G,A) = \{\emptyset\}$. Notice that algorithm [`A`]{} as described above correctly answers on such inputs. Checking whether $A$ empty and returning $\{\emptyset\}$ takes $O(n)$ time (depending how $A$ is given) and $O(1)$ space. We define $c(1)$ as an integer such that these steps take at most $c(1) \cdot n$ time and at most $c(1) \cdot n^2$ space on an input graph of order $n$. As $f(n,|\D(G,A)|,t) = n$ in this case, $P(1)$ holds.
\[lem:ind1\] For every integer $t \geq 1$, $P(t) \Rightarrow Q(t +1)$.
Let $t\geq 1$ and let us assume that the statement $P(t)$ holds (in particular, $p(t)$ is defined). Let $\mathcal{I} = (G, A, V_0, \dots, V_{p+1}, v_1, \dots, v_p, i, D^*)$ be an input of [`B`]{} such that $G[A]$ is $K_{t+1}$-free. Let us define $n = |G|$ and $d = |\D(G,A)|$. We review the description of [`B`]{} to show that $Q(t+1)$ holds. We first consider the auxiliary routine [`B’`]{}.
\[claim:auxi\] Given $\mathcal{I}$, [`B’`]{} enumerates $\C(D^*, i)$, with each output possibly repeated up to $n$ times, in time at most $q \cdot n
\cdot f(n, d, t)$ and space at most $q \cdot n^{\max(2, t)}$, for some constant $q$.
Let $S = V_{i+1}\setminus\{v_{i+1}\} \setminus N[D^*]$. Note that as $D^*$ dominates $V_i$, $S \subseteq N(v_{i+1})\cap V_{i+1}$. Notice that this set can be computed in $O(n^2)$ time and $O(n)$ space. Within the same time and space bounds we can also check whether $V_{i +
1} \subseteq N[D^*]$, a condition that we will use later.
Since $i <p$, we have $V_{i+1} \subseteq A$, from the definition of a peeling. In particular, $G[V_{i+1}]$ is $K_{t+1}$-free. As $S \subseteq
N(v_{i+1})\cap V_{i+1}$, we get that $G[S]$ is $K_t$-free. As we assume $P(t)$, we have the following.
\[rem:ih\] For any $S' \subseteq S$, a call to [`A`]{} on $(G,S')$ returns $\D(G,S')$ in time at most $p(t) \cdot f(n,|\D(G,S')|,t)$ and space at most $p(t) \cdot n^{\max(2, t)}$.
Comparing the steps of [`B’`]{} with the cases of Lemma \[lem:candidates-characterization\] and because of Remark \[rem:ih\], we observe that in the cases –, the output of [`B’`]{} is exactly $\C(D^*, i)$, without repetition. Set aside the time spent building $S$ and checking whether $D^*$ dominates $V_{i+1}$, the cases and can clearly be performed in $O(n)$ time and $O(1)$ space. In case , the call to [`A`]{} take time at most $$\begin{aligned}
& p(t) \cdot f(n, |\D(G,S)|,t)\\
=~& p(t) \cdot f(n, |\C(D^*, i)|,t)& \text{(by Lemma~\ref{lem:candidates-characterization})}\\
\leq~ & p(t) \cdot f(n, d, t) & \text{(by Corollary~\ref{cor:cibound})}
\end{aligned}$$ and space at most $p(t) \cdot n^{\max(2,t)}$, according to Remark \[rem:ih\]. We now assume that we are in case . Observe that then $t\geq 2$. According to Lemma \[lem:candidates-characterization\], every element of $\C(D^*, i)$ is output by [`A`]{} and every output of [`A`]{} belongs to $\C(D^*, i)$. However, it might be that the same set is output several times by [`B’`]{}. Notice that for any $w \in N[v_{i+1}]$ and two outputs $Q, Q'$ of [`A`]{} on $(G, S\setminus N[w])$, the sets $\{w\} \cup Q$ and $\{w\} \cup Q'$ are distinct as we assume that [`A`]{} enumerates $\D(G,S\setminus N[w])$ without repetition (see Remark \[rem:ih\]). Hence, any two repeated outputs of [`B’`]{} have the form $\{w\} \cup Q$ and $\{w'\} \cup Q'$ for two distinct $w,w' \in N[v_{i+1}]$ and some $Q\in \D(G,S\setminus N[w])$ and $Q'\in \D(G,S\setminus N[w'])$. As there are at most $n$ choices for $w$, we deduce that the same output is repeated at most $n$ times. Also, the remark at the end of the last case of the statement of Lemma \[lem:candidates-characterization\] together with Corollary \[cor:cibound\] imply the following.
\[rem:subsetS\] For every $w \in N[v_{i+1}]$, $|\D(G,S\setminus N[w])| \leq d$.
Regarding time and space complexity (again ignoring the time spent building $S$ and checking whether $D^*$ dominates $V_{i+1}$) we perform at most $n$ times (once for every choice of $w$) the following operations:
- the construction of $S \setminus N[w]$, in $O(n)$ time and space;
- a call to [`A`]{} on $(G, S \setminus N[w])$, in time at most $p(t) \cdot f(n, d, t)$ and space at most $p(t) \cdot n^{\max(2,t)}$, by remarks \[rem:ih\] and \[rem:subsetS\];
- for each set $Q$ among the at most $d$ outputs of [`A`]{}, a check whether holds, in $O(n^2)$ time and $O(n)$ space.
In total, the time complexity of these steps and those required to construct $S$ and to check whether $D^*$ dominates $V_{i+1}$ add up to: $$\begin{aligned}
& O(n) + n \cdot \left [ O(n) + p(t) \cdot f(n, d, t) + O\left (n^2 \cdot d
\right ) \right ]\nonumber\\
=~& O\left (p(t) \cdot n \cdot f(n, d, t) \right ) & \text{(as $t \geq 2$ in this case).} \label{eq:timebp}
\end{aligned}$$ Similarly, the space complexity can be upper-bounded by $O(p(t) \cdot
n^{\max(2,t)})$. Notice that these time and space bounds asymptotically dominate those obtained in the cases –. Hence, there is a constant $q$ such that [`B’`]{} runs in time at most $q \cdot n \cdot f(n,d,t)$ and space at most $q \cdot n^2$ on the considered input. This concludes the proof of the claim.
As proved in Lemma \[lem:avoid-repetitions\], the algorithm of Claim \[claim:auxi\] can be turned into an algorithm [`B`]{} that does not repeat outputs. That is, there is a constant $q(t+1)$ (depending on $q$) such that given $\mathcal{I}$, [`B`]{} runs in time at most $q(t+1) \cdot n^2\cdot f(n, d, t)^2$ and space at most $q(t+1) \cdot n^{\max(2,t)}$. Hence $Q(t+1)$ holds, as desired.
\[lem:ind2\] For every integer $t \geq 2$, $Q(t) \Rightarrow P(t)$.
Let us assume that for some integer $t\geq 2$, the statement $Q(t)$ holds (and in particular $q(t)$ is defined). Let $G$ be a graph and $A \subseteq V(G)$ be such that $G[A]$ is $K_t$-free. We set $n = |G|$ and $d = |\D(G,A)|$. By $Q(t)$, the enumeration of candidate extensions in $G(A)$ can be carried out by [`B`]{} in total time at most $$q(t)\cdot n^2 \cdot f(n, d, t-1)^2$$ and space at most $q(t) \cdot n^{\max(2,t)}$. According to Theorem \[thm:ordered-generation\], [`A`]{} then enumerates $\D(G,A)$ in time $$\begin{aligned}
& O(n^4\cdot d^2 + q(t)\cdot n^3 \cdot f(n, d, t-1)^2 \cdot d)\\
=~& O(n^4\cdot d^2 + q(t) \cdot f(n,d,t)) & \text{(thanks to the definition of $f$)}\\
=~& O(q(t) \cdot f(n,d,t)) & \text{(as $t\geq 2$)}
\end{aligned}$$ and space $O(q(t) \cdot n^{\max(2,t+1)})$. Therefore, there is a constant $p(t)$ (depending on $q(t)$) such that [`A`]{} runs on this input in time at most $p(t) \cdot f(n,d,t)$ and space at most $p(t) \cdot n^{\max(2,t)}$. This proves $P(t)$.
#### Concluding the proof.
By induction on $t$, using the base case $Q(1)$ provided by Lemma \[lem:ind1\] and the induction step that for every integer $t \geq 1$, $P(t)$ implies $P(t+1)$ provided by the combination of Lemmas \[lem:ind1\] and \[lem:ind2\], we conclude that for every integer $t\geq 1$, $P(t)$ holds. That is, the algorithm [`A`]{} has the properties claimed in the statement of the theorem.
We note that the complexity of the algorithm of Theorem \[thm:kt-free\] for $K_t$-free graphs could be sightly improved using Theorem \[thm:triangle-free\] as a base case, however that would not remove the exponential contribution of $t$ to the degree of the polynomial.
Variants of Kt-free graphs {#sec:variants}
==========================
We give output-polynomial algorithms for variants of $K_t$-free graphs relying on the algorithms and candidate characterizations of Sections \[sec:bt\], \[sec:triangle-free\] and \[sec:kt-free\].
Forbidding Kt+e
---------------
In this section we show how the algorithm of Theorem \[thm:kt-free\] on $K_t$-free graphs can be extended to the setting of $(K_t+e)$-free graphs.
\[thm:ktme\] There is an algorithm that, for every integer $t\geq 1$, enumerates minimal dominating sets in $(K_t+e)$-free graphs in output-polynomial time and polynomial space.
Let $t \in {\mathbb{N}}$ and let $G$ be a $(K_t+e)$-free graph. It is well-known that the minimal dominating sets of $G$ that do not induce edges are exactly the maximal independent sets of $G$. We can therefore enumerate these using the polynomial delay algorithm of Tsukiyama et al. [@tsukiyama1977new] for maximal independent sets. In the sequel we may thus focus on those minimal dominating sets of $G$ that induce at least one edge.
We show how to enumerate, for every edge $uv$ of $G$, the minimal dominating sets of $G$ that contain both $u$ and $v$. Let $A_{uv}=V(G) \setminus N[\{u,v\}]$ and observe that $G[A_{uv}]$ is $K_t$-free. First, we enumerate $G(A_{uv})$ using the algorithm of Theorem \[thm:kt-free\], which runs in output-polynomial time and polynomial space, as $t$ is fixed. For every $D\in \D(G,A_{uv})$ obtained from the aforementioned call, we output $D\cup\{u,v\}$ if it is a minimal dominating set of $G$, and discard $D$ otherwise. By Lemma \[lem:cand-ext-bound\] (applied for $H=G$ and $B = V(G)$) we have $
|\D(G,A_{uv})|\leq |\D(G)|
$. Hence, enumerating $\D(G,A_{uv})$ produces all those minimal dominating sets of $G$ that at least induce the edge $uv$ in time $\operatorname{poly}(n \cdot |\D(G)|)$ and space $\operatorname{poly}n$, where the degrees of these polynomials depend on $t$ (see Theorem \[thm:kt-free\]).
Now that we know how to enumerate minimal dominating sets that induce at least one particular edge, we can run the above routine for every edge of $G$ to enumerate all minimal dominating sets of $G$, possibly with repetitions. Observe that the same output can be repeated at most $|E(G)|$ times. Then, repetitions are avoided using Lemma \[lem:avoid-repetitions\] with ${\Sigma_{\rm in}}{}$ being the set of all graphs, ${\Sigma_{\rm out}}{}$ the set of all vertex sets, and $R$ the relation that associates every graph to its minimal dominating sets.
Forbidding Kt-e
---------------
Another interesting case is the one of $(K_t-e)$-free graphs. In this section we show how the characterization of Lemma \[lem:candidates-characterization\] can be used to enumerate candidate extensions in diamond-free graphs (which are $(K_t-e)$-free for $t=4$), which by Theorem \[thm:ordered-generation\] gives an output-polynomial algorithm enumerating minimal dominating sets in this class. We leave open the existence of such an algorithm in the case where $t\geq 5$.
In what follows, we consider a bicolored graph $G$ on $n$ vertices, with prescribed set $A\subseteq V(G)$ such that $G$ is diamond-free, together with a fixed peeling $(V_0,\dots, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1,\dots,v_p)$. Then we consider $$\begin{aligned}
& i \in {\left \{ 0, \dots, p-1 \right \}},
& D^* \in \D(G,V_i),\end{aligned}$$ and define $S=V_{i+1}\setminus\{v_{i+1}\}\setminus N[D^*]$ and $\C(D^*,i)$ as in Sections \[sec:bt\], \[sec:triangle-free\] and \[sec:kt-free\]. Note that contrarily to the triangle-free case and the $K_t$-free case considered above, we here require the whole graph $G$ to be diamond-free and not only $G[A]$. We start with an easy observation.
\[obs:clique-partition\] For every vertex $u$ of $G$, $G[N(u)]$ is $P_3$-free. Then $G[N(v_{i+1})]$, hence $G[S]$, can be partitioned into a disjoint union of cliques.
We will show how to minimally dominate one clique of $S$, then a disjoint union of cliques of $S$, and will conclude with the enumeration of $\C(D^*,i)$.
\[lem:clique-neighborhoods\] Let $K$ be a clique of $G[S]$ and $u$ be a vertex in $G-S$, $u\neq v_{i+1}$, that is adjacent to some vertex of $K$. If $u$ is adjacent to $v_{i+1}$ then it is complete to $K$. Otherwise it has exactly one neighbor in $K$.
If $u\in N(v_{i+1})$ then, as $G[N(v_{i+1})]$ is $P_3$-free and $K\subseteq N(v_{i+1})$, $u$ is complete to $K$. If $u$ is not adjacent to $v_{i+1}$ then it has exactly one neighbor in $K$, as otherwise $\{a,b,u,v_{i+1}\}$ would induce a diamond in $G$, for any two neighbors $a,b\in K$ of $u$.
\[lem:clique-domsetsenumeration\] Let $K$ be a clique of $G[S]$. Then $\D(G,K)$ can be enumerated in total time $O(n^2+ n\cdot |\D(G,K)|)$ and $O(n)$ space.
We describe an algorithm enumerating $\D(G,K)$ in the specified time and space. We first output $\{v_{i+1}\}$ as it is complete to $K$. We then output all vertices $u\in N(v_{i+1})$ such that $u\in K$ or $u$ is adjacent to some vertex of $K$. By Lemma \[lem:clique-neighborhoods\], these vertices are also complete to $K$. Then, for every $x\in K$, we compute the neighborhood of $x$ outside of $N(v_{i+1})$ in a time bounded by $O(n^2)$. By Lemma \[lem:clique-neighborhoods\], these neighborhoods are disjoint. At last, we enumerate the Cartesian products of these neighborhoods. This can clearly be done with linear delay and for a total time of $n\cdot |\D(G,K)|$ using $O(n)$ space as they are disjoint. Clearly, every element in such a Cartesian product is a minimal dominating set of $K$, and the described algorithm performs within the specified time and space bounds. The correctness of the algorithm follows from Lemma \[lem:clique-neighborhoods\].
\[lem:P3-free\] Let $W$ be a subset of $S$. Then $\D(G,W)$ can be enumerated in total time $O(n^{7}\cdot |\D(G,A)|^3)$ and $O(n^2)$ space.
We use the ordered generation described in Section \[sec:bt\]. The algorithm first computes a peeling $(U_1,\dots,U_{q+1})$ of $G(W)$ with vertex sequence $(u_1,\dots,u_q)$, in time $O(n^2)$ and space $O(n)$. Note that $N[u_1],\dots,N[u_q]$ is exactly the disjoint clique partition of $G[W]$. Given $j\in {\left \{ 0, \dots, q-1 \right \}}$ and $D^\circ\in \D(G,U_j)$, we define $\Cp(D^\circ, j)$ as the set of candidate extensions of $(D^\circ, j)$ with respect to the chosen peeling of $G(W)$ and we show how to enumerate $\Cp(D^\circ,j)$ in time $
O(n^6\cdot |\D(G,A)|^2)
$ and using $O(n)$ space.
We rely on the same characterization of candidate extensions that we use in the proof of Theorem \[thm:kt-free\], Lemma \[lem:candidates-characterization\]. Recall that this lemma allows us to consider four cases depending on whether $D^\circ$ dominates $U_{i+1}$, $U_{i+1}\setminus \{u_{i+1}\}$, $\{u_{i+1}\}$, or none of these three sets. If $U_{i + 1} \subseteq N[D^*]$, we output $\{\emptyset\}$. If $S=\emptyset$, we output $\{\{x\} \mid x\in N(u_{i+1})\}$. Otherwise, if $D^*\cap N(u_{i+1})\neq \emptyset$, we define $Y=U_{j+1}\setminus \{u_{j+1}\}\setminus N[D^\circ]$ and notice that $Y$ induces a clique in $G$. We then call the algorithm of Lemma \[lem:clique-domsetsenumeration\] on input $G$ and $Y$ to enumerate $\D(G,Y)$ in total time $$O(n^2+ n\cdot |\D(G,A)|)$$ and $O(n)$ space. In the remaining case, we iterate over $w \in N[u_{i+1}]$ and $Q \in \D(G, Y\setminus N[w])$ (obtained via a call to the algorithm of Lemma \[lem:clique-domsetsenumeration\]) and output $D \cup X$ if and only if the following holds: $$N[Q] \cap (S \cup \{u_{i+1}\})
\nsupseteq N[w] \cap (S \cup \{u_{i+1}\}).$$ In order to check this condition, we spend $O(n^2)$ time per element $Q \in \D(G, Y\setminus N[w])$, which sums up to $O(n^2\cdot |\D(G,A)|)$ spent per $w\in N[v_{i+1}]$, according to Lemma \[lem:cand-ext-bound\]. In total, the described algorithm enumerates $\Cp(D^\circ,i)$, possibly with repetitions, in time $O(n^3\cdot |\D(G,A)|)$ and using $O(n)$ space. Using Corollary \[cor:candexrep-to-dom\], we obtain an algorithm enumerating $\D(G,W)$ in time $O(n^{7}\cdot |\D(G,A)|^3)$, and using $O(n^2)$ space.
\[lem:diamond-free-candidates-enumeration\] There is an algorithm enumerating $\C(D^*,i)$, possibly with repetition, in total time $O(n^{8}\cdot |\D(G,A)|^3)$ and $O(n^2)$ space.
We conduct the same argument as in the previous lemma, and in the proof of Theorem \[thm:kt-free\]. Lemma \[lem:candidates-characterization\] allows us to consider four cases depending on whether $D^*$ dominates $V_{i+1}$, $V_{i+1}\setminus \{V_{i+1}\}$, $\{v_{i+1}\}$, or none of these three sets. We skip the first two trivial cases and jump to the third case. If $D^*\cap N(v_{i+1})\neq \emptyset$, we call the algorithm of Lemma \[lem:P3-free\] to enumerate $\D(G,S)$ in total time $$O(n^{7}\cdot |\D(G,A)|^3)$$ and $O(n^2)$ space. In the remaining case, we iterate over $w \in N[v_{i+1}]$ and $Q \in \D(G, S\setminus N[w])$ (obtained via a call to the algorithm of Lemma \[lem:P3-free\] as $S\setminus N[w]\subseteq S$) and output $D \cup X$ if and only if the following holds $$N[Q] \cap (S \cup \{v_{i+1}\})
\nsupseteq N[w] \cap (S \cup \{v_{i+1}\}).$$ As in the proof of Lemma \[lem:P3-free\], checking this condition costs $O(n^2)$ time per $Q \in \D(G, S\setminus N[w])$, for a total (asymptotically) unaffected running time of $O(n^{7}\cdot |\D(G,A)|^3)$ spent per $w\in N[v_{i+1}]$. In total, the described algorithm enumerates $\C(D^*,i)$, possibly with repetitions, in time $O(n^{8}\cdot |\D(G,A)|^3)$ and using $O(n^2)$ space.
As a consequence of Corollary \[cor:candexrep-to-dom\] and Lemma \[lem:diamond-free-candidates-enumeration\], we get the following.
\[thm:diamond-free\] There is an algorithm that, given a bicolored graph $G$ on $n$ vertices with prescribed set $A\subseteq V(G)$ such that $G$ is diamond-free, enumerates the set $\D(G,A)$ in time $$O(\operatorname{poly}(n) \cdot |\D(G,A)|^8)$$ and $O(n^3)$ space.
Note that when $A = V(G)$, we have $\D(G)=\D(G,A)$. Hence, Theorem \[thm:diamond-free\] implies the existence of an algorithm enumerating the minimal dominating sets in diamond-free graphs in output-polynomial time and using polynomial space, which is one of the two case of Theorem \[th:pawdiam\].
Paw-free graphs
---------------
We now consider the exclusion of a specific graph, the paw, and show admits an output-polynomial time algorithm in paw-free graphs.
In what follows, we consider a bicolored graph $G$ on $n$ vertices, with prescribed set $A\subseteq V(G)$ such that $G$ is paw-free, together with a fixed peeling $(V_0,\dots, V_{p+1})$ of $G(A)$ with vertex sequence $(v_1,\dots,v_p)$. Then we consider $$\begin{aligned}
& i \in {\left \{ 0, \dots, p-1 \right \}},
& D^* \in \D(G,V_i),\end{aligned}$$ and define $S=V_{i+1}\setminus\{v_{i+1}\}\setminus N[D^*]$ and $\C(D^*,i)$ as in Sections \[sec:bt\], \[sec:triangle-free\] and \[sec:kt-free\]. As in the previous section we stress that we require the whole graph $G$ to be paw-free, and not only $G[A]$. We start with an easy observation.
For every vertex $u$ of $G$, $G[N(u)]$ is $\overline{P_3}$-free. Hence $G[S]$ is a complete multipartite graph.
Note that if $S$ is an independent set, then the existence of an output-polynomial algorithm enumerating $\C(D^*,i)$ is given by Lemma \[lem:triangle-free-candidates-enumeration\]. In the next lemma, we consider the case where $S$ contains at least one edge. We denote by $I_1,\dots,I_q$ the complete multipartition of $G[S]$, where every $I_j$, $j\in {\left \{ 1, \dots, q \right \}}$ induces an independent set. Hence $q\geq 2$.
\[lem:paw-free-edge\] Let us assume that $S$ contains at least one edge, and let $u$ be a vertex of $G$, $u\neq v_{i+1}$, that has a neighbor in $S$. If $u$ is not adjacent to $v_{i+1}$, then it is complete to $S$. Otherwise, $u$ is complete to $S\setminus I_j$, for some $j\in {\left \{ 1, \dots, q \right \}}$.
Let $I_1,\dots,I_q$ be the complete multipartition of $G[S]$, where every $I_j$, $j\in {\left \{ 1, \dots, q \right \}}$ induces an independent set in $G$. As by hypothesis $S$ contains an edge, $q\geq 2$. Let us show the first case by contradiction. Let $u \in V(G) \setminus N[v_{i+1}]$ have a neighbor in $S$ and suppose that $u$ is not complete to $S$. Hence there are vertices $x \in S \cap N(u)$ and $ y\in S \setminus N(u)$. Note that $xy\not\in E(G)$ as otherwise $\{u, v_{i+1}, x, y\}$ induces a paw in $G$. Then $x,y\in I_j$ for some $j\in {\left \{ 1, \dots, q \right \}}$. Let $z \in S\setminus I_j$; such a vertex exists as $q \geq 2$ and it is complete to $\{v_{i+1}, x, y\}$ by definition of the $I_k$’s. Then either $uz \in E(G)$ and $\{u, v_{i+1}, y,z\}$ induces a paw, or $uz \notin E(G)$ and $\{u, v_{i+1}, y,z \}$ does, a contradiction.
We show the second case. If $u$ belongs to $S$ then it belongs to some $I_j$, $j\in{\left \{ 1, \dots, q \right \}}$ and is complete to $S\setminus I_j$, by definition of the $I_k$’s. We now assume $u \in N(v_{i+1}) \setminus S$. If there is no $j\in{\left \{ 1, \dots, q \right \}}$ such that $u$ is not complete to $S\setminus I_j$ then, as $q\geq 2$, $u$ it has at least two non-neighbors $x\in I_{j'}$ and $y\in I_{j''}$ for two different $j',j''\in {\left \{ 1, \dots, q \right \}}$. Then $\{u, v_{i+1}, x, y\}$ induces a paw in $G$, a contradiction.
\[lem:paw-free-candidates-enumeration\] There is an algorithm enumerating $\C(D^*,i)$ in total time $O(n^4 \cdot|\D(G,A)|)$ and $O(n^2)$ space.
In the case where $S$ induces an independent set, we use the algorithm of Lemma \[lem:triangle-free-candidates-enumeration\] to enumerate $\C(D^*,i)$ in time $$O(n^4\cdot|\D(G,A)|)$$ and $O(n^2)$ space. Otherwise, we know from Lemma \[lem:paw-free-edge\] that minimal dominating sets of $S$ are either of size at most two, or of the form $I_j$ for some $j\in {\left \{ 1, \dots, q \right \}}$. If $v_{i+1}\in N[D^*]$, that is if $S=V_{i+1}\setminus N[D^*]$, we try each of these sets and output those that minimally dominate $S$, in a time that is polynomially bounded by in $n$. This enumerates $\C(D^*,i)$ by definition. If $v_{i+1}\not\in N[D^*]$, we first output $I_j$ for every $j\in {\left \{ 1, \dots, q \right \}}$. Then, we iterate over every set $D\subseteq N[S\cup \{v_{i+1}\}]$ of size at most three and output those that minimally dominate $S$, in a time which is polynomially bounded by $n$. This will enumerate $\C(D^*,i)$ as if $X\in \C(D^*,i)$ then at most one vertex can have $v_{i+1}$ as a private neighbor and it follows from Lemma \[lem:paw-free-edge\] that at most two vertices can have a private neighbor in $S$. Notice that all of these steps have a running time bounded by $O(n^4 \cdot|\D(G,A)|)$, and using no more than $O(n^2)$ space.
As a consequence of Theorem \[thm:ordered-generation\] and Lemma \[lem:paw-free-candidates-enumeration\], we get the following.
\[thm:paw-free\] There is an algorithm that, given a bicolored graph $G$ on $n$ vertices with prescribed set $A\subseteq V(G)$ such that $G$ is paw-free, enumerates the set $\D(G,A)$ in time $$O(\operatorname{poly}(n) \cdot |\D(G,A)|^2)$$ and $O(n^3)$ space.
Note that when $A = V(G)$, we have $\D(G)=\D(G,A)$. Hence, Theorem \[thm:paw-free\] implies the existence of an algorithm enumerating the minimal dominating sets in paw-free graphs in output-polynomial time and using polynomial space, which is one of the two case of Theorem \[th:pawdiam\].
Technique limitations {#sec:beyond}
=====================
In this section, we discuss various obstacles that we detected in our attempts to improve our results or proofs.
A standard technique fails for bipartite graphs
-----------------------------------------------
A natural technique (sometimes called flashlight search or backtrack) to enumerate valid solutions to a given problem such as, for instance, sets of vertices satisfying a given property is to build them element by element. If during the construction one detects that the current partial solution cannot be extended into a valid one, then it can be discarded along with all the other partial solutions that contain it. Note that in order to apply this technique, one should be able to decide whether a given partial solution can be completed into a valid one. It turns out that for minimal dominating sets, this problem (that we will denote by ) is NP-complete [@kante2011enumeration], even when restricted to split graphs [@kante2015polynomial]. We show that it remains NP-complete in bipartite graphs.
\[thm:ext\] restricted to bipartite graphs is NP-complete.
As a consequence, is NP-complete in $(K_t + e)$-free graphs for $t \geq 3$. This suggests that the aforementioned technique is unlikely to be used to improve Theorem \[thm:op\].
This problem is known to be NP-complete for general graphs [@kante2011enumeration]. It has later been proved that the variant where we search for a minimal dominating set containing $A$, and avoiding a given vertex set $B$ remains intractable even on split graphs [@kante2015polynomial]. We show that is still hard for bipartite graphs and thus triangle-free graphs. As a consequence, one cannot expect to improve Theorem \[thm:op\] by testing if subsets of $V(G)$ can be extended into minimal dominating sets of $G$.
Since is NP-complete in the general case, it is clear that is in NP even when restricted to bipartite graphs. Let us now present a reduction from .
Given an instance $\mathcal{I}$ of with variables $x_1,\dots,x_n$ and clauses $C_1,\dots,C_m$, we construct a bipartite graph $G$ and a set $A\subseteq V(G)$ such that there exists a minimal dominating set containing $A$ if and only if there exists a truth assignment that satisfies all the clauses. The graph $G$ has vertex partition $(X,Y)$, defined as follows.
![A bipartite graph $G$ and a set $A\subseteq V(G)$ constructed from an instance of with variables $x_1,\dots,x_n$ and clauses $C_1,\dots,C_m$. Black vertices constitute the set $A$. Then $A$ can be extended into a minimal dominating set $D$ of $G$ if and only if there is a truth assignment of the variable satisfying all the clauses.[]{data-label="fig:extension-npc"}](extension-npc.pdf)
The first part $X$ contains two special vertices $u$ and $w$, and for every variable $x_i$, one vertex for each of the literals $x_i$ and $\neg x_i$. The second part $Y$ contains one vertex $y_{C_j}$ per clause $C_j$, one vertex $neg_{x_i}$ per variable $x_i$, and two special vertices $v$ and $z$. For every $i \in {\left \{ 1, \dots, n \right \}}$ we make $neg_{x_i}$ adjacent to the two literals $x_i$ and $\neg x_i$ and for every $j \in {\left \{ 1, \dots, m \right \}}$ we make $y_{C_j}$ adjacent to $u$ and to every literal $C_j$ contains. Finally, we add edges to form the path $uvwz$ and set $A=\{neg_{x_1},\dots,neg_{x_n},v,w\}$. Clearly this graph can be constructed in polynomial time from $\mathcal{I}$. The construction is illustrated in Figure \[fig:extension-npc\].
Let us show that $A$ can be extended into a minimal dominating set of $G$ if and only if $\mathcal{I}$ has a truth assignment that satisfies all the clauses. The proof is split into two claims. A *partial assignment* of $\mathcal{I}$ is a truth assignment of a subset of the variables $x_1, \dots, x_n$. Observe that a partial assignment may satisfy all the clauses ( the values of the non-assigned variables do not matter). A partial assignment that satisfies all the clauses is called a *minimal assignment* if no proper subset of the assigned variables admits such a partial assignment.
\[clm:domsat\] Let $S \subseteq \{x_1, \neg x_1, \dots, x_n, \neg x_n\}$ be a set containing at most one literal for each variable. Then $S$ minimally dominates $\{y_{C_1}, \dots, y_{C_m}\}$ if and only if its elements form a minimal assignment of $\mathcal{I}$.
Let $S$ be as above and let $j \in {\left \{ 1, \dots, m \right \}}$. Since $y_{C_j} \notin S$, the set $S$ contains a neighbor $x$ of $y_{C_j}$. By construction, $x$ is a literal appearing in $C_j$. Hence a partial assignment of the variables of $\mathcal{I}$ satisfying all its clauses is given by the literals present in $S$. Moreover, $x$ has a private neighbor $y_{C_{j'}}$, by minimality of $S$. The assignment given by $S$ is hence minimal: not specifying the value of the variable of $x$ would leave the clause $C_{j'}$ unsatisfied.
\[cl:minlit\] If $D$ is a minimal dominating set of $G$ containing $A$, then $D \setminus A \subseteq \{x_1, \neg x_1, \dots,\allowbreak{} x_n, \neg x_n\}$ and it contains at most one literal for each variable.
Notice that $\operatorname{Priv}(A,v) = \{u\}$. If $y_{C_j}$ belongs to $D$ for some $j \in {\left \{ 1, \dots, m \right \}}$, then $\operatorname{Priv}(D,v) = \emptyset$, a contradiction to the minimality of $D$. For similar reasons $u,z\notin D$. Hence $D\cap \{u,z, y_{C_1}, \dots, y_{C_m}\} = \emptyset$. Besides, for every $i \in {\left \{ 1, \dots, m \right \}}$, $D$ contains at most one of $x_i$ and $\neg x_i$, as otherwise $\operatorname{Priv}(D, neg_{x_i})$ would be empty, again contradicting the minimality of $D$. This proves the claim.
If $A$ can be extended into a minimal dominating set $D$ of $G$, then by combining the two claims above, we deduce that $\mathcal{I}$ has truth assignment that satisfies all clauses. Conversely, if $\mathcal{I}$ has such a truth assignment, then there is a set $S$ as in the statement of Claim \[clm:domsat\]. In $S \cup A$, every element of $S$ has a private neighbor, as a consequence of the minimality of $S$ and the fact that no element of $A$ has a neighbor among the clause variables. Besides, each of $neg_{x_1},\dots,neg_{x_n}$ has a private neighbor (because $S$ contains at most one of the two literals for each variable) and it is easy to see that the same holds for $v$ and $w$. Hence $S \cup A$ is a minimal dominating set of $G$.
Given an instance $\mathcal{I}$ of SAT, we constructed in polynomial time an instance $(G,A)$ of $\Dcs{}$ that is equivalent to $\mathcal{I}$. This proves that $\Dcs{}$ is NP-hard.
Limitations of the bicolored argument {#subsec:bic}
-------------------------------------
Let us present a brief argument of why enumerating the minimal dominating sets in a bicolored graph $G(A)$ is -hard if $A$ can contain an arbitrarily large clique and no restriction is put as to the structure of $G-A$ nor its interactions with $A$. In other words, we argue that can be reduced to the problem of enumerating the minimal dominating sets in a bicolored graph $G(A)$ where $A$ is a clique.
Because of Theorem \[thm:cobip-hard\], we know that enumerating the minimal dominating sets of a co-bipartite graph $G$ is -hard. However, note that free to disregard the minimal dominating sets consisting of exactly one vertex in each clique of the partition, every minimal dominating set is included in one of the two cliques. Let $A_1$ and $A_2$ be the two sides of this partition. Observe that as both $A_1$ and $A_2$ induce cliques, they satisfy any property that does not limit the size of the largest clique. Combined with the fact that minimal dominating sets consisting of exactly one vertex in each side of the partition are easy to enumerate, we obtain the desired conclusion.
Note however that this obstacle was circumvented in Theorem \[th:pawdiam\] by keeping track of what the forbidden structures in $G$ imply for the interactions between $G-A$ and $A$. Unfortunately, the arguments were quite ad hoc in nature and it is unclear how far they can be generalized.
This obstacle was bypassed in a different way in Theorem \[thm:ktme\], simply by first enumerating all the minimal dominating sets without a given structure, then using the fact that the structure appears in any remaining dominating set to guess where it does, and finally arguing that the vertices that remain to be dominated cannot induce an arbitrarily large clique. We now show that this technique is in fact very limited.
Limitations of enumerating all minimal dominating sets with a certain structure
-------------------------------------------------------------------------------
We present now a brief argument of why enumerating all $H$-free minimal dominating sets in a graph is -hard unless $H$ is a clique of size at most $2$.
The case where $H$ is not a clique is directly implied by the argument in Section \[subsec:bic\]. We now focus on the case where $H$ is a clique on at least $3$ vertices: it suffices to handle the case where $H$ is a triangle. In other words, we argue that can be reduced to the question of enumerating all triangle-free minimal dominating sets.
Consider a graph $G$. We build an auxiliary graph $G'$ by creating two copies $A$ and $B$ of $V(G)$, creating a vertex $u$, and setting $V(G')=A \cup B \cup \{u\}$. We set $A$ to induce a stable set, $B$ to induce a clique, and the vertex $u$ to be adjacent to all of $A$ and none of $B$. We set the edges between $A$ and $B$ to be such that a vertex in $A$ and a vertex in $B$ are adjacent if and only if the vertices of $G$ they are in bijection to are the same or are adjacent.
Let us consider what the structure of a minimal dominating set $D$ of $G'$ can be, and how easy it is to generate all minimal dominating sets of a given type:
1. $u \not\in D$. We generate all minimal dominating sets of the split graph $G'[A \cup B]$: this can be done in output-polynomial time according to Proposition \[prop:split-properties\]. For each such minimal dominating set, either the intersection with $A$ is non-empty and it is a minimal dominating set of $G'$, or it is empty and we can generate in polynomial-time all additions of a vertex of $A$ that would result in a minimal dominating set of $G'$, if any. Since the number of minimal dominating sets of $G'[A \cup B]$ with empty intersection with $A$ is polynomially bounded by the number of those with non-empty intersection (see Lemma \[lem:triangle-free-candidates-enumeration\], Inequality ), we can generate all minimal dominating sets of $G'$ not containing $u$ in output-polynomial time.
2. $D \cap B \neq \emptyset$ and $u \in D$. Then $|D \cap B|=1$, and for any vertex $v \in B$, the set $\{u,v\}$ is a minimal dominating set of $G'$.
3. $D \cap B = \emptyset$ and $u \in D$. All these minimal dominating sets are triangle-free. We note that there is a bijection between the minimal dominating sets of this type and the minimal dominating sets of $G$.
The first two cases are easy to generate in output-polynomial time. We note that, free again to disregard minimal dominating sets that are easy to generate, enumerating all triangle-free minimal dominating sets of $G'$ boils down to enumerating all minimal dominating sets of $G'$ that are included in $A \cup \{u\}$ and contain $u$. This is equivalent to enumerating all minimal dominating sets of $G$, hence the conclusion.
Note however that there is still hope for this technique when we assume some structure on the whole graph.
Perspectives for further research {#sec:concl}
=================================
In this paper, we investigated the enumeration of minimal dominating sets in graph classes forbidding an induced subgraph $H$. We gave algorithms that run in output polynomial time and polynomial space when $H$ is a clique, or more generally when $H =K_t +e$ and when $H$ is the paw or the diamond. We now discuss possible directions for future research. For simplicity, let us here denote by $\DomEnum{}(H)$ the problem restricted to $H$-free graphs.
The most natural continuation of our work is to seek output-polynomial time algorithms for $\DomEnum{}(H)$ for other choices of the graph $H$. We discuss a possible classification of the graphs $H$ depending whether $\DomEnum{}(H)$ admits an output-polynomial time algorithm, is -hard, or is not know to belong to one of these two cases. We stress that the two first cases may not be disjoint as it is currently an open problem whether admits an output-polynomial time algorithm in general. However, in the current state of the art, such a classification will highlight specific graph classes where the problem could be attacked more easily than in the general case.
Because of Theorem \[thm:cobip-hard\], if $H$ is such that co-bipartite graphs form a subclass of $H$-free graphs then $\DomEnum{}(H)$ is -hard. This includes the cases $H=C_t$ or $H=P_t$ with $t \geq 5$. This is also true for any graph $H$ that has an independent set of size at least three, in particular all graphs $H$ that have at least three connected components and graphs with two connected components where one component has one non-edge. Therefore, all the graphs $H$ with more than one connected component for which $\DomEnum{}(H)$ is not known to be -hard are of the form $H=K_p + K_q$ (where by $+$ we denote the disjoint union), for integers $p,q \geq 1$. We gave an output-polynomial time algorithm for the case where $p=2$ or $q=2$ in Theorem \[thm:ktme\] and leave open the existence of such algorithms for $p,q \geq 3$.
Let us now focus on connected choices of $H$. Besides the case where $H$ is a clique, that we addressed with Theorem \[thm:kt-free\], we settled the case where $H =K_t - e$ for $t=4$ (Theorem \[thm:diamond-free\]). For $t\in\{2,3\}$, $\DomEnum{}(H)$ is output-polynomial time solvable since $(K_t - e)$-free graphs then are, respectively, cliques and disjoint unions of cliques. To the best of our knowledge, it is currently unknown whether $\DomEnum{}(K_t - e)$ for $t \geq 5$ is -hard and whether it is output-polynomial time solvable. We also considered graphs $H$ of the form $(K_t-\{uv,vw\})$ for $t\geq 3$, , graphs obtained from a clique on $t$ vertices by removing two incident edges. When $t = 3$, $(K_t-\{uv,vw\})$-free graphs are exactly the complete multipartite graphs, for which an output-polynomial time algorithm can be obtained as in the proof of Lemma \[lem:paw-free-candidates-enumeration\]. We dealt with the case $t=4$ in Theorem \[thm:paw-free\] and leave open the cases of larger $t$.
Regarding the exclusion of specific graphs, we note that the status of $\DomEnum{}(P_t)$ is completely explored: either $t\leq 4$ and an output-polynomial time algorithm is known, or $t \geq 5$ and the problem is -hard, as noted above. Among graph classes defined by forbidding an induced cycle, we proved that $\DomEnum{}(C_3)$ is output-polynomial time solvable in Theorem \[thm:triangle-free\] and noted above that $\DomEnum{}(C_t)$ is -hard for $t\geq 5$, so only $\DomEnum{}(C_4)$ remains to be classified. The graph $C_4$ is also the only graph on at most 4 vertices for which $\DomEnum{}(H)$ has not been classified yet. Other graph classes that are closed by taking induced subgraphs and where no output-polynomial algorithm for neither -hardness proof are known include unit-disk graphs [@Kante2008; @GOLOVACH201630] and comparability graphs.
An other natural research direction is to optimize the running times of our algorithms or to prove that this is not possible. Theorem \[thm:ext\] suggests that no improvement of our results can be obtained using backtrack search. We leave as an open problem whether there are polynomial delay algorithms for in the cases that we considered.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank Paul Ouvrard for extensive discussions on the topic of this paper. We gratefully acknowledge support from Nicolas Bonichon and the Simon family for the organization of the $3^{\textrm{rd}}$ Pessac Graph Workshop, where part of this research was done. We also thank the organisers of the [Dagstuhl Seminar 18421](https://www.dagstuhl.de/18421) on algorithmic enumeration where some ideas present in this paper have been discussed. Last but not least, we thank Peppie for her unwavering support during the work sessions.
[KLM[[$^{+}$]{}]{}15]{}
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[^1]: A preliminary version of this article will appear in the proceedings of the 36^th^ Symposium on Theoretical Aspects of Computer Science (STACS 2019) [@bonamy2019DHRenum]. The first author has been supported by the ANR project GrR ANR-18-CE40-0032. The second author has been supported by the ANR project GraphEn ANR-15-CE40-0009. The last author has been supported by the ERC consolidator grant <span style="font-variant:small-caps;">Distruct</span>-648527.
[^2]: The complement of a bipartite graph.
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0.2in [**John Ellis**]{}$^1$, [**Keith A. Olive**]{}$^{2}$, [**Yudi Santoso**]{}$^{2}$ and [**Vassilis C. Spanos**]{}$^{2}$ 0.1in [*$^1$[TH Division, CERN, Geneva, Switzerland]{}\
$^2$[William I. Fine Theoretical Physics Institute,\
University of Minnesota, Minneapolis, MN 55455, USA]{}*]{}\
0.2in [**Abstract**]{}
Specific models of supersymmetry breaking predict relations between the trilinear and bilinear soft supersymmetry breaking parameters $A_0$ and $B_0$ at the input scale. In such models, the value of $\tan \beta$ can be calculated as a function of the scalar masses $m_0$ and the gaugino masses $m_{1/2}$, which we assume to be universal. The experimental constraints on sparticle and Higgs masses, $b \to s \gamma$ decay and the cold dark matter density $\Omega_{CDM} h^2$ can then be used to constrain $\tan
\beta$ in such specific models of supersymmetry breaking. In the simplest Polonyi model with $A_0 = (3 - \sqrt{3})m_0 = B_0 + m_0$, we find $11
\lappeq \tan \beta \lappeq 20$ ($\tan \beta \simeq 4.15$) for $\mu > 0$ ($\mu < 0$). We also discuss other models with $A_0 = B_0 + m_0$, finding that only the range $-1.9 \la A_0/m_0 \la 2.5$ is allowed for $\mu > 0$, and the range $1.25 \la A_0/m_0 \la 4.8$ for $\mu < 0$. In these models, we find no solutions in the rapid-annihilation ‘funnels’ or in the ‘focus-point’ region. We also discuss the allowed range of $\tan \beta$ in the no-scale model with $A_0 = B_0 = 0$. In all these models, most of the allowed regions are in the $\chi - {\tilde
\tau_1}$ coannihilation ‘tail’.
Introduction
============
One of the most important and least understood problems in the construction of supersymmetric models is the mechanism of supersymmetry breaking [@BIM]. Direct exploration of this may be far beyond our experimental reach for some considerable time, so we may have to rely on indirect information provided by measurements of the different soft supersymmetry-breaking parameters. Even here, so far we have no determinations, only limits obtained from accelerator experiments, cosmology and theoretical considerations. It is commonly assumed that the soft supersymmetry-breaking scalar masses $m_0$ have universal values at some GUT input scale, as do the gaugino masses $m_{1/2}$ and the trilinear soft supersymmetry-breaking parameters $A_0$, which is referred to as the constrained MSSM (CMSSM). One then frequently analyzes the impacts of the different phenomenological limits on the allowed values of $m_{1/2}$ and $m_0$ as functions of $\tan \beta$, the ratio of Higgs vacuum expectation values, assuming some default value of $A_0$ and determining the Higgs mixing parameter $\mu$ and the pseudoscalar Higgs mass $m_A$ by using the electroweak vacuum consistency conditions (see [@efgos] - [@hyperbolic] for recent studies of this type). The tree-level value of $m_A$ may be related to the bilinear soft supersymmetry-breaking parameter $B$, via $m_A^2 = - 2 B \mu / \sin 2 \beta$.
Specific models of supersymmetry breaking predict relations between these different soft supersymmetry-breaking parameters. For example, certain ‘no-scale’ models [@noscale] may predict $m_0 = 0$ at the Planck scale, and we have analyzed the extent to which this assumption is compatible with the phenomenological constraints, taking account of the possible running of $m_0$ between the Planck scale and the GUT scale [@eno5]. Here we analyze a different question, namely the consistency of some proposed relations between $m_0$, $A_0$ and $B_0$ which take the characteristic form A\_0 = m\_0, B\_0 = m\_0. \[hats\] A generic minimal supergravity model [@sugr2] prediction is that ${\hat B} = {\hat A} -1$ [@mark], and the simplest Polonyi model [@pol] predicts that $\vert {\hat A} \vert = 3 - \sqrt{3}$ [@bfs].
The first of the two relations (\[hats\]) may be used to replace an [*ad hoc*]{} assumption on the input value of $A_0$. The second imposes an important consistency condition on the value of $m_A$, which was otherwise treated as a dependent quantity that was not constrained [*a priori*]{}. For any given value of $m_{1/2}$ and $m_0$, this constraint is satisfied for only one specific value of $\tan \beta$. Therefore, the results of imposing the two constraints (\[hats\]) may conveniently be displayed in a single $(m_{1/2}, m_0)$ plane across which $\tan \beta$ varies in a determined manner. The phenomenological constraints on $m_{1/2}$ and $m_0$ can then be used to provide both upper and lower limits on the allowed values of $\tan \beta$.
In this paper, we analyze these constraints on $\tan \beta$ as functions of ${\hat A}$ in the generic scenario (\[hats\]), including the Polonyi case ${\hat A} = 3 - \sqrt{3}$ and other models with ${\hat A} = {\hat B}
+ 1$. In the Polonyi case, we find that $11 \lappeq \tan \beta \lappeq 20$ for $\mu > 0$, with only a small area in the $m_{1/2} - m_0$ plane with $\tan \beta \simeq 4.15$ surviving for $\mu < 0$. In general, we find consistent solutions for $-1.9 \la {\hat A} \la 2.25$ for $\mu > 0$ and $1.25 \la {\hat A} \la 4.8$ for $\mu < 0$. We also explore the range of $\tan \beta$ that is allowed in a no-scale scenario with $A_0 = B_0 = 0$ at the GUT scale. It should, however, be recalled that the no-scale boundary conditions [@noscale] were originally proposed to hold at the supergravity scale, which might be significantly above the GUT scale. In this case, renormalization-group running between these scales would generate ${\hat A}$ and ${\hat B} \ne 0$ at the GUT scale.
Models of Supersymmetry Breaking
================================
In this Section, we review briefly models that yield the characteristic patterns of supersymmetry breaking whose phenomenology we study later in the paper. We assume an $N = 1$ supergravity framework, interpreted as a low-energy effective field theory. This may be characterized by a K[ä]{}hler function $K$ that describes the kinetic terms for the chiral supermultiplets $\Phi \equiv (\zeta, \phi)$, where the $\zeta$ represent hidden-sector fields and the $\phi^i$ observable-sector fields, a holomorphic function $f(\Phi)$ that yields kinetic terms for the gauge supermultiplets $A_a$ as well as gauge couplings, and a holomorphic superpotential $W(\Phi)$. We assume the form of the gauge kinetic function $f$ to be such that the gaugino masses $m_{1/2}$ are universal at the GUT input scale, as are the gauge couplings.
So-called minimal supergravity theories have $K = \Sigma_i |\Phi^i|^2$, whereas no-scale models have non-trivial K[ä]{}hler functions such as $K =
- 3{\rm ln}(\zeta + \zeta^\dagger - \Sigma_j |\phi^j|^2)$. The scalar potential (neglecting any gauge contributions) is in general [@sugr2] V(,\^\*) = e\^K \[generalpot\] where we are working in Planck units. For minimal supergravity, we have $K^i =
{\phi^i}^* + {W^i}/W$, $K_i = \phi_i + W_i^*/W^*$, and $({K^{-1}})^j_i = \delta ^j_i$, and the resulting scalar potential is V(,\^\*) = e\^[ \_i [\^i]{}\^\*]{} . \[msgpot\] In this minimal case, the soft supersymmetry-breaking scalar masses $m_0$ are universal at the input GUT scale, with [@BIM] m\_0\^2 = m\_[3/2]{}\^2 + , \[msugra\] where $m_{3/2}$ is the gravitino mass and $\Lambda$ is the tree-level cosmological constant. If we further assume that the superpotential $W(\Phi)$ may be separated into pieces $F$ and $g$ that are functions only of observable-sector fields $\phi^i$ and hidden-sector fields $\zeta$, respectively, so that the superpotential parameters of the observable-sector fields do not depend on the hidden-sector fields, then the trilinear terms $A_0$ and bilinear terms $B_0$ are also universal, and [@BIM] B\_0 = A\_0 - m\_[3/2]{}. \[BA\] Finally, if we further assume that $\Lambda = 0$, then $m_0 = m_{3/2}$ and [@BIM] = - 1, \[BAhat\] which is one of the principal options we study below.
One of the primary motivations for the CMSSM, and for scalar mass universality in particular, comes from the simplest model for local supersymmetry breaking [@pol], which involves just one additional chiral multiplet $\zeta$ in addition to the observable matter fields $\phi_i$. We consider, therefore, a superpotential which is separable in this so-called Polonyi field and the $\phi_i$, and of the simple form g() = (+ ) \[polonyi\] with $\vert \beta \vert = 2 - \sqrt{3}$, ensuring that $\Lambda = 0$. The scalar potential in this model takes the form [@bfs] V & = & e\^[(||\^2 + ||\^2)]{} . \[cpot0\] We next expand the expression (\[cpot0\]) and drop terms that are suppressed by inverse powers of the Planck scale, which can be done simply by dropping terms of mass dimension greater than four. In the positive case, after inserting the vev for $\zeta$, $\langle \zeta \rangle = \sqrt{3} - 1$, we have [@bfs]: V & = & e\^[(4 - 2)]{}\
& = & e\^[(4 - 2)]{} |[F ]{}|\^2\
& & + m\_[3/2]{} e\^[(2 - )]{}( - F + h.c.) ) + m\_[3/2]{}\^2 \^\* , \[cpot\] which deserves some discussion.
First, up to an overall rescaling of the superpotential, $F \to
e^{\sqrt{3}-2} F$, the first term is the ordinary $F$-term part of the scalar potential of global supersymmetry. The next term, which is proportional to $m_{3/2}$, provides universal trilinear soft supersymmetry-breaking terms $A = (3 -
\sqrt{3}) m_{3/2}$ and bilinear soft supersymmetry-breaking terms $B = (2 - \sqrt{3}) m_{3/2}$, i.e., a special case of the general relation (\[BA\]) above between $B$ and $A$. Finally, the last term represents a universal scalar mass of the type advocated in the CMSSM, with $m_0^2 = m_{3/2}^2$, since the cosmological constant $\Lambda$ vanishes in this model, by construction.
As we have seen above, the generation of such soft terms is a rather generic property of low-energy supergravity models [@mark] and many of these conclusions persist when one generalizes the Polonyi potential. For example, if we choose $g(\zeta)$ so that [^1] $\langle g \rangle = \nu$, $\langle \partial g / \partial \zeta \rangle =
a^* \nu$, and $\langle \zeta \rangle = b$, the condition that $\Lambda =
0$ at $\zeta = b$ implies $|a + b|^2 = 3$. Substituting these expectation values in (\[cpot0\]), we find [@mark] that $A = b^* ( a+ b) \nu $ and once again $B = A - \nu$, but now with $A$ free. The constant $\nu$ determines the gravitino mass, and hence $m_0$, through: $m_0 = m_{3/2} = e^{{1 \over 2} b b^*} \nu$.
Another broad option for supersymmetry breaking is that provided by no-scale models [@noscale], of which the simplest example is K = - 3 [ln]{} ( + \^- \_i | \^i |\^2 ). \[noscale\] No-scale models have the universal values m\^2\_0 = 0, A\_0 = 0, B\_0 = 0 \[noscalesusyx\] at the input supergravity scale. The possibility that $m_0 = 0$ at the GUT scale has recently been studied [@eno5; @emy], and shown to be excluded by the phenomenological constraints. However, it was recalled that the input supergravity scale could be somewhat higher than the GUT scale, in which case one might find $m_0 \ne 0$ already at the GUT scale. Clearly the same could also be true for $A_0$ and $B_0$. However, the deviations from (\[noscalesusyx\]) are model-dependent, and we think it important to be aware of the phenomenological fate of the clear-cut $A_0 =
B_0 = 0$ option for supersymmetry breaking.
Electroweak Vacuum Conditions
=============================
Before discussing the phenomenological constraints on this model, we first show more precisely how the relation between $A$ and $B$ can be used to determine $\tan \beta$ when the radiative electroweak symmetry breaking conditions are applied.
In general, we start with the following set of input parameters defined at the GUT scale: $m_{1/2}$, $m_0$, $A_0$, $B_0$ and the Higgs mixing parameter $\mu_0$. By running the full renormalization-group equations (RGEs) down to the weak scale and minimizing the Higgs potential, one can solve for the Higgs vevs and masses or, equivalently, $M_Z$, $\tan \beta$, and $m_A$. At the tree level, these solutions take the simple form: M\_Z\^2 & = & [2 (m\_1\^2 + \^2 - (m\_2\^2 + \^2) \^2 ) (\^2 -1)]{}\
2 & = & [ - 2 B ]{}/(m\_1\^2 + m\_2\^2 + 2 \^2)\
m\_A\^2 & = & m\_1\^2 + m\_2\^2 + 2 \^2 \[treerel\] where $m_1$ and $m_2$ are the soft supersymmetry-breaking masses for the two Higgs doublets at the electroweak scale. However, since $M_Z$ is known, and because the full one-loop set of tadpole equations does not admit an analytical solution for $\tan \beta$, it is customary to use $M_Z$ and $\tan \beta$ as inputs and instead solve for $\mu$ and $B$: \^2 & = &\
B & = & -[1 2]{} (m\_1\^2 + m\_2\^2 + 2 \^2) 2 + \_B \[onelooprel\] where $\Delta_B$ and $\Delta_\mu^{(1,2)}$ are loop corrections [@Barger:1993gh; @deBoer:1994he; @Carena:2001fw], and here $m_{1,2} \equiv m_{1,2}(\mz)$. Since $\Delta_\mu$ depends on $\tan \beta$ and $\Delta_B$ depends on both $\mu$ and $\tan \beta$ in a nonlinear way, it is not possible to write down an analytical solution for $\tan \beta$. The above set of inputs and outputs defines the CMSSM.
In the types of models discussed in the previous section, we have specific GUT-scale boundary conditions on $B_0$, namely $B_0 = A_0 - m_0$ in minimal supergravity models or $B_0 = A_0 = 0$ in no-scale models. Therefore, we cannot treat the value of $B(M_Z)$ as a free parameter, and instead must solve numerically for $\tan \beta$. Thus, a given value of $m_{1/2}$, $m_0$, $A_0/m_0$, and $sgn(\mu)$ will correspond to a definite value for $\tan \beta$. When combined with the phenomenological constraints discussed below, we can determine for a particular model of supersymmetry breaking the allowed (and often quite restricted) values of $\tan \beta$.
Phenomenological Constraints on $m_{1/2}$ and $m_0$
===================================================
We apply the standard LEP constraints on the supersymmetric parameter space, namely $m_{\chi^\pm} > 104$ GeV [@LEPsusy], $m_{\tilde e} > 99$ GeV [@LEPSUSYWG_0101] and $m_h >
114$ GeV [@LEPHiggs]. The former two constrain $m_{1/2}$ and $m_0$ directly via the sparticle masses, and the latter indirectly via the sensitivity of radiative corrections to the Higgs mass to the sparticle masses, principally $m_{\tilde t, \tilde b}$ [^2]. We use the latest version of [FeynHiggs]{} [@FeynHiggs] for the calculation of $m_h$. We require the branching ratio for $b \rightarrow
s \gamma$ to be consistent with the experimental measurements [@bsg]. We also indicate the regions of the $(m_{1/2}, m_0)$ plane that are favoured by the BNL measurement [@newBNL] of $g_\mu - 2$ at the 2-$\sigma$ level, corresponding to a deviation of $(33.9 \pm 11.2) \times 10^{-10}$ from the Standard Model calculation of [@Davier] using $e^+ e^-$ data. We are however aware that this constraint is still under discussion and do not use it to constrain $\tan \beta$. All the $\mu > 0$ planes would be consistent with $g_\mu - 2$ at the 3-$\sigma$ level, whereas $\mu < 0$ is disfavoured even if one takes a relaxed view of the $g_\mu - 2$ constraint.
Finally, we impose the following requirement on the relic density of neutralinos $\chi$: $0.094 \le \Omega_\chi h^2 \le 0.129$, as suggested by the recent WMAP data [@wmap], in agreement with earlier indications. We recall that several cosmologically-allowed domains of the $(m_{1/2},
m_0)$ planes for different values of $\tan \beta$ have been discussed previously in the general CMSSM framework [@efgos] - [@efgosi], [@otherOmega] - [@hyperbolic]. One is a ‘bulk’ region at low $m_{1/2}$ and $m_0$, which has been squeezed considerably by the WMAP constraint on $\Omega_\chi h^2$. A second region is the $\chi - {\tilde
\tau_1}$ coannihilation ‘tail’ [@stauco; @moreco], which stretches to larger $m_{1/2}$, close to the boundary of the acceptable region where $m_\chi \le m_{\tilde
\tau_1}$. In the wake of WMAP, this ‘tail’ is now much narrower - because of the smaller range of $\Omega_\chi h^2$ - and shorter - because of the more stringent upper limit on $\Omega_\chi h^2$ [@eoss; @wmapothers]. A third region is the ‘funnel’ due to rapid $\chi \chi \to H, A$ annihilation that occurs at larger $m_0$ and $m_{1/2}$ [@efgosi; @funnel]. Finally, the fourth domain is the ‘focus-point’ region at large $m_0$, close to the boundary where radiative breaking of electroweak symmetry is no longer possible [@focus; @hyperbolic].
We see in the next Section that the ‘funnel’ and ‘focus-point’ regions are not present in the simple models of supersymmetry breaking introduced earlier, whilst the ‘bulk’ region is possible only for a very restricted range of $\tan \beta$. On the other hand, the coannihilation ‘tail’ generally remains permitted.
Examples of $(m_{1/2}, m_0)$ Planes
===================================
We display in Fig. \[fig:Polonyi\] the contours of $\tan \beta$ (solid blue lines) in the $(m_{1/2}, m_0)$ planes for selected values of ${\hat
A}$, ${\hat B}$ and the sign of $\mu$. Also shown are the contours where $m_{\chi^\pm} > 104$ GeV (near-vertical black dashed lines) and $m_h >
114$ GeV (diagonal red dash-dotted lines). The excluded regions where $m_\chi > m_{\tilde \tau_1}$ have dark (red) shading, those excluded by $b
\to s \gamma$ have medium (green) shading, and those where the relic density of neutralinos lies within the WMAP range $0.094 \le \Omega_\chi
h^2 \le 0.129$ have light (turquoise) shading. Finally, the regions favoured by $g_\mu - 2$ at the 2-$\sigma$ level are medium (pink) shaded.
As seen in panel (a) of Fig. \[fig:Polonyi\], when $\mu > 0$ and ${\hat
A} = -1.5$, close to its minimum possible value, the contours of $\tan
\beta$ rise diagonally from low values of $(m_{1/2}, m_0)$ to higher values, with higher values of $\tan \beta$ having lower values of $m_0$ for a given value of $m_{1/2}$. The $m_h = 114$ GeV contour rises in a similar way, and regions above and to the left of this contour have $m_h < 114$ GeV and are excluded. Therefore, only a very limited range of $\tan \beta \sim 4$ is compatible with the $m_h$ and $\Omega_{CDM} h^2$ constraints. At lower values of ${\hat A}$, the slope of the Higgs contour softens and even less of the parameter space is allowed. Below ${\hat A} \simeq -1.9$, the entire $m_{1/2} - m_0$ plane is excluded. When ${\hat
A}$ is increased to 0.75, as seen in panel (b) of Fig. \[fig:Polonyi\], both the $\tan \beta$ and $m_h$ contours rise more rapidly with $m_{1/2}$, and a larger range $ 9 \la \tan \beta \la 14$ is allowed [^3]. In the simplest Polonyi model with ${\hat A} = 3 - \sqrt{3}$ shown in panel (c) of Fig. \[fig:Polonyi\], we see that the $\tan \beta$ contours have noticeable curvature. In this case, the Higgs constraint combined with the relic density requires $\tan \beta
\gappeq 11$, whilst the relic density also enforces $\tan \beta \lappeq
20$ [^4]. Finally, in panel (d) of Fig. \[fig:Polonyi\], when ${\hat A} = 2.0$, close to its maximal value for $\mu > 0$, the $\tan \beta$ contours turn over towards smaller $m_{1/2}$, and only relatively large values $25 \la \tan \beta \la 35$ are allowed by the $b \to s \gamma$ and $\Omega_{CDM} h^2$ constraints, respectively.
In the case of $\mu < 0$, negative values of ${\hat A}$ are not allowed, and only a tiny area in the $(m_{1/2}, m_0)$ plane near the end point of the coannihilation tail around $m_{1/2} = 1000$ GeV is allowed in the positive Polonyi case ${\hat A} = 3 - \sqrt{3}$, as seen in panel (a) of Fig. \[fig:Polonyin\]. This is because the Higgs and $\Omega_{CDM} h^2$ constraints are barely compatible in this case, and allow only $\tan \beta
\simeq 4.15$. At larger values of ${\hat A}$, the allowed region is extended, as exemplified in panel (b) of Fig. \[fig:Polonyin\] for the case ${\hat A} = 2.0$, where a small region around $\tan \beta \simeq 5.5
- 5.7$ is allowed. This panel shows that, approximately, the value of $\tan \beta$ depends only on the ratio $m_{1/2}/m_0$ [@th].
There are several generic patterns in the results above that can be explained qualitatively, as follows. First, we notice that for any given value of $(m_{1/2}, m_0)$, $\tan \beta$ increases as ${\hat A}$ increases. The reason for this can be found by looking at the second equation of (\[treerel\]), and setting $A_0 = B_0 + m_0$. For large $\tan \beta$, $\sin 2 \beta \sim 1/\tan \beta$, so $B$ at the weak scale is inversely proportional to $\tan \beta$, at the tree level. In the $\mu > 0$ case, this tree-level value of $B$ is negative, so its value [*grows*]{} as $\tan \beta$ increases. While loop corrections are generally negative for $\mu > 0$, and RGE corrections to obtain $B(M_X)$ are positive, the monotonic growth of $B_0$ with $\tan \beta$ is preserved. Thus the resulting value of $B_0$, and hence also $A_0$, increases with $\tan
\beta$. In the $\mu < 0$ case, the tree-level value of $B$ is generally positive (the exception being when $m_1^2 + m_2^2 + 2 \mu^2 < 0$), and so its value [*decreases*]{} as $\tan \beta$ increases. However, there are some terms in the loop correction $\Delta_B$ that are proportional to $\mu \tan
\beta$ and flip the sign of $\Delta_B$ at a particular value of $\tan
\beta$, so that the full one-loop $B(M_W)$ is then again an increasing function of $\tan \beta$, and likewise $A_0$.
Using similar arguments, we can further understand the different behaviours of the $\tan\beta$ contours when $\mu$ is positive or negative with fixed ${\hat A}$, for example in the last panels in Fig. \[fig:Polonyi\] and Fig. \[fig:Polonyin\] for ${\hat A}=2$. To this end, look at the second equation in (\[onelooprel\]), bearing in mind that $\sin 2 \beta \sim 1/\tan \beta$. For $\mu>0$ and fixed $m_0$, as $m_{1/2}$ increases both $\Delta_B$ and the RGE corrections to $B$ increase, yielding a relatively constant value for $\tan\beta$ when the growth of the term $-\Delta_B$ almost compensates the positive RGE corrections. For large values of $m_{1/2}$, the RGE corrections take over, resulting in the bending of the $\tan\beta$ contours. On the other hand, for $\mu<0$, the flipping of the sign of $\Delta_B$ described in the paragraph above results in different behaviour. In this case, as $m_{1/2}$ increases with fixed $m_0$, $\tan\beta$ always decreases.
In panel (a) of Fig. \[fig:Polonyin\], the magnitude of the tree level value of $B$ at the weak scale increases with $m_0$, decreasing the value of $\tan
\beta$. However, the loop correction is also growing, tending to increase $\tan \beta$. We see from the figure that $\tan \beta$ is first decreasing and then increasing as $m_0$ is increased. This behaviour is different from panel (b) of Fig. \[fig:Polonyin\], where the tree level value of $B$ at the weak scale is decreasing with $m_0$, and dominates the determination of $\tan \beta$, which is now increasing monotonically.
At high values of ${\hat A}$ (and high $\tan \beta)$, the off-diagonal elements in the squark mass matrix become large at large $m_0$. Therefore, we find no solutions which are phenomenologically viable above a certain value of ${\hat A}$. This is because the regions where the LSP is the $\stau$ or the $\stop$ close off the parameter space [^5]. In fact, this feature is generic in the CMSSM as shown in Fig. 3 of [@efgo]. This effect is more severe at large $\tan\beta$, which further compounds the difficulty in going to large values of ${\hat A}$ in the type of models discussed here.
Finally, we note the absences of both the funnel and the focus-point regions. In the case of the funnel, this is due to the relatively small values of $\tan \beta$ allowed in the class of models considered here: we recall that the funnel region appears only for large $\tan \beta \gappeq 45$ for $\mu > 0$ and $\tan \beta \gappeq 30$ for $\mu < 0$ in the CMSSM.
To understand the absence of the focus-point region, we refer to [@hyperbolic], where it was shown that the position of the focus point is sensitive to the value of $A_0$. As $A_0$ is increased, the focus point is pushed up to higher values of $m_0$. Here, with $A_0 \propto m_0$, the focus-point region recedes faster than $m_0$ if ${\hat A}$ is large enough, and is therefore never encountered. For small ${\hat A}$, $\tan
\beta$ is small at large $m_0$, as shown in panel (b) of Fig. \[fig:Polonyi\], so we do not find a focus point in this case, either. In addition, as can be inferred from the small disconnected segment of the $\tan \beta = 10$ contour in the top left corner of panel (c), all the $\tan \beta$ contours loop back down to lower $m_0$ before reaching the focus-point region.
The above analysis shows that the ‘bulk’ $\Omega_{CDM} h^2$ region is almost completely excluded by the Higgs constraint, but a larger fraction would be allowed if we allowed a 2-GeV error in the CMSSM Higgs mass calculation, or if $m_t$ turns out to be significantly greater than $175$ GeV. Almost all the coannihilation ‘tail’ region is allowed. As remarked on above, there is no ‘funnel’ region at large $m_{1/2}$ and $m_0$, nor any ‘focus-point’ region at large $m_0$.
Bounds on $\tan \beta$
======================
It is clear from the previous figures that only limited ranges of $\tan
\beta$ are consistent with the phenomenological constraints within any given pattern of supersymmetry breaking. We display in Fig. \[fig:tanbeta\] the ranges of $\tan \beta$ allowed as a function of ${\hat A}$. For ${\hat B} = {\hat A} -1$ and $\mu > 0$, as shown by the solid lines, we see that the upper and lower limits on $\tan \beta$ both increase monotonically with ${\hat A}$. We find consistent solutions to all the phenomenological constraints only for - 1.9 < < 2.5, \[rangeA\] over which range 3.7 < 46. \[rangetb\] Generally speaking, the range of $\tan \beta$ for any fixed value of ${\hat A} < 0$ is very restricted, with larger ranges of $\tan \beta$ becoming allowed for ${\hat A} > 0$. In the specific case of the simplest Polonyi model with positive ${\hat A} = 3 - \sqrt{3}$, we find 11 < < 20, \[Polonyitb\] whereas the range in $\tan \beta$ for the negative Polonyi model with ${\hat A} = \sqrt{3} - 3$, is 4.4 – 4.6. Furthermore, the difference between the upper and lower limits on $\tan \beta$ never exceeds $\sim$ 14 for any fixed value of ${\hat A}$.
The corresponding results for $\mu < 0$ are 1.2 < < 4.8, \[rangeAn\] over which range 4 < 26. \[rangetbn\] The range of ${\hat A}$ is shifted, and the range of $\tan \beta$ reduced, as compared to the case of $\mu > 0$. In particular, the negative Polonyi model is disallowed and the positive version is allowed only for $\tan
\beta \sim 4.15$.
No-Scale Models
===============
We display in Fig. \[fig:noscale\] the results of a similar analysis for the no-scale case ${\hat A} = {\hat B} = 0$. For $\mu > 0$, the allowed range of $\tan \beta$ is 16 < < 30, \[tbpnoscale\] where the lower limit is provided by the Higgs search, and the upper limit is at the tip of the coannihilation ‘tail’. For $\mu < 0$, the same constraints allow just a small range around $\tan \beta \sim 4.8$. These two ranges are both shown as ‘error bars’ in Fig. \[fig:tanbeta\].
However, the other no-scale condition $m_0 = 0$ is not allowed for either sign of $\mu$, the minimum being $m_0 \simeq 62$ GeV for $\mu > 0$ and $\tan \beta \simeq 16$. The fact that $m_0 \ne 0$ is no surprise, since the same conclusion was reached previously without imposing the supplementary no-scale conditions ${\hat A} = {\hat B} = 0$ [@eno5]. However, as we have already pointed out, the no-scale boundary conditions should be interpreted as applying at the supergravity scale, so it is possible that $m_0, {\hat A}, {\hat B}$ all $\ne 0$, albeit small, at the GUT scale. We note that in this case, there is in fact a focus-point region at roughly the same position as in the CMSSM with $A_0 = 0$.
Conclusions
===========
We have shown in this paper that only a restricted range of $\tan \beta$ is allowed in any specific pattern of supersymmetry breaking. We have illustrated this point by discussions of minimal supergravity models with ${\hat A} = {\hat B} + 1$ and no-scale models with ${\hat A} = {\hat B} =
0$, but the same comment would apply to other models of supersymmetry breaking not discussed here. Within the class of minimal supergravity models, we have selected in particular the simplest Polonyi model with $\vert {\hat A} \vert = 3 - \sqrt{3}$, but also discussed models with other values of ${\hat A}$, finding a rather restricted range, in particular for $\mu < 0$.
One inference from our analysis is that an experimental determination of $\tan \beta$ could be a useful discriminator between different models of supersymmetry breaking. To understand the potential scope of this analysis tool, it would be necessary to study a wider class of models of supersymmetry breaking than those discussed here.
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[^1]: One could also consider models in which several fields $\zeta_i$ contribute to supersymmetry breaking.
[^2]: We assume as our default that $m_t = 175$ GeV.
[^3]: Note that the contours for given values of $\tan \beta$ always intersect the axis $m_0 =
0$ at the same value of $m_{1/2}$.
[^4]: The other Polonyi case with ${\hat A} = - 3 + \sqrt{3}$ (not shown) is very similar to panel (a) for ${\hat A} = - 1.5$, and has a very narrow allowed range of $\tan \beta \sim 4.5$.
[^5]: The neutralino-stop coannihilation region which occurs when $A_0$ is large in the small $(m_{1/2}, m_0)$ region [@stopco] does not appear in our analysis because $A_0$ is still too small.
|
---
abstract: 'We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,{\mathrm{k}})$ to its maximal compact subgroup $K=\mathrm{GL}(3,{\mathscr{R}})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.'
address:
- 'Department of Mathematics, University of Bristol, UK'
- 'Department of Mathematics and Statistics, University of Ottawa, Canada'
author:
- 'Peter S. Campbell'
- Monica Nevins
title: 'Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic field'
---
[^1]
Introduction
============
The complex representations of $p$-adic algebraic groups are of great interest, both in their own right and in what they can reveal through the Langlands program in number theory. The representation theory of $p$-adic groups also often mirrors the theory for real Lie groups, and it is especially interesting to see how analogous results will develop.
To this end, one goal is to examine the finer structure of representations by considering their restrictions to compact open subgroups. The theory of types promises that one can classify representations in the Bernstein decomposition by identifying among certain representations of compact open subgroups which ones they contain. In contrast, in the theory of real Lie groups, the *maximal* compact subgroups have a crucial role, encoding as they do all the topology of the group, and one classifies irreducible unitary representations by classifying the irreducible Harish-Chandra modules. Our interest is to explore to what extent information about the representations of the $p$-adic group resides in the maximal compact subgroup.
Representations of compact subgroups of $p$-adic groups are very tangible at a number of levels. Firstly, the representations of sufficiently small (exponentiable) compact open subgroups can all be constructed using Kirillov theory, as shown by Howe [@Howe]. Secondly, each compact open subgroup is pro-finite and consequently its representation theory is largely determined by the representation theory of Lie groups over finite local rings. Finally, any admissible representation of a $p$-adic group decomposes with finite multiplicity upon restriction to a compact open subgroup and so one can expect to recover information about the original representation by examining these constituents.
That said, the maximal compact subgroups are not exponentiable so Howe’s theory does not apply; and furthermore little information is known in general about the representation theory of Lie groups over local rings. Therefore one objective of this study is to provide some interchange between the representation theories of $p$-adic groups and of Lie groups over local rings.
Let ${\mathrm{k}}$ be a $p$-adic field and denote by ${\mathscr{R}}$ its integer ring. In this paper we consider the group ${G}= \mathrm{GL}(3,{\mathrm{k}})$ and let $K = \mathrm{GL}(3,{\mathscr{R}})$ be a maximal compact subgroup. In [@CNu], the authors considered unramified principal series representations and showed how their restriction to $K$ decomposed as per the double cosets in $K$ of smaller compact open subgroups $C_{{\mathtt{c}}}$ (defined in Section \[S:unram\]). In [@CNu], the added assumption that the inducing character was trivial implied that every double coset supported an intertwining operator of the representation, an assumption we relax here.
This paper is organized as follows. In Section \[S:unram\] we set our notation and recall some necessary results from [@CNu]. The key calculation for determining the decomposition is the determination of the double cosets in $C_{{\mathtt{c}}}\backslash K / C_{{\mathtt{d}}}$ which support intertwining operators for the restricted principal series representation; this is the main result in Section \[S:support\]. We go on to consider questions of irreducibility in Section \[S:counting\] and conclude with several examples to illustrate these decompositions in Section \[S:examples\].
The question of parameterizing double cosets of $B$ in $K$, and more generally of the subgroups $C_{(n,n,n)}$ in $K$, has been visited and solved by several authors with various goals in mind. In [@Onn] the goal was to look at which Bruhat decompositions would be independent of the characteristic of the residue field; the answer was that only $\mathrm{GL}(2,{\mathrm{k}})$ has this property. This implies, in particular, that the decomposition of principal series is essentially independent of $p$ for $\mathrm{GL}(2,{\mathrm{k}})$ (see [@Monica1; @Silberger]) but will depend on the properties of the residue field in all other cases. Several authors have considered related questions on the decomposition of representations of $p$-adic groups upon restriction to a maximal compact subgroup. These include the works on Silberger on $\mathrm{GL}(2,{\mathrm{k}})$ [@Silberger], the second author on $\mathrm{SL}(2,{\mathrm{k}})$ [@Monica1], and Bader and Onn [@Bader] on the Grassmann representation of $\mathrm{GL}(n,{\mathrm{k}})$. Gregory Hill has also constructed classes of representations of $\mathrm{GL}(n,{\mathscr{R}})$ in [@Hill]; a key part of his results was the determination of the double cosets of the subgroups $C_{(0,j,j)}$ in $K$.
Notation and Background {#S:unram}
=======================
Let ${\mathrm{k}}$ be a $p$-adic field of characteristic 0 and residual characteristic $p$. Let $q$ denote the number of elements in the residual field of ${\mathrm{k}}$. We assume throughout that $p>2$ and $q>3$. Denote the integer ring of ${\mathrm{k}}$ by ${\mathscr{R}}$ and the maximal ideal of ${\mathscr{R}}$ by ${\mathscr{P}}$. Choose a uniformizer $\pi$ and normalize the discrete valuation on ${\mathrm{k}}$ so that ${\mathrm{val}}(\pi)=1$.
Let ${G}= \mathrm{GL}(3,{\mathrm{k}})$ and let $K = \mathrm{GL}(3,{\mathscr{R}})$. Write ${{T_G}}$ for the diagonal torus in ${G}$ and ${{B_G}}$ for the upper triangular Borel subgroup. Write $T = {{T_G}}\cap K$ and $B = {{B_G}}\cap K$ for their intersections with $K$.
Principal series and Posets {#S:principalseries}
---------------------------
Let $\chi_{G}$ be a character, not necessarily unitary, of the torus ${{T_G}}$ and extend it trivially over the subgroup ${{B_G}}$; then the (normalized) induced representation $\phi_{G}= {\mathrm{Ind}}_{{B_G}}^{G}\chi_{G}$ is a principal series representation of ${G}$. We consider its restriction to $K$. Writing $\chi = \chi_{G}\vert_{T}$ and $\phi = \phi_{G}\vert_K$, we have that $\phi = {\mathrm{Ind}}_B^K \chi$ since $K$ is a good maximal compact. The principal series representation is called *ramified* if $\chi \neq {\mathbf{1}}$. The unramified case was considered in [@CNu].
Given a ramified character $\chi_{G}$ of ${{T_G}}$, we may write it as $\chi_{G}= (\chi_1,\chi_2,\chi_3)$ for characters $\chi_{i} : {\mathrm{k}}^{\times} \rightarrow{\mathbb{C}}^{\times}$. Recall that the *conductor* of a character $\chi_i$ of ${\mathrm{k}}^\times$ is the least $m \geq 0$ such that $1+{\mathscr{P}}^m \subseteq \ker(\chi_i)$; thus we make the convention that ${\mathrm{cond}}(\chi_i)=0$ if and only if $\chi_i\mid_{{\mathscr{R}}^\times} = {\mathbf{1}}$.
The use of normalized induction implies that ${\mathrm{Ind}}_{{B_G}}^{G}\chi \simeq {\mathrm{Ind}}_{{B_G}}^{G}\chi^w$ for any $w$ in the Weyl group of ${G}$, so we may reorder the characters $\chi_i$ in a convenient way. Moreover, if $\psi$ is a character of ${\mathrm{k}}^\times$ and $\psi \cdot \chi = (\psi\chi_1,\psi \chi_2,\psi \chi_3)$, then ${\mathrm{Ind}}_{{B_G}}^{G}\psi \cdot \chi = (\psi\circ \det) {\mathrm{Ind}}_{{B_G}}^{G}\chi$. It follows that we may assume that $\chi_1 = {\mathbf{1}}$ and that $$0 \leq M = {\mathrm{cond}}(\chi_2) \leq {\mathrm{cond}}(\chi_3) = N.$$ Then ${\mathrm{cond}}(\chi_1\chi_2^{-1}) = M$ and we may furthermore assume that ${\mathrm{cond}}(\chi_2\chi_3^{-1}) = {\mathrm{cond}}(\chi_1\chi_3^{-1}) = N$. Define ${{\mathtt{m}}} = (M,N,N)$. We will assume throughout that $\chi \neq {\mathbf{1}}$, so in particular $\chi_3 \neq {\mathbf{1}}$ and $N > 0$.
Let ${\mathtt{T}}= \{ {{\mathtt{c}}} = (c_1,c_2,c_3) \in {\mathbb{Z}}^3 \colon 0 \leq c_1, c_2 \leq c_3 \leq c_1+c_2\}$ and note that ${{\mathtt{m}}} = (M,N,N) \in {\mathtt{T}}$. Then ${\mathtt{T}}$ is a poset with ${{\mathtt{c}}} \preceq {{\mathtt{d}}}$ if $c_i \leq d_i$ for all $i$. We are particularly interested in the subposet ${{\mathtt{T}}_{{\mathtt{m}}}}= \{ {{\mathtt{c}}} \in {\mathtt{T}}\colon {{\mathtt{c}}} \succeq {{\mathtt{m}}}\}$.
Given ${{\mathtt{c}}} \in {\mathtt{T}}$, we define a subgroup $C_{{\mathtt{c}}}$ by $$C_{{\mathtt{c}}} = \left[ \begin{matrix} {\mathscr{R}}& {\mathscr{R}}& {\mathscr{R}}\\ {\mathscr{P}}^{c_1} & {\mathscr{R}}& {\mathscr{R}}\\
{\mathscr{P}}^{c_3} & {\mathscr{P}}^{c_2} & {\mathscr{R}}\end{matrix} \right] \cap K.$$ Then $C_{{\mathtt{c}}} \subseteq C_{{\mathtt{d}}}$ if and only if ${{\mathtt{c}}} \succeq {{\mathtt{d}}}$.
Let $K_n$ denote the $n$th principal congruence subgroup of $K$, that is, the normal subgroup of $K$ consisting of all those matrices which are equivalent to the identity matrix modulo ${\mathscr{P}}^n$. Then for all ${{\mathtt{c}}} \in {\mathtt{T}}$ we have $C_{{\mathtt{c}}} \supset K_{c_3}$.
Subrepresentations {#subrepresentations .unnumbered}
------------------
Let $\chi$ be the restriction to $T$ of a character of ${{T_G}}$, with the above conventions; then in particular $\chi_1 = {\mathbf{1}}$. If ${{\mathtt{c}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$, then we can extend $\chi$ to a character of $C_{{\mathtt{c}}}$, denoted $\chi_{{\mathtt{c}}}$ or simply $\chi$ if there is no possibility of confusion. Namely, given $g = (g_{ij}) \in C_{{\mathtt{c}}}$, we define $\chi_{{\mathtt{c}}}(g) = \chi_2(g_{22})\chi_3(g_{33})$. One verifies directly that this is multiplicative exactly when $c_1 \geq M$ and $c_2, c_3 \geq N$.
\[D:Uc\] For each ${{\mathtt{c}}}\in {{\mathtt{T}}_{{\mathtt{m}}}}$, set $U_{{\mathtt{c}}} = {\mathrm{Ind}}_{C_{{\mathtt{c}}}}^K \chi_{{\mathtt{c}}}$.
We have from [@CNu] that $\dim U_{{\mathtt{c}}} = (q+1)(q^2+q+1)q^{c_1+c_2+c_3-3}$ if $c_1c_2 >0$ and $\dim U_{{\mathtt{c}}} = (q^2+q+1)q^{2(c_1+c_2-1)}$ if exactly one of $c_1$ or $c_2$ is zero. The representation $U_{{\mathtt{c}}}$ is naturally a subrepresentation of $\phi$; in fact, it is contained in the subspace of $K_{c_3}$-fixed vectors of $\phi$. Consequently, one may also view $U_{{\mathtt{c}}}$ as a representation of the finite group $K/K_{c_3}$.
If ${{\mathtt{c}}} \succeq {{\mathtt{d}}}$ then we have $U_{{\mathtt{d}}} \subseteq U_{{\mathtt{c}}}$, so set $$V_{{\mathtt{c}}} = U_{{\mathtt{c}}} / \sum_{{{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}, {{\mathtt{d}}} \prec {{\mathtt{c}}}} U_{{\mathtt{d}}}.$$ This quotient can be identified with a summand of $\phi$. These summands are the building blocks of the decomposition of $\phi$ that we wish to study, so let us refine our description of $V_{{\mathtt{c}}}$.
Let ${{\mathtt{c}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$. If $c_1c_2 \neq 0$, define for each $i \in \{1,2,3\}$ the triple ${{\mathtt{c}}}_{\{i\}} = (c_1-\delta_{i1},c_2-\delta_{i2},c_3-\delta_{i3}) \in \mathbb{Z}^3$; if $c_1=0$ then only consider ${{\mathtt{c}}}_{\{3\}} = (0,c_2-1,c_2-1)$; and if $c_2=0$ then only consider ${{\mathtt{c}}}_{\{3\}} = (c_1-1,0,c_1-1)$. Set ${\mathscr{S}}_{{\mathtt{c}}} = \{ i \colon {{\mathtt{c}}}_{\{i\}} \in {{\mathtt{T}}_{{\mathtt{m}}}}\}$; then for all ${{\mathtt{d}}} \prec {{\mathtt{c}}}$ such that ${{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$, there is some $i\in {\mathscr{S}}_{{\mathtt{c}}}$ such that ${{\mathtt{d}}} \preceq {{\mathtt{c}}}_{\{i\}}$.
Further, let ${{\mathtt{c}}}_\emptyset = {{\mathtt{c}}}$ and for each non-empty $I \subseteq {\mathscr{S}}_{{\mathtt{c}}}$ define ${{\mathtt{c}}}_I = \max\{ {{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}\colon {{\mathtt{d}}} \preceq {{\mathtt{c}}}_{\{i\}} \text{ for all } i \in I\}$; this set contains ${{\mathtt{m}}}$ and so is nonempty. For example, if ${{\mathtt{m}}} = (0,0,0)$ and ${{\mathtt{c}}} = (2,3,4)$ then ${{\mathtt{c}}}_{\{1,2\}} = (1,2,3)$ since $(1,2,4) \notin {{\mathtt{T}}_{{\mathtt{m}}}}$. We have the following result from [@CNu].
\[T:calc\] For any ${{\mathtt{c}}}\in {{\mathtt{T}}_{{\mathtt{m}}}}$ we have $$\label{E:G1}
[V_{{{\mathtt{c}}}}] = \sum_{I \subseteq {\mathscr{S}}_{{{\mathtt{c}}}}} (-1)^{\vert I \vert }[U_{{{\mathtt{c}}}_{I}}],$$ where $[V]$ denotes the equivalence class of $V$ in the Grothendieck group of $K$.
Since the $U_{{\mathtt{c}}}$ are essentially induced representations of finite groups, the dimension ${\mathscr{I}}(U_{{\mathtt{c}}},U_{{\mathtt{d}}})$ of the space of intertwining operators between $U_{{\mathtt{c}}}$ and $U_{{\mathtt{d}}}$ is equal $\dim {\mathscr{H}}(\chi_{{\mathtt{c}}}, \chi_{{\mathtt{d}}})$ where $${\mathscr{H}}(\chi_{{\mathtt{c}}},\chi_{{\mathtt{d}}}) = \{ f \colon K \to \mathbb{C} \colon f(gkg') = \chi_{{\mathtt{c}}}(g) f(k) \chi_{{\mathtt{d}}}(g') \forall g \in C_{{\mathtt{c}}}, g' \in C_{{{\mathtt{d}}}}\}.$$ As an immediate corollary of the above theorem we therefore have an effective means of determining the number of intertwining operators between the various quotients $V_{{\mathtt{c}}}$.
\[C:intops\] Let ${{\mathtt{c}}},{{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$. Then the dimension of the space of intertwining operators between $V_{{\mathtt{c}}}$ and $V_{{\mathtt{d}}}$ is $${\mathscr{I}}(V_{{\mathtt{c}}},V_{{\mathtt{d}}}) = \sum_{I \subseteq {\mathscr{S}}_{{\mathtt{c}}}, J \subseteq {\mathscr{S}}_{{\mathtt{d}}}} (-1)^{\vert I \vert + \vert J \vert}\; {\mathscr{I}}(U_{{{\mathtt{c}}}_I}, U_{{{\mathtt{d}}}_J}).$$
Distinguished double coset representatives of $C_{{\mathtt{c}}}\backslash K / C_{{\mathtt{d}}}$ {#S:doublecosets}
-----------------------------------------------------------------------------------------------
We recall the parametrization of representatives for the double coset space $C_{{\mathtt{c}}}\backslash K / C_{{\mathtt{d}}}$, as given in [@CNu].
Let ${\mathtt{T}}^{1} = \{{{\mathtt{a}}} = (a_1,a_2,a_3) \in {\mathbb{Z}}^3 \colon 1 \leq a_1, a_2 \leq a_3 \}$. Given ${{\mathtt{c}}} \in {\mathtt{T}}$, define $\underline{{{\mathtt{c}}}} \in {\mathtt{T}}\cap {\mathtt{T}}^{1}$ by $\underline{c}_i = \max\{ c_i, 1\}$ for each $i$.
\[D:Tcd\] For any ${{\mathtt{c}}}, {{\mathtt{d}}} \in {\mathtt{T}}$, set $$\label{E:Tcddef}
{{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= \left\{ {{\mathtt{a}}} \in {\mathtt{T}}^{1} \colon {{\mathtt{a}}} \preceq \underline{{{\mathtt{c}}}}, {{\mathtt{a}}} \preceq \underline{{{\mathtt{d}}}} \; \text{and} \;
a_3 \leq \min\{a_1+\underline{c}_2,\underline{d}_1+a_2\} \right\}$$ with the following exceptions: $$\label{E:Tdefexcept}
{{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}=
\begin{cases}
\{(1,1,1)\} & \text{if ${{\mathtt{c}}}$ or ${{\mathtt{d}}}$ equals $(0,0,0)$;}\\
\left\{ (1,a,a) \colon a \leq \min\{c_2,d_2\} \right\} & \text{if $c_2d_2>0$ and $c_1=d_1=0$; and}\\
\left\{ (a,1,a) \colon a \leq \min\{c_1,d_1\} \right\} & \text{if $c_1d_1>0$ and $c_2=d_2=0$.}\\
\end{cases}$$
Next, for ${{\mathtt{a}}} \in {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ set $\min\{{{\mathtt{a}}}\} = \min\{a_1,a_2,a_3\}$. Then we define $$\begin{aligned}
\label{E:defacd}
{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}= \max&\{0, \min\{a_1,a_2, a_3-a_1, a_3-a_2, {{\mathtt{c}}}-{{\mathtt{a}}}, {{\mathtt{d}}}-{{\mathtt{a}}},\\
\notag & a_1+c_2-a_3, d_1+a_2-a_3\}\}\end{aligned}$$ and $$\label{E:defacdprime}
{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}' = \max\{0,\min\{d_3-a_3,c_3-a_3,c_1-a_1,d_2-a_2\}\} \geq {{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}.$$
For $x \in {\mathscr{R}}/{\mathscr{P}}^k$, define ${\mathrm{val}}(x) = \min\{ {\mathrm{val}}(y) \colon y+{\mathscr{P}}^k = x\}$; then $({\mathscr{R}}/{\mathscr{P}}^k)^\times = \{ x \in {\mathscr{R}}/{\mathscr{P}}^k \colon {\mathrm{val}}(x)=0 \}$. We set $${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}= \begin{cases}
({\mathscr{R}}/{\mathscr{P}}^{{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}})^\times & \text{if $a_1+a_2 \neq a_3$;}\\
\bigcup_{i=0}^{{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}'} (1+\pi^i{\mathscr{R}}^\times) \cap ({\mathscr{R}}/{\mathscr{P}}^{{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}+i})^\times \cap ({\mathscr{R}}/{\mathscr{P}}^{{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}'})^\times & \text{if $a_1+a_2=a_3$.}
\end{cases}$$ In other words, in this latter case, if ${\mathrm{val}}(x-1)=i> 0$ then $x$ and $y$ represent the same element of ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}$ exactly when ${\mathrm{val}}(x-y) \geq \min\{i+{{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}, {{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}'\}$.
Let ${{\mathtt{c}}},{{\mathtt{d}}} \in {\mathtt{T}}$. Enumerate the elements of $W \simeq S^3$ as $$W = \{1,s_1,s_2,s_1s_2,s_2s_1,w_0\}$$ where $s_i$ is the transposition $(i \ \ i+1)$ and $w_0$ is the longest element. Define a subset ${W_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ of $W$ as $${W_{{{\mathtt{c}}},{{\mathtt{d}}}}}= \begin{cases}
W & \text{if ${{\mathtt{c}}}, {{\mathtt{d}}} \succeq (1,1,1)$;}\\
\{1,s_1,w_0\} & \text{if $c_1d_1(c_2+d_2) >0$ and $c_2d_2 =0$;}\\
\{1,s_2,w_0\} & \text{if $c_1d_1 = 0$ and $(c_1+d_1)c_2d_2 > 0$;}\\
\{1,w_0\} & \text{if $c_1c_2=0$ and $d_1d_2=0$ but $(c_1+c_2)(d_1+d_2) > 0$;}\\
\{1\} & \text{if ${{\mathtt{c}}} = (0,0,0)$ or ${{\mathtt{d}}} = (0,0,0)$}.
\end{cases}$$
The following theorem is proven in [@CNu].
\[P:doubleCccosets\] Let ${{\mathtt{c}}},{{\mathtt{d}}} \in {\mathtt{T}}$. A complete set of distinct double coset representatives ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ of $C_{{\mathtt{c}}}\backslash K / C_{{\mathtt{d}}}$ is $${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= \bigcup_{w \in {W_{{{\mathtt{c}}},{{\mathtt{d}}}}}} {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w$$ where for $w \in {W_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ we define ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w$ as follows.
(i) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1 = \left\{t_{{{\mathtt{a}}},x} = \left[ \begin{matrix} 1 & 0 & 0 \\ \pi^{a_1} & 1 & 0\\
x\pi^{a_3} & \pi^{a_2} & 1\end{matrix} \right] \colon
{{\mathtt{a}}} \in {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}, x \in {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}\right\}$;
(ii) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_1} = \left\{ s_1^{(\alpha,\beta)} =\left[ \begin{matrix} 0 & 1 & 0\\ 1 & 0 & 0 \\ \pi^{\beta} & \pi^{\alpha} & 1 \end{matrix} \right] \colon \begin{array}{rcl}
1 \leq &\alpha& \leq \min\{\underline{d}_2,c_3\}\\
1 \leq &\beta& \leq \min\{\underline{c}_2,d_3\}\\
-c_1 \leq &\beta-\alpha& \leq d_1
\end{array} \right\};$
(iii) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_2} = \left\{s_2^{(\alpha,\beta)} = \left[ \begin{matrix} 1 & 0 & 0\\ \pi^{\beta} & 0 & 1 \\ \pi^{\alpha} & 1 & 0 \end{matrix} \right] \colon
\begin{array}{rcl}1 \leq &\alpha& \leq \min\{\underline{d}_1,c_3\}\\
1 \leq &\beta& \leq \min\{\underline{c}_1,d_3\}\\
-c_2 \leq &\beta-\alpha& \leq d_2 \end{array} \right\};$
(iv) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_1s_2} = \left\{ s_1s_2^{(\alpha)}=\left[ \begin{matrix} 0 & 0 & 1\\ 1 & 0 & 0 \\ \pi^{\alpha} & 1 & 0 \end{matrix} \right] \colon 1 \leq \alpha \leq \min\{d_1,c_2\}\right\};$
(v) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_2s_1} = \left\{ s_2s_1^{(\alpha)} = \left[ \begin{matrix} 0 & 1 & 0\\ 0 & \pi^{\alpha} & 1 \\ 1 & 0&0 \end{matrix} \right] \colon 1 \leq \alpha \leq \min\{c_1,d_2\}\right\}$;
(vi) ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{w_0} = \left\{w_0 = \left[ \begin{matrix} 0 & 0 & 1\\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} \right]\right\}$.
Determination of the set of double cosets supporting intertwining operators {#S:support}
===========================================================================
To understand the space of intertwining operators of the finite-dimensional representations $U_{{\mathtt{c}}}$ given in Definition \[D:Uc\], we construct bases for the spaces ${\mathscr{H}}(\chi_{{\mathtt{c}}},\chi_{{\mathtt{d}}})$. That is, for any ${{\mathtt{c}}}, {{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$ we must identify among the double cosets enumerated in Proposition \[P:doubleCccosets\] those which support intertwining operators of $U_{{\mathtt{c}}}$ with $U_{{\mathtt{d}}}$. Denote the subset of these cosets by ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\subseteq {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, and write $\mathcal{I}(U_{{\mathtt{c}}},U_{{\mathtt{d}}}) = \dim {\mathscr{H}}(\chi_{{\mathtt{c}}}, \chi_{{\mathtt{d}}}) = \vert {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\vert$. (Note that in the case that $\chi = {\mathbf{1}}$, which we continue to exclude, ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ and there is nothing to show.)
If ${{\mathtt{a}}} \in {\mathtt{T}}^{1}$ then define ${{\mathtt{a}}}^{{\mathrm{op}}} = (a_3-a_2, a_3-a_1, (a_3-a_1)+(a_3-a_2))$.
\[T:supp\] Let ${{\mathtt{c}}},{{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$, with ${{\mathtt{m}}} \succ (0,0,0)$ as before. Then a set of representatives for double cosets in $C_{{\mathtt{c}}}\backslash K /C_{{\mathtt{d}}}$ supporting elements of ${\mathscr{H}}(\chi_{{\mathtt{c}}},\chi_{{\mathtt{d}}})$ is $${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= \bigcup_{w \in{W_{{{\mathtt{c}}},{{\mathtt{d}}}}}} {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w$$ where the subsets ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w \subseteq {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w$ are defined as follows.
(i) ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$ is the set of all $t_{{{\mathtt{a}}},x}$ with ${{\mathtt{a}}} \in {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ and $x\in {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}$ such that one of the following holds:
(1) $a_1 \geq M$ and $a_2 \geq N$; or
(2) $a_1 < M$ and $a_2 \geq N$ and:
(a) $a_1+a_2 < a_3$ and $M \leq \min\{ {{\mathtt{c}}}-{{\mathtt{a}}}, {{\mathtt{d}}}-{{\mathtt{a}}}, c_2+a_1-a_3, d_1+a_2-a_3\}$; or
(b) $a_1+a_2 > a_3$ and $M \leq \min\{ {{\mathtt{c}}}-{{\mathtt{a}}}^{{\mathrm{op}}}, {{\mathtt{d}}}-{{\mathtt{a}}}^{{\mathrm{op}}}\}$; or
(c) $a_1+a_2 = a_3$ and $M \leq \min\{ {{\mathtt{c}}}-{{\mathtt{a}}}, {{\mathtt{d}}}-{{\mathtt{a}}}\}$ and ${\mathrm{val}}(x-1) \leq {{\mathtt{a}}}({{\mathtt{c,d}}})' - M$;
or
(3) $a_1 \geq N$ and $a_2 < N$ and the same conditions (a),(b),(c) with $M$ replaced by $N$;
(ii) ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_1}$ is the set of all $s_1^{(\alpha,\beta)}$ such that $$\begin{aligned}
N \leq& \alpha &\leq \min\{d_2,c_3\}-M,\\
N \leq& \beta & \leq \min\{c_2,d_3\}-M, \; \textrm{and} \\
M-c_1 \leq &\beta-\alpha &\leq d_1-M;\end{aligned}$$
(iii) ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_2}$ is the set of all $s_2^{(\alpha,\beta)}$ such that $$\begin{aligned}
N \leq &\alpha& \leq \min\{d_1,c_3\}-N, \\
N \leq &\beta& \leq \min\{c_1,d_3\}-N, \; \textrm{and}\\
N-c_2 \leq &\beta-\alpha& \leq d_2-N;\end{aligned}$$
(iv) and ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w = \emptyset$ for all other $w \in {W_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ and for any $w \notin {W_{{{\mathtt{c}}},{{\mathtt{d}}}}}$.
Let us first show that none of the cosets represented by elements of ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_1s_2} \cup {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_2s_1} \cup {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{w_0}$ can support intertwining operators. Choose an element $b \in {\mathscr{R}}^\times$ such that $\chi_3(b) \neq 1$. Set $g = {\mathrm{diag}}(b,1,1)$ and $g' = {\mathrm{diag}}(1,1,b)$; these are elements of $C_{{\mathtt{c}}}$ and $C_{{\mathtt{d}}}$ for any ${{\mathtt{c}}}, {{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$. One verifies that $g s_1s_2^{(\alpha)} = s_1s_2^{(\alpha)}g'$, $g w_0=w_0 g'$ and $s_2s_1^{(\alpha)}g = g's_2s_1^{(\alpha)}$, but that $\chi(g) \neq \chi(g')$. Consequently none of these representatives are in ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$.
From now on, let us adopt the notational convention that if $g = (g_{ij}) \in C_{{\mathtt{c}}}$ then $g_{21} = \gamma_{21}\pi^{c_1}$, $g_{32} = \gamma_{32}\pi^{c_2}$ and $g_{31} = \gamma_{31}\pi^{c_3}$; so $g' \in C_{{\mathtt{d}}}$ would have $g_{21}' = \gamma_{21}'\pi^{d_1}$, and so forth. Moreover, given a coset representative $h \in {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ and a pair of elements $g \in C_{{\mathtt{c}}}$ and $g' \in C_{{\mathtt{d}}}$ such that $g h = hg'$, we will call $(g,g')$ a *coset pair*.
Suppose now that $(g,g') \in C_{{\mathtt{c}}} \times C_{{\mathtt{d}}}$ are a coset pair for the representative $s_1^{(\alpha,\beta)}$. We determine directly that the matrix coefficients of $g$ and $g'$ satisfy $$\begin{aligned}
\label{E:s1equations}
\notag g_{22} &=& g_{33} -g_{23}\pi^\beta - \gamma_{21}'\pi^{d_1+\alpha-\beta}+\gamma_{32}\pi^{c_2-\beta} - \gamma_{31}'\pi^{d_3-\beta},\\
g_{22}' &=& g_{33} -g_{23}\pi^\beta -\gamma_{21}\pi^{c_1+\beta-\alpha} - \gamma_{32}'\pi^{d_2-\alpha} + \gamma_{31}\pi^{c_3-\alpha}, \\
\notag g_{33}' &=& g_{33} - g_{23}\pi^\beta - g_{23}'\pi^\alpha,\end{aligned}$$ with the remaining coefficients given by $$\begin{array}{rclrcl}
g_{11} &=& g_{22}' - g_{23}'\pi^{\alpha} \quad & g_{11}' &=& g_{22} + g_{23}\pi^\beta \\
g_{12} &=& \gamma_{21}'\pi^{d_1} - g_{23}'\pi^\beta \quad & g_{12}' &=& g_{23}\pi^\alpha + \gamma_{21}\pi^{c_1} \\
g_{13} &=& g_{23}' \quad & g_{13}' &=& g_{23}.
\end{array}$$ This allows us to compare $$\begin{aligned}
\chi_{{\mathtt{c}}}(g) &=&\chi_2(g_{22})\chi_3(g_{33})\\
&=& \chi_2(g_{33} -g_{23}\pi^\beta - \gamma_{21}'\pi^{d_1+\alpha-\beta}+\gamma_{32}\pi^{c_2-\beta} - \gamma_{31}'\pi^{d_3-\beta}) \chi_3(g_{33})\end{aligned}$$ with $$\begin{aligned}
\chi_{{{\mathtt{d}}}}(g') &=& \chi_2(g_{22}')\chi_3(g_{33}') \\
&=&\chi_2(g_{33} -g_{23}\pi^\beta -\gamma_{21}\pi^{c_1+\beta-\alpha} - \gamma_{32}'\pi^{d_2-\alpha} + \gamma_{31}\pi^{c_3-\alpha})\cdot \\
&& \chi_3(g_{33} - g_{23}\pi^\beta - g_{23}'\pi^\alpha).\end{aligned}$$ It follows that whenever $M \leq \min\{c_1 - \alpha+\beta, d_1+\alpha-\beta, c_2-\beta, d_2-\alpha, c_3-\alpha, d_3-\beta\}$ and $N \leq \min\{ \alpha, \beta\}$, then $\chi_{{\mathtt{c}}}(g) = \chi_{{{\mathtt{d}}}}(g')$, and so $s_1^{(\alpha,\beta)} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$. Conversely, when these inequalities are not satisfied, and additionally $\alpha, \beta \geq 1$, then we can use the relations above to construct a coset pair $(g,g')$ on which the characters do not agree. This proves part (ii); the proof of part (iii) is analogous and is omitted.
To prove part (i) of the theorem, suppose $t_{{{\mathtt{a}}},x} \in {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$ and let $(g,g') \in C_{{\mathtt{c}}}\times C_{{\mathtt{d}}}$ be a coset pair such that $g t_{{{\mathtt{a}}},x} = t_{{{\mathtt{a}}},x} g'$. To simplify notation, set $r_x = \pi^{a_1+a_2}-x\pi^{a_3}$. One calculates directly that the matrix coefficients of $g$ and $g'$ satisfy the relation $$\begin{aligned}
\label{E:taxrelation}
(g_{12}\pi^{a_1}+g_{13}\pi^{a_1+a_2} -g_{23}\pi^{a_2} )xr_x =&
- \gamma_{21}x\pi^{c_1+a_2} - \gamma_{21}'r_x\pi^{d_1+a_2-a_3} \\
\notag & + \gamma_{32}r_x\pi^{a_1+c_2-a_3}
+ \gamma_{32}'x\pi^{a_1+d_2} \\
\notag & +\gamma_{31}\pi^{c_3-a_3+a_1+a_2}
-\gamma_{31}'\pi^{d_3-a_3+a_1+a_2}\end{aligned}$$ and all other matrix coefficients are determined by the equations $$\begin{aligned}
g_{11}' &=& g_{22} + g_{23}x\pi^{a_3-a_1} + \gamma_{21}\pi^{c_1-a_1} - \gamma_{21}'\pi^{d_1-a_1}\\
g_{11} &=& g_{11}' -g_{12}\pi^{a_1} - g_{13}x\pi^{a_3} \\
g_{22}' &=& g_{22} -g_{12}\pi^{a_1} - g_{13}\pi^{a_1+a_2} + g_{23}\pi^{a_2}\\
g_{33}' &=& g_{22} - g_{12}r_x\pi^{-a_2} - \gamma_{32}\pi^{c_2-a_2} + \gamma_{32}'\pi^{d_2-a_2}\\
g_{33} &=& g_{33}' -g_{13}r_x + g_{23}\pi^{a_2},\end{aligned}$$ together with $g_{12}' = g_{12} +g_{13}\pi^{a_2}$, $g_{13}' = g_{13}$ and $g_{23}' = g_{23} - g_{13}\pi^{a_1}$. Note that in this case, as opposed to the one for $s_1^{(\alpha,\beta)}$ above, although any solution (with coefficients in ${\mathscr{R}}$) of gives a pair of matrices $(g,g')$ satisfying the relation $g t_{{{\mathtt{a}}},x} = t_{{{\mathtt{a}}},x} g'$, it must be additionally verified that $g$ and $g'$ are invertible in $K$.
Now, given a coset pair $(g,g')$ we have $$\label{E:chic}
\chi_{{\mathtt{c}}}(g) = \chi_2(g_{22}) \chi_3(g_{33}) = \chi_2(g_{22})\chi_3(g_{33}' -g_{13}r_x + g_{23}\pi^{a_2})$$ while $$\label{E:chicp}
\chi_{{\mathtt{d}}}(g') = \chi_2(g_{22} -g_{12}\pi^{a_1} - g_{13}\pi^{a_1+a_2} + g_{23}\pi^{a_2})\chi_3(g_{33}').$$ Hence these characters agree whenever $a_1 \geq M$ and $a_2 \geq N$, proving part (i)(1).
Now suppose $a_1$ and $a_2$ are both less than $N$. Choose a pair $(g_{12},g_{23}) \in {\mathscr{R}}\times {\mathscr{R}}$ of minimum valuation satisfying $g_{23}\pi^{a_2} = g_{12}\pi^{a_1}$, set $g_{13} = \gamma_{21} = \gamma_{21}' = \gamma_{31} = \gamma_{31}' = \gamma_{32} = \gamma_{32}' = 0$ and set $g_{22} = 1$. These are easily seen to define a coset pair $(g,g') \in C_{{\mathtt{c}}} \times C_{{\mathtt{d}}}$. Since ${\mathrm{val}}(g_{12}\pi^{a_1})={\mathrm{val}}(g_{23}\pi^{a_2}) = \max\{a_1,a_2\} < N = {\mathrm{cond}}(\chi_3)$, we have $\chi_{{\mathtt{c}}}(g) \neq \chi_{{{\mathtt{d}}}}(g')$ and it follows that $t_{{{\mathtt{a}}},x} \notin {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$.
There are exactly two cases left to consider: when $a_1 < M$ and $a_2 \geq N$, or when $a_1 \geq N$ and $a_2 < N$. Comparing and , and noting that $\max\{a_1, a_2\} \leq \min\{a_3, {\mathrm{val}}(r_x)\}$, we deduce that
(A) if $a_1 < M$ and $a_2 \geq N$, then $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$ if and only if ${\mathrm{val}}(g_{12}\pi^{a_1}) \geq M$ for all coset pairs $(g,g')$; and
(B) if $a_1 \geq N$ and $a_2 < N$, then $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$ if and only if ${\mathrm{val}}(g_{23}\pi^{a_2}) \geq N$ for all coset pairs $(g,g')$.
Consider case (A), that is, assume that $a_1 < M$ and $a_2 \geq N$. If ${\mathrm{val}}(g_{12}\pi^{a_1}) \geq a_2 \geq N$ then we are done; otherwise, the term with least valuation on the left hand side of is $g_{12}\pi^{a_1}xr_x$. Comparing with the right hand side, we deduce ${\mathrm{val}}(g_{12}\pi^{a_1}) + {\mathrm{val}}(r_x) \geq \alpha$ where $$\begin{aligned}
\alpha &=& \min\{c_1+a_2, d_1+a_2-a_3+{\mathrm{val}}(r_x), a_1+c_2-a_3+{\mathrm{val}}(r_x), a_1+d_2,\\
\notag && \quad \quad c_3-a_3+a_1+a_2, d_3-a_3+a_1+a_2\}.\end{aligned}$$ It follows that if $\alpha\geq M + {\mathrm{val}}(r_x)$, then $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ by (A) above. Restating this condition in the three cases $a_1+a_2<a_3$, $a_1+a_2>a_3$ and $a_1+a_2=a_3$ yields the conditions described in part (i)(2)(a,b,c) of the theorem.
Conversely, suppose $\alpha < M+{\mathrm{val}}(r_x)$ and set $g_{13}=g_{23}=0$. Choose a term of least possible valuation on the right hand side of ; set its coefficient (either $\gamma_{ij}$ or $\gamma_{ij}'$, for some $i>j$) to be $\pi^{a_1-\alpha}$ if $\alpha<a_1+{\mathrm{val}}(r_x)$ and $1$ otherwise. Then set the remaining coefficients of the right hand side of equal to zero and solve for $g_{12}$, which is now necessarily in ${\mathscr{R}}^\times$. Take $g_{22}=1$ and solve for the remaining coefficients. This results in a coset pair $(g,g') \in C_{{\mathtt{c}}} \times C_{{\mathtt{d}}}$ such that ${\mathrm{val}}(g_{12}\pi^{a_1}) < M$, so by (A) we conclude $t_{{{\mathtt{a}}},x} \notin {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, as required.
A similar argument establishes condition (i)(3) of the Theorem, following case (B), above.
Let us conclude this section by deriving some consequences of Theorem \[T:supp\]. The first, which is immediate, is a convenient restatement of the theorem in a special case.
\[C:nnnsupp\] Set ${{\mathtt{c}}}=(n,n,n)$ for $n \geq N$. The space of intertwining operators of $U_{{\mathtt{c}}} = V_\chi^{K_n}$ with itself has basis parametrized by $${\mathtt{S}}_{n} = \bigcup_{w \in W} {\mathtt{S}}_n^w$$ where
(i) ${\mathtt{S}}_n^1$ is the set of all $t_{{{\mathtt{a}}},x}$ such that $1 \leq a_1, a_2 \leq a_3 \leq n$, $x \in \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,c}}}}$ and one of conditions (1), (2) or (3) is met:
(1) ${{\mathtt{a}}} \succeq {{\mathtt{m}}}$; or
(2) $a_1<M$ and $a_2\geq N$ and:
(a) $a_1+a_2 < a_3 \leq n-M$, or
(b) $a_1+a_2 > a_3$ and $a_1+a_2 \geq M-n + 2a_3$, or
(c) $a_1+a_2 = a_3 \leq n-M$ and ${\mathrm{val}}(r_x) \leq n-M$; or
(3) $a_1 \geq N$, $a_2 < N$ and the same conditions (a),(b),(c), with $M$ replaced by $N$, are satisfied;
(ii) ${\mathtt{S}}_n^{s_1} = \{ s_1^{(\alpha,\beta)} \colon N \leq \alpha, \beta \leq n-M\}$;
(iii) ${\mathtt{S}}_n^{s_2} = \{ s_2^{(\alpha,\beta)} \colon N \leq \alpha, \beta \leq n-N\}$;
(iv) ${\mathtt{S}}_n^{s_1s_2} = {\mathtt{S}}_n^{s_2s_1} = {\mathtt{S}}_n^{w_0} = \emptyset$.
Our second corollary will be relevant for the purposes of calculating ${\mathscr{I}}(V_{{\mathtt{c}}}, V_{{\mathtt{d}}})$ in Section \[S:counting\].
\[C:consistency\] Suppose that ${{\mathtt{c}}},{{\mathtt{d}}},{{\mathtt{c}}}',{{\mathtt{d}}}' \in {{\mathtt{T}}_{{\mathtt{m}}}}$ with ${{\mathtt{c}}} \preceq {{\mathtt{c}}}'$ and ${{\mathtt{d}}} \preceq {{\mathtt{d}}}'$. Then ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\subseteq {\mathtt{S}}_{{\mathtt{c',d'}}}$.
By identifying ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}$ with a set of coset representatives from ${\mathscr{R}}^\times$ in a suitable manner, one easily sees that ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\subseteq \mathtt{R}_{{{\mathtt{c',d'}}}}$. Furthermore, it is clear that the list of constraints on elements of ${\mathtt{S}}$ in Theorem \[T:supp\] can only become less constrictive as ${{\mathtt{c}}}$ or ${{\mathtt{d}}}$ increase.
In particular, it makes sense to ask, for a given distinguished double coset representative $g\in \cup_{{{\mathtt{c,d}}}} {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, whether there exist ${{\mathtt{c}}},{{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$ for which $g \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$.
\[T:totalsupp\] The double cosets which support self-intertwining operators of $U_{{\mathtt{c}}}$, for some ${{\mathtt{c}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}$, are represented by $${\mathtt{S}}= \bigcup_{{{\mathtt{c}}},{{\mathtt{d}}} \in {{\mathtt{T}}_{{\mathtt{m}}}}} {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= \bigcup_{w \in W} {\mathtt{S}}^w$$ where
(i) ${\mathtt{S}}^1 = \{ t_{{{\mathtt{a}}},x} \colon a_3 \geq \max\{a_1,a_2\} \geq N, x \in {\mathscr{R}}^\times\}$;
(ii) ${\mathtt{S}}^{s_1} = \{ s_1^{(\alpha,\beta)} \colon \alpha, \beta \geq N\}$;
(iii) ${\mathtt{S}}^{s_2} = \{ s_2^{(\alpha,\beta)} \colon \alpha, \beta \geq N\}$; and
(iv) ${\mathtt{S}}^{s_1s_2} = {\mathtt{S}}^{s_2s_1} = {\mathtt{S}}^{w_0} = \emptyset$.
Moreover, up to identifying $t_{{{\mathtt{a}}},x}$ and $t_{{{\mathtt{a}}},y}$ whenever $x$ and $y$ have the same image in ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}$ for ${{\mathtt{c}}}$, ${{\mathtt{d}}}$ sufficiently large, these cosets are all distinct.
This follows from Corollary \[C:nnnsupp\] by allowing $n$ to grow without bound. Note that when ${{\mathtt{c}}} = {{\mathtt{d}}} = (n,n,n)$, we have simply ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}= \min\{ a_1, a_2, a_3-a_1, a_3-a_2, n-a_3\}$ and ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}' = n-a_3$.
Irreducibility {#S:counting}
==============
The results of the preceding section allow us to restate Corollary \[C:intops\] in terms of the sets ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$. That is, for any ${{\mathtt{c}}},{{\mathtt{d}}}\in {{\mathtt{T}}_{{\mathtt{m}}}}$, we have $$\mathcal{I}(V_{{{\mathtt{c}}}}, V_{{{\mathtt{d}}}})
=
\sum_{I\subseteq {\mathscr{S}}_{{{\mathtt{c}}}},\; J\subseteq {\mathscr{S}}_{{{\mathtt{d}}}}}
(-1)^{|I|+|J|} \;
\vert {\mathtt{S}}_{{{\mathtt{c}}}_I,{{\mathtt{d}}}_J} \vert.$$
The irreducibility of $U_{{\mathtt{m}}}=V_{{\mathtt{m}}}$ is known from Howe’s work [@Howe1 Theorem 1]. In this section, we demonstrate that this extends to many, but not all, of the quotients which are “extremal” in the sense that they have few immediate descendants in the poset ${{\mathtt{T}}_{{\mathtt{m}}}}$.
We retain the notation of the previous sections.
\[T:irred\] For each $n \in \mathbb{Z}$ with $N \leq n \leq N+M$, the $K$-module $V_{(M,N,n)}$ is irreducible.
Set ${{\mathtt{c}}} = (M,N,n)$. If $n=N$ then ${\mathscr{S}}_{{\mathtt{c}}}=\emptyset$; otherwise, ${\mathscr{S}}_{{\mathtt{c}}}$ is a singleton corresponding to the triple $(M,N,n-1)$. By Corollary \[C:intops\], and induction, it thus suffices to show that $\vert {\mathtt{S}}_{{{\mathtt{c,c}}}} \vert = n-N+1$.
If $M=0$ then ${W_{{{\mathtt{c}}},{{\mathtt{c}}}}}= \{1, w_0\}$ so ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^{s_1} = {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^{s_2} = \emptyset$. If $M>0$ then $\min\{c_2,c_3\}-M = N-M<N$ and $\min\{ c_1,c_3\}-N=M-N<N$ so again ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^{s_1} = {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^{s_2} = \emptyset$, regardless of the value of $n$. Thus ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}= {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^1$.
Now let $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^1$; so one of Theorem \[T:supp\](i)(1), (2) or (3) applies. If it were (2), then $a_1 < M$ and $a_2 \geq N$ imply that $M>0$ and $a_2 = N$ so neither case (a) nor case (c) could apply since $M > c_2-a_2 = 0$. Were case (b) to apply, then $M \leq \min\{ {{\mathtt{c-a}}}^{\mathrm{op}}\}$ would imply that $a_2=a_3 = N$ and so $N-(a_3-a_1) = a_1$ which is not greater than or equal to $M$, a contradiction. We similarly deduce that case (3) cannot apply. This leaves case (1), which consists of the elements $t_{{{\mathtt{a}}},x}$ with ${{\mathtt{a}}} = (M,N,m)$, $N \leq m \leq n$ and $x \in {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,c}}}}}$, each of which support an intertwining operator. For each such ${{\mathtt{a}}}$, we have ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{c}}}})}= {{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{c}}}})}' = 0$, so in fact $\vert {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,c}}}}}\vert = 1$. The desired conclusion follows.
\[T:otherirred1\] $V_{(m,n,n+m)}$ is irreducible for each $m \geq M$ and $n \geq N$.
We first consider the case that ${{\mathtt{c}}} = (m,n,m+n)$ with $m \geq 1$. Then ${\mathscr{S}}_{{{\mathtt{c}}}}$ is a singleton corresponding to ${{\mathtt{d}}}={{\mathtt{c}}}_{\{3\}}=(m,n,m+n-1)$. Hence ${\mathscr{I}}(U_{{\mathtt{c}}},V_{{\mathtt{c}}}) = {\mathscr{I}}(U_{{\mathtt{c}}},U_{{\mathtt{c}}}) -{\mathscr{I}}(U_{{\mathtt{c}}},U_{{\mathtt{d}}})$ and it suffices to show that $\vert {\mathtt{S}}_{{{\mathtt{c,c}}}} \setminus {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\vert =1$.
First note that since $c_1=d_1$ and $c_2=d_2$, and that both these are at most $d_3<c_3$, we have ${\mathtt{S}}_{{{\mathtt{c,c}}}}^w = {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^w$ for each $w \in W \setminus \{1\}$.
Next, note that ${\mathtt{T}}_{{\mathtt{c,c}}} \setminus {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ consists of the single element $(m,n,n+m) = {{\mathtt{c}}}$. Since $\vert \mathtt{X}^{{\mathtt{c}}}_{{{\mathtt{c,c}}}} \vert = 1$, there is a unique distinguished double coset of the form $t_{{{\mathtt{c}}},x} \in \mathtt{R}^1_{{{\mathtt{c,c}}}}$; it is clearly in ${\mathtt{S}}_{{{\mathtt{c,c}}}}$. We claim that this is the only element of ${\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{c}}}}}^1 \setminus {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$. Namely, let ${{\mathtt{a}}} \in {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$. Since $0 \leq d_3-a_3 = (d_1+d_2-1)-a_3 = (d_1+a_2-a_3)+(d_2-a_2)-1$, it must be true that $d_3-a_3 \geq \min\{d_1+a_2-a_3, d_2-a_2\}$ so necessarily ${{\mathtt{a(c,c)}}}={{\mathtt{a(c,d)}}}$. If furthermore $a_1+a_2=a_3$, then ${{\mathtt{a(c,c)}}}'={{\mathtt{a(c,d)}}}'$ by the same reasoning. Hence $\mathtt{X}^{{\mathtt{a}}}_{{\mathtt{c,c}}} = {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}$ for all such ${{\mathtt{a}}}$, as claimed.
We next claim that ${\mathtt{S}}_{{{\mathtt{c,c}}}}^1 \cap {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1 = {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$. Namely, given $t_{{{\mathtt{a}}},x} \in {\mathtt{S}}_{{{\mathtt{c,c}}}}^1 \cap {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$, since $d_3-a_3 \geq \min\{d_1+a_2-a_3, d_2-a_2\} = \min\{c_1+a_2-a_3, c_2-a_2\}$, we see that all of the conditions set out in Theorem \[T:supp\](i) are unchanged in passing from the pair $({{\mathtt{c,c}}})$ to the pair $({{\mathtt{c,d}}})$. Hence $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$.
This shows, for the case $m\geq 1$, that ${\mathtt{S}}_{{{\mathtt{c,c}}}}\setminus {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$ is a singleton from which we deduce the irreducibility of $V_{{\mathtt{c}}}$.
When ${{\mathtt{c}}} = (0,n,n)$, we have instead ${\mathscr{S}}_{{\mathtt{c}}} = \{3\}$ corresponding to ${{\mathtt{d}}} = {{\mathtt{c}}}_{\{3\}} = (0,n-1,n-1)$. We have ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$, ${\mathtt{S}}_{{{\mathtt{c,c}}}} = {\mathtt{S}}_{{{\mathtt{c,c}}}}^1$ and neither of the cases (i)(2) or (i)(3) of Theorem \[T:supp\] can apply. It thus follows easily that $\vert {\mathtt{S}}_{{{\mathtt{c,c}}}}\setminus {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\vert =1$ in this case as well.
\[T:otherirred2\] $V_{(m,n,\max\{n,m\})}$ is irreducible for each $m > M$ and $n > N$.
Suppose first that $m > \max\{M,1\}$ and $n > N$, and that $\max\{m,n\} = n$. Then ${{\mathtt{c}}} = (m,n,n)$ and ${\mathscr{S}}_{{\mathtt{c}}}= \{1,2\}$ with the corresponding triples ${{\mathtt{c}}}_{\{1\}}= (m-1,n,n)$, ${{\mathtt{c}}}_{\{2\}} = (m,n-1,n)$ and ${{\mathtt{c}}}_{\{1,2\}} = (m-1,n-1,n)$. We compute the alternating sum $$\label{E:icalc}
\mathcal{I}(U_{{\mathtt{c}}},V_{{\mathtt{c}}})=
\mathcal{I}(U_{{\mathtt{c}}},U_{{\mathtt{c}}})
- \mathcal{I}(U_{{\mathtt{c}}},U_{{{\mathtt{c}}}_{\{1\}}})
- \mathcal{I}(U_{{\mathtt{c}}},U_{{{\mathtt{c}}}_{\{2\}}})
+ \mathcal{I}(U_{{\mathtt{c}}},U_{{{\mathtt{c}}}_{\{1,2\}}})$$ as a sum of differences by defining $$\begin{aligned}
\mathcal{A}_0&=&{\mathtt{S}}_{{{\mathtt{c,c}}}} \setminus {\mathtt{S}}_{{{\mathtt{c}}},{{\mathtt{c}}}_{\{2\}}}\\
\mathcal{A}_1&=&{\mathtt{S}}_{{{\mathtt{c}}},{{\mathtt{c}}}_{\{1\}}} \setminus {\mathtt{S}}_{{{\mathtt{c}}},{{\mathtt{c}}}_{\{1,2\}}}.\end{aligned}$$ Thus we have $\mathcal{I}(U_{{\mathtt{c}}},V_{{\mathtt{c}}})=\vert \mathcal{A}_0 \vert - \vert
\mathcal{A}_1 \vert$. We use $({{\mathtt{d,d}}}')$ to denote either of the pairs $({{\mathtt{c}}},{{\mathtt{c}}}_{\{2\}})$ or $({{\mathtt{c}}}_{\{1\}},{{\mathtt{c}}}_{\{1,2\}})$, for ease of notation.
Suppose first that $s_1^{(\alpha,\beta)} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus{\mathtt{S}}_{{\mathtt{c,d'}}}$. Then, comparing the constraints on $\alpha$ and $\beta$ in Theorem \[T:supp\](ii) for ${{\mathtt{d}}}$ and ${{\mathtt{d}}}'$, we see that necessarily $\alpha =d_2-M=n-M$ and $$\max\{N,M-c_1+\alpha\}\leq \beta
\leq \min\{d_1-M+\alpha,c_2-M\}.$$ Since $d_1 \geq M$ by hypothesis, $c_2-M= n-M\leq \alpha+ d_1-M$ so these inequalities simplify to $\max\{N,n-m\} \leq \beta \leq n-M$. This constraint on the pair $(\alpha,\beta)$ is independent of the value of $d_1 \in \{m-1,m\}$ so $s_1^{(\alpha,\beta)} \in \mathcal{A}_0$ if and only if $s_1^{(\alpha,\beta)} \in \mathcal{A}_1$. Hence these cosets contribute nothing to the overall sum (\[E:icalc\]).
Now suppose that $s_2^{(\alpha,\beta)} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$. Then Theorem \[T:supp\](iii) implies that $\beta - \alpha \leq (c_1-N)-N$; but this bound is at most $d_2'-N$ since $c_1-N = n-N
\leq n-1 = d_2'$. Similarly, $d_1 \leq c_1$ and $\beta > 0$ together imply that $\beta-\alpha \geq N-c_1$, regardless of the value of $d_1\in \{m-1,m\}$. All other conditions on $(\alpha, \beta)$ being unchanged in passing from $({{\mathtt{c}}},{{\mathtt{d}}})$ to $({{\mathtt{c}}},{{\mathtt{d'}}})$, we deduce that $s_2^{(\alpha,\beta)} \in {\mathtt{S}}_{{{\mathtt{c,d'}}}}$. Hence none of these cosets appear in either $\mathcal{A}_0$ or $\mathcal{A}_1$.
Finally, consider distinguished coset representatives of the form $t_{{{\mathtt{a}}},x} \in {\mathtt{R}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$. First note that $${{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{T}}_{{{\mathtt{c,d'}}}} = \{(a_1,n,n) \colon 1 \leq a_1 \leq \underline{d}_1\},$$ and $\vert {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}\vert = \vert \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d'}}}}\vert = 1$, since $a_3-a_2=0$. Considering which of these are in ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, we deduce that these triples give rise to $m-M+1$ coset representatives in $\mathcal{A}_0$ and $m-M$ of them in $\mathcal{A}_1$.
Suppose now that ${{\mathtt{a}}} \in {\mathtt{T}}_{{\mathtt{c,d'}}}$. Since $0 \leq d_2'-a_2 = d_3-1-a_2 = (d_3-a_3)+(a_3-a_2)-1$, we have $d_2'-a_2 \geq \min\{d_3-a_3,a_3-a_2\}$ and so it follows that ${{\mathtt{a(c,d)}}} = {{\mathtt{a(c,d')}}}$. Similarly, if $a_1+a_2=a_3$ then $a_2<a_3$ implies that $d_2'-a_2 \geq d_3-a_3$ so ${{\mathtt{a(c,d)}}}'={{\mathtt{a(c,d')}}}'$. Hence for all ${{\mathtt{a}}} \in {\mathtt{T}}_{{\mathtt{c,d'}}}$ we have ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}= \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}'}$. So suppose $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1 \cap \mathtt{R}_{{\mathtt{c,d'}}}^1$. We first note that if $t_{{{\mathtt{a}}},x}$ falls under any of the conditions (2a), (2c), (3a) or (3c) of Theorem \[T:supp\], then the inequality $a_2<a_3$ implies $d_2' - a_2 \geq d_3-a_3$. Consequently, this condition is unchanged in passing from ${{\mathtt{d}}}$ to ${{\mathtt{d}}}'$ and so $t_{{{\mathtt{a}}},x} \in {\mathtt{S}}_{{\mathtt{c,d'}}}$. Similarly, if $t_{{{\mathtt{a}}},x}$ falls under condition (3b), then $a_2<a_3$ so $d_2'-(a_3-a_1) \geq d_3-(a_3-a_1)-(a_3-a_2)$; again we deduce $t_{{{\mathtt{a}}},x} \in {\mathtt{S}}_{{\mathtt{c,d'}}}$. So none of these occur in either $\mathcal{A}_0$ or $\mathcal{A}_1$.
On the other hand, if $t_{{{\mathtt{a}}},x}$ falls under condition (2b) for the pair $({{\mathtt{c,d}}})$, then it fails (2b) for the pair $({{\mathtt{c,d'}}})$ exactly when $a_3=a_2 \geq N$, $d_1 \geq M$, $d_2-(a_3-a_1)=M$ and $1 \leq a_1 < M$. Hence, noting also that this condition is independent of the choice of $d_1 \in \{m-1,m\}$, all such $t_{{{\mathtt{a}}},x}$ lie in both $\mathcal{A}_0$ and $\mathcal{A}_1$.
We deduce that $\vert \mathcal{A}_0 \vert - \vert
\mathcal{A}_1 \vert$ = 1 so the quotient $V_{{\mathtt{c}}}$ is indeed irreducible.
The case for $m \geq n$ follows by an analogous argument, where we interchange the roles of ${{\mathtt{c}}}_{\{1\}}$ and ${{\mathtt{c}}}_{\{2\}}$ throughout.
It only remains to show the case where $m = 1$ and $M=0$. In this case, ${{\mathtt{c}}} = (1,n,n)$ with $n > N \geq 1$ so we have ${{\mathtt{c}}}_{\{1\}} = (0,n,n)$, ${{\mathtt{c}}}_{\{2\}} = (1,n-1,n)$ and ${{\mathtt{c}}}_{\{1,2\}} = (0,n-1,n-1)$. Define $\mathcal{A}_0$ and $\mathcal{A}_1$ as above. Since $c_1-a_1 = 0$ for all ${{\mathtt{a}}} \in {{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, for any ${{\mathtt{d}}}$, and since cases (i)(2) and (i)(3) cannot occur, the analysis is much simplified from the above. We readily see that $\mathcal{A}_0 = \{ s_1^{(n,n-1)}, s_1^{(n,n)}, t_{(1,n,n),1} \}$ whereas $\mathcal{A}_1 = \{ t_{(1,n,n),1}, t_{(1,n-1,n),1} \}$. Thus we conclude again in this case that $V_{{\mathtt{c}}}$ is irreducible.
Recalling that $V^{K_n} \simeq U_{(n,n,n)}$ we immediately have the following Corollary.
\[C:maximal\] If $n>N$ then the quotient $V_{(n,n,n)}$ is the unique irreducible of maximal dimension in $V^{K_n}$.
The strict inequalities in Theorem \[T:otherirred2\] are necessary, as the following proposition shows.
\[P:red\]
(1) Let ${{\mathtt{c}}} = (M,n,n)$ with $n > N$. Then $${\mathscr{I}}(V_{{\mathtt{c}}},V_{{\mathtt{c}}}) = \begin{cases}
n-N+1 & \text{if $n < M+N$;}\\
M+1 & \text{if $n \geq M+N$}.
\end{cases}$$
(2) Let ${{\mathtt{c}}} = (n,N,n)$ with $n \geq N$. Then $${\mathscr{I}}(V_{{\mathtt{c}}},V_{{\mathtt{c}}}) = \begin{cases}
n-N+1 & \text{if $n < 2N$};\\
N+1 & \text{if $n \geq 2N$}.
\end{cases}$$
To prove part (1), let ${{\mathtt{c}}} = (M,n,n)$ with $M > 0$ and $n > N$. Then ${\mathscr{S}}_{{\mathtt{c}}} = \{2\}$ corresponding to the triple ${{\mathtt{c}}}_{\{2\}} = (M,n-1,n)$. We first compute ${\mathscr{I}}(U_{{\mathtt{c}}}, V_{{\mathtt{c}}}) = \vert {\mathtt{S}}_{{{\mathtt{c,c}}}} \setminus
{\mathtt{S}}_{{{\mathtt{c,c_{\{2\}}}}}} \vert$. For ease of notation, set $({{\mathtt{d}}},{{\mathtt{d}}}') = ({{\mathtt{c}}},{{\mathtt{c}}}_{\{2\}})$.
It is easy to see that $s_1^{(n-M,n-M)} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{S}}_{{{\mathtt{c,d}}}'}$ if $n-M \geq N$, whereas ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^{s_2} = {\mathtt{S}}_{{{\mathtt{c,d}}}'}^{s_2}=\emptyset$.
Of the elements in ${{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{T}}_{{{\mathtt{c,d'}}}} = \{ (a_1,n,n) \colon 1 \leq a_1 \leq M\}$, only $(M,n,n)$ gives rise to a representative in ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}$, and then exactly one, which we’ll denote $t_{(M,n,n),1}$.
For each ${{\mathtt{a}}} \in {\mathtt{T}}_{{{\mathtt{c,d'}}}}$, we have $a_2 < n$ and $a_3 \leq n$. Since ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}\leq \min\{a_3-a_2, n-a_3, d_2-a_2\}$ and $a_3 \leq d_2$ we deduce that if ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}\neq {{\mathtt{a}}}({{\mathtt{c,d}}}')$ then necessarily $a_3=n$ and $a_3=a_2$, a contradiction. Similarly, ${{{\mathtt{a}}}({{{\mathtt{c}}},{{\mathtt{d}}}})}'$ does not depend on the value of $d_2$. Hence ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}= \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d'}}}}$.
So suppose $t_{{{\mathtt{a}}},x} \in ({\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1 \cap \mathtt{R}_{{{\mathtt{c,d'}}}}) \setminus {\mathtt{S}}_{{{\mathtt{c,d'}}}}$. It does not fall under case (i)(1) of Theorem \[T:supp\] since this case is independent of ${{\mathtt{d}}}'$; nor can case (i)(3) occur since $a_1 \leq M$. In cases (i)(2)(a) and (c), the right hand side can depend on the value of $d_2 \in \{n-1,n\}$ if and only if $a_2 = a_3$, contradicting the hypotheses. In case (i)(2)(b), which holds only if $a_2 \geq N$, we must have that $a_3-a_2 = 0$ or else the right hand side is less than $M$. It follows that the right hand side depends on the value of $d_2$ exactly when $a_2=a_3\geq N$ and $n - (a_3-a_1) = M$; in each of these cases $\vert {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}\vert = \vert \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d'}}}} \vert =1$ and $t_{{{\mathtt{a}}},1} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{S}}_{{{\mathtt{c,d'}}}}$.
We conclude that when $M>0$ $$\begin{aligned}
{\mathtt{S}}_{{\mathtt{c,c}}} \setminus {\mathtt{S}}_{{{\mathtt{c,c}}}_{\{2\}}} &=
\{ t_{(M-k,n-k,n-k),1} \colon 0 \leq k \leq \min\{M-1,n-N\} \}\\
& \quad \cup \{ s_1^{(n-M,n-M)} \colon \text{if $n-M \geq N$}\}.\end{aligned}$$ A simpler analysis, which we consequently omit, allows us to further deduce that ${\mathtt{S}}_{{{\mathtt{c}}}_{\{2\}},{{\mathtt{c}}}} = {\mathtt{S}}_{{{{\mathtt{c}}}_{\{2\}}},{{{\mathtt{c}}}_{\{2\}}}}$ and so ${\mathscr{I}}(U_{{\mathtt{c}}},V_{{\mathtt{c}}}) = {\mathscr{I}}(V_{{\mathtt{c}}},V_{{\mathtt{c}}})$, and this has the value stated in the theorem.
When $M=0$, we have instead ${{\mathtt{c}}} = (0,n,n)$ and ${{\mathtt{c}}}_{\{3\}} = (0,n-1,n-1)$, and ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}= {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$. Since neither (i)(2) nor (i)(3) of Theorem \[T:supp\] can apply, and ${{\mathtt{a}}}({{\mathtt{d,d'}}}) = {{\mathtt{a}}}({{\mathtt{d,d'}}})'=0$ for all choices of ${{\mathtt{d}}},{{\mathtt{d}}}' \in \{ {{\mathtt{c}}}, {{\mathtt{c}}}_{\{3\}}\}$ and for all ${{\mathtt{a}}} \in {\mathtt{T}}_{{{\mathtt{d,d}}}'}$, we readily conclude that ${\mathtt{S}}_{{{\mathtt{c,c}}}} \setminus {\mathtt{S}}_{{{\mathtt{c,c}}}_{\{3\}}} = \{ t_{(1,n,n),1} \}$. Hence the quotient $V_{{\mathtt{c}}}$ is irreducible in this case.
To prove part (2), let ${{\mathtt{c}}} = (n,N,n)$ with $n \geq N$. Then ${\mathscr{S}}_{{\mathtt{c}}} = \{1\}$ with corresponding triple $(n-1,N,n)$. Reasoning as above, we deduce readily that $s_2^{(n-N,n-N)} \in {\mathtt{S}}_{{{\mathtt{c,c}}}} \setminus {\mathtt{S}}_{{{\mathtt{c,c}}}_{\{1\}}}$ whenever $n\geq 2N$ and that ${\mathtt{S}}^{s_1}_{{{\mathtt{c,c}}}} = {\mathtt{S}}^{s_1}_{{{\mathtt{c,c}}}_{\{1\}}}=\emptyset$.
Set $({{\mathtt{d}}},{{\mathtt{d}}}') = ({{\mathtt{c}}},{{\mathtt{c}}}_{\{1\}})$. Note that ${{\mathtt{T}}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{T}}_{{{\mathtt{c,d}}}'} = \{(n,a_2,n) \colon 1 \leq a_2 \leq N\}$ and each of these has $\vert {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}\vert = 1$. These triples thus give rise to only one coset in ${\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}^1$, namely that represented by $t_{(n,N,n),1}$.
Of those ${{\mathtt{a}}} \in {\mathtt{T}}_{{{\mathtt{c,d}}}'}$, one sees as above that ${\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}= \mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d'}}}}$. For such a triple ${{\mathtt{a}}}$, if $t_{{{\mathtt{a}}},x} \in {\mathtt{S}_{{{\mathtt{c}}},{{\mathtt{d}}}}}\setminus {\mathtt{S}}_{{{\mathtt{c,d'}}}}$ then it falls under case (i)(3)(b) of Theorem \[T:supp\] and we deduce as above that $N \leq a_1=a_3 \leq n$ and $n-a_3 = N-a_2$. These conditions further imply that $\vert {\mathtt{X}^{{\mathtt{a}}}_{{{\mathtt{c,d}}}}}\vert = 1$ meaning each such triple gives rise to a unique double coset.
We conclude that $$\begin{aligned}
{\mathtt{S}}_{{\mathtt{c,c}}} \setminus {\mathtt{S}}_{{{\mathtt{c,c}}}_{\{1\}}} &=
\{ t_{(n-k,N-k,n-k),1} \colon 0 \leq k \leq \min\{n-N,N-1\}\} \\
& \quad
\cup \{ s_2^{(n-N,n-N)} \colon \text{if $n-N \geq N$}\}.\end{aligned}$$ It is readily verified that ${\mathtt{S}}_{{{\mathtt{c}}}_{\{1\}},{{\mathtt{c}}}} = {\mathtt{S}}_{{{{\mathtt{c}}}_{\{1\}}},{{{\mathtt{c}}}_{\{1\}}}}$, and so ${\mathscr{I}}(U_{{\mathtt{c}}},V_{{\mathtt{c}}}) = {\mathscr{I}}(V_{{\mathtt{c}}},V_{{\mathtt{c}}})$. Counting the double cosets in the expression above yields part (2) of the theorem.
Examples {#S:examples}
========
We conclude the paper with two examples to illustrate the results in Section \[S:counting\].
\[Ex:1\]
$$\xymatrix{
& & \underset{1}{(4,4,4)}\ar[dr]\ar[dl] \\
& \underset{1}{(3,4,4)}\ar[dr]\ar[dl]& & \underset{1}{(4,3,4)}\ar[dr]\ar[dl] \\
\underset{3}{(2,4,4)}\ar[d]&&\underset{q-1}{(3,3,4)}\ar[drr]\ar[dll]\ar[d] & & \underset{3}{(4,2,4)}\ar[d] \\
\underset{1}{(2,3,4)}\ar[d]\ar[drr]&& \underset{1}{(3,3,3)}\ar[drr]\ar[dll] && \underset{1}{(3,2,4)}\ar[d]\ar[dll] \\
\underset{2}{(2,3,3)}\ar[drr]&& \underset{1}{(2,2,4)}\ar[d]& & \underset{2}{(3,2,3)}\ar[dll] \\
&& \underset{1}{(2,2,3)}\ar[d] \\
&& \underset{1}{(2,2,2)}\\
}$$
[ Reducibility of $V_{{\mathtt{c}}}$ for $V_\chi$ with $M=2$ and $N=2$. ]{}
[|ll|]{} Quotient & Dimension\
$V_{(4,4,4)}$ & $q^7(q-1)^2\alpha$\
$V_{(3,4,4)}, V_{(4,3,4)}$ & $q^6(q-1)^2\alpha$\
$V_{(3,3,4)}$ & $q^4(q-1)^3\alpha$\
$V_{(2,4,4)}, V_{(4,2,4)}$ & $q^6(q-1)\alpha$\
$V_{(2,3,4)}, V_{(3,3,3)}, V_{(3,2,4)}$ & $q^4(q-1)^2\alpha$\
$V_{(2,3,3)}, V_{(2,2,4)}, V_{(3,2,3)}$ & $q^4(q-1)\alpha$\
$V_{(2,2,3)}$ & $q^3(q-1)\alpha$\
$V_{(2,2,2)}$ & $q^3\alpha$\
\
[ Dimensions of $V_{{\mathtt{c}}}$ for $V_\chi$ with $M=2$ and $N=2$.]{}
Suppose that $M=N=2$ and let us consider the decomposition of $V^{K_4}$ under $K$. The values of $\mathcal{I}(V_{{\mathtt{c}}},V_{{\mathtt{d}}})$ are calculated using Corollary \[C:intops\] and Theorem \[T:supp\], with several values identified by Theorems \[T:irred\], \[T:otherirred1\] and \[T:otherirred2\] and Proposition \[P:red\]. The remaining computations were implemented in GAP [@GAP] and the results are represented schematically in Figure \[F:example1\], as follows.
Each triple ${{\mathtt{c}}}$ in Figure \[F:example1\] corresponds to the induced representation $U_{{\mathtt{c}}}$, and the number beneath it is the value of $\mathcal{I}(V_{{\mathtt{c}}},V_{{\mathtt{c}}})$. The arrows imply the partial order $\succeq$ on ${\mathtt{T}}$; hence the set of all components of the diagram below and including ${{\mathtt{c}}}$ may be identified with the whole of $U_{{\mathtt{c}}}$. For reference, we list in Table \[T:example1\] the dimensions of the quotients $V_{{\mathtt{c}}}$ occuring in Figure \[F:example1\]. These are calculated using Theorem \[T:calc\]. We abbreviate $\alpha = (q+1)(q^2+q+1)$.
Figure \[F:example1\] reveals several typical features of the $K$-representations $V_{{\mathtt{c}}}$. For example, we note that while many $V_{{\mathtt{c}}}$ are irreducible, several are not. Besides those identified by Proposition \[P:red\], for whom the number of intertwining operators grows at most linearly with $c_3$, there exist components such as $V_{(3,3,4)}$, for which the number of intertwining operators is a polynomial function of $q$. Such components occur more frequently in $V^{K_n}$ as $n$ increases, since they come into existence only when $\vert \mathtt{X}^{{\mathtt{a}}}_{{\mathtt{c,c}}} \vert$ is a polynomial in $q$, that is, when ${{\mathtt{a}}}({{\mathtt{c,c}}}) >0$.
\[Ex:2\]
$$\xymatrix{
\underset{2\; \dagger}{(1,4,4)}\ar[d] & \underset{q-2 \; \; \dagger}{(2,3,4)}\ar[dr]\ar[dl]\ar[d] & \underset{1}{(3,3,3)}\ar[dr]\ar[dl] & \underset{1}{(3,2,4)}\ar[d]\ar[dl] \\
\underset{1\; *}{(1,3,4)}\ar[d] & \underset{1}{(2,3,3)}\ar[dr]\ar[dl] & \underset{1 \; *}{(2,2,4)}\ar[d] & \underset{2}{(3,2,3)}\ar[dl] \\
\underset{2}{(1,3,3)}\ar[d] & & \underset{1}{(2,2,3)}\ar[dll]\ar[d] \\
\underset{1}{(1,2,3)}\ar[dr] & & \underset{1}{(2,2,2)}\ar[dl] \\
& \underset{1}{(1,2,2)}\\
}$$
[ Reducibility of and equivalences between $V_{{\mathtt{c}}}$ for $V_\chi$ with $M=1$ and $N=2$.]{}
[|ll|]{} Quotient & Dimension\
$V_{(1,4,4)}$ & $q^5(q-1)\alpha$\
$V_{(3,3,3)}, V_{(3,2,4)}$ & $q^4(q-1)^2\alpha$\
$V_{(2,3,4)}$ & $q^4(q-1)(q-2)\alpha$\
$V_{(1,3,4)}, V_{(2,2,4)}, V_{(3,2,3)}$ & $q^4(q-1)\alpha$\
$V_{(2,3,3)}$ & $q^3(q-1)^2\alpha$\
$V_{(1,3,3)}$ & $q^3(q-1)\alpha$\
$V_{(2,2,3)}$ & $q^2(q-1)^2\alpha$\
$V_{(1,2,3)}, V_{(2,2,2)}$ & $q^2(q-1)\alpha$\
$V_{(1,2,2)}$ & $q^2\alpha$\
\
[ Dimensions of $V_{{\mathtt{c}}}$ for $V_\chi$ with $M=1$ and $N=2$. ]{}
Consider a character $\chi$ for which $M=1$ and $N=2$. Figure \[F:example2\] describes a portion of the restriction to $K$ of $V_\chi$, namely, all subrepresentations $U_{{\mathtt{c}}}$ for which $V_{{\mathtt{c}}}$ has dimension of order $q^9$ or less. In terms of triples, this implies that we consider the elements ${{\mathtt{c}}} \in {\mathtt{T}}_{{\mathtt{m}}}$ for which $c_1+c_2+c_3 \leq 9$. Again, the number of intertwining operators between each pair of quotients is determined by Corollary \[C:intops\].
This example illustrates a phenomenon not present in Example \[Ex:1\]. For instance, there are two pairs of isomorphic irreducible representations: $V_{(1,3,4)} \simeq V_{(2,2,4)}$ (indicated by $*$ in Figure \[F:example2\]) and one of the two inequivalent irreducible summands of $V_{(1,4,4)}$ is isomorphic to exactly one of the irreducible summands of $V_{(2,3,4)}$ (indicated by $\dagger$ in Figure \[F:example2\]). Such pairs of isomorphic irreducibles, for distinct triples ${{\mathtt{c}}}, {{\mathtt{d}}} \in {\mathtt{T}}_{{\mathtt{m}}}$, can occur only when $c_3=d_3$, since otherwise the corresponding groups $C_{{\mathtt{c}}}$ and $C_{{\mathtt{d}}}$ lie in different levels of the filtration of $K$ by the normal subgroups $K_n$, which in turn would imply that $V_{{\mathtt{c}}}$ and $V_{{\mathtt{d}}}$ cannot intertwine as representations of $K$.
The dimensions of the representations in Figure \[F:example2\] are given in Table \[T:example2\]. We have again abbreviated $\alpha = (q+1)(q^2+q+1)$. We conjecture that $V_{(2,3,4)}$ in fact decomposes as a sum of $q-2$ distinct irreducibles, each of dimension equal to that of $V_{(1,3,4)}$, $V_{(2,2,4)}$ and $V_{(3,2,3)}$. This would be consistent with the remaining irreducible in $V_{(1,4,4)}$ having dimension equal to that of $V_{(3,3,3)}$ and $V_{(3,2,4)}$.
[9999]{}
Bader, Uri and Onn, Uri. On some geometric representations of $\mathrm{GL}_n(\mathcal{O})$, preprint. Campbell, P.S.; Nevins, M. Branching rules for unramified principal series representations of $\mathrm{GL}(3)$ over a $p$-adic field, preprint. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2004. (http://www.gap-system.org) Hill, G., On the nilpotent representations of ${\rm GL}\sb n(\mathcal{O})$. Manuscripta Math. 82 (1994), no. 3-4, 293–311. MR1265002 (95b:22040) Howe, R.E. On the principal series of ${\rm GL}_{n}$ over $p$-adic fields. Trans. Amer. Math. Soc. 177 (1973), 275–286. MR0327982 (48 \#6324) Howe, R.E. Kirillov theory for compact $p$-adic groups. Pacific J. Math. 73 (1977), no. 2, 365–381. MR0579176 (58 \#28314) Nevins, M. Branching rules for principal series representations of ${\rm SL}(2)$ over a $p$-adic field. Canad. J. Math. 57 (2005), no. 3, 648–672. MR2134405 Onn, U., Prasad, A., and Vaserstein, L. A note on Bruhat decomposition of $\mathrm{GL}(n)$ over local principal ideal rings, Comm. Algebra 34 (2006), no. 11, 4119–4130. MR2267575 Silberger, A.J. Irreducible representations of a maximal compact subgroup of ${\rm pgl}\sb{2}$ over the $p$-adics. Math. Ann. 229 (1977), no. 1, 1–12. MR0463366 (57 \#3318)
[^1]: This research is supported by grants from NSERC and from the Faculty of Science of the University of Ottawa. The first author would also like to acknowledge the support of the Centre de Recherches en Mathématiques (CRM) while at the University of Ottawa.
|
---
abstract: 'We study *near-alternating links* whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of essential surfaces and the hyperbolic volume of their complements: we show that the Strong Slope Conjecture is true for near-alternating knots with spanning Jones surfaces, their colored Jones polynomials admit stable coefficients, and the stable coefficients provide two-sided bounds on the volume of the knot complement. We also discuss extensions of these results to their Murasugi sums and a class of highly twisted links.'
address: 'Department of Mathematics, University of Texas, Austin TX 78712'
author:
- Christine Ruey Shan Lee
bibliography:
- 'references.bib'
title: 'Jones slopes and coarse volume of near-alternating links'
---
Introduction
============
Since the discovery of the Jones polynomial and related quantum knot invariants, a central problem in quantum topology has been to understand the connection between those invariants and the geometry of the knot complement. An important example of these quantum invariants is the colored Jones polynomial, which assigns a sequence $\{J_K(v, n)\}_{n=2}^{\infty}$ of Laurent polynomials from the representation theory of $U_q(\mathfrak{sl}_2)$ to a link $K\subset S^3$, and contains the Jones polynomial as the first term of the sequence, see Definition \[defn:cp\]. Conjectures such as the Volume Conjecture [@Kas97; @MM01; @MM02] and the Strong Slope Conjecture [@Gar11; @KT15] predict that the colored Jones polynomial is closely related to the hyperbolic geometry and the topology of surfaces of the knot complement.
Much evidence for this relationship comes from the class of *semi-adequate* links. These are a class of links satisfying a diagrammatic condition, see Definition \[defn:adequate-diagram\]. An adequate knot satisfies the Strong Slope Conjecture, see Conjecture \[conj:slopes\], and certain stable coefficients of their colored Jones polynomial give volume bounds on the complement of an adequate knot [@DL07; @FKP08; @FKP13]. For these results, a key component is the existence of *essential* spanning surfaces, see Deifnition \[defn:essential\], along which the complement may be decomposed into simpler geometric components. Such surfaces have also been shown to be fundamental to the characterization of alternating knots [@Jos15; @Ho15] and adequate knots [@Kal16].
In this paper, we are motivated by the question of when we can expect the Slope Conjecture and coarse volume bounds to be realized by spanning surfaces from state surfaces of the knot diagram beyond adequate links. Our answer to this question in this paper is the introduction of the class of *near-alternating* links, to be defined below in Definition \[defn:near-alternating\]. For a near-alternating knot, we compute its Jones slopes, show that there exist essential spanning surfaces in its exterior realizing the Strong Slope Conjecture, and we prove that the first, second, penultimate, and the last coefficient of its colored Jones polynomial are stable. If the near-alternating diagram is prime, twist-reduced, and highly twisted with more than 7 crossings in each twist region, then the link is hyperbolic by [@FKP08], and we show that these stable coefficients provide coarse volume bounds for the link exterior. These results closely mirror those for adequate links, and we show that near-alternating links are not adequate, thus they form a strictly new class.
We also consider extensions of these results to more general classes of links in this paper. The first direction for extension is motivated by Murasugi sums of knots, which is a method for producing link diagrams that can produce all link diagrams. A *near $A$-adequate link* is obtained by taking a certain Murasugi sum of a near-alternating link diagram with an $A$-adequate link diagram. We compute a Jones slope for a near-$A$ adequate knot and find a spanning Jones surface for the slope. For the second direction, we consider the class of highly twisted knots with multiple ($\geq 2$) negative twist regions. We show that with certain restrictions on the negative twist regions, a highly twisted knot that is *sufficiently positively-twisted*, which means that a sufficient number of positive crossings are added to every positive twist region, satisfies the Strong Slope Conjecture with stable first, second, penultimate, and last coefficient giving coarse volume bounds on the knot complement.
We give the necessary definitions in order to state the main results below. We shall always consider a link $K\subset S^3$. The indices $i, j, k$ should be considered independently in each instance unless explicitly stated otherwise.
Near-alternating link
---------------------
Let $G$ be a finite, weighted planar graph in $S^2$. For each edge $e$ of $G$ let $\omega_e \in \mathbb{Z}$ be the weight. We may replace each vertex $v$ of $G$ with a disk $D^2$ and each edge $e$ with a twisted band $B$ consisting of $|\omega_e|$ right-handed (positive) or left-handed (negative) half twists if $\omega_e>0$, or if $\omega_e<0$, respectively. We denote the resulting surface by $F_G$ and consider the link diagram $D = \partial(F_G)$. All link diagrams $D$ may be represented as $\partial(F_G)$ for some finite, weighted planar graph $G$.
A *path* in a weighted graph $G$ with vertex set $V$ and a weighted edge set $E$ is a finite sequence of distinct vertices $v_1, v_2, \ldots, v_k$ such that $(v_i, v_{i+1}) \in E$ for $i = 1, 2, \cdots, k-1$. We define the *length* of a path $W$ as $$\ell(W):= 2+\sum_{i=1}^{k-1} (|\omega_i|-2),$$ where $\omega_i$ is the weight of the edge $(v_i, v_{i+1})$ in $W$.
A graph $G$ is said to be *$2$-connected* if it does not have a vertex whose removal results in a disconnected graph, such a vertex is called a *cut vertex*.
\[defn:near-alternating\] We say that a non-split link diagram $D$ is *near-alternating* if $D = \partial(F_G)$, where $G$ is a 2-connected, finite, weighted planar graph without one-edged loops with a single negative edge $e=(v, v')$ of weight $r<0$, such that $|r| \geq 2$. In addition, the graph $G^e$ obtained by deleting the edge $e$ from $G$ satisfies the following conditions.
(a) Let $\omega$ be the minimum of $\ell(W)$ taken over all paths $W$ in $G^e$ starting at $v$ and ending at $v'$ and let $t$ be the total number of such paths. Then $t>2$, and $$\frac{\omega}{t} > |r|.$$
(b) The graph $G^e$ remains 2-connected, and $\partial(F_{G^e})$ is prime. We also require that the diagram $D_r = \partial{F_{G/e}}$, where $G/e$ is the graph obtained from $G$ by contracting the edge $e$, be adequate.
A link $K$ is said to be *near-alternating* if it admits a near-alternating diagram, see Figure \[fig:exneara\] for an example and the conventions for a negative or a positive twist region.
A pretzel link $P(\frac{1}{t_1}, \frac{1}{t_2}, \ldots, \frac{1}{t_m})$ is near-alternating if $t_1 <0$, $t_i >0$ for all $1<i\leq m$, and $$\frac{\min_{1<i\leq m} \left\{ t_i \right\}}{m-1} > |t_1|.$$
Strong Slope Conjectures
------------------------
Let $D$ be a link diagram. A *Kauffman state* $\sigma$ is a choice of replacing every crossing of $D$ by the $A$- or $B$-resolution as in Figure \[fig:abres\], with the (dashed) segment recording the location of the crossing before the replacement.
Applying a Kauffman state results in a set of disjoint circles called *state circles*. We form a $\sigma$-*state graph* $s_{\sigma}(D)$ for each Kauffman state $\sigma$ by letting the resulting state circles be vertices and the segments be edges. The *all-$A$* state graph $s_A(D)$ comes from the Kauffman state which chooses the $A$-resolution at every crossing of $D$. Similarly, the *all-$B$* state graph $s_B(D)$ comes from the Kauffman state which chooses the $B$ resolution at every crossing of $D$.
Let $$h_n(D) = -(n-1)^2c(D) -2(n-1)|s_A(D)| + \omega(D) ((n-1)^2+2(n-1)), \text{ where } \label{eq:lowerbound}$$ $c(D)$ is the number of crossings of $D$, $\omega(D) = c_+(D)-c_-(D)$ is the writhe of $D$ with an orientation, and $|s_A(D)|$ is the number of vertices in the all-$A$ state. We can now state the main result of this paper.
Let $d(n)$ be the minimum degree of $J_K(v, n)$, the *$n$th colored Jones polynomial of $K$.*
\[thm:degree\] Let $K \subset S^3$ be a link admitting a near-alternating diagram $D$ with a single negative twist region of weight $r<0$ and let $h_n(D)$ be defined by , then $$d(n) = h_n(D) - 2r((n-1)^2+(n-1)).$$
Note that the case for many 3-tangle pretzel knots with a near-alternating diagram was already shown in [@LV], and the degree of the Jones polynomial of pretzel knots was computed in [@HTY00] for certain pretzel knots which are mostly not near-alternating.
Theorem \[thm:degree\] proves the Strong Slope Conjecture for near-alternating knots which we now describe. An orientable and properly embedded surface $S \subset S^3 \setminus K$ is *essential* if it is incompressible, boundary-incompressible, and non boundary-parallel. If $S$ is non-orientable, then $S$ is *essential* if its orientable double cover in $S^3\setminus K$ is essential in the sense as defined.
\[defn:essential\] Let $S$ be an essential and orientable surface with non-empty boundary in $S^3 \setminus K$. A fraction $\frac{p}{q} \in \mathbb{Q} \cup \{\frac{1}{0}\}$ is a *boundary slope* of $K$ if $p\mu + q\lambda$ represents the homology class of $\partial S$ in $\partial N(K)$, where $\mu$ and $\lambda$ are the canonical meridian and longitude basis of $\partial N(K)$. The boundary slope of an essential non-orientable surface is that of its orientable double cover.
Garoufalidis showed in [@Gar11] that since the colored Jones polynomial is $q$-holonomic [@GL05], the functions $d(n)$ and $d^*(n)$, where $d^*(n)$ is the maximum degree of $J_K(v, n)$, are *quadratic quasi-polynomials* viewed as functions from $\mathbb{N} \rightarrow \mathbb{N}$. This means that there exist integers $p_K$, $C_K \in \mathbb{N}$ and rational numbers $a_j, b_j, c_j, a^*_j, b^*_j, c^*_j$ for each $0\leq j < p_K$, such that for all $n > C_K$, $$d(n) = a_jn^2+ b_jn + c_j \text{ if } n = j \pmod{p_K},$$ and $$d^*(n) = a^*_jn^2+ b^*_jn + c^*_j \text{ if } n = j \pmod{p_K}.$$
We consider the sets $js_K:= \{a_j\}$ and $js^*_K:= \{a^*_j\}$. An element $\frac{p}{q} \in js_K \cup js^*_k$ is called a *Jones slope*. We also consider the sets $jx_K := \{\frac{b_j}{2} \}$ and $jx^*_K:= \{\frac{b^*_j}{2}\}$. We may now state the Strong Slope Conjecture.
[([@Gar11; @KT15])]{}\[conj:slopes\] Given a Jones slope of $K$, say $\frac{p}{q} \in js_K$, with $q>0$ and $(p, q)=1$, there is an essential surface $S\subset S^3\setminus K$ with $|\partial S|$ boundary components such that each component of $\partial S$ has slope $\frac{p}{q}$, and $$-\frac{\chi(S)}{|\partial S|q} \in jx_K.$$ Similarly, given $\frac{p^*}{q^*} \in js^*_K$ with $q^*>0$ and $(p^*, q^*)=1$, there is an essential surface $S^*\subset S^3\setminus K$ with $|\partial S^*|$ boundary components such that each component of $\partial S^*$ has slope $\frac{p^*}{q^*}$, and $$\frac{\chi(S^*)}{|\partial S^*|q^*} \in jx^*_K.$$
An essential surface in $S^3\setminus K$ satisfying the conditions described in the conjecture is called a *Jones surface*.
The difference in our convention from [@Gar11; @KT15] is that in this paper the asterisk $*$ indicates the corresponding quantity from the maximum degree, rather than the minimum degree, of the $n$th colored Jones polynomial $J_K(v, n)$, while $d(n)$ indicates the corresponding quantities from the minimum degree. Also, instead of substituting $v = \frac{1}{A^4}$ we substitute $v = \frac{1}{A}$ for the colored Jones, see Definition \[defn:cp\] for our choice of the normalization convention.
The Strong Slope Conjecture is currently known for alternating knots [@Gar11], adequate knots [@FKP13], which is a generalization of alternating knots by Definition \[defn:adequate-diagram\], iterated $(p, q)$-cables of torus knots and iterated cables of adequate knots [@KT15], and families of 3-tangle pretzel knots [@LV]. It is also known for all knots with up to 9 crossings [@Gar11; @KT15; @Howie] and an infinite family of arborescent non-Montesinos knots [@HD17]. The Slope Conjecture is also known for 2-fusion knots [@GR14].
In the context of the Strong Slope Conjecture, Theorem \[thm:degree\] says that $js_K= \{-2c_-(D)-2r\}$ and $jx_K=\{c(D)-|s_A(D)|+r\}$. The surface realizing $js_K$ and $jx_K$ from Theorem \[thm:degree\] is a state surface corresponding to a Kauffman state constructed as follows.
Given a Kauffman state $\sigma$ on a link diagram $D$, we may form the *$\sigma$-state surface*, denoted by $S_{\sigma}(D)$, by filling in the disjoint circles in $s_{\sigma}(D)$ with disks, and replacing each segment recording the previous location of the crossing by half-twisted bands. See Figure \[fig:abressurface\].
For a near-alternating knot $K$ with $\partial F_G = $ a diagram $D$ of $K$, the surface $F_G$ is essential by [@OR12 Theorem 2.15] and is given by the state surface $S_{\sigma}(D)$ where $\sigma$ chooses the $B$-resolution on the $|r|$ crossings corresponding to the single edge with negative weight $r$ in $G$, and the $A$-resolution everywhere else. We compute the boundary slope and Euler characteristic of this surface and show that it matches with $js_K$ and $jx_K$.
\[thm:jsurface\] Let $K\subset S^3$ be a link admitting a near-alternating diagram $D = \partial (F_G)$ with a single negative twist region of weight $r<0$, then the surface $F_G$ is essential with 1 boundary component such that each component has slope $-2c_-(D)-2r$ and $$-\chi(S) = c(D)-|s_A(D)|+r.$$
To see the Jones surface $S^* \subset S^3\setminus K$ with boundary slope $\frac{p^*}{q^*}$ matching $js^*_K$ and $\frac{\chi(S^*)}{|\partial S^*|q^*}$ matching $jx^*_K$, we use the fact that a near-alternating link is *$B$-adequate*, see Lemma \[lem:nabad\], as defined below.
\[defn:adequate-diagram\] A link diagram $D$ is *$A$-adequate* (resp. *$B$-adequate*) if its all-$A$ (resp. all-$B$) state graph $s_A(D)$ (resp. $s_B(D)$) has no one-edged loops. A link $K$ is *semi-adequate* (*$A$- or $B$-adequate*) if it admits a diagram that is $A$- or $B$-adequate. If a link $K$ admits a diagram that is both $A$- and $B$-adequate, then we say that $K$ is adequate.
Note that alternating knots form a subset of adequate knots.
Let $$h^*_n(D) = (n-1)^2c(D) +2(n-1)|s_B(D)| + \omega(D) ((n-1)^2+2(n-1)). \label{eq:upperbound}$$
It is well known that for any link diagram $D$, we have $h_n(D)\leq d(n)$, $d^*(n) \leq h^*_n(D)$ and the first equality is achieved when $D$ is $A$-adequate, while the second equality is achieved when $D$ is $B$-adequate. This follows from [@LT88], [@Lic97 Lemma 5.4], and [@FKP13]. Therefore, if $K$ is $A$-adequate (resp. $B$-adequate) then there is a single Jones slope in $js_K$ (resp. in $js^*_K$).
If $D$ admits an $A$-(resp. $B$-)adequate diagram, then [@Oza11] implies that the all-$A$ (resp. all-$B$) state surface is essential. An all-$A$ or all-$B$ state surface was shown by [@FKP13] to realize $js_K, jx_K$, or $js^*_K, jx^*_K$, respectively. As noted, a near-alternating diagram is $B$-adequate, so the all-$B$ state surface of $D$ realizes the Jones slope of $js^*_K$ and $jx^*_K$. The surface $F_G$ and the all-$B$ state surface of a near-alternating diagram verify the Strong Slope Conjecture for these knots.
Numerical evidence, particularly those from 3-string pretzel knots [@LV] and fusion knots [@GR14], suggests that the graphical conditions imposed on a near-alternating knot diagram are the best possible to ensure that Jones slope are integral and realized by state surfaces. In other words, if a knot diagram $D = \partial(F_G)$ where $G$ is a 2-connected, finite, weighted planar graph without one-edged loops with a single negative edge of weight $r<0$, so that the quantities $\omega$ and $t$ still make sense, we expect that $\frac{\omega}{t} \leq r$ implies that the Jones slope is rational, or, it is not realized by a state surface. We will address this in a future project.
Generalization to Murasugi sums
-------------------------------
We extend Theorem \[thm:degree\] by restricting to certain *Murasugi sums*, or *planar star product*, of a near-alternating diagram with an $A$-adequate diagram. We consider a general version of the planar star product (Murasugi sum) of two link diagrams $D_1$ and $D_2$.
Let $s_{A}(D_1)$ and $s_{A}(D_2)$ be the all-$A$ state graphs of two links diagrams $D_1$ and $D_2$, respectively. If we glue $s_{A}(D_1)$ and $s_{A}(D_2)$ along a vertex, we obtain a new graph called the *star product* of $D_1$ and $D_2$. The new graph uniquely determines a link diagram which we denote by $D$. We say that $D$ is a *Murasugi sum*, or *planar star product* of $D_1$ and $D_2$, and we write $D = D_1 \star D_2$.
Theorem \[thm:degree\] generalizes with some restrictions on the Murasugi sum.
\[thm:Murasugi\] Suppose $K$ is a link with a diagram $D(K) = D_1 \star D_2$ that is a Murasugi sum of a near-alternating diagram $D_1$ with a single negative twist region of weight $r<0$, and an $A$-adequate diagram $D_2$, such that the circle in $s_A(D_1)$ along which the Murasugi sum is formed has no one-edged loops, then $$d(n) = h_n(D) - 2r((n-1)^2+(n-1)).$$
An essential spanning surface $F$ for $K$ with boundary slope $-2c(D)-2r$ and realizing $jx_K$ may be formed by taking the Murasugi sum of two spanning surfaces $F_1$, $F_2$ for the links $L_1$ and $L_2$ as follows [@Oza11].
Let $F$ be a spanning surface for a link $K$. Suppose that there exists a 2-sphere $S$ decomposing $S^3$ into two 3-balls $B_1, B_2$ such that $F\cap S$ is a disk. Put $F_i = F\cap B_i$ for $i = 1, 2$. Then we say that $F$ has a *Murasugi decomposition* into $F_1$ and $F_2$ and denote it by $F = F_1 \star F_2$. Conversely, we say that $F$ is obtained from $F_1$ and $F_2$ by a Murasugi sum along a disk $F\cap S$.
We use the fact that the Murasugi sum of two essential surfaces is essential by [@Ga85; @Oza11] to show the following.
Suppose a knot $D(K) = D_1 \star D_2$ is a Murasugi sum of a near-alternating diagram $D_1$ and an $A$-adequate diagram $D_2$, such that the circle in $s_A(D_1)$ along which the Murasugi sum is formed has no one-edged loops. Let $G$ be the 2-connected, weighted planar graph from which we obtain $D_1=\partial(F_G)$. The Jones slope $js_K=\{-2c_-(D)-2r\}$ is realized by a Murasugi sum $S$ of the surface $F_G$ and the all-$A$ state surface for $D_2$, and $$-\chi(S) = jx_K=\{c(D)-|s_A(D)|+r\}.$$
As for the question of whether a near-alternating knot can admit an $A$-adequate diagram, we show, using the Kauffman polynomial, that a near-alternating knot cannot admit a diagram that is both $A$- and $B$-adequate.
\[thm:naknoadequate\] A near-alternating knot does not admit an adequate diagram.
It is an interesting question whether the colored Jones polynomial can be used to obstruct the existence of an $A$-adequate diagram for a near-alternating knot. The criterion from [@Lee16] may be applied if there is information restricting the number of positive crossings in a diagram. We will pursue this question in a future project.
We would like to remark that by [@Aetal92 Theorem 3.1], every near-alternating link admits an almost-alternating diagram, and it is not known whether every almost-alternating link is semi-adequate.
Stable coefficients and Coarse volume
-------------------------------------
Let $\alpha_{i, n}$ be the coefficient of $v^{d(n)+4i}$ of the *reduced* colored Jones polynomial $\hat{J}_K(v, n):=J_K(v, n)/J_{\vcenter{\hbox{\includegraphics[scale=.05]{circ.png}}}}(v, n)$, where $J_{\vcenter{\hbox{\includegraphics[scale=.05]{circ.png}}}}(v, n)$ is the $n$th colored Jones of the unknot, and let $\alpha'_{i, n}$ be the coefficient of $v^{d^*(n)-4i}$, so that $\alpha_{0, n}, \alpha_{1, n}, \alpha'_{1, n}, \alpha'_{0, n}$ are the first, second, penultimate, and last coefficient of $\hat{J}_K(v, n)$, respectively.
Let $i\geq 0$, the first $i$th coefficient (resp. last $i$th coefficient) of the reduced colored Jones polynomial is *stable* if $\alpha_{i, j} = \alpha_{i, i}$ (resp. $\alpha'_{i, j} = \alpha'_i$) for all $j\geq i$.
It is known that for an adequate knot, the first and last $i$th coefficient are stable for all $i \geq 0$ [@Arm]. The cases $i=0$ and $i=1$ have first been shown by [@DL06]. They also give explicit formulas for the coefficients from the all-$A$ and all-$B$ state graphs of an adequate diagram of a knot. These have been used to give a two-sided volume bound for alternating knots [@DL07]. Futer, Kalfagianni, and Purcell have these coefficients to give two-sided bounds on the volume of a hyperbolic, adequate knot [@FKP13]. These results establish that for an adequate knot, the stable coefficients of the colored Jones polynomial are *coarsely related* to the volume of the knot as defined below.
Let $f, g: Z \rightarrow \mathbb{R}_+$ be functions from some (infinite) set $Z$ to the non-negative reals. We say that $f$ and $g$ are *coarsely related* if there exist universal constants $C_1\geq 1$ and $C_2\geq 0$ such that $$C_1^{-1}f(x)-C_2 \leq g(x) \leq C_1f(x) + C_2 \ \ \ \forall x\in Z.$$
The Coarse Volume Conjecture [@FKP13 Question 10.13] asks whether there exists a function $B(K)$ of the coefficients of the colored Jones polynomials of every knot $K$, such that for hyperbolic knots, $B(K)$ is coarsely related to hyperbolic volume $vol(S^3 \setminus K)$. Here the infinite set $Z$ is taken to be the set of hyperbolic knots.
We show that a near-alternating knot has stable first, second, last, and penultimate coefficients which are determined by state graphs of a near-alternating diagram. We give a two-sided bound on the volume of a highly twisted, near-alternating knot based on these coefficients. To simplify notation we will just write $\alpha_n$ for $\alpha_{0, n}$, $\beta_n$ for $\alpha_{1, n}$, $\alpha'_n$ for $\alpha'_{0, n}$, and $\beta'_n$ for $\alpha'_{1, n}$.
Let $\mathbb{G}$ be a graph without one-edged loops, an edge $e = (v, v')$ is called *multiple* if there is another edge $e' = (v, v')$ in $\mathbb{G}$. The *reduced graph* of $\mathbb{G}$, denoted by $\mathbb{G}'$, is obtained from $\mathbb{G}$ keeping the same vertices but replacing each set of multiple edges between a pair of vertices $v, v'$ by a single edge. The *first Betti number* of a graph, denoted by $\chi_1(\mathbb{G})$, is the number $v-e+k$, where $v$ is the number of vertices of $\mathbb{G}$, $e$ is the number of edges of $\mathbb{G}$, and $k$ is the number of connected components of $\mathbb{G}$.
\[thm:tail\] Let $K$ be a link admitting a near-alternating diagram $D = \partial (F_G)$, where $G$ is a finite 2-connected, weighted planar graph with a single negatively-weighted edge of weight $r < 0$. The first and second coefficient, $\alpha_n, \beta_n$, respectively, of the reduced colored Jones polynomial $\hat{J_K}(v, n)$ of a near-alternating link $K$ are stable. Write $\alpha = \alpha_n$ and $\beta = \beta_n $. We have $|\alpha| = 1$ and $|\beta| = \begin{cases} \chi_1(s_{\sigma}(D)') + 1 \text{ if $|r|=2$} \\
\chi_1(s_{\sigma}(D)') \text{ if $|r|>2$ }
\end{cases}$, where $\sigma$ is the Kauffman state corresponding to the state surface $F_G$ and $\chi_1(s_{\sigma}(D)')$ is the first Betti number of the reduced graph of $s_{\sigma}(D)$. The last and penultimate coefficient, $\alpha'_n, \beta'_n$, respectively, are also stable, and we write $\alpha' = \alpha'_n$ and $\beta' = \beta'_n$. We have $|\alpha'|=1$ and $|\beta'| = \chi_1(s_{B}(D)')$.
If $K$ is such that the near-alternating diagram is prime and twist-reduced with more than 7 crossings in each twist region, then $K$ is hyperbolic, and $$.35367(|\beta|+|\beta'| -1) < vol(S^3\setminus K) < 30v_3(|\beta|+|\beta'| + M - 2),$$ for a constant $M\geq 0$. Here $v_3\approx 1.0149$ is the volume of a regular ideal tetrahedron. In other words, stable coefficients of $K$ are coarsely related to the hyperbolic volume of $S^3\setminus K$.
The second stable coefficient $\beta$ is computed in terms of the Euler characteristic of the state surface $F_G = S_{\sigma}(D)$ in a formula similar to those given in [@DL06; @DL07] for adequate knots. Numerical experiments suggest that more coefficients of the reduced colored Jones polynomial should be stable. However, we do not pursue this question in this paper. For the two-sided bound on volume, we use volume estimates based on the twist numbers of a knot developed in [@FKP08] using the works of Adams, Agol, Lackenby, and Thurston. For other examples of volume estimates for links admitting different types of diagrams, see [@BMPW15] and [@Gia15; @Gia16].
Sufficiently positively-twisted links
-------------------------------------
In the final section of the paper we generalize these results to links with a diagram $D = \partial F_G$ where $G$ has more than one negative edge, so $D$ has multiple negative twist regions. We consider the effect of adding full positive twists to the positive twist regions of $D$.
For a weighted planar graph $G$ let $G^-$ denote the sub-graph of $G$ consisting of the negative edges of $G$, and let $G^-_c$ be a connected component of $G^-$.
\[thm:eventual\]Let $K\subset S^3$ be a knot with a prime, twist-reduced diagram $D=\partial (F_G)$, where $G$ is 2-connected, the graph $G^{\{e\}}$ obtained by deleting all the negatively-weighted edges $\{e_i\}$ from $G$ remains 2-connected, and each connected component $G^-_c$ of $G$ is a single negative edge $e_i$. In addition, the diagram $D' = \partial F_{G/\{e\}}$, where $G/\{e\}$ is the graph obtained from $G$ by contracting along each edge $e_i$, is adequate. Assume that $D$ has $tw(D) \geq 2$ twist regions, and that each region contains at least 7 crossings. Let $K_m$ be the knot obtained from $K$ by adding $m$ full twists on two strands to every positive twist region. There exists some integer $0 < M_K < \infty$ such that for all $m > M_K$,
(i) $K_m$ is hyperbolic,
(ii) the Strong Slope Conjecture is true for $K_m$ with spanning Jones surfaces, and
(iii) the coefficients $\alpha = \alpha_n$, $\beta = \beta_n$, $\alpha' = \alpha'_n$, and $\beta' = \beta'_n$ are stable. They give the following two-sided volume bounds for $S^3\setminus K_m$: $$|\beta|+|\beta'|+M+2(R-1) \leq vol(S^3\setminus K_m) \leq |\beta|+|\beta'|+M-1 + \frac{R-1}{3},$$ for some constant $M$, where $R$ is the number of maximal negative twist regions in $D$.
In this theorem it is not determined whether $M$ is always non-negative or always non-positive in the two-sided bound, while $R$ is always positive. An example of a highly twisted link from a graph $G$ satisfying the graphical constraint of the theorem is shown below in Figure \[fig:htwist\].
Organization {#organization .unnumbered}
------------
In Section \[sec:prelim\], we give a definition of the colored Jones polynomial in terms of skein theory and summarize elementary results needed for Theorem \[thm:degree\], which is proven in Section \[sec:jslope\]. In Section \[sec:jsurface\], we prove Theorem \[thm:jsurface\] by computing the boundary slope and the Euler characteristic of $F_G$, and we generalize a part of Theorem \[thm:degree\] to Murasugi sums of a near-alternating knot and an $A$-adequate knot by proving Theorem \[thm:Murasugi\] and its corollary. We show Theorem \[thm:naknoadequate\], which says that a near-alternating knot is not adequate in Section \[sec:nadequate\]. Finally, we compute stable coefficients and give a coarse volume bound to prove Theorem \[thm:tail\] in Section \[sec:cvolume\]. In Section \[sec:sptl\], we prove Theorem \[thm:eventual\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
This is a side project that grew out of a project with Roland van der Veen. I would like to thank him for our conversations which made this spin-off possible. I would also like to thank Efstratia Kalfagianni, Stavros Garoufalidis, and Oliver Dasbach for their comments and encouragement on this work, and for their hospitality during my visits. Lastly, I would like to thank Mustafa Hajij for interesting discussions on stability properties of the colored Jones polynomial, Adam Lowrance for pointing out that near-alternating knots are almost-alternating, and Joshua Howie for interesting conversations on the Strong Slope Conjecture.
Graphical skein theory {#sec:prelim}
======================
We follow the approach of [@Lic97] in defining the Temperley-Lieb algebra. The following formulas are also found in [@MV94]. Let $F$ be an orientable surface with boundary which has a finite (possibly empty) collection of points specified on $\partial F$. A link diagram on $F$ consists of finitely many arcs and closed curves on $F$ such that
- There are finitely many transverse crossings with an over-strand and an under-strand.
- The endpoints of the arcs form a subset of the specified points on $\partial F$.
Two link diagrams on $F$ are isotopic if they differ by a homeomorphism of $F$ isotopic to the identify. The isotopy is required to fix $\partial F$.
\[defn:skein\] Let $A$ be a fixed complex number. The *linear skein* $\mathcal{S}(F)$ of $F$ is the vector space of formal linear sums over $\mathbb{C}$ of isotopy classes of link diagrams in $F$ quotiented by the relations
(i) $D \sqcup \vcenter{\hbox{\includegraphics[scale=.10]{circ.png}}} = (-A^2-A^{-2}) D, $
(ii) $ \vcenter{\hbox{\includegraphics[scale=.2]{crossing1.png}}} = A^{-1} \ \vcenter{\hbox{\includegraphics[scale=.2]{crossing2.png}}} \ + A \ \vcenter{\hbox{\includegraphics[scale=.2]{crossing3.png}}} \ .$
We consider the linear skein $\mathcal{S}(D, n)$ of the disc $D$ with $2n$-points specified on its boundary. For $D_1, D_2 \in {\mathcal{S}}(D,n)$, there is a natural multiplication operation $D_1\cdot D_2$ defined by identifying the top boundary of $D_1$ with the bottom boundary of $D_2$. This makes $\mathcal{S}(D, n)$ into an algebra $TL_n$, called the *Temperley-Lieb algebra*. The algebra $TL_n$ is generated by crossingless matchings $1_n, e^{1}, \ldots, e^{n-1}$ of $2n$ points of the form shown in Figure \[fig:TLgen\].
We will denote $n$ parallel strands, the identity $1_n$, also by $|_n$.
Suppose that $A^4$ is not a $k$th root of unity for $k\leq n$. There is an element ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n$ in $TL_n$ called the *Jones-Wenzl idempotent*, which is uniquely defined by the following properties. For the original reference where the projector was defined and studied, see [@Wen87].
(i) ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n \cdot e^i = e^i \cdot {\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}=0$ for $1 \leq i \leq n-1$. \[list:prop1\]
(ii) ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n -1 $ belongs to the algebra generated by $\{e^1, e^2,\ldots, e^{n-1}\}$.
(iii) ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n \cdot {\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n = {\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n$,
(iv) Let $\mathcal{S}(S^1 \times I)$ be the linear skein of the annuli with no points marked on its boundaries. The image of ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n$ in $\mathcal{S}(S^1 \times I)$ obtained by joining the $n$ boundary points on the top with the those at the bottom is equal to $$\triangle_n = (-1)^n[n] \cdot \text{the empty diagram on $S^1\times I$},$$ \[list:prop4\]
where $[n]$ is the *quantum integer* defined by $$[n]:= \frac{A^{2(n+1)} - A^{-2(n+1)}}{A^{2}-A^{-2}}.$$
From the defining properties, the Jones-Wenzl idempotent also satisfies a recursion relation and two other identities as indicated in Figures \[fig:jw1\], \[fig:jw2\], and \[fig:jw3\].
$$\label{eq:jwrecursive}
\centering
\def\svgwidth{.9\columnwidth}
\PandocStartInclude{jwrecursion.pdf_tex}\PandocEndInclude{input}{337}{32}$$
$$\label{eq:jwidentity}
\def\svgwidth{.5\columnwidth}
\PandocStartInclude{jwidentity.pdf_tex}\PandocEndInclude{input}{344}{31}$$
$$\label{eq:jwidentity2}
\def\svgwidth{.5\columnwidth}
\PandocStartInclude{jwidentity2.pdf_tex}\PandocEndInclude{input}{352}{32}$$
\[defn:cp\] Let $D$ be a diagram of a link $K\subset S^3$ with $k$ components. For each component $D_i$ for $i \in \{1,\ldots, k\}$ of $D$ take an annuli $A_i$ via the blackboard framing. Let $f: \mathcal{S}(S^1\times I) \rightarrow \mathcal{S}(\mathbb{R}^2)$ be the map that sends an element of $\mathcal{S}(S^1\times I)$ to each $A_i$ in the plane. For $n\geq 2$, the *$n$th unreduced colored Jones polynomial* $J_K(v; n)$ may be defined by substituting $A = v^{-1}$ into the bracket portion of $$J_K(v, n) := ((-1)^{n-1}v^{(n^2-1)})^{\omega(D)} \left\langle f\left({\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_{n-1} \right) \right \rangle.$$ This definition of the colored Jones polynomial follows the convention of [@KT15], except that their $A$ is such that $v = A^{-4}$, and we do not multiply by an extra $(-1)^{n-1}$. Note that this gives $J_{\vcenter{\hbox{\includegraphics[scale=.05]{circ.png}}}}(v, n+1) = (-1)^n[n]$ as the normalization.
The Kauffman bracket here is extended by linearity and gives the polynomial multiplying the empty diagram after reducing the diagram via skein relations. The skein $f\left({\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_{n-1} \right)$ is the $n-1$ blackboard cable of $D$ decorated by a Jones-Wenzl idempotent, which we will denote by $D^{n-1}_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}$ from now on.
Let
We can use the identities indicated in Figure \[fig:fusion\] and \[fig:untwisting\] to simplify the bracket $\left\langle f\left(\vcenter{\hbox{\includegraphics[scale=.15]{jwprojc.png}}} \right) \right\rangle.$
A triple of non-negative integers $a, b, c$ is called *admissible* if $a+b+c$ is even and $|a-b|\leq c \leq a+b$.
$$\label{eq:fusion}
\PandocStartInclude{fusion.pdf_tex}\PandocEndInclude{input}{386}{27}$$
$$\label{eq:untwisting}
\PandocStartInclude{untwisting.pdf_tex}\PandocEndInclude{input}{395}{31}$$
Let $a, b, c$ be admissible, let $\theta(a, b, c)$ be the bracket of the skein shown in Figure \[fig:theta\].
$ \theta(a, b, c):= \vcenter{\hbox{\PandocStartInclude{theta.pdf_tex}\PandocEndInclude{input}{406}{60}}}$
[[@Lic97 Lemma 14.5].]{} Let $\triangle_n!:= \triangle_1 \cdot \triangle_2 \cdot \cdots \triangle_n$ and $\triangle_0! = 1$. Also let $x = \frac{a+b-c}{2}, z = \frac{a+c-b}{2},$ and $ y = \frac{b+c-a}{2}$, then $\theta(a, b, c)$ is given explicitly by the following formula. $$\theta(a, b, c):= \frac{\triangle_{x+y+z}!\triangle_{x-1}! \triangle_{y-1}! \triangle_{z-1}!}{\triangle_{y+z-1}!\triangle_{z+x-1}!\triangle_{x+y-1}!}$$
Let $\deg{f}$ be the maximum degree of a Laurent polynomial $f \in \mathbb{Z}[A, A^{-1}]$. We will mainly be concerned with the degree of the terms in the formulas above. For convenience, we will list the degrees of $\triangle_c$, $d(a, b, c)$, and $\theta(a, b, c)$ here. They are obtained by examining the formulas. $$\begin{aligned}
\label{eq:degs}
\deg{\triangle_c} &= 2c, \text{ and } \notag \\
\deg{\theta(a, b, c)} &= a + b + c.\end{aligned}$$
We will be using the following lemma from [@Arm].
Let ${\mathcal{S}}$ be a crossing-less diagram decorated by Jones-Wenzl idempotents ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n$, consider the skein $\overline{{\mathcal{S}}}$ obtained from ${\mathcal{S}}$ by replacing each of the idempotents by the identity $|_n$, so $\overline{{\mathcal{S}}}$ consists of disjoint circles. The skein ${\mathcal{S}}$ is called *adequate* if no circle in $\overline{{\mathcal{S}}}$ passes through any of the regions previously decorated by an idempotent more than once.
\[lem:jwad\] Let ${\mathcal{S}}\in {\mathcal{S}}(\mathbb{R}^2)$ be a skein decorated by Jones-Wenzl idempotents ${\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}_n$, and $\overline{{\mathcal{S}}}$ be the skein obtained by replacing each Jones-Wenzl idempotent by the identity element $ |_n$, then $$\deg\langle {\mathcal{S}}\rangle \leq \deg\langle \overline{{\mathcal{S}}} \rangle.$$ If ${\mathcal{S}}$ is a crossing-less skein that is adequate, then $$\deg\langle {\mathcal{S}}\rangle = \deg\langle \overline{{\mathcal{S}}} \rangle.$$
We also use an additional identity from [@MV94].
\[lem:jwid3\] For $y\geq 1$,
$$\PandocStartInclude{jwidentity3.pdf_tex}\PandocEndInclude{input}{442}{32}$$
The slight difference with [@MV94] in the coefficient multiplying the right-hand side is due to their slightly different convention for the quantum integer.
Jones slopes {#sec:jslope}
============
We prove Theorem \[thm:degree\] in this Section. Let $H_n(D) = -h_{n+1}(D)+\omega(D)(n^2+2n)$. We will only deal with the Kauffman bracket from now on with the variable $A$. Theorem \[thm:degree\] then follows from the following theorem.
\[thm:bracketdegree\] If $K$ is a near-alternating link with a single negative twist region of weight $r<0$, then $$\deg \langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = H_n(D) + 2r(n^2+n).$$
Overview
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Our main strategy is to find a suitable state sum for $\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle$ which has a degree-dominating term. If $D$ is near-alternating, we may simplify the sum and disregard many of the terms whose skein evaluates to zero in the Kauffman bracket. This is done in Section \[subsec:simplifyss\]. In Section \[subsec:degree-dominating\], we highlight the term in the state sum which will be shown to be degree-dominating. The most laborious step of the proof comes from bounding the degree of a term coming from another state $\sigma$ in the state sum. We do this in Section \[subsec:sigmab\], where we first estimate the crossings on which $\sigma$ chooses the $B$-resolution by Lemma \[lem:count\]. The reason why this gives a bound on the degree is given by Lemma \[lem:sigmab\]. This leads to the important corollary, Lemma \[lem:wcount\], which we can apply to the case where $D$ is a near-alternating diagram to bound the degree of the term in the state sum corresponding to $\sigma$. Finally in Section \[subsec:complete\] we put the estimates together to finish the proof of Theorem \[thm:bracketdegree\]. Upon first reading the reader may skip the proof of Lemma \[lem:count\] to get a sense of how it is applied.
Simplifying the state sum {#subsec:simplifyss}
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Let $D$ be a near-alternating link diagram, which means that it has a single negative twist region of weight $r<0$. We fix $n$. Given the skein $ D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}$, slide the idempotents along the link strands and make copies until there are four idempotents framing the negative twist region. See Figure \[fig:framenegt\] below.
By the fusion and untwisting formulas, we may fuse the two strands of the negative twist region and get rid of the crossings. This results in a sum over the fusion parameter $a$ such that the triple $a, n, n$ is admissible. For a fixed $a$ consider a Kauffman state $\sigma$ on the set of remaining crossings. Applying $\sigma$ results in a skein ${\mathcal{S}}^a_{\sigma}$ that is the disjoint union of a connected component $J^a_{\sigma}$ decorated by Jones-Wenzl idempotents with circles. Let $$\begin{aligned}
{\text{sgn}}(\sigma) &= \# \text{ of crossings on which $\sigma$ chooses the $A$-resolution} \\
&- \# \text{ of crossings on which $\sigma$ chooses the $B$-resolution}. \end{aligned}$$
We have $$\begin{aligned}
\label{eq:gssum}
\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle &= \sum_{\sigma, \ a \ : \ a, \ n, \ n \text{ admissible }} \frac{\triangle_a}{\theta(n, n, a)} ((-1)^{n-\frac{a}{2}}A^{2n-a+n^2-\frac{a^2}{2}})^{r} A^{sgn(\sigma)} \langle {\mathcal{S}}^{a}_{\sigma} \rangle. \\
\intertext{To simplify notation let $d(a, r)=r(2n-a+n^2-\frac{a^2}{2})$, and we write }
\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle &= \sum_{\sigma, \ a \ : \ a, \ n, \ n \text{ admissible }} \frac{\triangle_a}{\theta(n, n, a)} (-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle J^a_{\sigma} \ \sqcup \text{ disjoint circles}\rangle. \label{eq:statesum}\end{aligned}$$
After isotopy, we may assume that $J^a_{\sigma}$ has the form shown in Figure \[fig:case\], since other states evaluate to 0 by the Kauffman bracket with a cup/cap composed with an idempotent.
We say that the Kauffman state $\sigma$ has *$c$ split strands*, if after isotoping $J^a_{\sigma}$ to the form in Figure \[fig:case\], there are $2c$ split strands connecting the top and bottom pairs of Jones-Wenzl idempotents.
To further reduce the number of terms to consider in the sum of , we prove the following lemma.
\[lem:localzero\] Consider a skein ${\mathcal{S}}$ with the following local picture.
The skein is zero if $\frac{a}{2}-c > 0$.
If $\frac{a}{2}-c > 0$, then $n-c-x>0$. When $n-c-x > 0$, the skein is not adequate since we have a circle passing through the same idempotent twice, see Figure \[fig:localh\] for an example. Note also that $y = z = \frac{c}{2} = n-x$.
Now if $x$ is zero, we can slide the top two idempotents down to the bottom one by and get a cap composed with a idempotent which gives 0 for the skein. When $x \not=0$, we show by induction on $x$ that every term in the sum of the skein from repeatedly expanding the idempotent via has a cap composed with an idempotent after sliding by . Thus, every term in the sum is zero and the skein is zero.
Suppose $x =1$, there are two idempotents and therefore four terms in the sum from expanding via $\eqref{eq:jwrecursive}$. This takes care of the base case: For any $n, c$ such that $n-c-1 > 0$, we have that ${\mathcal{S}}= 0$.
Now suppose that $x=k+1$ and we have that every term when $x = k$ evaluates to 0 by the induction hypothesis for any $n-c-k>0$. We expand the pair of idempotents to get the panel of four figures in Figure \[fig:induction\].
The first three figures clearly reduce to that of the case $x=k$ and $n-1-c-(x-1)>0$. We simplify the last figure by Lemma \[lem:jwid3\]. This is shown in Figure \[fig:induction2\].
If $x-2=0$, then we are done. Otherwise, we again expand the top pair of idempotents to get another panel of 4 figures as shown in Figure \[fig:induction3\].
The first three cases reduce to the case $x = k-1$ with $n-2-c-(x-2)>0$. For the last one we repeat the step of Figure \[fig:induction2\] using Lemma \[lem:jwid3\] to keep reducing $x$. Then, expand the top part repeatedly as in the Figure \[fig:induction3\] and apply the induction hypothesis to smaller $x$, so that we can look at the last figure in the panel to determine whether we need to apply the step of Figure \[fig:induction2\] again. We repeat these last two steps until $x$ goes to 0.
By Lemma \[lem:localzero\], we have that becomes $$\begin{aligned}
\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle &= \sum_{\sigma, \ a \ : \ a, \ n, \ n, \text{ admissible }} \frac{\triangle_a}{\theta(n, n, a)}(-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle J^a_{\sigma} \sqcup \text{ disjoint circles}\rangle \\
&= \sum_{\sigma, \ a \ : \ a, \ n, \ n, \text{ admissible}, \ \frac{a}{2} \leq c} \frac{\triangle_a}{\theta(n, n, a)} (-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle J^a_{\sigma} \sqcup \text{ disjoint circles}\rangle.\end{aligned}$$
Now let $$\deg(\sigma, a) := \text{deg} \left( \frac{\triangle_a}{\theta(n, n, a)}(-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle J^a_{\sigma} \sqcup \text{disjoint circles} \rangle \right).$$
The degree-dominating term in the state sum {#subsec:degree-dominating}
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Consider the state $\sigma_A$ which chooses the $A$-resolution at all the crossings (recall that the state is applied on the remaining crossings of $D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}$ after getting rid of the negative twist region using the fusion and the untwisting formulas). We have that ${\mathcal{S}}_{\sigma_A}^a$ has 0 split strands and thus $\langle J^a_{\sigma_{A}} \rangle = 0$ for all values of $a$ except $a=0$. A simple computation using Lemma \[lem:jwad\] shows $$\deg(\sigma_A, 0) = H_n(D) + 2r(n^2+n).$$ The strategy to prove Theorem \[thm:bracketdegree\] is then to show that $$\deg(\sigma, a) < \deg(\sigma_A, 0)$$ for any other Kauffman state $\sigma$ and $a$.
Given $a$ and $\sigma$ with $c$ split strands such that $\frac{a}{2} \leq c$, the skein $J^a_{\sigma}$ is adequate, and thus by Lemma \[lem:jwad\] and , $$\label{eqn:aeqc}
\deg(\sigma, a) = a-2n+ d(a, r) + {\text{sgn}}(\sigma) + \deg \langle \overline{{\mathcal{S}}^{a}_{\sigma}} \rangle,$$ where $\overline{{\mathcal{S}}^{a}_{\sigma}}$ is the skein obtained from ${\mathcal{S}}^{a}_{\sigma}$ by replacing all the idempotents with the identity. From this we can see that if $\frac{a}{2} < c$ then $\overline{{\mathcal{S}}^{a}_{\sigma}}$ has fewer circles than $\overline{{\mathcal{S}}^{2c}_{\sigma}}$ so we may assume that $\frac{a}{2} = c$, see Figure \[fig:casefuntwist\].
In order to compare $\deg(\sigma, 2c)$ with $\deg(\sigma_A, 0)$, we use the concept of a sequence of states.
Crossings on which a state $\sigma \not= \sigma_A$ chooses the $B$-resolution {#subsec:sigmab}
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In this section we characterize the set of crossings on which a state $\sigma$ which is not $\sigma_A$ with $c>0$ split strands chooses the $B$-resolution. We describe this by proving that certain states are included in a sequence of states from $\sigma_A$ to $\sigma$. The terminology of a sequence of states appears in [@Lic97].
A *sequence* $s$ of states starting at $\sigma_1$ and ending at $\sigma_f$ on a set of crossings in a skein ${\mathcal{S}}$ is a finite sequence of Kauffman states $\sigma_1, \ldots, \sigma_f$, where $\sigma_{i}$ and $\sigma_{i+1}$ differ on the choice of the $A$- or $B$-resolution at only one crossing $x$, so that $\sigma_{i+1}$ chooses the $B$-resolution at $x$ and $\sigma_i$ chooses the $A$-resolution.
Let $s=\{\sigma_1, \ldots, \sigma_f \}$ be a sequence of states starting at $\sigma_1$ and ending at $\sigma_f$. Choosing the $B$-resolution at a crossing $\vcenter{\hbox{\includegraphics[scale=.2]{crossing1.png}}}$ corresponds to locally replacing $\vcenter{\hbox{\includegraphics[scale=.2]{crossing3.png}}}$ by $\vcenter{\hbox{\includegraphics[scale=.2]{crossing2.png}}}$. In each application from $\sigma_i$ to $\sigma_{i+1}$ either two circles of $\overline{{\mathcal{S}}_{\sigma_i}}$ merge into one or a circle of $\overline{{\mathcal{S}}_{\sigma_i}}$ splits into two. When two circles merge into one as the result of changing the $A$-resolution to the $B$-resolution, the number of circles of the skein decreases by 1 while the sign of the state decreases by 2. More precisely, let ${\mathcal{S}}_{\sigma}$ be the skein resulting from applying the Kauffman state $\sigma$, we have $${\text{sgn}}(\sigma_{i+1}) + \deg \langle \overline{{\mathcal{S}}_{\sigma_{i+1}}}\rangle = {\text{sgn}}(\sigma_{i}) + \deg \langle \overline{{\mathcal{S}}_{\sigma_{i}}}\rangle -4,$$ when a pair of circles merges from $\sigma_{i}$ to $\sigma_{i+1}$.
The above gives the following lemma which allows us to bound the degree of a skein ${\mathcal{S}}_{\sigma_f}$ from applying a Kauffman state $\sigma_f$, by considering how many pairs of circles are merged in a sequence of states from $\sigma_1=\sigma_A$ to $\sigma_f$.
\[lem:sigmab\] Let ${\mathcal{S}}$ be a skein with crossings and $s = \sigma_1, \ldots, \sigma_f$ be a sequence of Kauffman states on the crossings of ${\mathcal{S}}$. If $g$ is the number of pairs $(\sigma_i, \sigma_{i+1})$ in $s$ such that $\sigma_{i+1}$ merges circles in $\sigma_i$ in $s$, then $$\label{eq:mergelower}
{\text{sgn}}(\sigma_{f}) + \deg \langle \overline{{\mathcal{S}}_{\sigma_{f}}} \rangle = {\text{sgn}}(\sigma_{1}) + \deg \langle \overline{{\mathcal{S}}_{\sigma_{1}}} \rangle -4g.$$
We use this to obtain an upper bound of $\deg(\sigma, a)$ where $a=2c$ by considering a sequence starting at $\sigma_A$ and ending at $\sigma$. We use the technical concept of the *flow* of a Kauffman state through the edge of a walk.
Let $x$ be a crossing and $x^n$ be the $n$-cable. Represent $x^n$ so that it is a skein in ${\mathcal{S}}(D, 2n)$ and oriented as in the first figure of Figure \[fig:skeinflow\]. Consider a Kauffman state $\sigma$ on $x^n$. We say that $\sigma$ has $k \leq n$ strands *flowing through* the crossing $x$ if applying $\sigma$ to $x^n$ results in $2k$ arcs connecting $2k$ points on the top and the bottom. See Figure \[fig:skeinflow\] for an example.
This is not a new concept. Works involving elements in the Temperley-Lieb algebra have defined for an arbitrary element of $TL_{m, n}$ the quantity which captures the number of strands $k$ of a skein in $TL_{m, n}$ that connects $k$ points from the top to $k$ points in the bottom and called this quantity different names. For example, see [@Hog14] where the quantity is called the *through-degree*, and [@Roz12], where the quantity is measured by the *width-deficit*. As far as the author is aware there does not seem to be standard terminology for this quantity. The focus in this paper with this definition is on a crossing in a specific twist region, and we count pairs of arcs rather than the number of arcs from top to bottom.
Notations and conventions for graphical representations {#notations-and-conventions-for-graphical-representations .unnumbered}
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The following technical lemma, Lemma \[lem:count\], allows us to understand a sequence $s$ from $\sigma_A$ to $\sigma$, if $\sigma$ has strands flowing through a crossing. It is necessary to first establish some notations and labelling conventions.
Firstly, we orient $(D, 2n)$ as shown in Figure \[fig:markings\] and identify it with $[-1, 1] \times [-1, 1]$. For a crossing $x$ oriented as in Figure \[fig:skeinflow\], let $U_1, \ldots, U_n$ be the set of arcs between the $2n$ points on the top edge of the disk, innermost first, from the all-$A$ state on the set of crossings $x^n$. Similarly we have the lower arcs $L_1, \ldots, L_n$. The arcs cut up the disk into regions containing segments corresponding to the all-$A$ state on the crossings of $x^n$. Let $C^u_i$ be the set of crossings whose corresponding segments in the all-$A$ state are between $U_i$ and $U_{i+1}$. Similarly we have $C_i^{\ell}$, and the set of edges between $U_n$ and $L_n$ is denoted by $C^u_n = C^{\ell}_n$. See Figure \[fig:markings\] for an illustration of these markings.
From this, we will represent a Kauffman state $\sigma$ on $x^n$ by taking the all-$A$ state of $x^n$. Recall that this consists of the all-$A$ state circles and edges (dashed segments) corresponding to taking the $A$-resolution at every crossing. We make the following modification in order to represent $\sigma$:
1. If $\sigma$ chooses the $B$-resolution at a crossing, replace the corresponding segment in the all-$A$ state by a red solid edge.
2. Remove all other edges from the state.
This representation will allow us to consider intersections of arcs in ${\mathcal{S}}(D, 2n)$ with the skein resulting from applying $\sigma$ to $x^n$. This is the $\sigma$-state on $x^n$ with the segments removed, so it only consists of state circles. In particular, in this representation consisting of black arcs and red edges, intersection of an arc with a black arc counts as one intersection with the skein, and an intersection of an arc with a red edge counts as two.
With the orientation on the square $(D, 2n)$, it should be clear what we mean by an edge being on the left/right to another edge. This also explains what it means for a crossing in $x^n$ to be on the left/right of another crossing. We will frequently not distinguish between the crossing and the corresponding edge whenever we are merely concerned with their relative positions.
\[lem:count\] Let ${\mathcal{S}}$ be a skein with crossings, but without Jones-Wenzl idempotents, $\sigma$ be a Kauffman state on ${\mathcal{S}}$, and let $x^n$ be an $n$-cabled crossing in ${\mathcal{S}}$, with $x_\sigma$ the result of applying $\sigma$ to the crossings in $x^n$.
(a) If $\sigma$ has $k$ strands flowing through $x$, then $\sigma$ chooses the $B$-resolution on a set of $k^2$ crossings $C_{\sigma}$ of $x^n$, where $C_{\sigma} = \cup_{i=n-k+1}^n (u_i \cup \ell_i)$ is a union of crossings $u_i \subseteq C^u_i$ and $\ell_i \subseteq C^{\ell}_i$, such that
- $u_i$, $\ell_i$ each has $k-n+i$ crossings for $n-k+1\leq i \leq n$.
- For each $n-k+2\leq i \leq n$, and a pair of crossings in $u_i$ (resp. $\ell_i$) whose corresponding segments $c, c'$ in the all-$A$ state of $x^n$ are adjacent, there is a crossing $c''$ in $u_{i-1}$ (resp. $\ell_{i-1}$), where the end of the segment corresponding to $c''$ on $U_{i}$ (resp. $L_i$) lies between the ends of $c$ and $c'$.
(b) Consider a sequence $s= \{\sigma_1, \ldots, \sigma_f = \sigma\}$ of Kauffman states restricted to $x^n$, where $\sigma_1$ is a Kauffman state which chooses the $A$-resolution at every crossing in $x^n$. Suppose that in ${\mathcal{S}}_{\sigma_1}$, the 2$n$ arcs connecting the top $2n$ points belong to $n$ circles disjoint from the $2n$ arcs connecting the bottom $2n$ points, which also belong to $n$ circles. Then $g\geq \frac{k(k+1)}{2}$, where $g$ is the number of pairs $(\sigma_i, \sigma_{i+1})$, $1\leq i \leq f-1$, in $s$ such that $\sigma_{i+1}$ merges a pair of circles in $\sigma_i$.
As an example, if $n=3$ and $\sigma$ flows through a crossing $x$ with 2 strands, then $\sigma$ chooses the $B$-resolution on a subset of crossings of the form as shown in Figure \[fig:flowconfig\]. There may be other crossings on which $\sigma$ chooses the $B$-resolution, but the claim is that there must be a *subset* of crossings on which $\sigma$ chooses the $B$-resolution of the form as described in the lemma.
In $C_{\sigma} = \cup_{2}^{3} (u_i\cup\ell_i)$, we have that $u_3=\ell_3$ contains 2 crossings and $u_2, \ell_2$ each containing 1 crossing. The segment in the all-$A$ state of $x^n$ corresponding to the crossing in $u_2$ has an end on $U_3$ between the ends of the segments corresponding to the two crossings in $u_3$. The same is true of the crossing in $\ell_2$. The total number of crossings in $C_{\sigma}$ is then $4=k^2$, which makes the total number of crossings of $x^n$ on which $\sigma$ chooses the $B$-resolution to be $\geq 4$.
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#### **Proof of (a)**
For a Kauffman state $\sigma$ which has $k$ strands flowing through a crossing $x$, if we draw a line from the left end of the square to the right end, it must have $\geq 2k$ intersections with the curves resulting from applying the state. Isotope link strands so that the set of crossings $C^{\ell}_i$ for $1\leq i <n$ is between the horizontal lines at height $h =-\frac{n-i}{n}$ and $h = -\frac{n-i+1}{n}$. Similarly, isotope link strands so that the set of crossings $C^{u}_i$ for $1\leq i <n$ is between the horizontal lines at height $h=\frac{n-i}{n}$ and $h = \frac{n-i+1}{n}$. Now we isotope the crossings of $C^u_n = C^{\ell}_n$ so that it is between $h=-\frac{1}{n}$ and $h=\frac{1}{n}$, see Figure \[fig:line\].
Beginning with the set of crossings $C^{u}_n$, we see that $\sigma$ must choose the $B$-resolution on $k$ crossings, since the horizontal line $H$ at $h=0$ must intersect the resulting skein at least $2k$ times. Now isotope $H$ so that it enters and exits the region containing the crossings in $C^{u}_{n-1}$. Then for $C^{u}_{n-1}$, $\sigma$ must choose the $B$-resolution on a set of $k-1$ crossings in $\frac{2}{n}<h<\frac{3}{n}$, since a pair of vertical lines provides 2 intersections with a horizontal line between the two heights bounding the set of crossings in $C^{u}_{n-1}$. We repeat this argument for $C^{u}_{i}$ for $n-k+1\leq i < n-1$, isotoping $H$ to enter and exit the region bounding crossings of $C^{u}_i$ each time and noting that $H$ would already have $2(n-i)$ intersections with the strands of the skein. Then for each $i$, $\sigma$ must choose the $B$-resolution on $k-(n-i)$ crossings in $C^u_i$.
The same argument works by symmetry when we consider lines intersecting the lower crossings $C^{\ell}_i$. Taking the sum over $n-k+1\leq i < n-1$, we have a lower bound for the total number of crossings of $x^n$ on which $\sigma$ chooses the $B$-resolution. $$k + 2\sum_{i=1}^{k-1} i = k^2.$$
For the second part of the claim, we first prove that we can find a set of crossings $C_{\sigma}'$ of $x^n$ on which $\sigma$ chooses the $B$-resolution, where $C'_{\sigma} = \cup_{i=n-k+1}^n (u'_i \cup \ell'_i)$ is a union of crossings $u'_i \subseteq C^u_i$ and $\ell'_i \subseteq C^{\ell}_i$, such that
- $u'_i$, $\ell_i'$ each has two crossings for $n-k+1 < i \leq n$, and one crossing for $i = n-k+1$.
- The two crossings in $u'_n = \ell'_n$ are furtherest possible in the sense that the two segments corresponding to the crossings in the all-$A$ state are furtherest possible i.e., every crossing in $C^{u}_n = C^{\ell}_n$ on which $\sigma$ chooses the $B$-resolution lies between. For each $n-k+1\leq i <n$, the end(s) of the segment(s) corresponding to the crossing(s) in $u'_i$ (resp. $\ell'_i$) on $U_{i+1}$ (resp. $L_{i+1}$) lie(s) between the two segments corresponding to the crossings in $u'_{i+1}$ (resp. $\ell'_{n+1}$). For $n-k+1 < i < n$, if we take two crossings for $u'_i$ (resp. $\ell'_i$), then they are the furtherest possible satisfying the above conditions. See Figure \[fig:firstset\] for an illustration of these requirements.
For $i=n$, we know that $H$ has to intersect at least $2k$ points. Therefore, the number of crossings in $C^u_n = C^{\ell}_n$ on which $\sigma$ chooses the $B$-resolution is at least $k$, and we may take the two furtherest crossings for the set $u'_n \cup \ell'_n$. (There is nothing to prove if $k=1$, because then we can just take one crossing for $u'_n \cup \ell'_n$ and we have the set $C'_{\sigma}$, which will also satisfy the conditions for $C_{\sigma}$.) For $i = n-1, n-2, \ldots n-k+1$, if there are not two crossings in $C^u_i$ for which the ends of the corresponding segments on $U_{i+1}$ lie between the segments from the crossings $u'_n$, then we can isotope $H$ such that it has fewer than $2k$ intersections with the skein, see Figure \[fig:firstsetH\] below.
We argue this by assuming that the sets were already inductively constructed for $i+1$, and we would like to pick a set of crossings in $C^n_{i}$. If $i \not= n-k+1$, then Figure \[fig:firstsetH\] shows the isotopy that will result in fewer than $2k$ intersections between $H$ and the skein from $\sigma$, assuming that there are no crossings in $C^u_i$ whose corresponding segments lie between those of the crossings in $C^u_{i+1}$. For $i\geq n-k + 2$, there has to be at least 4 intersections of $H$ with the skein in the region between $U_{i}$ and $U_{i+1}$, since $H$ will have at most $2(n-i)$ intersections before entering/exiting. This gives at least two crossings in $C^u_{i}$ on which $\sigma$ chooses the $B$-resolution whose corresponding segments are between those of $u'_{i+1}$ . If $i = n-k+1$ then we require at least 2 intersections, hence the single crossing that we can pick for $u_{n-k+1}'$. The argument for $\ell_i'$ is completely symmetric.
To complete the rest of $(a)$, we add crossings to $C'_{\sigma}$ to get a set $C_{\sigma}$ which satisfies the remaining requirements. The set of two crossings in $u'_{n-k+2}$ (resp. $\ell'_{n-k+2}$) certainly satisfies the conditions of $(a)$ since there is a single crossing of $u'_{n-k+1}$ (resp. $\ell'_{n-k+1}$), the end of whose corresponding segment lies between those of the two crossing on $U_{n-k+2}$ (resp. $L_{n-k+2}$).
We describe the algorithm for constructing $C_{\sigma}$ inductively for $i > n-k+2$. Let $u'_i$ (resp. $\ell'_i$) be such that $|u'_i|= 2$ (resp. $|\ell'_i|=2$). We label the “left" crossing in $u'_i$ (resp. $\ell'_i$) with a $-$ and the “right" crossing in $u'_i$ (resp. $\ell'_i$) with a $+$. So $-x$ denotes a left crossing, for example, and $-u'_i$ denotes all the left crossings in $u'_i$.
Algorithm for constructing $C_{\sigma}$ {#algorithm-for-constructing-c_sigma .unnumbered}
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At the very beginning we have $C'_{\sigma}$ which has already been constructed.
1. Consider the difference $k - |u'_n|$, if this difference is 0 then terminate. $C'_{\sigma}$ is already a set of edges which satisfies the assumptions of part (a) of the lemma. Set $C_{\sigma} = C'_{\sigma}$.
2. For $i = n$, $n-1 \ldots$, $n-k+1$, set $C = k-|u'_i|-(n-i)$. We assume inductively that $C'_{\sigma}$ satisfies the following for $n-k+1 \leq j \leq n$:
(i) An edge in $-u'_j$ with two edges above and below to the left of it, is the leftmost possible for all edges to the right of the two edges. Similarly, An edge in $+u'_j$ with two edges above and below to the right of it, is the rightmost possible for all edges to the left of the two edges.
(ii) Let $\pm p$ be the midpoint of an edge whose corresponding crossing, say $x$, is in $\pm u'_j$ , then there are two arcs $H^-$ and $H^+$, where $H^-$ starts at (-1, 0) and ends at $-p$, and $H^+$ starts at $p$ and ends at $(1, 0)$. The numbers of intersections of $H^+$ and $H^-$ with $x_{\sigma}$ are given by:
If $x\in -u'_j$, $$\label{eqn:intersect1} |H^- \cap x_{\sigma}| = 2 \left( \# \text{ of crossings to the left of $x$ in $u'_j$ } \right) + (n-j).$$ If $x\in +u'_j$, $$\label{eqn:intersect2} |H^+ \cap x_{\sigma}| = 2 \left( \# \text{ of crossings to the right of $x$ in $u'_j$} \right) + (n-j).$$
That these assumptions are valid follows from Lemma \[lem:conditions\]. Before we prove it, we proceed with the algorithm with those assumptions.
- Let $-x$ be the rightmost edge in $-u_i'$. There is an edge $x'$ in $C'_{\sigma}$ above in $u'_{i-1}$ and another edge $x''$ below it in $u'_{i+1}$. Both of these edges are to the right of $-x$. There are only a few cases for the edges in $C^u_{i}$ to the right of $-x$, but whose ends on $U_i$ and $U_{i+1}$ are not to the right of both $x'$ and $x''$, respectively. They are shown as dashed edges in the following figure.
Let $-p$ be the midpoint of $-x$ and $-p'$ be the midpoint between $U^{i}$ and $U^{i+1}$ immediately to the right of both $x'$ and $x''$ and to the left of any crossings in $C^u_{i}$ on which $\sigma$ chooses the $B$-resolution to the right of both $x'$ and $x''$. Either we can draw an arc from the left of $-p$ to $-p'$ that only has 2 intersections with $x_{\sigma}$, see Figure \[fig:skeinHcomb2\], or, there are two choices for the existence of an edge $y$ in either $C^u_{i+1}$ or $C^u_{i-1}$. This is shown in Figure \[fig:skeinHcomb3\].
Without loss of generality we will just assume that it is in $C^u_{i-1}$ where we have the edge $y$, and we consider the rightmost such edge. Now we consider $-x_1$ which is the nearest edge in $u'_{i-1}$ to the left of $y$. Let $p_1$ be the midpoint of $-x_1$ and $p'_1$ be the point between $y$ and the nearest edge $z_1=x'$ in $u'_{i-1}$ to the right of $y$. Again, we see if we can draw an arc from the left of $p_1$ to $p'_1$ that only has 2 intersections with $U_{i-1}$. If not, there exists another $y_1$ which obstructs this. We repeat the same steps with $y_1$ to obtain a necessarily finite sequence of edges $y, y_1, \ldots, y_m$. For $y_m$ we draw an arc from $p_m$ to $p'_m$ that has only 4 intersections with $x_{\sigma}$ (including the intersection with $-x_m$), then we connect $p'_j$ with $p'_{j-1}$ for each $j$ with an arc that is parallel to the rest of $y$’s and to the left of the $z_j$’s, see Figure \[fig:skeinHcomb4\] below.
There is only a single intersection of the arc between $p'_j$ and $p'_{j-1}$ with $x_{\sigma}$ because of assumption (i). Thus we get an arc from $p_m$ to $-p'$ that has $m+4$ intersections with $x_{\sigma}$. Now $$\begin{aligned}
& \# \text{ of edges in $u'_{i-j}$ before $p_j$} \\
&= (\# \text{ of edges in $u'_i$ before $p_1$}) - m. \end{aligned}$$
Using assumption (ii) on $p_m$, we get an arc from $(-1, 0)$ to $-p'$ with the number of intersections with $x_{\sigma}$ as in . The arc $H$ which is the union of the arc $H_1$ from $(-1, 0)$ to $p_m$, and the arc $H_2$ from $p_m$ to $-p'$ has the number of intersections with $x_{\sigma}$ given by $$|H\cap x_{\sigma}| = 2\cdot \left( \# \text{ of crossings to the left of $-p'$ in $u'_i$ } \right) + n-i.$$
Similarly, with the same argument replacing $-$ with $+$, “right" with “left," and “left" with “right", we can get another arc $H'$ from $(1, 0)$ to $p'$ that has the number of intersections given by $$|H'\cap x_{\sigma}| = 2\cdot \left( \# \text{ of crossings to the right of $p'$ in $u'_i$ } \right) + n-i.$$ Now consider the straight line segment $L$ from $-p'$ to $p'$. If $\sigma$ does not choose the $B$-resolutoin on any crossing in $C^u_{i}$ between $-p'$ and $p'$, then we get an arc $H'' = H\cup L\cup H'$ that has $\leq 2(k-1)$ intersections with $x_{\sigma}$, which is a contradiction. We add this crossing to $u'_i$ and move on to the next $i$ in the iteration.
- This is similar to the case when $C=1$. The arguments are the same except that at the last stage we can add a furtherest pair of edges, each marked with $-$ and $+$ for left and right, to $u'_i$. After this we move onto the next $i$ in the iteration.
3. We repeat from Step (1) until $k-|u'_n| = 0$.
Running the same algorithm for $\ell'_n$ with the obvious adjustment by symmetry gives us $C_{\sigma}$.
\[lem:conditions\] Every iteration of $C'_{\sigma}$ through the algorithm satisfies conditions $(i)$ and $(ii)$.
For the first iteration of $C'_{\sigma}$, condition (i) is vacuously true. For a crossing in $-u'_{i}$, the arc as shown satisfies condition (ii). The same arc by reflection also works for a crossing in $+u'_i$.
For each subsequent iteration of $C'_{\sigma}$, the edges added are specifically chosen to satisfy both $(i)$ and $(ii)$.
#### **Proof of (b)**
We apply the $B$-resolution first on the set of crossings with structure as described in part (a). We see that the overall degree of the skein decreases by $2k^2+2k$ from the all-$A$ state. Divide by $4$ and we get statement (b) of the lemma.
Let $G$ be the 2-connected, weighted planar graph where $D = \partial (F_G)$ and recall that $G^e$ is $G$ with the negative edge $e=(v, v')$ deleted. Let $W$ be a path in $G^e$ from $v$ to $v'$. For a positively-weighted edge $\epsilon$ of $G^e$ corresponding to a positive twist region $T$ in $D = \partial(F_G)$, orient the twist region as an element in ${\mathcal{S}}(D, 2n)$, so that all the crossings are as in, Figure \[fig:skeinflow\] we say that a state $\sigma$ in the state sum of on $D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}$ *flows through* $\epsilon$ with $k$ strands if the skein in ${\mathcal{S}}(D, 2n)$ resulting from applying $\sigma$ to the twist region has $2k$ arcs connectin $2k$ points on the top and the bottom.
A consequence of Lemma \[lem:count\] is the following.
\[lem:wcount\] Let $D$ be a link diagram and $G$ be a 2-connected, weighted planar graph such that $D = \partial (F_G)$. Let $\epsilon = (v, v')$ be an edge in $G$ corresponding to a maximal positive twist region with $\omega$ crossings, and $\sigma$ is a Kauffman state that flows through $(v, v')$ with $k$ strands, then $\sigma$ restricted to $T^n$ (the $n$-cable of the crossings in $T$) chooses the $B$-resolution on at least $\omega k^2$ crossings and merges at least $(\omega-2)\frac{k(k+1)}{2}$ circles.
If $\sigma$ flows through the edge $\epsilon$ with $k$ strands than it flows through every crossing in the twist region $T$ represented by $\epsilon$ with at least $k$ strands. We apply Lemma \[lem:count\](a) and just add up the number of crossings on which $\sigma$ chooses the $B$-resolution over each $x^n$ for a crossing $x \in T$. This gives that $\sigma$ chooses the $B$-resolution on at least $\omega k^2$ crossings. In a twist region with $\omega$ crossings we have that in the all-$A$ state on $T^n$ there are $(\omega-2)$ sets of $n$ disjoint circles. Thus we can apply part (b) of Lemma \[lem:count\] at least $\omega - 2$ times.
Proof of Theorem \[thm:bracketdegree\] {#subsec:complete}
--------------------------------------
Now we complete the proof of Theorem \[thm:bracketdegree\]. Recall that from Section \[subsec:simplifyss\] we have $$\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = \sum_{\sigma, \ a \ : \ a, \ n, \ n, \text{ admissible }, \ \frac{a}{2} \leq c} \frac{\triangle_a}{\theta(a, n, n)}(-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle J^a_{\sigma} \sqcup \text{ disjoint circles}\rangle,$$ and we would like to show that $$\deg(\sigma, a) < \deg(\sigma_A, 0),$$ where $\deg(\sigma_A, 0) = H_n(D) + 2r(n^2+n)$, and $\deg(\sigma, a)$ is the maximum degree of a term indexed by $\sigma, a$ in the state sum of $\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}\rangle$. Recall also that $c$ is the number of split strands of $\sigma$ and that by we need only to consider states $\sigma$ with parameter $a$ such that $\frac{a}{2}=c$.
If $\sigma$ is a state with $c=0$ that is not the all-$A$ state, then it must choose the $B$-resolution at a crossing outside of the maximal twist region of negative weight $r$, which will merge at least one pair of circles compared to the all-$A$ state. Hence, a sequence $s$ from $\sigma_A$ to $\sigma$ for $a=0$ contains at least one pair of states that merges a pair of circles. This implies that $$\deg(\sigma, 0) \leq \deg(\sigma_A, 0) - 4,$$ so $$\deg(\sigma, 0) < \deg(\sigma_A, 0).$$
If $\sigma$ is a state with $c > 0$, then the skein ${\mathcal{S}}_{\sigma}^a = J^a_{\sigma} \sqcup (\text{disjoint circles})$ can be decomposed along a square $(D, 2n)$ containing the Jones-Wenzl idempotents as shown in the following figure, so that we get two skeins ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ in ${\mathcal{S}}(D, 2n)$.
Now in ${\mathcal{S}}_2$ with $\sigma$ applied we have at least $2c$ strands connecting the $2c$ points at the top to the $2c$ points at the bottom on the boundary of the disk $D$.
Let $D = \partial (F_G)$ be a near-alternating link diagram and $G^e$ be the graph obtained from $G$ by deleting the single edge $e=(v, v')$ of negative weight $r$. Let $t$ be the total number of paths from $v$ to $v'$ in $G^e$ and let $k_i$ be the number of strands with which the state $\sigma$ flows through a path $W_i$ for $1\leq i \leq t$. We have $$\sum_{i=1}^{t} 2k_i \geq 2c.$$
We can construct a sequence $s$ from $\sigma_A$ to $\sigma$ by changing the resolution from $A$-to $B$- on the set of crossings $x^n$ for each crossing $x$ outside the maximal negative twist region.
The first part of the sequence consists of changing the resolutions on the crossings corresponding to the twist regions in $W_1$, the second part of the sequence consists of changing the resolutions on the crossings corresponding to the twist regions in $W_2$, and so on until we change the choices of resolution on $W_t$.
Now for each walk $W_i$ with $k_i$ strands flowing through we have a sequence $\{\sigma^i_1, \ldots \sigma^i_f\}$ which changes the $A$-resolution to the $B$-resolution on at least $\ell (W_i)k^2$ crossings and merges at least $(\ell(W_i)-2)k$ pairs of circles by Lemma \[lem:wcount\]. Let $$\deg(\overline{\sigma_A}, 2c) := \text{deg} \left( \frac{\triangle_{2c}}{\theta(n, n, 2c)}(-1)^{rn-rc} A^{d(2c, r) + sgn(\sigma)} \langle \overline{J^{2c}_{\sigma}} \sqcup \text{disjoint circles} \rangle \right).$$ Let $\omega_i = \ell(W_i)$, and recall $\omega:= \min_{1\leq i\leq t} \left\{ \omega_i \right\}$, we have $$\begin{aligned}
\deg(\sigma, 2c) &\leq \deg(\overline{\sigma_A}, 2c) - \left( \sum_{i=1}^{t} (\omega - 2)(2k_i^2+2k_i)+4k_i^2 \right) - r\frac{(2c)^2}{2} - (2c)(r-1) \\
&\leq \deg(\sigma_A, 0) - 2c - r\frac{(2c)^2}{2} - (2c)(r-1). \end{aligned}$$ Since $\sum_{i=1}^t 2k_i \geq 2c$, we may assume that $\sum_{i=1}^{t} 2k_i = 2c$, so the $k_i$’s form a partition of $c$. The following lemma shows that we may replace it by a minimal partition.
Let $P = \{n_1, \cdots, n_t\}$ be an integer partition of $n$ where the $n_i$’s may be zero, so $n = n_1 + \cdots + n_t$. We say that a partition of $n$ into $t$ parts is a *minimal partition*, denoted by $P_m$, if it has the minimal $m = \max_{1\leq i \leq t} n_i$ out of all partitions of $n$ into $t$ parts.
\[lem:part\] Fix $n$ and $t$. A minimal partition $P_m=\{m_1, \ldots, m_t \}$ of $n$ into $t$ parts is unique up to rearrangement of indices. If $P=\{n_1, \ldots n_t \}$ is another partition of $n$ into $t$ parts, then $$\sum_{i=1}^t m_i^2 \leq \sum_{i=1}^t n_i^2.$$
A minimal partition $P_m$ may be constructed as follows. If $n\leq t$ then the partition has $m_1=m_2=\cdots =m_n = 1$ and $m_{n+1} = m_{n+2}=\cdots m_{t} = 0$. If $n>t$, let $j = n \pmod{t}$. The partition $P_m$ has $m_1=m_2=\cdots = m_j = \floor{n/t}+1$ and $m_{j+1} = m_{j+2} = \cdots = m_{t} = \floor{n/t}.$ The partition is minimal, since we may obtain any other partition of $n$ into $t$ parts from $P_m$ by subtracting 1’s from a non-zero summand and adding 1 to any other. Similarly, it is unique up to arrangement.
For the statement that $\sum_{i=1}^t m_i^2 \leq \sum_{i=1}^t n_i^2$, there is nothing to prove if $P = P_m$. Let $m' = \max_{1\leq i \leq t} n_i$ and $m = \max_{1\leq i \leq t} m_i$. Since $P_m$ is minimal and unique up to rearrangement we can assume that $m'> m$ and $m' = n_1$ for $P$, $m = m_1$ for $P_m$. Suppose $m' = m + k$ for some integer $k > 0$. This means that we may write $$P = \{m_1+k, m_2-k_2, \ldots, m_t-k_t\},$$ where $k_2, \ldots, k_t \geq 0$ and $k_2 + \cdots + k_t = k$. Now we have $$\begin{aligned}
\sum_{i=1}^t n_i^2 &= (m_1+k)^2 + (m_2-k_2)^2+\cdots + (m_k-k_t)^2. \\
&= (m_1^2 + 2m_1k + k^2) + \sum_{i=2}^t (m_i^2 - 2m_ik_i +k_i^2) \\
&= \left( \sum_{i=1}^t m_i^2 \right) + 2m_1k + k^2 + \sum_{i=2}^t (-2m_ik_i + k_i^2). \\
\intertext{Now}
& 2m_1k+k^2 + \sum_{i=2}^t (-2m_ik_i + k_i^2) \geq 2m_1k + k^2-2m_1k + \sum_{i=2}^t (k_i)^2 \geq 0.
\end{aligned}$$ This concludes the proof of the lemma.
Finally, replacing $\{k_i\}$ by a minimal partition $P_m = \{m_1, \ldots, m_t\}$, we have $$\deg(\sigma, 2c) \leq \deg(\sigma_A, 0) - \left(\sum_{i=1}^{t} (\omega - 2)(2m_i^2+2m_i) + 4m_i^2\right)- \left( r\frac{(2c)^2}{2} + 2cr \right).$$ If $|r| < \frac{\omega}{t}$ with $|r| \geq 2$ and $t> 2$, then the difference $$- \left(\sum_{i=1}^{t} (\omega - 2)(2m_i^2+2m_i)+4m_i^2\right)- \left( r\frac{(2c)^2}{2} + 2cr \right) \label{eqn:inequality}$$ is negative, so $$\deg(\sigma, 2c) < \deg(\sigma_A, 0)$$ for every other Kauffman state $\sigma$ with $c> 0$ split strands. Since we also know this inequality for $\sigma$ with $c=0$ split strands, this shows that $\deg \langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = \deg(\sigma_A, 0)$ and finishes the proof of the theorem.
Boundary slope and Euler characteristic {#sec:jsurface}
=======================================
In this section we verify that there exists an essential spanning surface which realizes the Jones slope $js_K = \{-2c_-(D) -2r\}$ and the quantity $jx_K=\{c(D)-|s_A(D)|+r \}$ of a near-alternating link $K$ computed in Section \[sec:jslope\]. This is the statement of Theorem \[thm:jsurface\].
Let $D$ be a near-alternating diagram, the surface $F_G$, such that $D = \partial (F_G)$ for a 2-connected, weighted planar graph $G$ as in Definition \[defn:near-alternating\], is called a *pretzel surface*. It is shown to be essential under certain conditions on the graph $G$ in [@OR12].
[[[@OR12 Theorem 2.15]]{}]{} \[thm:pretzele\] Let $G$ be a 2-connected planar graph in $S^2$ with edges $e_1, \ldots, e_n$ having weights $\omega_1, \ldots, \omega_n \in \mathbb{Z}$.
1. If $|\omega_i| \geq 3$ for all $i$, then the surface $F_G$ is essential.
2. If $\omega_1 \leq -2$ and $\omega_i \geq 2$ for $i = 2, \ldots, n$, and the surface $F_G$ is not essential, then $G$ has an edge, say $e_2$, that is parallel to $e_1$ (i.e., $e_2$ is another edge on the same pair of vertices as $e_1$) such that $\omega_1=-2$ and $\omega_2 = 2$ or $3$.
Note that the original wording of the theorem in [@OR12] says “algebraically incompressible and boundary incompressible" instead of “essential."
The surface $F_G$ is clearly also a state surface from the state that chooses the $B$-resolution on all the crossings in the single negative twist region of $D$, and the $A$-resolution on all the rest of the crossings. A formula for the boundary slope of a state surface is given by the following lemma.
\[lem:stateslope\] Let $D$ be a diagram of an oriented knot $K$, and let $\sigma$ be a state of $D$. Then the state surface $S_{\sigma}(D)$ has as its boundary slope $$2c_+^B(\sigma)-2c_-^A(\sigma),$$ where $c_+^B(\sigma)$ is the number of positive crossings where the $B$-resolution is chosen, and $c_-^A(\sigma)$ is the number of negative crossings where the $A$-resolution is chosen.
If $K$ is a near-alternating knot, we can apply Theorem \[thm:pretzele\] to show that $F_G$ is an essential surface for $K$. If the maximal negative twist region of weight $r<0$ in a near-alternating diagram $D$ of $K$ has $r=-2$, the only way the surface $F_G$ is not essential is if $G$ has an edge $e_2$, that is parallel to $e_1$ corresponding to the negative twist region, such that $e_2$ has weight $2$ or $3$. However, the condition on the diagram being near-alternating implies that if an edge is parallel to $e_1$, then it must have more than 6 crossings, since it would give a path in $G^{e_1}$ between the same pair of vertices, and we require that the weight of such a path be greater than $2t$ while $t>2$.
We verify that $F_G$ is indeed a Jones surface realizing the Jones slope $js_K$ and $jx_K$ from Theorem \[thm:degree\] by computing its boundary slope and Euler characteristic. This will complete the proof of Theorem \[thm:jsurface\].\
#### **Boundary slope**
Now that we know that a pretzel surface for a near-alternating knot is essential, we compute its boundary slope. A pretzel surface comes from the state $\sigma$ which chooses the $B$-resolution at each crossing in the negative twist region, and this is the only difference between $\sigma$ and the all-$A$ state. Either all these crossings are positive, or they are all negative. We use Lemma \[lem:stateslope\] to compare the boundary slope of this state to the boundary slope of the all-$A$ state which is $2c_+^B(\sigma_A)-2c_-^A(\sigma_A)=0-2c_-^A(\sigma_A)=-2c_-(D)$. Suppose the crossings in the twist region are positive, then we get $2c_+^B(\sigma)-2c_-^A(\sigma)=2(c_+^B(\sigma_A)-r)-2c_-^A(\sigma_A)=-2c_-(D)-2r$ as the boundary slope. If the crossings in the twist region are negative, we also get $2c_+^B(\sigma)-2c_-^A(\sigma)=2c_+^B(\sigma_A)-2(c_-^A(\sigma_A)+r)=-2c_-(D) - 2r$ for the boundary slope, and we are done.\
#### **Euler characteristic**
It is clear that the Euler characteristic of the surface is $\chi(S_A(D)) - r = (|s_A(D)|-r)-c(D) = -(c(D)-|s_A(D)|+r) = jx_K$.
\[lem:nabad\] A near-alternating link is $B$-adequate.
Apply the $B$-resolution to all the crossings in a near-alternating diagram $D$, we see that the all-$B$ state graph of $D$ is given by the dual graph of $G^e$ with $r-1$ vertices attached from the single negative twist region. Since $r\geq 2$, each of the segments from the crossings in the negative twist region connects a pair of distinct vertices in $s_B(D)$, so if $D$ is not $B$-adequate, then $\partial(F_{G^e})$ is not $B$-adequate. Since $G^e$ is required to be 2-connected, and $\partial(F_{G^e})$ is an alternating diagram, we conclude that $\partial(F_{G^e})$ is reduced. Otherwise, a vertex of the edge corresponding to the nugatory crossing would be a cut vertex, contradicting the assumption that $G^e$ is 2-connected. Thus, $\partial(F_{G^e})$ is adequate by [@Lic97 Proposition 5.3] and we have a contradiction to $\partial(F_{G^e})$ being not $B$-adequate.
Murasugi sums
-------------
We prove Theorem \[thm:Murasugi\] in this section. The proof is very similar to that of Theorem \[thm:degree\].
Let $D = D_1 \star D_2$ be a Murasugi sum of a near-alternating diagram $D_1=\partial(F_G)$ and an $A$-adequate diagram $D_2$ along a state circle which is not the one with one-edged loops in $D_1$. To compute the colored Jones polynomial we may decorate the maximal negative twist region of $D$ from $D_1$ with 4 Jones-Wenzl projectors as indicated in Figure \[fig:framenegt\], and remove the negative crossings using the fusion and the untwisting formulas.
We have $$\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = \sum_{\sigma, \ a \ : \ a, \ n, \ n \text{ admissible }} \frac{\triangle_a}{\theta(n, n, a)} ((-1)^{n-\frac{a}{2}}A^{2n-a+n^2-\frac{a^2}{2}})^{r} A^{sgn(\sigma)} \langle {\mathcal{S}}^{a}_{\sigma} \rangle,$$ where ${\mathcal{S}}^a_{\sigma}$ is a disjoint union of a connected component $J^a_{\sigma}$ decorated by the projectors and disjoint circles resulting from applying $\sigma$. We can also assume that $J^a_{\sigma}$ may be isotoped to the form shown in Figure \[fig:case\], with $c$ the number of split strands for a Kauffman state $\sigma$ defined the same way.
We similarly define $$\deg(\sigma, a) := \text{deg} \left( \frac{\triangle_a}{\theta(n, n, a)}(-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle {\mathcal{S}}^a_{\sigma} \rangle \right),$$ where $$d(a, r) = r(2n-a+n^2-\frac{a^2}{2}).$$
It is also straightforward to see that $$\deg(\sigma_A, 0) = H_n(D) + 2r(n^2+n).$$
We have the inequality $$\deg(\sigma, 2c) \leq \deg(\sigma_A, 0) - \left(\sum_{i=1}^{t} (\omega - 2)(2m_i^2+2m_i) + 4m_i^2\right)- \left( r\frac{(2c)^2}{2} + 2cr \right),$$ where $P=\{m_1, \ldots, m_t\}$ is a minimal partition of $n$ into $t$ parts, and $t$ is the total number of distinct paths from $v$ to $v'$ in $G^e$.
Let $S$ be the state circle in $s_A(D_1)$ along which the Murasugi sum $D = D_1\star D_2$ is formed. Let $\sigma'$ be the state obtained from $\sigma$ by changing the resolution from $B$-to $A$- for crossings of $D$ contained within the state circle $S$. Then, $$\deg(\sigma, 2c) \leq \deg(\overline{\sigma'}, 2c),$$ where $$\deg(\overline{\sigma'}, 2c):= \text{deg} \left( \frac{\triangle_a}{\theta(n, n, a)}(-1)^{rn-r\frac{a}{2}} A^{d(a, r) + sgn(\sigma)} \langle \overline{{\mathcal{S}}^a_{\sigma}} \rangle \right).$$
Now $\sigma'$ restricted to $D_1$ defines a skein ${\mathcal{S}}^a_{\sigma'}(D_1)$ in the state sum expansion of $\langle (D_1)^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle$ with $c$ split strands. Thus we obtain the same inequality as : $$\deg(\sigma', 2c) = \deg(\overline{\sigma'}, 2c) \leq \deg(\sigma_A, 0) - \left(\sum_{i=1}^{t} (\omega - 2)(2m_i^2+2m_i) + 4m_i^2\right)- \left( r\frac{(2c)^2}{2} + 2cr \right).$$
It follows immediately that $\deg \langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = \deg(\sigma_A, 0)= H_n(D)+2r(n^2+n)$, since if $\frac{\omega}{t}>r$, then the difference
$$- \left(\sum_{i=1}^{t} (\omega - 2)(2m_i^2+2m_i) + 4m_i^2\right)- \left( r\frac{(2c)^2}{2} + 2cr \right) < 0.$$ Thus $\deg(\sigma, 2c) < \deg(\sigma_A, 0)$.
The state surface $S_{\sigma}(D)$ of $D$ obtained from the state which chooses the $B$-resolution on all the crossings in the negative twist region and the $A$-resolution for the rest of the crossings is a Jones surface realizing the Jones slope $js_K=\{-2c_-(D)-2r\}$ and the corresponding linear term $jx_K=\{c(D)-|s_A(D)|+r\}$.
It is easy to see using Lemma \[lem:stateslope\] that the boundary slope and the Euler characteristic of $S_{\sigma}(D)$ match $js_K$ and $jx_K$. It remains to show that it is essential.
The surface $S_{\sigma}(D)$ is a Murasugi sum $S_1 \star S_2$, where $S_1$ is the pretzel surface for the near-alternating $D_1$, and $S_2$ is the all-$A$ state surface for the $A$-adequate $D_2$. By [@Oza11; @OR12], these are both essential. This implies that $S_{\sigma}(D) = S_1\star S_2$ is essential by the following result concerning the essentiality of Murasugi sums of essential surfaces.
Note that in the literature, when Murasugi sum is mentioned, it usually refers to the Murasugi sum of Seifert surfaces of the respective knots. Gabai’s work [@Ga85] establishes important geometric properties for the Murasugi sums of orientable surfaces in the knot complement. He has shown that a Murasugi sum of orientable essential surfaces is essential. Ozawa establishes the following important generalization to Murasugi sums of (not necessarily) orientable surfaces.
\[lem:msume\] If $F_1$ and $F_2$ are essential, then $F= F_1\star F_2$ is also essential.
Note that the original wording of Lemma \[lem:msume\] in [@Oza11] says “$\pi_1$-essential" instead of “essential."
It is immediate by Lemma \[lem:msume\] that $S_{\sigma}(D)$ is essential. Thus we prove Theorem \[thm:Murasugi\]. Note that the diagram $D=D_1\star D_2$ is not guaranteed to be $B$-adequate.
Near-alternating knots are not adequate {#sec:nadequate}
=======================================
We show that a near-alternating knot does not admit an adequate diagram. The criterion for an adequate knot from the colored Jones polynomial is the following result due to Kalfagianni [@Kal16]. For large enough $n$ let $$s_1(n)n^2 + s_2(n)n + s_3(n) = d^*(n)-d(n).$$
\[thm: adequatecrit\] For a knot $K$ let $c(K)$ and $g_T(K)$ denote the crossing number and the Turaev genus of $K$, respectively. The knot $K$ is adequate if and only if for some $n > n_K$, we have $$s_1(n) = 2c(K), \text{ and } s_2(n) = 4-4g_T(K) -2c(K).$$ Furthermore, every diagram of $K$ that realizes $c(K)$ is adequate and it also realizes $g_T(K)$.
We will begin by proving the analogue of [@LT88 Lemma 8] concerning the Kauffman polynomial for a near-alternating knot. Recall that the Kauffman two-variable polynomial $\Lambda_D(a, z)$ is defined uniquely by the following [@Lic97 Theorem 15.5]
- $\Lambda(\vcenter{\hbox{\includegraphics[scale=.10]{circ.png}}})=1$, where $\vcenter{\hbox{\includegraphics[scale=.10]{circ.png}}}$ is the standard diagram of the unknot.
- $\Lambda(a, z)$ is unchanged by Reidemeister moves of Type II and III on the diagram D.
- $\Lambda(\vcenter{\hbox{\includegraphics[scale=.15]{loop.png}}}) = a \Lambda(\vcenter{\hbox{\includegraphics[scale=.15]{unloop.png}}})
$.
- $$\label{eqn:k2poly}
\Lambda(\vcenter{\hbox{\includegraphics[scale=.25]{over.png}}}) + \Lambda(\vcenter{\hbox{\includegraphics[scale=.25]{under.png}}}) = z(\Lambda(\vcenter{\hbox{\includegraphics[scale=.2]{crossing2.png}}}) + \Lambda(\vcenter{\hbox{\includegraphics[scale=.2]{crossing3.png}}})).$$
A diagram which locally differs in one of the four pictures in is denoted by $D_+$, $D_-$, $D_0$, and $D_{\infty}$, respectively.
We will need the following useful results by Thistlethwaite [@Thi88].
\[thm:kconnect\] Let $D$ be a $c(D)$-crossing link diagram which is a connected sum of link diagrams $D_1, \ldots, D_k$. Let $\Lambda(a, z) = \sum_{r, s} u_{rs} a^rz^s$ for $D$, and let $b_1, \ldots, b_k$ be the lengths of the longest bridges of $D_1, \ldots, D_k$, respectively. Then for each non-zero coefficient $u_{rs}$, $|r|+s\leq c(D)$ and $s\leq c(D)-(b_1 + \cdots + b_k)$.
\[thm:coefflambda\] Let $D$ be a connected, alternating diagram with $c(D)\geq 3$ crossings, and let $G$ be the graph associated with the black-and-white coloring of the regions of $D$ for which the crossings of $D$ all have positive sign. Let $\Lambda_D(a, z) = \sum p_s(a) z^s$, and let $\chi_G(x,y) = \sum v_{rs} x^ry^s.$ Then $$\begin{aligned}
p_{n-1}(a)&= v_{1, 0}a^{-1} + v_{0, 1}a, \text{ and } \\
p_{n-2}(a)&= v_{2, 0}a^{-2} + (v_{2, 0} + v_{0, 2}) + v_{0, 2} a^2. \end{aligned}$$
In fact, Thistlethwaite remarks immediately following this theorem in [@Thi88] that the coefficient $p_{n-1}(a)$ may be written as $\kappa(a+a^{-1})$ with $\kappa$ strictly positive if $D$ is a prime, alternating diagram with at least two crossings.
We prove a mild generalization of [@LT88 Lemma 8] using the same argument which applies in the setting of near-alternating diagrams.
\[lem:gennearalt\] Let $D$ be a near-alternating diagram with a maximal negative twist region of weight $r < 0$ with $|r|\geq 2$. Then, the $z$-degree of $\Lambda(a, z)$ of $D$ is $c(D)-2$, and the coefficient of $z^{n-2}$ in $\Lambda(a, z)$ of $D$ is nonzero.
We induct on $|r| \geq 2$. Note that if $D$ is a near-alternating diagram with a negative twist region of weight $r<0$, $|r|\geq 2$, then the same diagram with the maximal negative twist region replaced by a negative twist region of 2 crossings is still near-alternating. Thus it is valid to consider the base case with $|r|=2$. For $|r|=2$, switching the top crossing in the twist region with weight $r$ results in an alternating diagram $D_-$ isotopic to one with $c(D)-2$ crossings by a Type II Reidemeister move. By Theorem \[thm:kconnect\], we see that the $z$-degree of $\Lambda(a, z)$ for $D_-$ is strictly less than $c(D)-2$. One of the nullifications of this crossing results in a non-alternating diagram, say $D_0$, with $c(D)-1$ crossings and a bridge of length 3. Again by Theorem \[thm:kconnect\], the $z$-degree of $\Lambda_{D_0}$ is at most $n-4$. The other nullification produces a removable kink and results in a prime $(c(D)-2)$-crossing alternating diagram $D_{\infty}$ (this was required by condition (b) in Definition \[defn:near-alternating\] defining a near-alternating diagram). Applying Theorem \[thm:coefflambda\] and the subsequent remark, we get that the $z^{c(D)-3}$ term of $\Lambda(a, z)$ for $D_{\infty}$ has coefficient $\kappa a^{\pm 1}(a^{-1}+a)$ with $\kappa > 0$. Plug this into the defining relation for $\Lambda(a, z)$ based on $D_+=D$, $D_-$, $D_0$, and $D_{\infty}$, we get that the coefficient of $z^{n-2}$ in $\Lambda(a, z)$ for $D$ is the same as the coefficient of $z^{n-3}$ in $\Lambda(a, z)$ for $D_{\infty}$, which is nonzero. This takes care of the base case. For $|r| > 2$, $D_0$ is a near-alternating diagram with $|r|-1$ negative crossings in the negative twist region, and that is where we apply the inductive hypothesis. We get $$\Lambda(D_+) + \underbrace{\Lambda(D_-)}_{\text{$z$-degree $\leq c(D)-3$}} = z(\underbrace{\Lambda(D_0)}_{\text{$z$-degree $= c(D)-3$}} + \underbrace{\Lambda(D_{\infty})}_{\text{$z$-degree $\leq c(D)-4$}}).$$ This shows that $\Lambda(D)=\Lambda(D_+)$ has $z$-degree determined by the $z$-degree of $\Lambda(D_0)$ with the same coefficient. After multiplying $\Lambda(D_0)$ by $z$, we finish the proof of the Theorem.
Using Theorem \[thm: adequatecrit\], Theorem \[thm:kconnect\], Theorem \[thm:coefflambda\], and Lemma \[lem:gennearalt\], we prove Theorem \[thm:naknoadequate\], which we restate here.
A near-alternating knot does not admit an adequate diagram.
Given a knot $K$ with a near-alternating diagram $D$ of a negative twist region of weight $r<0$ such that $|r|>2$, suppose that $K$ also admits a non-alternating, adequate diagram $D_A$. Then $c(D_A) = c(D)+r$ by Theorem \[thm: adequatecrit\], and $D_A$ has a bridge of length $\geq 2$. But this contradicts Lemma \[lem:gennearalt\] by Theorem \[thm:kconnect\], since Lemma \[lem:gennearalt\] implies that the $z$-degree of $\Lambda(a, z)$ for $D$ is $c(D)-2$, but Theorem \[thm:kconnect\] applied to $\Lambda(a, z)$ for $D_A$ would imply that $\Lambda(a, z)$ has $z$-degree $\leq c(D)-r-2$. This is because $D$ and $D_A$ are related by a sequence of Type I, II, and III Reidemeister moves, but a Type I Reidemeister move only affects the $a$-degree of $\Lambda(a, z)$, while the Type II and III moves leave $\Lambda(a, z)$ invariant. Thus the only other possibility is that it admits a reduced, alternating diagram with $c(D)+r$ crossings which also leads to a contradiction to Lemma \[lem:gennearalt\], since $|r| \geq 2$.
Stable coefficients and volume bounds {#sec:cvolume}
=====================================
In this section we prove Theorem \[thm:tail\], which we reprint here.
Let $K$ be a link admitting a near-alternating diagram $D = \partial (F_G)$, where $G$ is a finite 2-connected, weighted planar graph with a single negatively-weighted edge of weight $r < 0$. The first and second coefficient, $\alpha_n, \beta_n$, respectively, of the reduced colored Jones polynomial $\hat{J_K}(v, n)$ of a near-alternating link $K$ are stable. Write $\alpha = \alpha_n$ and $\beta = \beta_n $. We have $|\alpha| = 1$ and $|\beta| = \begin{cases} \chi_1(s_{\sigma}(D)') + 1 \text{ if $|r|=2$} \\
\chi_1(s_{\sigma}(D)') \text{ if $|r|>2$ }
\end{cases}$, where $\sigma$ is the Kauffman state corresponding to the state surface $F_G$ and $\chi_1(s_{\sigma}(D)')$ is the first Betti number of the reduced graph of $s_{\sigma}(D)$. The last and penultimate coefficient, $\alpha'_n, \beta'_n$, respectively, are also stable, and we write $\alpha' = \alpha'_n$ and $\beta' = \beta'_n$. We have $|\alpha'|=1$ and $|\beta'| = \chi_1(s_{B}(D)')$.
If $K$ is such that the near-alternating diagram is prime and twist-reduced with more than 7 crossings in each twist region, then $K$ is hyperbolic, and $$.35367(|\beta|+|\beta'| -1) < vol(S^3\setminus K) < 30v_3(|\beta|+|\beta'| + M - 2),$$ for a constant $M\geq 0$. Here $v_3\approx 1.0149$ is the volume of a regular ideal tetrahedron. In other words, stable coefficients of a highly-twisted, near-alternating link $K$ are coarsely related to the hyperbolic volume of $S^3\setminus K$.
Note that since a near-alternating link $K$ with a near-alternating diagram $D$ is $B$-adequate, if we write the $n$th-reduced colored Jones polynomial $\hat{J}_K(v, n) = J_K(v, n)/J_{\vcenter{\hbox{\includegraphics[scale=.05]{circ.png}}}}(v, n)$ as $$\label{eqn:scoeff} \hat{J}_K(v, n) = \alpha_n v^{\hat{d}(n)} + \beta_n v^{\hat{d}(n)+4} + \cdots + \beta'_n v^{\hat{d}^*(n) -4} + \alpha'_n v^{\hat{d}^*(n)} ,$$ where $\hat{d}(n)$ is the minimum degree and $\hat{d}^*(n)$ is the maximum degree of $\hat{J}_K(v, n)$, respectively, then $|\beta'_n|= \chi(s_B(D)')$ and $|\alpha'_n|= 1$ by [@DL06 Theorem 3.1]. So what we need to determine is $|\alpha_n|$ and $|\beta_n|$.
Stability of coefficients
-------------------------
We shall use the following result from [@DL06].
[[@DL06 Theorem 3.1]]{} \[thm:dlcoeff\] Let $D$ be an $A$-adequate link diagram. Write $\hat{J}_K(v, n)$ as in . Then we have for all $n$, $$|\alpha_n| = 1 \text{ and } |\beta_n| = \chi_1(s_A(D)').$$
Note that the original statement is for a knot diagram but the argument extends without problem to a link diagram.
From the proof of Theorem \[thm:bracketdegree\] we see that the skein in the state sum realizing the degree comes from the state $\sigma_A$ which restricts to the $A$-resolution on crossings outside of the maximal negative twist region. The first coefficient $\alpha_n$ is just the first coefficient of the skein ${\mathcal{S}}^0_{\sigma_A}$ from $D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}}$ realizing the degree, so $|\alpha_n| = 1$. For the second coefficient $\beta_n$, as long as $\frac{\omega}{t}>|r|$ with $|r|, t \geq 2$, the inequality implies that no skein ${\mathcal{S}}^a_{\sigma}$ from another state $\sigma$ with $c > 0$ split strands contributes to the 2nd coefficient. Therefore, we need only to consider the contribution of other skeins $\sigma$ with $c=0$ split strands.
For a skein with $0$ split strands we may remove the $r$ half twists on $n$ strands on the portion of the skein decorated by idempotents, so $$\sum_{a \ : \ a, \ n, \, n \text{ admissible }}\langle {\mathcal{S}}^a_{\sigma} \rangle = A^{r(n^2+2n)}\langle {\mathcal{S}}'_{\sigma} \rangle,$$ where ${\mathcal{S}}'_{\sigma}$ is the new skein. In a process similar to that in [@DL06], we consider Kauffman states (now on all the crossings of ${\mathcal{S}}'_{\sigma}$) which chooses the $B$-resolution on a single crossing corresponding to a segment in $\sigma_A$ between a pair of circles in the state graph $s_A(\overline{{\mathcal{S}}'_{\sigma}})$. Let $D_r$ be the reduced, alternating diagram obtained from $D$ by removing from $s_A(D)$ the edges corresponding to the crossings in the negative twist region of $D$, then recovering a link diagram by reversing the application of the all-$A$ Kauffman state. See Figure \[fig:dr\] below for an example.
We know $D_r$ is reduced because of condition (b) on $D$ of a near-alternating diagram, since $D_r = \partial(F_{G/e})$ where $G/e$ is the $G$ with the negative edge $e$ contracted. There is a bijection between the set of Kauffman states of $D_r$ which contribute to the 1st and 2nd coefficient of $\langle D_r \rangle$ and the set $SC:=\{{\mathcal{S}}'_{\sigma}: \sigma \text{ chooses the $B$-resolution on a single crossing of } c(D^n)\setminus r^n \}$ by
$$\sigma \in SC \mapsto \sigma \text{ on } D_r$$ This implies that the 2nd coefficient of the sum $$\sum_{\sigma \text{ with } c = 0, \ \sigma \in SC} \langle {\mathcal{S}}^a_{\sigma} \rangle = \sum_{\sigma \text{ with } c = 0, \ \sigma\in SC} A^{r(n^2+2n)}\langle {\mathcal{S}}'_{\sigma} \rangle$$ is equal to the 2nd coefficient of the colored Jones polynomial of the link with the diagram $D_r$. Since $D_r$ is adequate, we may apply Theorem \[thm:dlcoeff\] to $D_r$. This gives that the 2nd coefficient of its colored Jones polynomial is given by $e'_r-v_r+1$, where $e'_r$ is the number of edges in the reduced all-$A$ state graph of $D_r$ and $v_r$ is the number of vertices of $s_A(D)'$. We compare this to the data from $D$, where $e'$ is the number of edges of the reduced graph $s_{\sigma}(D)'$ and $v$ is the number of vertices in $s_{\sigma}(D)'$. $$\begin{aligned}
|\beta_n| &= e_r'-v_r+1 = \begin{cases} &e'-1-(v+r)+1 = e'-v+2, \text{ if } |r| = 2. \\ \notag
&e'+r-(v+r)+1 = e'-v+1, \text{ if } |r| > 2. \end{cases}
\intertext{So we have}
|\beta_n| &= \begin{cases} &|\chi_1(s_{\sigma}(D)')| + 1 \text{ if } |r|=2 \\
&|\chi_1(s_{\sigma}(D)')| \text{ if } |r| > 2. \end{cases} \label{eq:2ndcoeff}\end{aligned}$$ The stability of these coefficients follows from the stability of the 1st and 2nd coefficient for the link represented by $D_r$ since the computation was done independent of $n$.
Two-sided volume bounds from stable coefficients $\alpha$, $\beta$, $\alpha'$, and $\beta'$
-------------------------------------------------------------------------------------------
The following theorem from [@FKP08] provides volume bounds on a hyperbolic link complement based on the number of twist regions in a diagram of the link.
\[thm:twestimate\] Let $K\subset S^3$ be a link with a prime, twist-reduced diagram $D$. Assume that $D$ has $tw(D)>2$ twist regions, and that each region contains at least 7 crossings. Then $K$ is a hyperbolic link satisfying $$0.70735 (tw(D)-1) < vol(S^3\setminus K) < 10v_3(tw(D)-1),$$ where $v_3\approx 1.0149$ is the volume of a regular ideal tetrahedron.
\[thm:twadequate\] Let $K$ be a link in $S^3$ with an adequate diagram $D$ such that every twist region of $D$ contains at least 3 crossings. Then $$\frac{1}{3} tw(D) + 1 \leq |\beta| + |\beta'| \leq 2 tw(D).$$
We use Theorem \[thm:twestimate\] and Theorem \[thm:twadequate\] to relate the number of twist regions of a diagram to the stable coefficients $\alpha, \beta, \alpha',$ and $\beta'$, obtained in the previous section. In particular we show the following:
\[lem:twistestimate\] Let $K$ be a link with a near-alternating diagram that is prime and twist-reduced with at least 3 crossings in every twist region of $D$. Then $$\frac{tw(D)-1}{3} + 1 \leq |\beta| + |\beta'|+M -1 \leq 2(tw(D)-1),$$ for a constant $M \geq 0$.
Recall that if we remove the crossings corresponding to the negative twist region of weight $r$ by choosing the $A$-resolution at each of the negative crossings, then we have a reduced alternating diagram by assumption, which we will denote by $D_r$. For $D_r$ we can immediately apply Theorem \[thm:twadequate\]. Let $e'_{A'}, v_{A'}$ be the number of edges and vertices in the reduced all-$A$ state graph of $D_r$, and $e'_{B'}, v_{B'}$ be the number of edges and vertices in the reduced all-$B$ state graph of $D_r$. In particular we get $$\begin{aligned}
&e'_{A'} + e'_{B'} -v_{A'} - v_{B'} + 2 \leq 2tw(D'),
\intertext{ and}
&e'_{A'} + e'_{B'} -v_{A'} - v_{B'} + 2 \geq \frac{tw(D')}{3} + 1.
\intertext{Since $D$ has at least one more twist region than $D'$, this gives}
&e'_{A'} + e'_{B'} -v_{A'} - v_{B'} + 2 \leq 2(tw(D)-1),
\intertext{and}
&e'_{A'} + e'_{B'} -v_{A'} - v_{B'} + 2 \geq \frac{tw(D)-1}{3} + 1.
\intertext{Note that $D$ is $B$-adequate and we assume that $|r|>2$. Let $e'$ be the number of edges in $s_{\sigma}(D)$ and $v = |s_{\sigma}(D)|$. Using the result \eqref{eq:2ndcoeff} above on $|\beta|$ we get}
|\beta| + |\beta'| &= e'-v + e'_B - v_B + 2. \\
\intertext{Substituting for quantities from $D'$, }
&= e'_{A'} - v_{A'} + 1 + e'_{B'}-M-r-(v_{B'}-(r+1)) +1,
\intertext{where $M=e'_{B'} - e'_B-r\geq 0$. So then}
&|\beta| + |\beta'|+M -1 \leq 2(tw(D)-1) \\
&|\beta| + |\beta'| + M -1 \geq \frac{tw(D)-1}{3}+1.\end{aligned}$$
Lemma \[lem:twistestimate\] combined with Theorem \[thm:twestimate\] gives the corollary.
Let $K$ be a link with a near-alternating diagram $D$ that is prime and twist-reduced. Assume that $D$ has $tw(D) > 2$ twist regions and that each region contains at least 7 crossings. Then $K$ is a hyperbolic link satisfying $$.35367(|\beta|+|\beta'| -1 ) < vol(S^3\setminus K) < 30v_3(|\beta|+|\beta'|+M-2),$$ for some constant $M \geq 0$.
Highly twisted links {#sec:sptl}
====================
We prove Theorem \[thm:eventual\] by adapting the proof of Theorem \[thm:degree\] to a highly twisted link diagram with multiple negative twist regions. We recall Theorem \[thm:eventual\] for convenience.
For a weighted planar graph $G$ let $G^-$ denote the sub-graph of $G$ consisting of the negative edges of $G$, and let $G^-_c$ be a connected component of $G^-$.
Let $K\subset S^3$ be a knot with a prime, twist-reduced diagram $D=\partial (F_G)$, where $G$ is 2-connected, the graph $G^{\{e\}}$ obtained by deleting all the negatively-weighted edges $\{e_i\}$ from $G$ remains 2-connected, and each connected component $G^-_c$ of $G$ is a single negative edge $e_i$. In addition, the diagram $D' = \partial F_{G/\{e\}}$, where $G/\{e\}$ is the graph obtained from $G$ by contracting along each edge $e_i$, is adequate. Assume that $D$ has $tw(D) \geq 2$ twist regions, and that each region contains at least 7 crossings. Let $K_m$ be the knot obtained from $K$ by adding $m$ full twists on two strands to every positive twist region. There exists some integer $0 < M_K < \infty$ such that for all $m > M_K$,
(i) $K_m$ is hyperbolic,
(ii) the Strong Slope Conjecture is true for $K_m$ with spanning Jones surfaces, and
(iii) the coefficients $\alpha = \alpha_n$, $\beta = \beta_n$, $\alpha' = \alpha'_n$, and $\beta' = \beta'_n$ are stable. They give the following two-sided volume bounds for $S^3\setminus K_m$: $$|\beta|+|\beta'|+M+2(R-1) \leq vol(S^3\setminus K_m) \leq |\beta|+|\beta'|+M-1 + \frac{R-1}{3},$$ for some constant $M$, where $R$ is the number of maximal negative twist regions in $D$.
We illustrate the graphical conditions of Theorem \[thm:eventual\] in the Figure \[fig:htwistg\] below.
(i) It follows from [@FKP08 Theorem 1.2] that $K_m$ is hyperbolic for any $m\geq 1$, since we started with a prime, twist-reduced diagram $D$ with $tw(D) > 2$ and more than 7 crossings in every twist region.
(ii) We prove a generalized version of Theorem \[thm:degree\] to obtain the Jones slope $js_{K_m}$ and the linear term $jx_{K_m}$. First we consider the following quantities on a diagram $D = \partial(F_G), $ where $G$ is as before a 2-connected, weighted planar graph.
- The number of maximal negative twist regions in $D$.
- For $1\leq i \leq R$, let $r_i$ be the weight of an edge corresponding to a maximal negative twist region of $D$. Define $$r:= \min_{1\leq i \leq R} \{r_i \}.$$
- Let $G^{e_1, \ldots, e_R}=G^{\{e\}}$ be the graph $G$ with edges $\{e_i\}_{i=0}^{R}$ corresponding to the negative twist regions removed. Let $G^-$ be the sub-graph consisting of all the negative edges of $G$. For a connected component $G^-_{i}$ of $G^-$ for a single $e_i = (v_i, v_i')$. Define $t_i$ be the number of paths in $G^{\{e_i\}}$ between the pair of vertices $v_i, v_i'$, then let $$t:= \max_{1\leq i \leq R} t_i.$$
- We define the length of a path $W = v_1, v_2, \ldots, v_k$ slightly differently as $$\ell(W)= 2+ \sum_{\text{positive edges $\epsilon$ in $W$}} |(\epsilon|-2),$$ where $|\epsilon|$ is the weight of the edge. Let $\omega_i$ be the minimum length of all walks between $v_i$ and $v_i'$ in $G^{e_i}$. Then $$\omega = \min_{1\leq i\leq R} \omega_i.$$
\[thm:multidegree\] Let $G$ be a 2-connected, weighted planar graph such that every component of $G^-$ is a single negative edge of $G$, and let $D = \partial(F_G)$. If $$\frac{\omega}{t} > rR, \label{eq:multi}$$ then $$\label{eqn:multic}
d(n) = h_n(D) - 2\left(\sum_{i=1}^R r_i \right)((n-1)^2+(n-1)).$$
Note that we can always take $m$ large enough such that $K_m$ with diagram $D_m$ satisfies , and this result implies that $js_{K_m} = \{-2c_-(D_m)-2\left(\sum_{i=1}^R r_i \right) \}$ and $jx_{K_m}=\{c(D_m)-|s_A(D_m)|+ \left(\sum_{i=1}^R r_i \right)\}$ for $m$ sufficiently large.
We consider the diagram $D^n$ decorated with a Jones-Wenzl idempotent and slide and double the idempotents until there are four idempotents framing every maximal negative twist region. Apply the fusion formula to each of these twist regions and expand over Kauffman states on the set of remaining crossings, we have that
$$\langle D^n_{{\vcenter{\hbox{\includegraphics[scale=.1]{jwproj.png}}}}} \rangle = \sum_{\sigma, \ a_i: \ a_i, \ n, \ n \text{ admissible }} \left( \prod_{i=1}^{R} \frac{\triangle_{a_i}}{\theta(n, n, a_i)} ((-1)^{n-\frac{a_i}{2}}A^{2n-a_i+n^2-\frac{a_i^2}{2}})^{r} A^{sgn(\sigma)} \right) \langle {\mathcal{S}}^{a_1, a_2, \ldots, a_R}_{\sigma} \rangle.$$
The skein ${\mathcal{S}}^{a_1, a_2, \ldots, a_R}$ is similarly the disjoint union of (possibly more than one) connected components decorated by the Jones-Wenzl projects, and circles. For each Kauffman state $\sigma$ we similarly consider the number of split strands $c_i$ for each negative twist region $T_1, \ldots, T_R$. The difference between this case and the one considered for near-alternating knots is that the split strands might go through the Jones-Wenzl projectors from other negative twist regions. This is our reason for considering a slightly different definition of the length of a path which disregards the negative twist regions. Since, in applying Lemma \[lem:wcount\], only the positive twist regions contribute to the decrease in the degree. See Figure \[fig:htwiststd\] for an example of this behavior.
Define $$\text{deg}(\sigma, a_1, \ldots, a_R):= \text{deg} \left( \prod_{i=1}^{R} \frac{\triangle_{a_i}}{\theta(n, n, a_i)} ((-1)^{n-\frac{a_i}{2}}A^{2n-a_i+n^2-\frac{a_i^2}{2}})^{r} A^{sgn(\sigma)} \right) \langle {\mathcal{S}}^{a_1, a_2, \ldots, a_R}_{\sigma} \rangle.$$
Let $\sigma_A$ be the all-$A$ state that chooses the $A$-resolution. A simple computation shows that
$$\text{deg}(\sigma_A, 0, \ldots, 0) = H_n(D) + 2\left(\sum_{i=1}^R r_i \right)(n^2+n).$$
For Kauffman states $\sigma \not=\sigma_A$ with $c_1, \ldots, c_R$ split strands for each twist region, we have the following inequality.
$$\begin{aligned}
&\deg(\sigma, a_1, \ldots, a_R) \\
&\leq \deg(\sigma_A, 0, \ldots, 0) - \sum_{i=1}^R \frac{1}{R}\sum_{j=1}^{t} \left((\omega - 2)(2k_{i,j}^2+2k_{i,j})+4k_{i,j}^2 \right) + \sum_{i=1}^R \left(r\frac{(2c_i)^2}{2} + (2c_i)(r+1)\right),
\intertext{where $\{k_{i, 1}, \ldots, k_{i, t}\}$ forms a minimal partition of $c_i$ for $1\leq i \leq R$ in $t$ parts.}\end{aligned}$$
We see that as long as the inequality of is satisfied, $$\deg(\sigma, a_1, \ldots, a_R) < \deg(\sigma_A, 0, \ldots, 0).$$ Thus the degree is realized by choosing the all-$A$ state for crossings outside of these negative twist regions. This proves Theorem \[thm:multidegree\].
By Lemma \[lem:localzero\] and counting the number of circles in $\overline{{\mathcal{S}}_{\sigma}^{a_1, \ldots, a_R}}$, it suffices to prove the inequality for $\deg(\sigma, a_1, \ldots, a_R)$ where $a_i = c_i$. Let $c = \sum_i c_i$ be the total number of split strands.
To compare the degree $\deg(\sigma, a_1, \ldots, a_R)$ to $\deg(\sigma_A, 0, \ldots, 0)$, we sum over the decrease in $\deg(\sigma, a_1, \ldots, a_R)$ by the number of strands with which $\sigma$ flows through a path $W$ in $G^{e_i}$ between a pair of vertices $v_i, v_i'$. If $\sigma$ has $c_i$ split strands for a twist region $T_i$ for a negatively-weighted edge $e_i=(v_i, v_i')$, then it must flow through paths between $v_i$ and some other vertex $v_i'$ with a total of $c_i$ strands. We divide by $R$ to account for possible over-counting for different twist regions sharing the same split strands.
A Jones surface is given by the state surface $S$ where we choose the $B$-resolution at all the crossings in a negative twist region, and $A$-resolution for the rest. The resulting surface is the pretzel surface for $D$, which is therefore essential by Theorem \[thm:pretzele\]. Using the procedure given by Lemma \[lem:stateslope\] to compute the boundary slope and the Euler characteristic of the surface will match it to $js_{K_m}$ and $jx_{K_m}$. This combined with the fact that $K_m$ is $B$-adequate by the following lemma, which generalizes Lemma \[lem:nabad\], gives the Strong Slope Conjecture.
Let $G$ be a finite, 2-connected, planar graph without one-edged loops and let $G^-$ be the sub-graph of negative edges of $G$. If $G\setminus G^-$ is 2-connected, then the diagram $D = \partial(F_G)$ is $B$-adequate.
$G\setminus G^-$ is a connected graph consisting entirely of positive edges, so $D^+ = \partial(F_{G\setminus G^-})$ is alternating. It is adequate if it is reduced, which is ensured by $G\setminus G^-$ being 2-connected, since otherwise the nugatory crossing would give a cut vertex. Now the maximal negative twist regions in the $B$-resolution gives a string of state circles corresponding to the bigons of the twist region connected by segments in the all-$B$ state graph. As long as the number of crossings in the twist region is $\geq 2$, it is not possible for the crossings of these twist regions to give one-edged loops with the choice of the $B$-resolution. Thus no edge in $G=G^- \cup G\setminus G^-$ can be a one-edged loop.
(iii) Again if $m$ is large enough, we can completely ignore the contribution to the 2nd coefficient of the $n$th colored Jones polynomial from states other than the state which chooses the all-$A$ state on all crossings outside of the negative twist regions in $L$. Untwisting those negative twists gives an adequate diagram $D'$ with the same 2nd coefficient for the colored Jones polynomial. Compute the coefficient in terms of the first Betti number of $s_A(D')$ and apply Theorem \[thm:twadequate\] to $D'$, we similarly have $$\frac{tw(D')}{3}+1 \leq e'_{A'}+e'_{B'}-v_{A'}-v_{B'} + 2 \leq 2tw(D').$$
Now let $M = (e'_{B'}-v_{B'}+1) - (e'_{B}-v_{B}+1)$, then $$\begin{aligned}
\frac{tw(D')}{3}+1 &\leq |\beta|+|\beta'|+M \leq 2tw(D'), \\
\intertext{which implies}
\frac{tw(D)-R}{3}+1 &\leq |\beta|+|\beta'|+M \leq 2(tw(D)-R).
\intertext{Now Theorem \ref{thm:twestimate} gives}
|\beta|+|\beta'|+M+2(R-1) &\leq 2(tw(D)-1) \\
\frac{tw(D)-1}{3} &\leq |\beta|+|\beta'|+M-1 + \frac{R-1}{3}.
\intertext{Combined with Theorem \ref{thm:twestimate}, this gives the two-sided volume bound}
|\beta|+|\beta'|+M+2(R-1) &\leq vol(S^3\setminus K_m) \leq |\beta|+|\beta'|+M-1 + \frac{R-1}{3}.\end{aligned}$$
|
---
abstract: 'We study the effect of antiferromagnetic longitudinal coupling on the one-dimensional transverse field Ising model with nearest-neighbour couplings. In the topological phase where, in the thermodynamic limit, the ground state is twofold degenerate, we show that, for a finite system of $N$ sites, the longitudinal coupling induces $N$ level crossings between the two lowest lying states as a function of the field. We also provide strong arguments suggesting that these $N$ level crossings all appear simultaneously as soon as the longitudinal coupling is switched on. This conclusion is based on perturbation theory, and a mapping of the problem onto the open Kitaev chain, for which we write down the complete solution in terms of Majorana fermions.'
author:
- Grégoire Vionnet
- Brijesh Kumar
- Frédéric Mila
title: Level crossings induced by a longitudinal coupling in the transverse field Ising chain
---
The topological properties of matter are currently attracting a considerable attention [@kane; @zhang]. One of the hallmarks of a topologically non trivial phase is the presence of surface states. In one dimension, the first example was the spin-1 chain that was shown a long time ago to have a gapped phase [@haldane] with two quasi-degenerate low-lying states (a singlet and a triplet) on open chains [@kennedy]. These low-lying states are due to the emergent spin-1/2 degrees of freedom at the edges of the chains which combine to make a singlet ground state with an almost degenerate low-lying triplet for an even number of sites, and a triplet ground state with an almost degenerate low-lying singlet when the number of sites is odd. In that system, the emergent degrees of freedom are magnetic since they carry a spin 1/2, and they can be detected by standard probes sensitive to local magnetisation such as NMR [@tedoldi].
In fermionic systems, a topological phase is present if the model includes a pairing term (as in the mean-field treatment of a p-wave superconductor), and the emergent degrees of freedom are two Majorana fermions localised at the opposite edges of the chain [@kitaev]. Their detection is much less easy than that of magnetic edge states, and it relies on indirect consequences such as their impact on the local tunneling density of states [@mourik; @nadj-perge], or the presence of two quasi-degenerate low-lying states in open systems. In that respect, it has been suggested to look for situations where the low-lying states cross as a function of an external parameter, for instance the chemical potential, to prove that there are indeed two low-lying states [@sarma].
In a recent experiment with chains of Cobalt atoms evaporated onto a Cu${}_2$N/Cu(100) substrate [@toskovic], the presence of level crossings as a function of the external magnetic field has been revealed by scanning tunneling microscopy, which exhibits a specific signature whenever the ground state is degenerate. The relevant effective model for that system is a spin-1/2 XY model in an in-plane magnetic field. The exact diagonalisation of finite XY chains has indeed revealed the presence of quasi-degeneracy between the two lowest energy states, that are well separated from the rest of the spectrum, and a series of level crossings between them as a function of the magnetic field [@dmitriev]. Furthermore, the position of these level crossings is in good agreement with the experimental data. It has been proposed that these level crossings are analogous to those predicted in topological fermionic spin chains, and that they can be interpreted as a consequence of the Majorana edge modes [@mila].
The topological phase of the XY model in an in-plane magnetic field is adiabatically connected to that of the transverse field Ising model, in which the longitudinal spin-spin coupling (along the field) is switched off. However, in the transverse field Ising model, the two low-lying states never cross as a function of the field, as can be seen from the magnetisation curve calculated by Pfeuty a long time ago [@pfeuty], and which does not show any anomaly. The very different behaviour of the XY model in an in-plane field in that respect calls for an explanation. The goal of the present paper is to provide such an explanation, and to show that the presence of $N$ level crossings, on a chain of $N$ sites, is generic as soon as an antiferromagnetic longitudinal coupling is switched on. To achieve this goal, we have studied a Hamiltonian which interpolates between the exactly solvable transverse field Ising (TFI) and the longitudinal field Ising (LFI) chains. The approach that best accounts for these level crossings turns out to be an approximate mapping onto the exactly solvable Kitaev chain, which contains all the relevant physics. In the Majorana representation, the level crossings are due to the interaction between Majorana fermions localised at each end of the chain.
The paper is organized as follows. In section \[sec:model\], we present the model and give some exact diagonalisation results on small chains to get an intuition of the qualitative behaviour of the spectrum. We show in section \[sec:PT\] that perturbation theory works in principle but is rather limited because of the difficulty to go to high order. We then turn to an approximate mapping onto the open Kitaev chain via a mean-field decoupling in section \[sec:MF\]. The main result of this paper is presented in section \[sec:Maj\], namely the explanation of the level crossings in a Majorana representation. Finally, we conclude with a a quick discussion of some possible experimental realisations in section \[sec:exp\].
Model {#sec:model}
=====
We consider the transverse field spin-1/2 Ising model with an additional antiferromagnetic longitudinal spin-spin coupling along the field, i.e. the Hamiltonian $$\label{eqHspin}
H=J_x \sum_{i=1}^{N-1}S_i^x S^x_{i+1} + J_z \sum_{i=1}^{N-1}S^z_{i} S^z_{i+1}
- h \sum_{i=1}^N S^z_{i}$$ with $J_z \geq 0$ [^1]. This model can be seen as an interpolation between the TFI model ($J_z=0$) and the LFI model ($J_x=0$). The case $J_z=J_x$ corresponds to the effective model describing the experiment in Ref. , up to small irrelevant terms [^2]. Since we will be mostly interested in the parameter range $0\leq J_z\leq J_x$, we will measure energies in units of $J_x$ by setting $J_x=1$ henceforth. The spectrum of the Hamiltonian in Eq. is invariant under $h\to -h$ since the Hamiltonian is invariant if we simultaneously rotate the spins around the $x$-axis so that $S^z_i \to -S^z_i~\forall i$. Hence, we will in most cases quote the results only for $h \geq 0$.
The TFI limit of $H$ can be solved exactly by Jordan-Wigner mapping onto a chain of spinless fermions [@pfeuty]. In the thermodynamic limit, it is gapped with a twofold degenerate ground state for $h < h_c=1/2$, and undergoes a quantum phase transition at $h=h_c$ to a non-degenerate gapped ground state for $h>h_c$. The twofold degeneracy when $h<h_c$ can be described by two zero-energy Majorana edge modes [@kitaev]. As a small positive $J_z$ is turned on, there is no qualitative change in the thermodynamic limit, except that $h_c$ increases with $J_z$. Indeed, the model is then equivalent to the ANNNI model in a transverse field which has been extensively studied before, see for example [@chakrabarti; @jalal]. A second order perturbation calculation in $1/h$ yields $h_c =1/2+(3/4)J_z+O(J_z^2) $ for small $J_z$ and $h_c=1/2+J_z +O(1/J_z)$ for large $J_z$ [@rujan; @hassler]. Since, for $J_z \gtrsim 1$, there are other phases arising [@hassler], we shall mostly consider $J_z \lesssim 1$ in the following in order to stay in the phase with a degenerate ground state.
For a finite size chain, the twofold degeneracy of the TFI model at $0<h<h_c$ is lifted and there is a small non-vanishing energy splitting $\epsilon = E_1-E_0$ between the two lowest energy states, where the $E_k$ are the eigenenergies and $E_k \leq E_{k+1} ~ \forall k$. This splitting is exponentially suppressed with the system length, $\epsilon \sim \exp(-N/\xi)$ [@kitaev]. These two quasi-degenerate states form a low energy sector separated from the higher energy states. The spectrum for $J_z=0$ and $N=3$ is shown in Fig. \[fig1\]a. For $J_z > 0$, the splitting $\epsilon$ has an oscillatory behaviour and vanishes for some values of $h$. For $N=3$, it vanishes once for $h>0$. See the spectrum for $J_z=0.5$ and $J_z=1$ in Figs \[fig1\]b-c. As $J_z$ becomes large, there is no low energy sector separated from higher energy states any more. In the LFI limit, $J_z\to\infty$, the eigenstates have a well defined magnetisation in the $z$-direction and the energies are linear as a function of $h$, see Fig. \[fig1\]d. In this limit, the level crossings are obvious. As the field is increased, the more polarised states become favoured, which leads to level crossings.
[ ![Exact diagonalisation spectrum as a function of $h$ for $N=3$ with $J_z=0$ (a), $J_z=0.5$ (b), $J_z=1$ (c) and $J_z=10$ (d). []{data-label="fig1"}](fig1 "fig:") ]{}
[ ![Exact diagonalisation spectrum relative to the ground state energy, $E_k-E_0$, (a), two lowest energies $E_0$ and $E_1$ (c) and magnetisation $M=-\dd E_0/\dd h$ (e) as a function of $h$ for $N=6$ with $J_z=0$ (TFI limit). The plots in (b), (d) and (f) show the same for $J_z=0.75$. []{data-label="fig2"}](fig2 "fig:") ]{}
The plots in Fig. \[fig1\] are instructive for very small $N$ but become messy for larger chains. In Figs \[fig2\]a-b, we show the spectrum relative to the ground state energy, i.e. $E_k-E_0$, of a chain of $N=6$ sites for $J_z=0$ and $J_z=0.75$. The energies $E_0$ and $E_1$ are plotted in Figs \[fig2\]c-d for the same parameters. The structure of the spectrum is similar to the $N=3$ case, except that now $\epsilon$ vanishes at three points for $h>0$. In general, there are $N$ points of exact degeneracy where the splitting $\epsilon$ vanishes since the spectrum is symmetric under $h\to -h$. This is shown in Fig. \[fig3\] for $2\leq N\leq 8$. For $N$ even, there are $N/2$ level crossings for $h>0$, and for $N$ odd, there are $(N-1)/2$ level crossings for $h>0$ and one at $h=0$.
As shown in Figs \[fig2\]e-f, the level crossings lead to jumps in the magnetisation $M(h)=-\dd E_0/\dd h$. The number of magnetisation jumps turns out to be independent of $J_z$ for $0<J_z<\infty$, as illustrated in Fig. \[fig4\]. In the LFI limit, most of the jumps merge together at $h=J_z$, with an additional jump persisting for even $N$ at $h=J_z/2$ [^3]. In this large $J_z$ region, however, there is no quasi-degeneracy and the magnetisation jumps indicate level crossings but no oscillation in contrast to the small $J_z$ region. Since there are no level crossings in the TFI limit, one might expect the number of crossings to decrease as $J_z$ decreases. However, the exact diagonalisation results do not support this scenario, and hint to all level crossings appearing at the same time as soon as $J_z\neq 0$. This is a remarkable feature that we shall explain in the following.
A useful equivalent representation of the Hamiltonian in Eq. in terms of spinless fermions is obtained by applying the Jordan-Wigner transformation used to solve exactly the TFI model [@pfeuty], $$\label{eqJW}
\begin{cases}
S_i^x = \frac{1}{2}(c_i\dag + c_i)\exp\left({{\mathrm{i}}}\pi\sum_{j<i}c_j\dag c_j\right)\\
S_i^y = \frac{1}{2{{\mathrm{i}}}}(c_i\dag - c_i)\exp\left({{\mathrm{i}}}\pi\sum_{j<i}c_j\dag c_j\right)\\
S_i^z=c_i\dag c_i - \frac{1}{2},
\end{cases}$$ which yields $$\begin{split} \label{eqHfermions}
H &= \frac{1}{4} \sum_{i=1}^{N-1}(c_i\dag-c_i)(c_{i+1}\dag+c_{i+1})
- h \sum_{i=1}^N (c_i\dag c_i - \frac{1}{2})\\&\quad
+ J_z \sum_{i=1}^{N-1}(c_i\dag c_i
- \frac{1}{2})(c_{i+1}\dag c_{i+1} - \frac{1}{2})
\end{split}$$ where the $c_i,c_i\dag$ are fermionic annihilation and creation operators. This is the Hamiltonian of a spinless p-wave superconductor with nearest-neighbour density-density interaction. As for the simpler TFI model, the Hamiltonian is symmetric under a $\pi$-rotation of the spins around the $z$-axis, $S^x_{i}\to -S^x_{i}$ and $S^y_{i}\to -S^y_{i}$ in the spin language. This leads to two parity sectors given by the parity operator $$P=e^{{{\mathrm{i}}}\pi \sum_{j=1}^N c_j\dag c_j} = (-2)^N S_1^z\cdots S_N^z.$$ In other words, the Hamiltonian does not mix states with even and odd number of up spins, or equivalently with even and odd number of fermions. The ground state parity changes at each point of exact degeneracy, and thus alternates as a function of the magnetic field for $J_z > 0$. This can be understood qualitatively by looking at Fig. \[fig2\]f. The magnetisation plateaus are roughly at $M=0,1,2,3$. Hence to jump from one plateau to the next, one spin has to flip, thus changing the sign of the parity $P$.
[ ![Exact diagonalisation energy splitting $\epsilon=E_1-E_0$ between the two lowest energy states as a function of $h$ for several $N$ and $J_z=0.75$. There are $N$ level crossings since the spectrum is symmetric under $h\to -h$. []{data-label="fig3"}](fig3 "fig:") ]{}
[ ![Exact diagonalisation magnetisation $M$ as a function of $h$ and $J_z$ for $N=6$. The red dashed lines indicate the discontinuities. []{data-label="fig4"}](fig4 "fig:") ]{}
Perturbation theory {#sec:PT}
===================
As a first attempt to understand if the $N$ level crossings develop immediately upon switching on $J_z$, we treat the $V=J_z \sum_{i=1}^{N-1}S^z_{i} S^z_{i+1}$ term as a perturbation to the exactly solvable transverse field Ising model. One may naively expect that degenerate perturbation theory is required since the TFI chain has a quasi-twofold degeneracy at low field. Fortunately, the two low-energy states live in different parity sectors [@pfeuty] that are not mixed by the perturbation $V$. We can therefore apply the simple Rayleigh-Schrödinger perturbation theory in the range of parameters we are interested in, i.e. $J_z \lesssim 1$.
[ ![Energy splitting at zeroth (a), first (b), second (c) and third (d) order (blue solid lines) compared to the exact diagonalisation result (black dashed lines) for $N=6$ and $J_z=0.25$. []{data-label="fig5"}](fig5 "fig:") ]{}
[ ![Energy splitting at third order in perturbation theory for several $N$ and $J_z=0.25$. Since the spectrum is symmetric under $h\to -h$, there are $N$ level crossings for $N\leq 7$ and $7$ level crossings for $N \geq 7$. []{data-label="fig6"}](fig6 "fig:") ]{}
Writing $A_i=c_i\dag + c_i$ and $B_i=c_i\dag - c_i$, the perturbation can be rewritten as $V=({J_z}/{4}) \sum_{i=1}^{N-1}B_iA_iB_{i+1}A_{i+1}$. The unperturbed eigenstates are $\ket{m} = \Upsilon\dag_m\ket{0}$ where $\ket{0}$ is the ground state and the $\Upsilon\dag_m$ are a product of the creation operators corresponding to the Bogoliubov fermions. The matrix elements are then $$\begin{split}
\braket{n|V|m}=\frac{J_z}{4}\sum_{i=1}^{N-1}
\braket{0|\Upsilon_n B_iA_iB_{i+1}A_{i+1}\Upsilon\dag_m|0}
\end{split}$$ which can be computed by applying Wick’s theorem, similarly to how correlation functions are found in [@lieb]. We computed the effect of $V$ up to third order, with the basis of virtual states slightly truncated, namely by keeping states with at most three Bogoliubov fermions. Since the more fermions there are in a state, the larger its energy, we expect this approximation to be excellent.
As shown in Fig. \[fig5\], the number of crossings increases with the order of perturbation, and to third order in perturbation, the results for $N=6$ sites are in qualitative agreement with exact diagonalisations. From the way level crossings appear upon increasing the order of perturbation theory, one can expect to induce up to $2m+1$ level crossings if perturbation theory is pushed to order $m$, see Fig. \[fig6\]. So these results suggest that the appearance of level crossings is a perturbative effect, and that, for a given size $N$, pushing perturbation theory to high enough order will indeed lead to $N$ level crossings for small $J_z$. However, in practice, it is impossible to push perturbation theory to very high order. Indeed, the results at order 3 are already very demanding. So, these pertubative results are encouraging, but they call for an alternative approach to actually prove that the number of level crossings is indeed equal to $N$, and that these level crossings appear as soon as $J_z$ is switched on.
Fermionic mean-field approximation {#sec:MF}
==================================
[ ![Self-consistent mean-field parameters $\mu$ (blue solid lines and crosses), $t$ (red dotted lines and dots) and $\Delta$ (yellow dashed lines and squares) as a function of $h$ for $J_z=0.75$ and $N=6$ (a), as a function of $J_z$ for $h=0.4$ and $N=6$ (b) and as a function of $N$ for $h=0.4$ and $J_z=0.75$ (c). []{data-label="fig7"}](fig7 "fig:") ]{}
In the fermionic representation, Eq. $\eqref{eqHfermions}$, there is a quartic term that cannot be treated exactly. Here, we approximate it by mean-field decoupling. In such an approximation, one assumes the system can be well approximated by a non-interacting system (quadratic in fermions) with self-consistently determined parameters. For generality, we decouple the quartic term in all three mean-field channels consistent with Wick’s theorem, $$\begin{split}
\label{eqMF}
c\dag_{i}&c_{i}c\dag_{i+1}c_{i+1} \approx \\
&\braket{c\dag_{i}c_{i}} c\dag_{i+1}c_{i+1} + \braket{c\dag_{i+1}c_{i+1}}
c\dag_{i}c_{i} - \braket{c\dag_{i}c_{i}} \braket{c\dag_{i+1}c_{i+1}} \\
- &\braket{c\dag_{i}c\dag_{i+1}} c_{i}c_{i+1} - \braket{c_{i}c_{i+1}}
c\dag_{i}c\dag_{i+1} +\braket{c\dag_{i}c\dag_{i+1}} \braket{c_{i}c_{i+1}} \\
+&\braket{c\dag_{i}c_{i+1}} c_{i}c\dag_{i+1} +\braket{c_{i}c\dag_{i+1}}
c\dag_{i}c_{i+1} - \braket{c\dag_{i}c_{i+1}} \braket{c_{i}c\dag_{i+1}}.
\end{split}$$ Here, $\braket{.}$ denotes the ground state expectation value. The $3N-2$ self-consistent parameters $\braket{c\dag_{i}c_{i}}$, $\braket{c\dag_{i}c\dag_{i+1}}$ and $\braket{c\dag_{i}c_{i+1}}$ can be found straightforwardly by iteratively solving the quadratic mean-field Hamiltonian.
As it turns out, it is more instructive to consider only three self-consistent parameters. To do so, we solve the mean-field approximation of the translationally invariant Hamiltonian ($c_{N+1}=c_1$), $$\begin{split}
H' &= \sum_{i=1}^{N} \left\lbrace \frac{1}{4}
(c_i\dag-c_i)(c_{i+1}\dag+c_{i+1}) - h(c_i\dag c_i - \frac{1}{2}) \right.
\\ &\left. + J_z (c_i\dag c_i - \frac{1}{2})(c_{i+1}\dag c_{i+1} -
\frac{1}{2}) \right\rbrace \\&\approx \sum_{i=1}^{N} \left\lbrace
-\mu c_i\dag c_i +( t c_{i+1}\dag c_i + {\rm h.c.} )
- (\Delta c_{i+1}\dag c\dag_i + {\rm h.c.} ) \right\rbrace
\\& \quad + {\rm const},
\end{split}$$ where $\mu=h+J_z(1-2\braket{c\dag_{i}c_{i}})$, $t=1/4 - J_z\braket{c\dag_{i}c_{i+1}}$ and $\Delta=1/4-J_z\braket{c_{i}c_{i+1}}$ are determined self-consistently. These parameters are found to be real, and are shown in Fig. \[fig7\] as a function $h$, $J_z$ and $N$.
Using these self-consistent parameters, the Hamiltonian in Eq. is then approximated by the following mean-field problem on an open chain: $$\begin{split} \label{eqHkit}
H_{\rm MF} &= -\sum_{i=1}^N \mu \left(c\dag_{i}c_{i}-\frac{1}{2}\right)
\\&\quad + \sum_{i=1}^{N-1} \left[\left(t c\dag_{i+1}c_{i} + {\rm h.c.}\right)
- \left( \Delta c\dag_{i+1}c\dag_{i} + {\rm h.c. }\right) \right],
\end{split}$$ up to an irrelevant additive constant [^4].
[ ![Critical fields, $h_{\rm crit}$, where the degeneracy is exact, as a function of $J_z$ in the self-consistent mean-field approximation (blue crosses) compared to the exact diagonalisation result (red squares) for (a) $N=6$ and (b) $N=7$. []{data-label="fig8"}](fig8 "fig:") ]{}
[ ![Energy splitting $\epsilon=E_1-E_0$ as a function of $h$ in the self-consistent mean-field approximation (blue solid line) compared to the exact diagonalisation result (red dashed line) for $J_z=0.5$ and (a) $N=6$ and (b) $N=7$. []{data-label="fig9"}](fig9 "fig:") ]{}
Since the self-consistent parameters are almost independent of the system size (see Fig. \[fig7\]c), the boundaries are not very important and the bulk contribution is determinant. This partly justifies the approximation of playing with the boundary conditions to get the approximate model with just three self-consistent parameters. This approximation is also justified by the great quantitative agreement with the exact diagonalisation results for the critical fields for $J_z \lesssim 0.8$ (see Fig. \[fig8\]), and to a lesser extent for the energy splitting $\epsilon=E_1-E_0$ between the two lowest energy states, see Fig. \[fig9\].
For $N$ odd, the degeneracy at $h=0$ is protected by symmetry for any $J_z$ in the Hamiltonian . Indeed, under the transformation $S_i^z \to -S_i^z~\forall i$, the parity operator transforms as $P\to (-1)^N P$. Hence, for $N$ odd and $h=0$, the ground state has to be twofold degenerate. As can be seen in Fig. \[fig8\]b, the critical field $h=0$ at low $J_z$ evolves to a non-zero value for large $J_z$, thus showing that this symmetry is broken by the mean-field approximation . The discrepancy is, however, small for $J_z \lesssim 0.8$ as can also be seen in Fig. \[fig9\]b.
We observe from Fig. \[fig7\]a that as a function of magnetic field, the parameters $t$ and $\Delta$ are almost constant, whereas $\mu$ is almost proportional to $h$. Thus, we can understand the physics of the level oscillations by forgetting about the self-consistency and considering $\mu$, $t$ and $\Delta$ as free parameters, i.e. by studying the open Kitaev chain [@kitaev], where the level crossings happen as $\mu$ is tuned. Compared to the TFI model for which $\Delta=t$, the main effect of $J_z>0$ is to make $0 < \Delta < t$, which, as we shall see in the next section, is the condition to see level oscillations.
Such a mapping between the two lowest lying energy states of the interacting Kitaev chain and of the non-interacting Kitaev chain can be made rigorous for a special value of $h>0$, provided the boundary terms in equation are slighty modified [@katsura]. But this particular exact case misses out on level-crossing oscillations.
Level oscillations and Majorana fermions {#sec:Maj}
========================================
We define $2N$ Majorana operators $\gamma'_i,~\gamma_i''$ as: $$\label{eqMaj}
\begin{cases}
\gamma'_{i} = c_{i}+ c\dag_{i}\\
\gamma''_{i} = -{{\mathrm{i}}}(c_{i} - c\dag_{i})
\end{cases}$$ which satisfy ${\gamma'}_i\dag=\gamma'_i$, ${\gamma''}_i\dag=\gamma''_i$, $\lbrace \gamma'_i,\gamma''_j\rbrace=0$ and $\lbrace \gamma'_i,\gamma'_j\rbrace =
\lbrace \gamma''_i,\gamma''_j\rbrace=2\delta_{ij}$. Since the $\mu,~t,~\Delta$ are real, the $H_{MF}$ of Eq. reads $$\begin{split}
H_{\rm MF}=& \frac{{{\mathrm{i}}}}{2}\sum_{i=1}^{N-1}
\left[ -(t+\Delta)\gamma''_{i}\gamma'_{i+1}
+ (t-\Delta )\gamma'_{i}\gamma''_{i+1} \right] \\
& -\frac{{{\mathrm{i}}}\mu}{2} \sum_{i=1}^N \gamma'_{i}\gamma''_{i}=
\frac{{{\mathrm{i}}}}{2} \sum_{i,j=1}^{N} \gamma'_iM_{ij}\gamma''_j.
\end{split}$$ From the singular value decomposition of $M$, we write $M=U\Sigma V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma={\rm diag}(\epsilon_1,\ldots , \epsilon_N)$ with real $\epsilon_i$ and $|\epsilon_i|\leq |\epsilon_{i+1}|~\forall i$. Thus, the Hamiltonian reads $$\begin{split}
H_{\rm MF} &= \frac{{{\mathrm{i}}}}{2} \sum_{i,j,k=1}^{N}
\gamma'_i U_{ik}\epsilon_k V^T_{kj}\gamma''_j
=\frac{{{\mathrm{i}}}}{2} \sum_{k=1}^{N} \epsilon_k \tilde \gamma'_k \tilde \gamma''_k \\
&=\sum_k \epsilon_k (\eta\dag_{k}\eta_{k}-\frac{1}{2})
\end{split}$$ where $$\tilde \gamma'_k = \sum_{i=1}^N \gamma'_i U_{ik}, \qquad \tilde
\gamma''_k = \sum_{i=1}^N \gamma''_i V_{ik}$$ are the rotated Majorana operators, and the $\eta_k = \frac{1}{2} (\tilde \gamma'_{k}+{{\mathrm{i}}}\tilde \gamma''_{k}) $ are fermionic annihilation operators corresponding to the Bogoliubov quasiparticles.
As derived in Appendix, in general the Majorana operators, $\tilde \gamma'_k$ and $\tilde\gamma''_k$, are of the form $$\label{eqtildegammageneral}
\begin{split}
\tilde \gamma'_k = \sum_j (a_+ x_+^j + b_+ x_+^{N+1-j}
+ a_- x_-^j + b_- x_-^{N+1-j} ) \gamma'_{j}\\
\tilde \gamma''_k = \sum_j ( a_+ x_+^{N+1-j} + b_+ x_+^j
+a_- x_-^{N+1-j} + b_- x_-^j) \gamma''_j{}
\end{split}$$ where the $x_\pm$, $a_\pm$ and $b_\pm$ are functions of the energy $\epsilon_k$ which is quantised in order to satisfy the boundary conditions. On can easily solve numerically the nonlinear equation for the $\epsilon_k$. Here, we will instead focus on a simple analytical approximation for $\tilde \gamma'_1,~\tilde\gamma''_1$ and $\epsilon_1$ which works well to discuss the level crossings, and is equivalent to the Ansatz given in [@kitaev].
From Eqs. and , we see that for $\epsilon=0$, we have either $a_\pm=0$ or $b_\pm=0$. Without loss of generality, we can choose $b_\pm(\epsilon=0) = 0$. Since we expect $\epsilon_1 \ll 1$, we approximate $$\label{eqapp1}
b_\pm (\epsilon_1) \approx b_\pm (0)=0$$ and $$\label{eqapp2}
x_\pm (\epsilon_1) \approx x_\pm(0) =
\frac{\mu \pm \sqrt{\mu^2-4t^2+4\Delta^2}}{2(t+\Delta)},$$ which yields $$\label{eqapp3}
\begin{split}
\tilde \gamma'_1 &\approx \sum_j (a_+ x_+^j + a_- x_-^j ) \gamma'_{j}\\
\tilde \gamma''_1 &\approx\sum_j ( a_+ x_+^{N+1-j} +a_- x_-^{N+1-j} )\gamma''_j
\end{split}$$ with $\sum_j (a_+ x_+^j + a_- x_-^j )^2=1$. The boundary conditions now read
\[eqbc12\] $$\begin{aligned}
a_+ + a_- &=& 0 \label{eqbc1} \\
a_+ x_+^{N+1} + a_- x_-^{N+1} &=& 0 \label{eqbc2}\end{aligned}$$
and in general cannot be both satisfied unless $\epsilon_1=0$ exactly.
If $|x_\pm|<1$, $\tilde\gamma'_1$ is localised on the left side of the chain with its amplitude $\sim e^{-j/\xi}$ as $j\gg 1$ with $\xi=-1/\ln (\max(|x_+|,|x_-|))$. Furthermore, $\tilde\gamma''_1$ is related to $\tilde\gamma'_1$ by the reflection symmetry $j\to N+1-j$. Thus, in the thermodynamic limit, the boundary condition is irrelevant and $\epsilon_1 \to 0$ as $N\to \infty$. Similarly, if $|x_\pm| >1$ the boundary condition becomes irrelevant in the thermodynamic limit. However, if $|x_+|>1$ and $|x_-|<1$, or $|x_+|<1$ and $|x_-|>1$, then $\tilde\gamma'_1$, $\tilde\gamma''_1$ have significant weight on both sides of the chain and both boundary conditions and remain important in the thermodynamic limit. Hence, the approximation $\epsilon_1 \approx 0$ is bad, indicating a gapped system. As discussed in [@kitaev], for $|\mu|<2|t|$ we have either $|x_\pm|<1$ or $|x_\pm|>1$ which yields $\epsilon_1=0$ in the thermodynamic limit. This is the topological phase with a twofold degenerate ground state. For a finite system, however, the boundary conditions and are in general not exactly satisfied and the system is only quasi-degenerate with a gap $\epsilon \sim e^{-N/\xi}$. For $|\mu|>2|t|$, either $|x_+|>1$ and $|x_-|<1$, or $|x_+|<1$ and $|x_-|>1$, and the system is gapped.
In the topological phase, $|\mu|<2|t|$, there are parameters for which the boundary conditions can be exactly satisfied even for $N<\infty$ and thus $\epsilon_1=0$ exactly. In such a case, there is an exact zero mode even for a finite chain. This was previously discussed in Ref. [@kao], as well as in [@hedge] where a more general method that applies to disordered systems is described. If $x_\pm \in \mathbb{R}$, it is never possible to satisfy the boundary conditions and therefore the quasi-gap is always finite, $\epsilon_1 \neq 0$. However, if $x_+=re^{{{\mathrm{i}}}\phi} \not \in \mathbb{R}$, Eq. yields $x_-=x_+^*$ and $(x_+^{N+1} - x_-^{N+1}) \propto r^{N+1}\sin[(N+1)\phi]$. Thus it may happen for specific parameters that $\epsilon_1=0$ exactly. This degeneracy indicates a level crossing. The phase $\phi$, defined for $|\mu|<\mu_c=2\sqrt{t^2-\Delta^2}$, is given by $$\tan{\phi} = \sqrt{(\mu_c/\mu)^2-1}.$$ It thus goes continuously from $\phi(\mu=0^+)=\pi/2$ to $\phi(\mu\to\mu_c) \to 0$. Hence, there are critical chemical potentials, $0 \leq \mu_{\lceil N/2\rceil} < \ldots < \mu_m <\ldots<\mu_1<\mu_c$, such that $\phi(\mu=\mu_m)=\frac{\pi m}{N+1}$ (see Fig. \[fig10\]a). For these critical $\mu_m$, the system is exactly degenerate, i.e. $\epsilon_1=0$. In the TFI limit, we have $\Delta=t$ and $\mu_c=0$, thus there are no level crossings.
[ ![(a) Phase $\phi(\mu)$ of $x_+=re^{{{\mathrm{i}}}\phi}$ within the approximation for several $\Delta$ with $t=1$ and $N=6$. The horizontal black dotted lines indicate the values $\phi=\frac{\pi m}{N+1}$. (b) Splitting $E_1-E_0=|\epsilon_1|$ in the Kitaev chain calculated exactly solving numerically the full self-consistent equations described in Appendix (blue solid line) and with the analytical approximate result in Eq. (red dashed line) for $N=6$, $t=1$ and $\Delta=0.3$. []{data-label="fig10"}](fig10 "fig:") ]{}
For $|\mu|<2|t|$, writing $x_+=re^{{{\mathrm{i}}}\phi}$ with $r>0$, we have $$\label{eqapprox3}
\begin{split}
\epsilon_1 &=\Sigma_{11}=(U^T M V)_{11} \\
&\approx 4(t+\Delta) a_+^2 r^{N+2} \sin(\phi) \sin[(N+1)\phi],
\end{split}$$ where we used the approximations , and the boundary condition \[respectively \] when $t\Delta>0$ (respectively $t\Delta<0$), since in this case $|x_\pm|<1$ (respectively $|x_\pm|>1$). Note that $\phi(-\mu) = \phi(\mu)-\pi$, and thus $\epsilon_1$ is an odd function of $\mu$ for odd $N$ and an even function of $\mu$ for even $N$. Since $\epsilon_1$ changes sign whenever $\sin((N+1)\phi)=0$, the degeneracy points indicate level crossings. This approximate description works extremely well, as shown in Fig. \[fig10\]b for $\Delta=0.3t$. Because $\phi$ takes all the values in $]0,\pi/2]$ for $0 < \mu <\mu_c$, and in $]-\pi,-\pi/2]$ for $-\mu_c < \mu < 0$, there are either exactly $N$ level crossings as a function of $\mu$ if $0<\mu_c\in \mathbb{R}$, i.e. if $|\Delta|<|t|$, and no zero level crossing otherwise. At the points of exact degeneracy, $b_\pm(\epsilon=0)=0$, the zero-mode Majorana fermions are localised on opposite sides of the chain. When the degeneracy is not exact, however, $b_\pm (\epsilon\neq 0) \neq 0$ and the zero-mode Majorana fermions mix together to form Majoranas localised mostly on one side but also a little bit on the opposite side.
In the XY model in an out-of-plane magnetic field, which is equivalent to the non-interacting Kitaev chain [@lieb], these level crossings lead to an oscillatory behaviour of the spin correlation functions [@barouch]. In the context of p-wave superconductors, the level oscillations described above also arise in more realistic models and are considered one of the hallmarks of the presence of topological Majorana fermions [@sarma; @loss]. Although it is still debated whether Majorana fermions have already been observed, strong experimental evidence for the level oscillations was reported in [@markus].
Coming back to the mean-field Hamiltonian of Eq. , we can get the phase $\phi$ within the approximation , i.e. the phase of $x_+(\epsilon=0)$, as a function of the physical parameters $h,~J_z$ since we know how the self-consistent parameters $\mu,~t,~\Delta$ depend on them. We plot in Fig. \[fig11\] the phase $\phi$ as a function of $h$ for several $J_z$ which yields a good qualitative understanding of the sudden appearance of $N$ level crossings as soon as $J_z > 0$. As previously discussed, the self-consistent parameters are almost independent of $N$ and therefore the curves $\phi(h)$ are almost independent of $N$ as well. The main effect of $N$ is to change the condition $\phi(\mu=\mu_m)=\frac{\pi m}{N+1}$ for the boundary condition in Eq. to be satisfied and thus for the system to be exactly degenerate.
[ ![Phase $\phi(h)$ of $x_+(\epsilon=0)$ based on the self-consistent parameters $\mu,~t,~\Delta$ of the mean-field decoupling for several $J_z$ and (a) $N=6$, (b) $N=9$, (c) $N=12$. The horizontal black dotted lines indicate the values $\phi=\frac{\pi m}{N+1}$. []{data-label="fig11"}](fig11 "fig:") ]{}
Summary {#sec:exp}
=======
The main result of this paper is that the level crossings between the two lowest energy eigenstates of the XY chain in an in-plane magnetic field are more generally a fundamental feature of the transverse field Ising chain with an antiferromagnetic longitudinal coupling howsoever small. These points of level crossings (twofold degeneracy) correspond to having Majorana edge modes in a Kitaev chain onto which the problem can be approximately mapped. The level crossings of the XY chains have been observed experimentally in [@toskovic] by scanning tunneling microscopy on Cobalt atoms evaporated onto a Cu${}_2$N/Cu(100) substrate. By varying the adsorbed atoms and the substrate, it should be possible to vary the easy-plane and easy-axis anisotropies, and thus to explore the exact degeneracy points for various values of the longitudinal coupling. The possibility to probe the two-fold degeneracy of this family of spin chains is important in view of their potential use for universal quantum computation [@loss2]. Besides, one could also realise the spinless fermionic Hamiltonian in an array of Josephson junctions as described in [@hassler]. The advantage of this realisation is that it allows a great flexibility to tune all the parameters of the model. We hope that the results of the present paper will stimulate experimental investigations along these lines.
We acknowledge Somenath Jalal for useful discussions and the Swiss National Science Foundation for financial support. B.K. acknowledges the financial support under UPE-II and DST-PURSE programs of JNU.
Majorana solutions of the Kitaev chain {#secMajWF}
======================================
To solve the Kitaev chain , we need to find the singular value decomposition of $$M=\begin{pmatrix}
-\mu & \tau_- & 0 & \cdots\\
\tau_+ & -\mu & \tau_- & 0 &\cdots \\
0 &\tau_+ & -\mu & \tau_- & 0 & \cdots \\
& & \ddots & \ddots & \ddots & \\
&\cdots & 0 & \tau_+ & -\mu & \tau_- \\
& & \cdots & 0 & \tau_+ & -\mu
\end{pmatrix}$$ with $\tau_\pm = t\pm\Delta$, i.e. find orthogonal matrices $U$, $V$ and a real diagonal matrix $\Sigma$ such that $M=U\Sigma V^T$. Writing $\vec u_k$ and $\vec v_k$ the $k^{\rm th}$ columns of $U$ and $V$ respectively, they satisfy $$\label{eqsvd}
\begin{cases}
M\vec v_k &= \epsilon_k \vec u_k \\
\vec u_k^T M &= \epsilon_k \vec v_k^T.
\end{cases}$$
Let’s find two unit-norm column vectors $\vec u,~\vec v$ and $\epsilon$ such that $M\vec v=\epsilon \vec u$ and $\vec u^TM=\epsilon \vec v^T$. First we forget about the normalisation and boundary conditions and focus on the secular equation. Setting the components of $\vec u,~\vec v$ as $u_j=a x^j$ and $v_j = b x^j$, we have $$\begin{split}
M \vec v &= \frac{b}{a}\frac{\tau_+ -\mu x +\tau_- x^2}{x}
\vec u+ {\rm b.t.} \\
\vec u^T M &=\frac{a}{b}\frac{\tau_- -\mu x +\tau_+ x^2}{x}
\vec v^T + {\rm b.t.}
\end{split}$$ where b.t. stands for boundary terms. Hence, $u$ and $v$ satisfy the secular equation provided $$\label{eqratio}
\frac{b}{a}=\sqrt{\frac{\tau_- -\mu x +\tau_+ x^2}
{\tau_+ -\mu x +\tau_- x^2}}$$ and $$\label{eqEps}
\epsilon = \frac{1}{x}\sqrt{(\tau_- -\mu x +\tau_+ x^2)
(\tau_+ -\mu x +\tau_- x^2)}.$$ Because of the reflection symmetry $j \to N+1-j$, if $x$ is a solution of equation for some $\epsilon$, then $1/x$ is also a solution. Assuming $\epsilon$ known, the solutions are $x_\pm$, $1/x_\pm$ and satisfy $$\begin{split}
0 &= \epsilon^2 x^2 - (\tau_- -\mu x +\tau_+ x^2)
(\tau_+ -\mu x +\tau_- x^2) \\
&\propto (x-x_+)(x-1/x_+)(x-x_-)(x-1/x_-)
\end{split}$$ which by identification yields, writing $\rho_\pm = x_\pm + 1/x_\pm$, $$\label{eqxpm}
\begin{split}
x_\pm &= \frac{1}{2}\left(\rho_\pm +\sqrt{\rho_\pm^2 - 4}\right),\\
\rho_\pm &= \frac{\mu t \pm \sqrt{(t^2-\Delta^2)\epsilon^2 +
\Delta^2(\mu^2-4t^2+4\Delta^2)}}{t^2-\Delta^2}.
\end{split}$$ Taking into account the reflection symmetry, the general form of the components of $\vec u,~\vec v$ is thus $$\label{equv}
\begin{split}
u_j = a_+ x_+^j + b_+ x_+^{N+1-j} + a_- x_-^j + b_- x_-^{N+1-j} \\
v_j = a_+ x_+^{N+1-j} + b_+ x_+^j +a_- x_-^{N+1-j} + b_- x_-^j
\end{split}$$ with the ratios $b_+/a_+$ and $b_-/a_-$ given by equation with $x=x_+$ and $x=x_-$ respectively.
Furthermore, we have the boundary conditions $$\label{eqbc}
\begin{split}
a_+ + b_+x_+^{N+1} + a_- + b_-x_-^{N+1}&= 0 \\
a_+ x_+^{N+1} + b_+ + a_-x_-^{N+1} + b_-&= 0
\end{split}$$ which set the ratio $a_-/a_+$ and give the quantisation condition on the energies $\epsilon_k$. The last degree of freedom, say $a_+$, is then set by normalising $\vec u$ (from equation , $\| \vec u\| = \| \vec v \|$).
Note that for the special cases $t= \Delta$ and $\mu=0$, we have $\tilde \gamma'_1 = \gamma'_1$ and $\tilde \gamma''_1=\gamma''_N$ with $\epsilon_1=0$. We have a similar result for $t=-\Delta$ and $\mu=0$. For these two cases, the general formalism described above does not apply since it yields $x_\pm=0,\pm\infty$.
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[^1]: This Hamiltonian is equivalent to an XY model in an in-plane magnetic field, but we chose to rotate the spins around the $x$-axis so that we recover the usual formulations of the TFI and LFI models as special cases.
[^2]: In Ref. it is explained that the $\pm 3/2$ doublet of the spin 3/2 Cobalt adatoms can be projected out by a Schrieffer-Wolff transformation due to the strong magnetic anisotropy. The resulting effective spin 1/2 model is the one of equation with $J_x=J_z$ and additional nearest-neighbour out-of-plane and next-nearest-neighbour in-plane Ising couplings. These additional terms do not lead to qualitative changes because the model is still symmetric under a $\pi$-rotation of the spins around the $z$-axis, and since their coupling constants are small ($\sim 0.1J_x$) they have only a small quantitative effect in exact diagonalisation results.
[^3]: In the LFI model the lowest energy with a given magnetisation is $E_{0,M= 0} =-J_z(N-1)/4$ and $E_{0,M\neq 0} = E_{0,0}+ J_z(|M|-1/2) - Mh$. Thus for even $N$, the ground state has $M=0$ for $0<h<J_z/2$, $M=1$ for $J_z/2<h<J_z$ and $M=N/2$ for $h>J_z$, whereas for odd $N$ the ground state has $M=1/2$ for $0<h<J_z$ and $M=N/2$ for $h>J_z$.
[^4]: In the periodic chain used to get the mean-field parameters, there is a level crossing when $\mu=2t$. To get good agreement with exact diagonalisation results and avoid a small discontinuity, we need to compute the expectation values in the state adiabatically connected to the ground state at $\mu<2t$. Thus for $\mu>2t$, the $\braket{.}$ are not computed in the ground state, but in the first excited state. Since all the level crossings arise for $\mu < 2t$, this has no influence on the following discussion.
|
---
abstract: 'In this paper we generalize the concept of the $q$-plate allowing arbitrary functions of both the radial and the azimuthal variables, and simulate their effect on uniformly polarized beams in the far-field regime. This gives a tool for achieving beams with hybrid states of polarization (SoPs), and alternative phase and intensity distributions. We propose an experimental device based on a liquid crystal on silicon (LCoS) display for emulating these generalized $q$-plates and show a new application that takes advantage of the pixelated nature of this kind of devices for representing discontinuous elements resulting from the random combination of two different $q$-plates, i.e. multiplexed $q$-plates.'
address:
- '$^{1}$ Facultad de Ciencias Exactas y Naturales, Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina.'
- '$^{2}$ Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina.'
author:
- 'M Vergara$^{1,2}$ and C Iemmi$^{1,2}$'
title: 'Generalized and multiplexed $q$-plates with radial and azimuthal dependence'
---
=4
\[sec:intro\] Introduction
==========================
Vortex beams carrying orbital angular momentum (OAM) have proven to be useful in a large number of applications ranging from classical implementations such as optical communications [@wang2012], microscopy [@furhapter2005] micro-manipulation [@bowman2011] and micro-machines design [@asavei2009], to the realization of quantum information protocols in high dimensional Hilbert spaces [@molina2004] and multilevel quantum key distribution [@mirho2013]. On the other hand vector beams, characterized by showing a non uniform distribution of the SoP, have been widely studied because of their tight focusing properties [@quabis], besides its potential application to communications [@cheng], optical tweezers [@woerde], quantum entanglement [@gabriel] and more.
While light propagates through a homogeneous and isotropic medium, SoP and vorticity are separately conserved; but they may be coupled in presence of an anisotropic and inhomogeneous media. In 2006 Marrucci et al. introduced for this purpose the $q$-plate, which consist of a half-wave retarder where the principal axis rotates with the azimuth angle [@marrucci2]. Its matrix representation in the Jones formalism has the form $$\begin{aligned}
M_q(\theta) =
\left( \begin{array}{cc}
\cos(2q\theta) & \sin(2q\theta) \\
\sin(2q\theta) & -\cos(2q\theta) \end{array} \right),\end{aligned}$$ where $2q$ is the times the retarder’s principal axis gives a whole turn around the center of the element.
Although in the first years the design of the $q$-plates was mainly oriented to the conversion of spin angular momentum to OAM, over time these elements evolved towards the objective of obtaining vector and vortex beams conformed by complex superposition of SoPs and OAMs.
$q$-plates are typically inhomogeneous and anisotropic devices where the spin to orbital conversion (STOC) is related to the Pancharatnam-Berry phase. Even though they are highly versatile elements, with many potential applications in the field of singular optics, different approaches extending the concept of $q$-plates were proposed in order to obtain greater flexibility in the design and diversity of responses. Some of them are based on metasurfaces [@Devlin896] which allow the combined use of the dynamic and geometric phases. Others create $q$-plates with different $q$ values depending on the region of the element [@ji], or make use of spatial light modulators (SLMs) to design $q$-plates with a non linear dependence of the azimuthal coordinate for binary codification [@holland]. In a recent paper [@vergara19] we propose the use of a generalized $q$-plate, allowing in its design non linear functions of the azimuthal coordinate for giving place to the generation of novel vortex and vector beams, and show a device based on a parallel aligned LCoS capable of achieving such distributions, hence emulating the generalized $q$-plate.
Here we propose the simulation of an element that rises from allowing arbitrary modulations of the polarization field of a beam, and in both polar coordinates $r$ and $\theta$, in such a way that we are able to explore complex vector and vortex beams, with novel polarization and phase structures. On the other hand we take the pixelated nature of the LCoS display as an advantage and propose a scheme for emulating multiplexed generalized $q$-plates, defined as discontinuous random combinations of two different $q$-plates, hence achieving superposition of vector or vortex beams with different $q$ value. Superposition of vortex beams carrying OAM have shown multiple applications, for example for creating arbitrary OAM *qudit* states for quantum information [@schulz13], for optical trapping and micromanipulation using residual OAM resulting from a superposition [@tao06], or for optical communications [@anguita14]; while superposition of vector beams has been used for 3D polarization control [@li12], improved interferometry [@lerman09], and more.
The Jones matrix that describes this kind of generalized $q$-plate is $$\begin{aligned}
M_{\Phi}(r, \theta) =
\left( \begin{array}{cc}
\cos[2\Phi(r, \theta)] & \sin[2\Phi(r, \theta)] \\
\sin[2\Phi(r, \theta)] & -\cos[2\Phi(r, \theta)] \end{array} \right),
\label{eq:$q$-general}\end{aligned}$$ and it represents a half wave plate in which the director axis angle is an arbitrary function $\Phi(r,\theta)$.
When a linearly polarized beam passes through such an element, it becomes a vector beam with a structured polarization pattern in which the azimuth of the polarization ellipses varies as a function $2\Phi(r,\theta)$. On the other hand, when impinging with a circularly polarized beam, the phase of the beam becomes $2\Phi(r,\theta)$ while the polarization inverts its sense of rotation.
This way, generalized $q$-plates allow both the generation of vortex beams with phase singularities carrying OAM and vector beams with polarization singularities, showing many potential applications in the field of singular optics. As seen in a previous work [@vergara19], interesting effects arise when these fields are propagated towards the far field regime.
In section \[sec:simulation\] we show results of simulating the effect of some of these generalized $q$-plates on uniformly polarized input beams, in the far field approximation. In section \[sec:multiplex\] we show how to create superposition of vector or vortex beams by means of multiplexing different $q$-plates in the same element. This can be seen as a discontinuous generalized $q$-plate. Pixel by pixel modulation offered by SLMs makes this approach possible for experimental implementation. The main conclusions are given in section \[sec:conclus\].
\[sec:simulation\] Simulation of generalized q-plates with radial and azimuthal dependence
==========================================================================================
Beams created from arbitrary functions $\Phi$ of the azimuthal and radial coordinates may show a variety of interesting effects in their amplitude, phase and polarization structure. In this section we show some examples of the different behaviors found. We simulated the effect of generalized $q$-plates of the form $$\begin{aligned}
\Phi(r,\theta) = \Phi_r(r) + \Phi_{\theta}(\theta),\end{aligned}$$ on uniformly polarized beams, in the far field approximation. We used for the sake of simplicity the same functions $\Phi_{\theta}$ as in [@vergara19], and only polynomial functions for $\Phi_r$.
As a first example, figure \[fig:CLespiralV\] shows the results obtained in the far field for the generalized $q$-plate function $\Phi(r,\theta) = -{q_r}\pi(r/r_0)^{p_r} + {q_t}(2\pi)^{1-{p_t}}\theta^{p_t}$, when input light is vertically polarized. Regarding the azimuthal dependence, this function shows polynomial growth with power $p_t$ and total variation $q_t2\pi$. Radial function shows polynomial decreasing from the center with power $p_r$, reaching a value of $-q_r\pi$ when $r = r_0$, being $r_0$ the plate radius. This function describes a family of spiral distributions like those shown in the first row of figure \[fig:CLespiralV\].
![(Color online) Some particular results for plates with polynomial growth both in $r$ and $\theta$. First row shows the generalized $q$-plate matrix argument $2\Phi(r,\theta)$ (modulo $2\pi$), the other three rows show the intensity and polarization distributions, the azimuth, and the form factor, respectively, in the far field regime. Input polarization is vertical.[]{data-label="fig:CLespiralV"}](figure1.png){width="0.5\columnwidth"}
When $p_t = 1$ (linear in $\theta$), cylindrical symmetry in polarization and intensity distributions is preserved. In the case shown in the first column of figure \[fig:CLespiralV\], the result is a cylindrical vector beam with a central singularity with topological charge $q_t = 1/2$, which is measured as the times the polarization ellipse mayor axis gives a whole turn around the beams axis, this angle is refereed to as the polarization azimuth. In this case, in addition, intensity and polarization azimuth varies radially. For instance, along the first concentric intensity maximum the polarization is radial, while along the next minimum it is azimuthal.
Fourth row on figure \[fig:CLespiralV\] shows the form factor, $f = b/a$, which is the ratio between the minor $b$ and mayor $a$ axis of the polarization ellipse, and whose sign is defined negative for right-handed, and positive for left-handed sense of rotation. In order to plot the polarization ellipses we used a color code based on form factor. In a neighborhood near $f = 0$ we considered polarization to be linear (green), and around $f = \pm1$ we considered polarization to be circular (blue), in any other case the polarization is elliptical (red). Yellow contours on the intensity images delimit areas with intensities below $0.5\%$ of maximum value.
There is a wide variety of vector beams that can be created by changing the four parameters in the expression of $\Phi$. When increasing the power $p_t$ we observed the same behaviour as reported in [@vergara19], central singularity splits into several singularities with lower topological charge, and the linearly polarized vector beam becomes a beam with hybrid SoP. In the case where $q_r=2$, $p_r=2$, $q_t=\frac{1}{2}$ and $p_t=2$, there is a central intensity minimum around which polarization vector rotates, with topological charge 1. This singularity takes place between two C-points with topological charge $-\frac{1}{2}$, this is an isolated point of circular polarization with topological charge defined as above. There are two more C-points near these, with inverse topological charge, in such a way that total topological charge remains $q_t=1/2$. The degree of freedom in $r$ allows to modulate the spacial distribution of intensity, and thus the position of polarization singularities and critical points.
Even though in the Fourier plane we obtained hybrid states of polarization, it is worth to mention that these beams carry no OAM. When a linearly polarized beam passes through a generalized $q$-plate it becomes a vector beam with linear polarization with orientation given by the function $\Phi(r,\theta)$ [@vergara19]. During propagation, diffraction phenomenon creates hybrid SoPs, but OAM is conserved. For creating OAM from polarization structure it is necessary to achieve radial or azimuthal variation of the form factor, in the object plane [@wang2010]. This way, these beams carry only polarization topological charge, in the form of polarization vortexes and C-points. Regions of elliptical polarization with right and left sense of rotation are balanced in every case.
![(Color online) Some particular results for plates with polynomial growth both in $r$ and $\theta$. First row shows the generalized $q$-plate matrix argument $2\Phi(r,\theta)$ (modulo $2\pi$), the other two rows show the intensity and polarization distributions, and the phase distribution, respectively, in the far field regime. Input polarization is left circular.[]{data-label="fig:CLespiralL"}](figure2.png){width="0.5\columnwidth"}
Figure \[fig:CLespiralL\] shows results for the same cases than figure \[fig:CLespiralV\], when input light is left circularly polarized. Intensity minima match with form factor maximum points of the case with linearly polarized input, thus polarization singularities are “replaced” by phase singularities. A vertically polarized beam can be described as the balanced superposition of left and right circularly polarized beams, and after passing through the generalized $q$-plate, left circular polarization turns right, and vice versa. Then, it is reasonable that when input light is left circularly polarized, regions of the output beam corresponding to left circular maxima show no intensity. In this case, when growth in $\theta$ is non-linear, output intensity, phase and polarization distributions lose the rotation symmetry. These phase singularities can be seen as vortexes carrying OAM with topological charge $\pm1$ depending on the phase gradient, the total OAM of the beam is given by the sum of the contributions, which gives $2q_t$.
![(Color online) Some particular results for plates with polynomial growth in $r$ and sinusoidal variation in $\theta$. First row shows the generalized $q$-plate matrix argument $2\Phi(r,\theta)$ (modulo $2\pi$), the other three rows show the intensity and polarization distributions, the azimuth, and the form factor, respectively, in the far field regime. Input polarization is vertical in all cases.[]{data-label="fig:CLsinpolV"}](figure3.png){width="0.5\columnwidth"}
Another case explored is that of a polynomial growth in the radial variable with a sinusoidal variation in the azimuthal one. The simulated function is $\Phi(r,\theta) = -q_r\pi(r/r_0)^{p_r} -(\pi/2)[\cos(q_t\theta)-1]$. Figures \[fig:CLsinpolV\] and \[fig:CLsinpolL\] show the results in this case, in a way analogous to the previous figures. When input light is vertically polarized, for odd $q_t$ values there appear form factor critical points (maxima and minima in these cases), which match with singular points in the azimuth, giving place to C-points with topological charges $\pm\frac{1}{2}$. On the other hand for even $q_t$ values, form factor is uniformly 0 (linear polarization) and azimuth singularities arise matching the intensity zeroes. Since azimuthal modulation of the plate is sinusoidal, every singularity needs to “annihilate" another one with opposite charge, adding a total topological charge of 0. This way we get structured light with no net topological charge, neither from phase nor polarization.
![(Color online) Some particular results for plates with polynomial growth in $r$ and sinusoidal variation in $\theta$. First row shows the generalized $q$-plate matrix argument $2\Phi(r,\theta)$ (modulo $2\pi$), the other two rows show the intensity and polarization distributions, and the phase distribution, respectively, in the far field regime. Input polarization is left circular in all cases.[]{data-label="fig:CLsinpolL"}](figure4.png){width="0.5\columnwidth"}
When input light is left circularly polarized there appear $2q_t$ phase singularities (vortexes). When $q$ is odd, these singularities match with half of the C-points mentioned in the case with linearly polarized input, and when $q$ is even they match with every polarization singularity. The topological charges in every case add up to 0, consistent with the sinusoidal modulation in $\theta$.
To implement experimentally these elements we propose a compact device that emulates the effect of the generalized $q$plates described above making use of the phase modulation provided by a phase only SLM. The proposed setup, sketched in figure \[fig:dispositivo\], uses a commercially available parallel aligned reflective liquid crystal on silicon (PA-LCoS) display, which introduces a programmable phase modulation to one linear component of the field (let us suppose that the director of the LC molecules is horizontally oriented).
For our purpose, the first half of the SLM is programmed with a phase modulation $\psi = 2\Phi(r,\theta)$ and the second half with a phase modulation $-\psi = -2\Phi(r,\theta)$. The quarter wave plate QWP2 is oriented at $45^\circ$ respect to the LC director, introducing a $-90^\circ$ rotation of the polarization vector due to the double passage. This way, the Jones matrix describing the effect of the SLM is $$\begin{aligned}
M_{\textnormal{SLM}}(r, \theta) =
\left( \begin{array}{cc}
0 & -ie^{-i2\Phi(r,\theta)} \\
ie^{i2\Phi(r,\theta)} & 0 \end{array} \right).\end{aligned}$$ Then, the phase programmed in the first half of the SLM is added to the vertical component of the field, and the phase programmed in the second half to the horizontal component. Quarter wave plate QWP1, oriented at $45^\circ$, and QWP3 oriented at $-45^\circ$, transform the input and output beams accordingly in order to emulate the behaviour of a generalized $q$-plate, this is, adding the respective phase modulation to circularly polarized orthogonal components of the input field. Jones matrix of the whole device can be written as $$\begin{aligned}
\eqalign{
M &= QWP(-45^\circ)*M_{\textnormal{SLM}}(r, \theta)*QWP(45^\circ)\\
&= \left( \begin{array}{cc}
-\cos[2\Phi(r, \theta)] & -\sin[2\Phi(r, \theta)] \\
-\sin[2\Phi(r, \theta)] & \cos[2\Phi(r, \theta)] \end{array} \right)\\
&= e^{i\pi}M_{\Phi}(r, \theta).}\end{aligned}$$
This matrix representation coincides with that of the generalized $q$-plate (\[eq:$q$-general\]), up to a global phase factor, then emulating all the expected behaviors. In addition, since it is possible to program the PA-LCoS pixel by pixel, it allows to make modifications at video rates, adding flexibility in the design of the mimicked plates. Lens L2 is useful for measuring the output beam at different propagation distances, between near and far field regimes, it can be removed for observing directly the intensity obtained at the exit of the device.
![(Color online) Experimental compact device proposed for emulating generalized $q$-plates using a reflective PA-LCoS.[]{data-label="fig:dispositivo"}](figure5.png){width="0.5\columnwidth"}
\[sec:multiplex\] Multiplexed q-plates and beam superposition
=============================================================
The possibility of using arbitrary functions $\Phi$ in the definition of the generalized $q$-plate and the ability of implementing these in SLMs, which allow pixel by pixel modulation, gives the ability of multiplexing various plates in the same device simultaneously, generating superposition of vector or vortex beams coming from different functions.
The use of a SLM leads to represent the function $\Phi$ (as a discontinuous version of itself) onto an array of square elements (pixels), which can take discrete phase values. Multiplexing can be achieved by selecting randomly two complementary sets of pixels, and representing on each set a different function. This random multiplexing scheme has been used previously to increase depth of focus of diffractive lenses [@iemmi2006]. This feature is possible due to the pixelated structure of the SLM and can not be accomplished by conventional $q$-plate devices.
As a simple example, we show the result of combining linear $q$-plates, where $\Phi(\theta) = q\theta$, with different $q$ values, onto a hypothetical array of $100\times100$ pixels. The functions for $q_1=1/2$ and $q_2=1$ are shown in figure \[fig:multiplex\], together with the discontinuous $\Phi(r,\theta)$ resultant from multiplexing both. Results for other different combinations are shown in figure \[fig:multiV\] for input linear polarization and figure \[fig:multiL\] for input left circular polarization. The superposition of beams coming from each set of pixels is obtained. This scheme can generate superposition of alternative vector or vortex beams with arbitrary topological charges. The relative weight of the components in the superposition can be varied by changing the size of the set of pixels assigned to each component, this can be easily done given the flexibility and speed provided by the use of SLMs. The number of superimposed beams can be increased, but in practice it is limited by the resolution of the SLM.
![$2\Phi$ functions for $q=1/2$, $q=1$ and the multiplexed $q$-plate that combines $q_1=1/2$ and $q_2=1$.[]{data-label="fig:multiplex"}](figure6.png){width="0.5\columnwidth"}
Fraunhofer field coincides with the Fourier transform of the field at the $q$-plate plane. For a function $g(r,\theta)$ separable in polar coordinates this can be written in terms of an infinite sum of weighted Hankel transforms [@goodman], $$\begin{aligned}
\mathcal{F}\left\lbrace g(r,\theta)\right\rbrace = \sum_{k=-\infty}^{\infty} c_k (-i)^k \exp(ik\phi) \mathcal{H}_k \left\lbrace g_R (r)\right\rbrace,
\label{eq:hankel}\end{aligned}$$ where $c_k$ is a complex coefficient and $\mathcal{H}_k$ is the Hankel transform operator of order $k$, $$\begin{aligned}
\mathcal{H}_k \left\lbrace g_R (r)\right\rbrace = 2\pi \int_{0}^{\infty} r g_{R} (r) J_k(2\pi r \rho) dr,\end{aligned}$$ being $J_k$ the $k$th-order Bessel function of the first kind, and $g(r,\theta) = g_R (r)g_{\Theta} (\theta)$.
Far field coming from a $q$-plate with even $2q$ value only shows terms with even $k$ value in the expression of (\[eq:hankel\]), so the factor $(-i)^k = \pm 1$. When $2q$ is an odd number, only terms with odd $k$ value appear, then $(-i)^k = \pm i$. In the case of superposition $q$-plates with same parity, both contributions’ phase factor differ in an even multiple of $\pm \pi/2$, then generating uniform linear polarized vector beams with polarization singularities in the intensity minima, caused by destructive interference where added polarization vectors are parallel and have opposite phases. On the other hand, if the combined $2q$ values have different parity, the contributions’ phase factor differ in an odd multiple of $\pm \pi/2$, hence C-points take place at spots where added polarization vectors are orthogonal, and intensity does not show destructive interference.
![(Color online) Far field diffraction resulting from multiplexed $q$-plates when input beam is vertically polarized.[]{data-label="fig:multiV"}](figure7.png){width="0.5\columnwidth"}
![(Color online) Far field diffraction resulting from multiplexed $q$-plates when input beam is left circularly polarized.[]{data-label="fig:multiL"}](figure8.png){width="0.5\columnwidth"}
Higher $q$ value $q$-plates create donut shaped beams with larger radii, so in the superposition, the inner region of the beam shows the structure of the one created by the lower $q$ value $q$-plate, while in the outer region the higher $q$ value predominates.
When input polarization is left circular (\[fig:multiL\]), only remains in the far field the right circular contribution to the beams shown in figure \[fig:multiV\]. Intensity minima in these cases are phase vortex carrying OAM, as expected.
The scheme we propose not only allows the use of generalized and multiplexed $q$-plates with arbitrary functions (of which only a few examples are given here as a demonstration), but also gives speed an flexibility in the implementation, due to the use of a phase only LCoS display with high spatial resolution and video rate operation. This opens a wide range of possibilities for the creation of alternative vector and vortex beams.
\[sec:conclus\] Conclusion
==========================
We proposed a generalization of the concept of $q$-plate, allowing in its definition non-linear functions of both radial and azimuthal variables, for creating novel vector or vortex beams, depending on the input state of polarization. We simulated the effect of these kind of element on uniformly vertical and left circular polarized beams.
In the far field regime, it is found that when losing linearity in the azimuthal variable, the conventional central singularity of vector/vortex beams divides into several singularities of minimum topological charge. In the cases where the input light is linearly polarized, the output beam can exhibit, either C-points with topological charge $\pm\frac{1}{2}$, as well as other types of critical points of the form factor, or dark polarization singularities (flowers/webs). Distribution of left and right circular polarization regions are symmetrical. Circularly polarized input beams, result in the appearance of phase vortexes, carrying OAM with topological charge $\pm1$. Two examples of function were shown, where the net polarization topological charge is finite or zero respectively, despite of the local polarization structure. The intensity profiles and singularity distributions in each case depends on the particular chosen function $\Phi(r,\theta)$, giving the chance to model distributions of any optical singularity known. This gives a tool for achieving beams with hybrid SoPs, and novel intensity distributions.
We applied this generalization to the creation of multiplexed $q$-plates, defined by discontinuous functions, consisting on different $q$-plates encoded on complementary sets of randomly picked pixels from a SLM, obtaining in the far field regime the superposition of vortex/vector beams generated by the individual $q$-plates involved, a result that can not be achieved by conventional $q$-plate devices, and has potential application in many fields including quantum and classical communications, interferometry, optical trapping and micromanipulation, and singular optics in general.
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---
author:
- |
[, [Ebrahim Ghorbani]{}]{}\
\
\
\
\
title: '****'
---
> **Abstract**
>
> = 0 mm
>
> The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. It is proved that $E(G)\geq 2(n-\chi(\overline{G}))\ge 2({{\rm ch}}(G)-1)$ for every graph $G$ of order $n$, and that $E(G)\ge 2{{\rm ch}}(G)$ for all graphs $G$ except for those in a few specified families, where $\overline{G}$, $\chi(G)$, and ${{\rm ch}}(G)$ are the complement, the chromatic number, and the choice number of $G$, respectively.
>
> [: Energy, choice number.]{}
>
> : 05C15, 05C50, 15A03.
[**1. Introduction**]{}
All the graphs that we consider in this paper are finite, simple and undirected. Let $G$ be a graph. Throughout this paper the [*order*]{} of $G$ is the number of vertices of $G$. If $\{
v_1, \ldots, v_n\}$ is the set of vertices of $G$, then the [*adjacency matrix*]{} of $G$, $A=[a_{ij}]$, is an $n\times{n}$ matrix where $a_{ij}=1$ if $v_i$ and $v_j$ are adjacent and $a_{ij}=0$ otherwise. Thus $A$ is a symmetric matrix with zeros on the diagonal, and all the eigenvalues of $A$ are real and are denoted by $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$. By the eigenvalues of $G$ we mean those of its adjacency matrix. The [*energy*]{} $E(G)$ of a graph $G$ is defined as the sum of the absolute values of all eigenvalues of $G$, which is twice the sum of the positive eigenvalues since the sum of all the eigenvalues is zero. For a survey on the energy of graphs, see [@gut].
For a graph $G$, the [*chromatic number*]{} of $G$, denoted by $\chi(G)$, is the minimum number of colors needed to color the vertices of $G$ so that no two adjacent vertices have the same color. Suppose that to each vertex $v$ of a graph G we assigned a set $L_v$ of $k$ distinct elements. If for any such assignment of sets $L_v$ it is possible, for each $v\in V(G),$ to choose $\ell_v\in L_v$ so that $\ell_u\neq\ell_v$ if $u$ and $v$ are adjacent, then $G$ is said to be [*$k$-choosable*]{}. The [*choice number*]{} ${{\rm ch}}(G)$ of $G$ is the smallest $k$ such that $G$ is $k$-choosable.
We denote by $A_{n,t}$, $1\leq t \le n-1$, the graph obtained by joining a new vertex to $t$ vertices of the complete graph $K_n$. If we add two pendant vertices to a vertex of $K_n$, the resulting graph has order $n+2$ and we denote it by $B_n$.
In [@agz], it is proved that apart from a few families of graphs, $E(G)\geq 2\max(\chi(G), n-\chi(\overline{G}))$ (see the following theorem). Our goal in this paper is to extend this result to the choice number of graphs.
\[ab\] Let $G$ be a graph. Then $E(G)<2\chi(G)$ if and only if $G$ is a union of some isolated vertices and one of the following graphs:\
(i) the complete graph $K_n$;\
(ii) the graph $B_n$;\
(iii) the graph $A_{n,t}$ for $n\leq7$, except when $(n,t)=(7,4)$, and also for $n\geq8$ and $t\in \{1, 2, n-1 \}$;\
(iv) a triangle with two pendant vertices adjacent to different vertices.
The following is our main result.
\[main\] Let $G$ be a graph. Then $E(G)<2\,{{\rm ch}}(G)$ if and only if $G$ is a union of some isolated vertices and one of the following graphs:\
(i)–(iv) as in Theorem \[ab\];\
(v) the complete bipartite graph $K_{2,4}$.
[**2. Proofs**]{}
In this section we present a proof for Theorem \[main\]. To do so we need some preliminaries.
A well-known theorem of Nordhaus and Gaddum [@nor] states that for every graph $G$ of order $n$, $\chi(G)+\chi(\overline{G})\leq n+1$. This inequality can be extended to the choice number. The graphs attaining equality are characterized in [@dgm]. It is proved that there are exactly three types of such graphs defined as follows.
- A graph $G$ is of [*type $F_1$*]{} if its vertex set can be partitioned into three sets $S_1, T, S_2$ (possibly, $S_2=\emptyset$) such that $S_1\cup S_2$ is an independent set of $G$, every vertex of $S_1$ is adjacent to every vertex of $T$, every vertex of $S_2$ has at least one non-neighbor in $T$, and $|S_1|$ is sufficiently large that the choice number of the induced subgraph on $T\cup S_1$ is equal to $|T| + 1$. This implies that ${{\rm ch}}(G) = |T| + 1$ also. Theorem 1 of [@gmm] states that if $T$ does not induce a complete graph, then $|S_1|\ge |T|^2$; we will use this result later.
- A graph is of [*type $\bar F_1$*]{} if it is the complement of a graph of type $F_1$.
- A graph is of [*type $F_2$*]{} if its vertex set can be partitioned into a clique $K$, an independent set $S$, and a 5-cycle $C$ such that every vertex of $C$ is adjacent to every vertex of $K$ and to no vertex of $S$.
\[ch\] (a) [[@ert]]{} ${{\rm ch}}(G) + {{\rm ch}}(\overline{G})\le n + 1$ for every graph $G$ of order $n$.\
(b) [[@dgm]]{} Equality holds in (a) if and only if $G$ is of type $F_1$, $\bar F_1$ or $F_2$.
\[1\] For every graph $G$ of order $n$, $$E(G)\ge
2(n-\chi(\overline G))\ge2(n-{{\rm ch}}(\overline G))\ge2({{\rm ch}}(G)-1).$$
[As remarked in [@agz], the first inequality follows from Theorem 2.30 of [@fav], which states that $n-\chi(\overline{G})\leq
\lambda_1(G)+\cdots+\lambda_{\chi(\overline{G})}(G)$. The second inequality holds because ${{\rm ch}}(G)\ge \chi(G)$ for every graph $G$, and the third inequality holds by Theorem \[ch\]$(a)$.]{}
\[chrm\] For every graph $G$, ${{\rm ch}}(G)\leq
\lambda_1(G)+1$.
[Wilf ([@w], see also [@spec p. 90]) proved that every graph $G$ has a vertex with degree at most $\lambda_1(G)$, and so does every induced subgraph of $G$. He deduced from this that $\chi(G)\le \lambda_1(G)+1$, and the same argument also proves that ${{\rm ch}}(G)\le\lambda_1(G)+1$.]{}
\[2k2\] Suppose $G$ has $2K_2$ as an induced subgraph. Then $E(G)\ge2{{\rm ch}}(G)$.
[By the Interlacing Theorem (Theorem 0.10 of [@spec]), $\lambda_2(G)\ge\lambda_2(2K_2)=1$, and so $E(G)\ge2(\lambda_1(G)+\lambda_2(G))\ge2(\lambda_1(G)+1)\ge2{{\rm ch}}(G)$ by Lemma \[chrm\].]{}
We are now in a position to prove Theorem \[main\].
**Proof of Theorem \[main\]**. Let $G$ be a graph such that $E(G) < 2{{\rm ch}}(G)$. We may assume that $G$ has at least one edge, since otherwise $G$ is the union of some isolated vertices and $K_1$, which is permitted by $(i)$ of Theorem \[main\]. Since removing isolated vertices does not change the value of $E(G)$ or ${{\rm ch}}(G)$, we may assume that $G$ has no isolated vertices. If ${{\rm ch}}(G) + {{\rm ch}}(\overline G) \le n$, then $E(G) \ge 2{{\rm ch}}(G)$ by Lemma \[1\]; this contradiction shows that ${{\rm ch}}(G) + {{\rm ch}}(\overline G) =
n + 1$, which means that $G$ has one of the types $F_1$, $\bar F_1$ and $F_2$ by Theorem \[ch\]$(b)$. We consider these three cases separately.
**Case 1**. $G$ has type $F_1$. Then $G$ has $G[T]\vee \overline{K}_k$ as an induced subgraph, where $G[T]$ is the subgraph induced by $G$ on $T$, $k=|S_1|$, and $\vee$ denotes ‘join’. Let $|T|=t$, so that ${{\rm ch}}(G)=t+1$. If $G[T]$ is a complete graph, then $\chi(G)=t+1={{\rm ch}}(G)$, so that $E(G)<2\chi(G)$ and $G$ is one of the graphs listed in Theorem \[ab\]. So we may assume that $G[T]$ is not a complete graph. In this case, as remarked after the definition of type $F_1$, $k=|S_1|\ge|T|^2\geq t^2$. Thus $$\lambda_1(G[T]\vee \overline{K}_k)\geq
\lambda_1(K_{t,t^2})=t\sqrt{t}\geq t+1,$$ provided $t\geq3$; since ${{\rm ch}}(G)=t+1$, we have $E(G)\geq 2\,{{\rm ch}}(G)$. So we may assume that $t\leq 2$, then $G[T]=\overline{K}_2$ and $k\geq t^2=4$. For $k\geq5$, we have $\lambda_1(K_{2,k})\geq \sqrt{10}>3={{\rm ch}}(K_{2,k})$, thus $E(G)\geq 2\,{{\rm ch}}(G)$. So we may assume that $k=4$. If $G\neq
K_{2,4}$, then either $|S_1|\ge5$ or $|S_2|>0$; thus $G$ has either $K_{2,5}$ or $H$ as an induced subgraph, where $H$ is formed from $K_{2,4}$ by adding an extra vertex joined to one of the vertices of degree $4$. We have $E(K_{2,5})=2\sqrt{10}>6$. The graph $H$ has a $P_4$ as an induced subgraph so $\lambda_2(H)\ge\lambda_2(P_4)>0.6$. On the other hand $\lambda_1(H)\ge\lambda_1(K_{2,4})=2\sqrt{2}$. Therefore $E(H)>2(2\sqrt{2}+0.6)>6$. Hence $E(G)>6=2\,{{\rm ch}}(G)$ if $G\ne K_{2,4}$. Therefore $G=K_{2,4}$.
**Case 2**. $G$ has type $\bar F_1$. So $\overline G$ is of type $F_1$ with the associated partition $\{S_1, T, S_2\}$. Let $t=|T|$ and $k=|S_1|$. If $\overline G[T]$ is not a complete graph, then $k\ge t^2>1$ as in Case 1; hence $G$ has $2K_2$ as an induced subgraph, which gives a contradiction by Lemma \[2k2\]. So $\overline G[T]$ is a complete graph.Let $J$ be the set of those vertices of $T$ that are adjacent to all vertices of $S_2$ in $G$. Let $v$ be a vertex of $S_1$. Then $G$ is a graph of type $F_1$ with the associated partition $\{S_1', T', S_2'\}$, in which $$\begin{array}{ll}
S_1'=\{v\},~
T'=S_2\cup (S_1\setminus\{v\}),~ S_2'=T, & \hbox{if $k\ge2$;} \\
S_1'=J\cup\{v\},~ T'=S_2,~ S_2'=T\setminus J, & \hbox{if $k=1$.}
\end{array}$$ Therefore the result follows by Case 1.
**Case 3**. $G$ has type $F_2$. Thus $G$ has a 5-cycle as an induced subgraph. So $\lambda_2(G)+\lambda_3(G)\geq
\lambda_2(C_5)+\lambda_3(C_5)>1$. Hence, by Lemma \[chrm\], we obtain $$E(G)\geq 2(\lambda_1+\lambda_2+\lambda_3)>2(1+\lambda_1)\geq
2{{\rm ch}}(G).$$$\Box$
[**Acknowledgement.**]{} The authors are indebted to the Institute for Studies in Theoretical Physics and Mathematics (IPM) for support; the research of the first author was in part supported by a grant from IPM (No. 86050212). They are also grateful to the referee for her/his helpful suggestions.
[mm]{} S. Akbari, E. Ghorbani, S. Zare, Some relations between rank, chromatic number, and energy of graphs, Discrete Math., to appear.
D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Theory and Applications, third ed., Johann Ambrosius Barth, Heidelberg, 1995.
S. Dantas, S. Gravier, F. Maffray, Extremal graphs for the list-coloring version of a theorem of Nordhaus and Gaddum, Discrete Appl. Math. 141 (2004) 93–101.
P. Erdős, A.L. Rubin, H. Taylor, Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Congress. Numer. 26 (1980) 125–157.
O. Favaron, M. Mahéo, J.-F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti-II), Discrete Math. 111 (1993) 197–220.
S. Gravier, F. Maffray, B. Mohar, On a list-coloring problem, Discrete Math. 268 (2003) 303–308.
I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue. A. Wassermann (eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, 196–211.
E.A. Nordhaus, J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175–177.
H.S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967) 330–332.
|
---
abstract: 'We produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor, using an original generalized cross product on totally symmetric tensors. This allows us to formulate coordinate-free conditions using polynomial covariant tensors for identifying all the symmetry classes of the Elasticity tensor and prove that these conditions are both necessary and sufficient. Besides, we produce a new minimal set of 297 generators for the invariant algebra of the Elasticity tensor, using these tensorial covariants.'
address:
- 'Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, 94235, Cachan, France'
- 'Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, 94235, Cachan, France'
- 'Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, 94235, Cachan, France'
- 'Sorbonne Université, UMPC Univ Paris 06, CNRS, UMR 7190, Institut d’Alembert, F-75252 Paris Cedex 05, France & Univ Paris Sud 11, F-91405 Orsay, France'
author:
- 'M. Olive'
- 'B. Kolev'
- 'R. Desmorat'
- 'B. Desmorat'
title: Characterization of the symmetry class of an Elasticity tensor using polynomial covariants
---
Introduction {#sec:intro}
============
The problem of determining a minimal set of generators for the invariant algebra of the Elasticity tensor was solved recently [@OKA2017]. This definitively answered an old academic question: assuming that one could measure the components of the Elasticity tensors of two materials, can one decide by finitely many calculations, whether the two materials are identical as elastic materials, i.e. that the two tensors are related by a rotation?
Close to this problem, is another academic question: *can one decide by finitely many calculations what is the symmetry class of a given Elasticity tensor?*
In the specific case of the Elasticity tensor, it is known [@FV1996] that there are exactly *eight symmetry classes*: isotropic, cubic, transversely isotropic, trigonal, tetragonal, orthotropic, monoclinic and triclinic. This problem has a long history, recalled by Forte and Vianello in [@FV1996]. These authors have definitively clarified the mathematical problem about the symmetry classes of an Elasticity tensor and removed the link with crystallographic point groups which was extremely confusing and lead to the false assumption that there were ten, rather than eight, symmetry classes [@Fed1968; @CM1987; @HDP1991]. These eight classes were confirmed in 2001, using an alternative approach [@CVC2001], where symmetry planes rather than rotations play the central role. Note however that this approach, using symmetry planes, cannot be generalized to find the symmetry classes of higher order tensorial representations due to the fact that not all closed subgroups o f ${\mathrm{O}}(3)$ can be generated by plane reflections (see [@ODKD2020]). Finally, in 2014, a definitive and systematic way to determine the symmetry classes of any finite dimensional representation of the groups ${\mathrm{SO}}(2)$, ${\mathrm{SO}}(3)$, ${\mathrm{O}}(2)$ or ${\mathrm{O}}(3)$ was formulated (see [@Oli2014; @OA2013; @OA2014a; @Oli2019]). This method uses *clip’s tables* and the decomposition of the representation into *irreducible components*, a strategy which was initiated in the nineteens [@CLM1990; @CG1994; @CG1996].
Nevertheless, determining explicitly the symmetry class of a given Elasticity tensor is not an easy task and has been the subject of many researches, using different means. Moreover, the problem becomes even more complicated if one consider that, in *real life*, a measured Elasticity tensor (assuming that one can access to all of its components) is subject to experimental errors and has therefore no symmetry but is nevertheless *close* to a given theoretical tensor with a *given symmetry* [@GTP1963; @MN2006].
Concerning this problem, we would like first to cite the excellent work of François and coauthors [@Fra1995; @FGB1998] who performed a deep experimental and numerical study of the problem using acoustic measurements on an hexagonal testing sample of a raw material. The problem is then addressed numerically by testing how far is a plane reflection $s_{{\pmb{n}}}$ from a symmetry of the given experimental Elasticity tensor ${\mathbf{E}}$ (see also [@DKS2011]). Scanning a large range of directions ${\pmb{n}}$ lead to build a *pole figure*, which is a graphical representation over an hemisphere (representing all the unit vectors ${\pmb{n}}$ up to $\pm 1$) of the distance between ${\mathbf{E}}$ and its transformed $s_{{\pmb{n}}}\star {\mathbf{E}}$ by $s_{{\pmb{n}}}$. This gives qualitative information about the possible number of symmetry planes of the material. This pole figure can then be used to initialize an optimization algorithm to produce the “nearest” tensor with the expected symmetry.
Besides these experimental and numerical approaches, the literature is abundant about formulations of coordinate-free criteria to characterize Elasticity tensors which have *exactly* a given symmetry class.
Some authors [@BBS2007] have used the Kelvin representation [@Kelvin1; @Kelvin2; @Ryc1984] of the Elasticity tensor to achieve this goal. They have formulated necessary and sufficient conditions involving the multiplicity of the $6$ eigenvalues of the Kelvin representation and of the eigenvalues of its eigenvectors (the *eigenstrains*, which are in fact second order tensors). Such criteria are however very sensitive to rounding errors: one needs first to find the roots of a degree 6 polynomial and then of several polynomials of degree 3 (for each Kelvin eigenmodes) which depend on these roots.
Other approaches make use of the harmonic decomposition $({\mathbf{H}}, {\mathbf{d}}^{\prime}, {\mathbf{v}}^{\prime}, \lambda, \mu)$ of the Elasticity tensor ${\mathbf{E}}$, where ${\mathbf{d}}^{\prime}$ and ${\mathbf{v}}^{\prime}$ are respectively the deviatoric part of the dilatation tensor ${\mathbf{d}}=\operatorname{tr}_{12} {\mathbf{E}}$ and the Voigt tensor ${\mathbf{v}}=\operatorname{tr}_{13} {\mathbf{E}}$. For instance, following [@CM1987], some authors [@Cow1989; @Jar1994; @Bae1998a; @CVC2001] have extracted information about the symmetry class of ${\mathbf{E}}$ using ${\mathbf{d}}^{\prime}$ and ${\mathbf{v}}^{\prime}$. However, if this works well for certain orthotropic or monoclinic tensors ${\mathbf{E}}$, there are still many cases where the information on the symmetry class is not carried by the pair ${\mathbf{d}}^{\prime}, {\mathbf{v}}^{\prime}$. Indeed, there exist orthotropic and monoclinic tensors ${\mathbf{E}}$ for which ${\mathbf{d}}^{\prime}$ and ${\mathbf{v}}^{\prime}$ vanish.
In the same spirit, but to avoid loosing the information contained in the harmonic fourth-order component ${\mathbf{H}}$, Baerheim [@Bae1998b] has used the *harmonic factorization* introduced by Sylvester [@Syl1909] (see also [@Bac1970] and [@OKDD2018] for a more modern treatment). This factorization allows to decompose an harmonic tensor of order $n$ as an $n$-tuple of vectors, the so-called *Maxwell multipoles* [@Bac1970]. Baerheim has formulated criteria on the multipoles to characterize the different symmetry classes of ${\mathbf{E}}$. The difficulties with this approach is that the multipoles are not uniquely defined [@OKDD2018] and that the only way to obtain them is to solve a polynomial equation of degree $2n$ (hence, one degree eight and two degree four polynomials for the Elasticity tensor).
More recently, in [@AKP2014], the authors have suggested to reconsider the question in the general framework of *Real Algebraic Invariant Theory*. They have used a generating set of the invariant algebra of *fourth-order harmonic tensors* ${\mathbb{H}}^{4}({\mathbb{R}}^{3})$ proposed in [@BKO1994] to characterize the symmetry classes of a tensor ${\mathbf{H}}\in{\mathbb{H}}^{4}({\mathbb{R}}^{3})$, writing down polynomial equations and inequations involving the generators of the invariant algebra. However, these relations become increasingly complicated when the symmetry group becomes smaller and only the cubic, transversely isotropic, tetragonal, trigonal and orthotropic classes have been characterized this way. Note also that the same approach has already been used by Vianello [@Via1997] in 1997 for the full 2D Elasticity tensor, where formulas are considerably much simpler than in 3D.
In this paper, we propose to give a definitive answer to this classification problem for the full Elasticity tensor, using *polynomial covariants* rather than invariants (and avoiding, this way, increasing complexity). The answer is furnished by theorem \[thm:main\], which is our main result. In some sense, our result is particularly simple, since one needs only to check that some polynomial functions defined on the components of the Elasticity tensor vanish (we do not need to solve any algebraic equation).
In order to obtain these results, we have been lead to formulate rigorously what is the *covariant algebra* of a given representation ${\mathbb{V}}$ of the rotation group ${\mathrm{SO}}(3)$ (even if this terminology is already well-known in classical invariant theory of binary forms [@Olv1999]). It was also necessary to introduce a generalization of the cross-product for totally symmetric tensors. Using these tools, we were able to explicit a minimal set of *70 generators for the covariant algebra of ${\mathbb{H}}^{4}({\mathbb{R}}^{3})$* in . These fundamental covariants are the cornerstone which has allowed us to characterize first the symmetry class of a tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}({\mathbb{R}}^{3})$ and then of a full Elasticity tensor ${\mathbf{E}}$.
A by-product of these achievements is the production of a new set of minimal generators for the invariant algebra of the Elasticity tensor, using the covariants in and the covariant tensor operations of section \[sec:covariant-tensor-operations\]. These generators are given in and shall be more useful for the mechanical community than the original invariants furnished in [@OKA2017], which were described using *transvectants* [@Olv1999].
Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------
The paper is organized as follows. In , we provide basic definitions and recall the link between totally symmetric tensors and homogeneous polynomials. In , we recall the basic covariant operations on tensors and introduce the *generalized cross-product* between totally symmetric tensors. The is devoted to the definition of polynomial covariants of a linear representation and basic facts about the covariant algebra. A minimal generating set of 70 polynomial covariants for ${\mathbb{H}}^{4}({\mathbb{R}}^{3})$ is provided in . The symmetry classes are introduced in section \[sec:symmetry-classes\] and a way to compute them is provided. The provides several lemmas which connect the dimension of covariant spaces of order one and two to their symmetry class. In , several criteria which restrict the symmetry class of one or several totally symmetric tensors, using polynomial covariants are formulated. The characterization of the symmetry class of a fourth-order harmonic tensor ${\mathbf{H}}$ using polynomial covariants is given in and the result for a full Elasticity tensor ${\mathbf{E}}$ is given in section \[sec:Ela-symmetry-classes\]. In addition, three appendices are provided. In , we recall the basics about the spaces of binary forms of degree $n$, ${\mathrm{S}_{n}}$ (which are models for irreducible representations of ${\mathrm{SL}}(2,{\mathbb{C}})$) and we relate the invariant algebra of ${\mathrm{S}_{2n}}\oplus{\mathrm{S}_{2}}$ to the covariant algebra of ${\mathrm{S}_{2n}}$. In , we explain how we have been able to compute a minimal set of generators for the covariant algebra of ${\mathbb{H}}^{4}({\mathbb{R}}^{3})$ using the knowledge of a minimal set of generators for the covariant algebra of ${\mathrm{S}_{8}}$. Finally, in , we provide a new minimal set of 297 generators for the invariant algebra of the Elasticity tensor using the tensorial covariants provided in section \[sec:H4-covariant-algebra\].
Symmetric and harmonic tensors {#sec:sym-harm-tensors}
==============================
Let ${\mathbb{T}}^{n}({\mathbb{R}}^{3})$ be the vector space of $n$-th order tensors on the Euclidean space ${\mathbb{R}}^{3}$. Thanks to the Euclidean product, we do not have to distinguish between *upper* and *lower* indices. Therefore, an $n$-th order tensor may always be considered as a $n$-linear mapping $${\mathbf{T}}: {\mathbb{R}}^{3} \times \dotsb \times {\mathbb{R}}^{3} \to {\mathbb{R}}, \qquad ({\pmb{x}}_{1},\dotsc,{\pmb{x}}_{n}) \mapsto {\mathbf{T}}({\pmb{x}}_{1},\dotsc,{\pmb{x}}_{n}).$$ The subspace ${\mathbb{S}}^{n}({\mathbb{R}}^{3})$ of totally symmetric tensors can be identified with the vector space ${\mathcal{P}_{n}}({\mathbb{R}}^{3})$ of homogeneous polynomials of degree $n$. This isomorphism generalizes, to higher order tensors, the well-known connection between quadratic forms and symmetric bilinear forms obtained by *polarization* (see [@OKDD2018] for more details).
For instance, the polynomial representation of a totally symmetric fourth-order tensor ${\mathbf{S}}= (S_{ijkl})$ is given by $$\begin{gathered}
S_{1111}\,x^{4} + S_{2222}\,y^{4} + S_{3333}\,z^{4} + 12S_{1122}\,x^{2}y^{2} + 12S_{1133}\,x^{2}z^{2}
\\
+ 12S_{2233}\,y^{2}z^{2} + 4S_{1222}\,xy^{3} + 4S_{1112}\,yx^{3} + 4S_{1113}\,zx^{3} + 4S_{1333}\,xz^{3}
\\
+ 4S_{2223}\,zy^{3} + 4S_{2333}\,yz^{3} + 6S_{1123}\,yzx^{2} + 6S_{1223}\,xzy^{2} + 6S_{1233}\,xyz^{2}.
\end{gathered}$$
Contracting two indices $i,j$ on a totally symmetric tensor ${\mathbf{S}}$ does not depend on the particular choice of the pair $i,j$. Thus, we can refer to this contraction without any reference to a particular choice of indices. We will denote this contraction as $\operatorname{tr}{\mathbf{S}}$, which is a totally symmetric tensor of order $n-2$ and is called the *trace* of ${\mathbf{S}}$.
An $n$-th order totally symmetric and *traceless* tensor will be called an *harmonic tensor* and the subspace of ${\mathbb{S}}^{n}({\mathbb{R}}^{3})$ of harmonic tensors will be denoted by ${\mathbb{H}}^{n}({\mathbb{R}}^{3})$ (or simply ${\mathbb{H}}^{n}$, if there is no ambiguity).
In the correspondence between totally symmetric tensors and homogeneous polynomials, a *traceless* totally symmetric tensor ${\mathbf{H}}$ corresponds to an *harmonic polynomial* ${\mathrm{h}}$ (*i.e.* with vanishing Laplacian: $\triangle {\mathrm{h}}= 0$) and this justifies the appellation of *harmonic tensor*. The space of homogeneous harmonic polynomials of degree $n$ will be denoted by ${\mathcal{H}_{n}}({\mathbb{R}}^{3})$.
The natural action of the special orthogonal group ${\mathrm{SO}}(3)$ (or the full orthogonal group ${\mathrm{O}}(3)$) on ${\mathbb{R}}^{3}$ induces the tensorial representation $\rho_n$ on ${\mathbb{T}}^{n}({\mathbb{R}}^{3})$, defined by $$(\rho_n(g)({\mathbf{T}}))({\pmb{x}}_{1},\dotsc,{\pmb{x}}_{n})=(g \star {\mathbf{T}})({\pmb{x}}_{1},\dotsc,{\pmb{x}}_{n}) : = {\mathbf{T}}(g^{-1} {\pmb{x}}_{1},\dotsc,g^{-1} {\pmb{x}}_{n}),$$ where ${\mathbf{T}}\in {\mathbb{T}}^{n}({\mathbb{R}}^{3})$ and $g \in {\mathrm{SO}}(3)$. Under this linear representation, the subspaces ${\mathbb{S}}^{n}({\mathbb{R}}^{3})$ and ${\mathbb{H}}^{n}({\mathbb{R}}^{3})$ are invariant. Moreover, ${\mathbb{H}}^{n}({\mathbb{R}}^{3})$ is *irreducible* [@GSS1988] (its only invariant subspaces are itself and the null space).
Every finite dimensional representation ${\mathbb{V}}$ of the rotation group ${\mathrm{SO}}(3)$ can be decomposed into a direct sum of irreducible representations, each of them being isomorphic to an harmonic tensor space ${\mathbb{H}}^{n}({\mathbb{R}}^{3})$, by an equivariant isomorphism.
An alternative model for the irreducible representations of ${\mathrm{SO}}(3)$ is furnished by the spaces of harmonic polynomials ${\mathcal{H}_{n}}({\mathbb{R}}^{3})$, where the action of ${\mathrm{SO}}(3)$ on polynomials is given by $(g \star {\mathrm{p}})({\pmb{x}}) : = {\mathrm{p}}(g^{-1} {\pmb{x}})$.
Every homogeneous polynomial of degree $n$ can be decomposed [@OKDD2018] as the following: $$\label{eq:polynomial-harmonic-decomposition}
{\mathrm{p}}= {\mathrm{h}}_{0} + {\mathrm{q}}\,{\mathrm{h}}_{1} + \dotsb + {\mathrm{q}}^{r}{\mathrm{h}}_{r},$$ where ${\mathrm{q}}= x^{2} + y^{2} + z^{2}$, $r = [n/2]$ – with $[\cdot]$ integer part – and ${\mathrm{h}}_{k}$ is a harmonic polynomial of degree $n-2k$.
Given a homogeneous polynomial ${\mathrm{p}}$, the highest order component in , namely ${\mathrm{h}}_{0}$, which is uniquely defined, is called the *harmonic projection* of ${\mathrm{p}}$ and denoted $({\mathrm{p}})_{0}$.
Covariant operations on tensors {#sec:covariant-tensor-operations}
===============================
In this section, we will introduce three operations on tensors, which *commute with the action of the rotation group* and are thus called *covariant operations*. The first one is the *symmetric tensor product*.
The symmetric tensor product between two tensors ${\mathbf{T}}^{1} \in {\mathbb{T}}^{p}({\mathbb{R}}^{3})$ and ${\mathbf{S}}^{2} \in {\mathbb{T}}^{q}({\mathbb{R}}^{3})$ is defined as $${\mathbf{T}}^{1}\odot{\mathbf{T}}^{2} : = ({\mathbf{T}}^{1} \otimes {\mathbf{T}}^{2})^{s} \in {\mathbb{S}}^{p + q}({\mathbb{R}}^{3}),$$ where the total symmetrisation of a tensor ${\mathbf{T}}\in {\mathbb{T}}^{n}({\mathbb{R}}^{3})$, noted ${\mathbf{T}}^{s} \in {\mathbb{S}}^{n}({\mathbb{R}}^{3})$, is defined as $${\mathbf{T}}^{s}({\pmb{x}}_{1},\dotsc,{\pmb{x}}_{n}) : = \frac{1}{n!}\sum_{\sigma \in \mathfrak{S}_{n}} {\mathbf{T}}({\pmb{x}}_{\sigma(1)},\dotsc,{\pmb{x}}_{\sigma(n)})$$ where $\mathfrak{S}_{n}$ is the symmetric group on $n$ letters.
When *restricted to totally symmetric tensors*, the polynomial counterpart of the symmetric tensor product is just the usual product of polynomials. This product is thus associative and commutative. It is equivariant relative to either the rotation group ${\mathrm{SO}}(3)$ and the full orthogonal group ${\mathrm{O}}(3)$.
The *harmonic decomposition* of an homogenous polynomial of degree $n$ leads thus to the following harmonic decomposition of a totally symmetric tensor ${\mathbf{S}}\in {\mathbb{S}}^{n}({\mathbb{R}}^{3})$: $$\label{eq:symmetric-harmonic-decomposition}
{\mathbf{S}}= {\mathbf{H}}_{0} + {\mathbf{q}}\odot {\mathbf{H}}_{1} + \dotsb + {\mathbf{q}}^{\odot r-1}\odot {\mathbf{H}}_{r-1}+ {\mathbf{q}}^{\odot r} \odot{\mathbf{H}}_{r},$$ where ${\mathbf{H}}_{k}$ is an harmonic tensor of degree $n-2k$. In this formula, ${\mathbf{q}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ is the *Euclidean metric tensor* (which writes as ${\mathbf{q}}=(\delta_{ij})$ in any orthonormal basis) and ${\mathbf{q}}^{\odot k}$ means the symmetrized tensorial product of $k$ copies of ${\mathbf{q}}$.
The second one is the contraction between two tensors ${\mathbf{T}}^1\in{\mathbb{T}}^{p}({\mathbb{R}}^{3})$ and ${\mathbf{T}}^{2}\in{\mathbb{T}}^{q}({\mathbb{R}}^{3})$ over one or several subscripts. This operation uses the Euclidean structure represented by the canonical Euclidean metric tensor ${\mathbf{q}}= (q_{ij})$ and its inverse ${\mathbf{q}}^{-1} = (q^{ij})$. It is defined as follows: $$({\mathbf{T}}^1 \overset{(r)}{\cdot} {\mathbf{T}}^{2})_{i_{1} \dotsb i_{p-r}j_{r+1} \dotsb j_{q}} = q^{i_{p-r+1}j_{1}} \dotsm q^{i_{p}j_{r}} T^1_{i_{1}\dotsb i_{p}} \, T^{2}_{j_{1} \dotsb j_{q}}.$$ The $r$-contraction of two tensors is an ${\mathrm{O}}(3)$-equivariant mapping $${\mathbb{T}}^{p}({\mathbb{R}}^{3}) \times {\mathbb{T}}^{q}({\mathbb{R}}^{3}) \to {\mathbb{T}}^{p+q-2r}({\mathbb{R}}^{3}),$$ and for $n=p=q$, the $n$-contraction corresponds to the canonical scalar product on ${\mathbb{T}}^{n}({\mathbb{R}}^{3})$.
In an *orthonormal basis* $({\pmb{e}}_{i})$, we have $$\begin{aligned}
({\mathbf{T}}^1 \cdot {\mathbf{T}}^{2})_{i_{1} \dotsb i_{p-1}j_{2} \dotsb j_{q}} & = T^1_{i_{1}\dotsb i_{p-1}k} \, T^{2}_{kj_{2} \dotsb j_{q}}, \\
({\mathbf{T}}^1 \operatorname{:}{\mathbf{T}}^{2})_{i_{1} \dotsb i_{p-2}j_{3} \dotsb j_{q}} & = T^1_{i_{1}\dotsb i_{p-2}kl} \, T^{2}_{klj_{3} \dotsb j_{q}}, \\
({\mathbf{T}}^1 \operatorname{\raisebox{-0.25ex}{\vdots}}{\mathbf{T}}^{2})_{i_{1} \dotsb i_{p-3}j_{4} \dotsb j_{q}} & = T^1_{i_{1}\dotsb i_{p-3}klm} \, T^{2}_{klmj_{4} \dotsb j_{q}}.
\end{aligned}$$
The *symmetric $r$-contraction* between two totally symmetric tensors ${\mathbf{S}}^{1} \in {\mathbb{S}}^{p}({\mathbb{R}}^{3})$ and ${\mathbf{S}}^{2} \in {\mathbb{S}}^{q}({\mathbb{R}}^{3})$ is defined as $$({\mathbf{S}}^1 \overset{(r)}{\cdot} {\mathbf{S}}^{2})^{s}.$$
The polynomial counterpart of the symmetric $r$-contraction is obtained as follows. If ${\mathbf{S}}^1, {\mathbf{S}}^{2}$ correspond respectively to the polynomials ${\mathrm{p}}_{1}, {\mathrm{p}}_{2}$, then, $({\mathbf{S}}^1 \overset{(r)}{\cdot} {\mathbf{S}}^{2})^s$ corresponds to the polynomial $${\mathrm{p}}= \frac{(p-r)!}{p!}\frac{(q-r)!}{q!} \sum_{k_{1}+k_{2}+k_{3}=r} \frac{r!}{k_{1}!k_{2}!k_{3}!}\frac{\partial^r {\mathrm{p}}_{1}}{\partial x^{k_{1}}\partial y^{k_{2}}\partial z^{k_{3}}}\frac{\partial^r {\mathrm{p}}_{2}}{\partial x^{k_{1}}\partial y^{k_{2}}\partial z^{k_{3}}}.$$
The third covariant operation is the *generalized cross product*, which extends the standard cross product between vectors of ${\mathbb{R}}^{3}$ to symmetric tensors of arbitrary order.
\[def:generalized-cross-product\] The generalized cross product (or *Lie-Poisson product*) between two totally symmetric tensors ${\mathbf{S}}^{1} \in {\mathbb{S}}^{p}({\mathbb{R}}^{3})$ and ${\mathbf{S}}^{2} \in {\mathbb{S}}^{q}({\mathbb{R}}^{3})$ is defined as $${\mathbf{S}}^{1} \times {\mathbf{S}}^{2} : = - \left({\mathbf{S}}^{1}\cdot \pmb \varepsilon \cdot {\mathbf{S}}^{2}\right)^s
\in {\mathbb{S}}^{p + q -1}({\mathbb{R}}^{3}).$$ where $\pmb \varepsilon$ is the *Levi–Civita* tensor. In any orthonormal basis, we get $$({\mathbf{S}}^{1}\times{\mathbf{S}}^{2})_{i_{1}\dotsb i_{p+q-1}} := (\varepsilon_{i_{1}jk}S^{1}_{ji_{2}\dotsb i_{p}}S^{2}_{ki_{p+1} \dotsb i_{p+q-1}})^{s}$$
The generalized cross product is skew-symmetric: $${\mathbf{S}}^{2} \times {\mathbf{S}}^{1} = -{\mathbf{S}}^{1} \times {\mathbf{S}}^{2}.$$ Its polynomial counterpart is (up to a scaling factor) the *Lie–Poisson bracket* on ${\mathfrak{so}}^{*}(3,{\mathbb{R}})$, the dual of the Lie algebra of the rotation group (isomorphic to ${\mathbb{R}}^{3}$). More precisely, if ${\mathrm{p}}_{1}, {\mathrm{p}}_{2}$ are the polynomial representatives of ${\mathbf{S}}^{1},{\mathbf{S}}^{2}$, then the polynomial representative of ${\mathbf{S}}^{1} \times {\mathbf{S}}^{2}$ is $$\frac{1}{pq} \{{\mathrm{p}}_{1}, {\mathrm{p}}_{2}\}_{LP} = \frac{1}{pq}\det({\pmb{x}},\nabla {\mathrm{p}}_{1}, \nabla {\mathrm{p}}_{2}).$$ This product is equivariant relative to the rotation group ${\mathrm{SO}}(3)$ but not to full orthogonal group ${\mathrm{O}}(3)$. In that later case, we get $$(g \star {\mathbf{S}}^{1}) \times (g \star {\mathbf{S}}^{2}) = (\det g) \left(g \star({\mathbf{S}}^{1} \times {\mathbf{S}}^{2})\right).$$
\[rem:Sxq=0\] Note that if ${\mathbf{q}}$ is the Euclidean metric tensor, then ${\mathbf{S}}\times {\mathbf{q}}= 0$ for every totally symmetric tensor ${\mathbf{S}}$ (indeed, the radial function ${\mathrm{q}}= x^{2} + y^{2} + z^{2}$ is a *Casimir function* for the Lie-Poisson bracket on ${\mathfrak{so}}^{*}(3,{\mathbb{R}})$). In particular, ${\mathbf{S}}\times {\mathbf{a}}= {\mathbf{S}}\times {\mathbf{a}}'$ for every symmetric *second-order* tensor ${\mathbf{a}}$, where $$\label{eq:dev}
{\mathbf{a}}^\prime={\mathbf{a}}-\frac{1}{3} \operatorname{tr}({\mathbf{a}})\, {\mathbf{q}}$$ is the deviatoric (*i.e.* harmonic) part of ${\mathbf{a}}$.
Polynomial covariants {#sec:polynomial-covariants}
=====================
Let ${\mathbb{V}}$ be a finite dimensional representation of a group $G$. The linear action of $G$ on ${\mathbb{V}}$ extends naturally to the algebra ${\mathbb{R}}[{\mathbb{V}}]$ of real polynomial functions defined on ${\mathbb{V}}$ by $$(g\star {\mathrm{p}})({\pmb{v}}) : = {\mathrm{p}}(g^{-1}\star {\pmb{v}}), \qquad {\mathrm{p}}\in {\mathbb{R}}[{\mathbb{V}}], \, g \in G.$$ A polynomial ${\mathrm{p}}\in {\mathbb{R}}[{\mathbb{V}}]$ is *invariant* if $g\star {\mathrm{p}}= {\mathrm{p}}$ for all $g\in G$. The set ${\mathbb{R}}[{\mathbb{V}}]^{G}$, also noted ${\mathbf{Inv}}({\mathbb{V}})$, of all invariant polynomials is a sub-algebra of ${\mathbb{R}}[{\mathbb{V}}]$, called the *invariant algebra* of ${\mathbb{V}}$. In [@KP2000], Kraft and Procesi have generalized the concept of invariants in the following way.
Given two representations ${\mathbb{V}}$ and ${\mathbb{W}}$ of a group $G$, we define ${\mathrm{Pol}}({\mathbb{V}},{\mathbb{W}})$ to be the space of polynomial mappings ${\mathrm{p}}$ from ${\mathbb{V}}$ to ${\mathbb{W}}$ (*i.e* each component function is a polynomial expression of the components of ${\pmb{v}}\in {\mathbb{V}}$, and such in any basis). A *polynomial covariant of ${\mathbb{V}}$ of type ${\mathbb{W}}$* is a $G$-equivariant polynomial mapping ${\mathrm{p}}: {\mathbb{V}}\to {\mathbb{W}}$, which means that $${\mathrm{p}}(g\star {\pmb{v}}) = g\star {\mathrm{p}}({\pmb{v}}), \qquad \forall {\pmb{v}}\in {\mathbb{V}}, \, \forall g \in G.$$
The problem with this definition is that the set ${\mathrm{Pol}}({\mathbb{V}},{\mathbb{W}})^{G}$, of polynomial covariant of ${\mathbb{V}}$ of type ${\mathbb{W}}$ is only a vector space and not an algebra. We will therefore extend this definition as follows.
\[def:covariant-algebra\] Let ${\mathbb{V}}, {\mathbb{W}}$ be finite dimensional representations of a group $G$. The *covariant algebra of ${\mathbb{V}}$ of type ${\mathbb{W}}$*, noted ${\mathbf{Cov}}({\mathbb{V}},{\mathbb{W}})$, is defined as the invariant algebra $${\mathbb{R}}[{\mathbb{V}}\oplus {\mathbb{W}}^{*}]^{G},$$ where ${\mathbb{W}}^{*}$ is the dual vector space of ${\mathbb{W}}$.
We can define similarly, $\mathbf{Con}({\mathbb{V}},{\mathbb{W}})$, the *contravariant algebra of ${\mathbb{V}}$ of type ${\mathbb{W}}$* as ${\mathbb{R}}[{\mathbb{V}}\oplus {\mathbb{W}}]^{G}$. However, if ${\mathbb{W}}$ and ${\mathbb{W}}^{*}$ are equivalent representations (for instance if the representation ${\mathbb{W}}$ is unitary), we do not have to distinguish between these two algebras which are canonically isomorphic.
Note that the covariant algebra ${\mathbf{Cov}}({\mathbb{V}},{\mathbb{W}})$ has a natural bi-graduation. It is the direct sum of the finite dimensional vector spaces ${\mathbf{Cov}}_{d,k}({\mathbb{V}},{\mathbb{W}})$ of bi-homogeneous polynomial $p({\pmb{v}},\omega)$:
- of total degree $d$ in ${\pmb{v}}\in {\mathbb{V}}$, called the **degree** of the covariant,
- and, of total degree $k$ in $\omega \in {\mathbb{W}}^{*}$, called the **order** of the covariant.
Furthermore, the subspace of covariants of order $0$ is identical to the invariant algebra of ${\mathbb{V}}$.
The vector space of polynomial covariants ${\mathrm{Pol}}({\mathbb{V}},{\mathbb{W}})^{G}$ can thus be identified with $${\mathbf{Cov}}_{1}({\mathbb{V}},{\mathbb{W}}) = \bigoplus_{k=0}^{+\infty}{\mathbf{Cov}}_{k,1}({\mathbb{V}},{\mathbb{W}}),$$ the vector space of first-order covariants.
In this paper, we will only be interested when $G = {\mathrm{SO}}(3)$ and ${\mathbb{W}}$ is the Euclidean space ${\mathbb{R}}^{3}$ (in which case, we do not have to make any difference between the covariant and the contravariant algebras), and we will set $${\mathbf{Cov}}({\mathbb{V}}) := {\mathbb{R}}[{\mathbb{V}}\oplus {\mathbb{R}}^{3}]^{{\mathrm{SO}}(3)}.$$ An element ${\mathrm{p}}\in {\mathbf{Cov}}({\mathbb{V}})$ is thus a polynomial which can be written as $${\mathrm{p}}({\pmb{v}},{\pmb{x}}) = \sum_{i,j,k} p_{ijk}({\pmb{v}})x^{i}y^{j}z^{k},$$ where each coefficient $p_{ijk}({\pmb{v}})$ is a polynomial function of ${\pmb{v}}$ and such that $${\mathrm{p}}(g \star {\pmb{v}}, {\pmb{x}}) = {\mathrm{p}}({\pmb{v}},g^{-1} \star {\pmb{x}}),$$ for all ${\pmb{v}}\in {\mathbb{V}}$, ${\pmb{x}}\in {\mathbb{R}}^{3}$ and $g \in {\mathrm{SO}}(3)$.
Any homogeneous polynomial covariant of ${\pmb{v}}\in {\mathbb{V}}$ of degree $d$ and of type ${\mathbb{S}}^{k}({\mathbb{R}}^{3})$ can thus be identified with a polynomial in ${\mathbf{Cov}}_{d,k}({\mathbb{V}})$.
One fundamental result, obtained in the nineteenth century, is that the invariant and covariant algebras of a finite dimensional representation of a compact group is finitely generated.
The covariant algebra ${\mathbf{Cov}}({\mathbb{V}})$ is finitely generated, i.e. there exists a finite set $\mathcal{B} : = {\left\{{\mathrm{p}}_{1},\dotsc,{\mathrm{p}}_s\right\}}$ in ${\mathbf{Cov}}({\mathbb{V}})$ such that $${\mathbf{Cov}}({\mathbb{V}}) = {\mathbb{R}}[{\mathrm{p}}_{1},\dotsc,{\mathrm{p}}_s].$$ Moreover, one can always find such a system where the ${\mathrm{p}}_{j}$ are bi-homogeneous, both in ${\pmb{v}}\in {\mathbb{V}}$ and ${\pmb{x}}\in {\mathbb{R}}^{3}$.
A set of generators $\mathcal{B}$ for ${\mathbf{Cov}}({\mathbb{V}})$ is called an *integrity basis*. An integrity basis $\mathcal{B}$ is *minimal* if no proper subset of it is an integrity basis.
A minimal integrity basis for ${\mathbf{Cov}}({\mathbb{S}}^{2}({\mathbb{R}}^{3}))$ is provided by three invariants $\operatorname{tr}({\mathbf{a}})$, $\operatorname{tr}({\mathbf{a}}^{2})$ and $\operatorname{tr}({\mathbf{a}}^{3})$, three order $2$ covariants ${\mathbf{q}}$, ${\mathbf{a}}$ and ${\mathbf{a}}^{2}$ and one order $3$ covariant ${\mathbf{a}}\times {\mathbf{a}}^{2}$.
\[rem:cardinal-minimal-basis\] Of course, a minimal integrity basis is not unique. However, its cardinality $n({\mathbb{V}})$ is a constant. To see this, as in [@DL1985/86], set $${\mathbf{Cov}}^{ + }({\mathbb{V}}) : = \sum_{d + m>0} {\mathbf{Cov}}_{d,m}({\mathbb{V}}),$$ which is an ideal of the graded algebra ${\mathbf{Cov}}({\mathbb{V}})$. Then $({\mathbf{Cov}}^{ + }({\mathbb{V}}))^{2}$ is the space of covariants which can be written as a sum of reducible covariants. For each $(d,m)$ such that $d + m>0$, let $\delta_{d,m}$ be the codimension of $({\mathbf{Cov}}^{ + }({\mathbb{V}}))^{2}_{d,m}$ in ${\mathbf{Cov}}_{d,m}({\mathbb{V}})$. Since ${\mathbf{Cov}}({\mathbb{V}})$ is finitely generated, there exists an integer $p$ such that $\delta_{d,m} = 0$ for $d + m \ge p$ and we can define $$n({\mathbb{V}}) : = \sum_{d,m} \delta_{d,m}.$$ Then, any *minimal integrity basis* is of cardinal $n({\mathbb{V}})$. As far as we know, there is no way to obtain the constant $n({\mathbb{V}})$ but to compute a minimal basis.
A minimal integrity basis for the covariant algebra of H4 {#sec:H4-covariant-algebra}
=========================================================
In this section, we propose to describe a minimal integrity basis for ${\mathbf{Cov}}({\mathbb{H}}^{4})$. As detailed in , ${\mathbf{Cov}}({\mathbb{H}}^{4})$ is connected with the invariant algebra ${\mathbf{Inv}}({\mathrm{S}_{8}}\oplus{\mathrm{S}_{2}})$ (theorem \[thm:decomplexification\]), where ${\mathrm{S}_{n}}$ is the space of binary forms of degree $n$ (see ). This algebra is itself connected to the covariant algebra of the binary form of degree 8, ${\mathbf{Cov}}({\mathrm{S}_{8}})$ (theorem \[thm:basis-for-S2n-oplus-S2\]). A minimal covariant basis for ${\mathbf{Cov}}({\mathrm{S}_{8}})$ is known at least partially since 1880 and was first produced by von Gall [@vGal1880] (see also [@Bed2008; @Cro2002; @Oli2017; @OKA2017]). These results have been used to obtain degrees and orders of a minimal basis for ${\mathbf{Cov}}({\mathbb{H}}^{4})$ which are given in .
degree / order 0 1 2 3 4 5 6 7 9 \# Cum
---------------- --- ---- ---- ---- --- --- --- --- --- ---- ----- --
0 - - 1 - - - - - - 1 1
1 - - - - 1 - - - - 1 2
2 1 - 1 - 1 - 1 - - 4 6
3 1 - 1 1 1 1 1 1 1 8 14
4 1 - 2 1 1 2 1 1 1 10 24
5 1 1 2 2 1 3 - 1 - 11 35
6 1 1 2 3 1 1 - - - 9 44
7 1 2 2 3 - - - - - 8 52
8 1 2 2 2 - - - - - 7 59
9 1 3 1 - - - - - - 5 64
10 1 2 - - - - - - - 3 67
11 - 2 - - - - - - - 2 69
12 - 1 - - - - - - - 1 70
Tot 9 14 14 12 6 7 3 3 2 70
: Degrees and orders of a minimal covariant basis for ${\mathbf{Cov}}({\mathbb{H}}^{4})$[]{data-label="tab:deg-ord-cov-H4"}
Once we know the information provided in , we have a lot of freedom in the choice of an explicit minimal basis. Checking that a system of 70 arbitrary covariants satisfying the requirements of is a minimal integrity basis requires moreover the knowledge of the *Hilbert series* [@Spr1980] $$H(z,t): = \sum_{d,k\geq 0} a_{d,k}z^dt^k,$$ which encodes the dimension $a_{d,k}$ of each finite dimensional vector space ${\mathbf{Cov}}_{d,k}({\mathbb{H}}^{4})$. However, the Hilbert series $H(z,t)$ is a rational function which can be computed *a priori* [@Spr1980; @Spr1983; @LP1990; @Stu2008; @Bed2009; @Bed2011], using the Molien-Weyl formula [@Stu2008]: $$H(z,t) = \int_{{\mathrm{SO}}(3)} \frac{1}{\det (I - t\rho_{1}(g))}\frac{1}{\det (I - z\rho_{4}(g))} \, d\mu(g)$$ where $d\mu$ is the Haar measure on ${\mathrm{SO}}(3)$ (see [@Ste1994 Section 4.1]), $\rho_{1}$ is the standard representation of ${\mathrm{SO}}(3)$ on ${\mathbb{R}}^{3}$ and $\rho_{4}$ is the representation of ${\mathrm{SO}}(3)$ on ${\mathbb{H}}^{4}$. Thus, for each module ${\mathbf{Cov}}_{d,k}({\mathbb{H}}^{4})$ where $(d,k)$ appears in , we have checked inductively on $n = d+k$ that adding new covariants of immediate superior degree/order to the subspace generated by reducible covariants of lower order/degree, we obtain a vector space of dimension $a_{d,k}$.
\[thm:H4-covariants\] The polynomial covariant algebra of ${\mathbb{H}}^{4}$ is generated by a minimal basis of 70 homogeneous covariant polynomials, which degree/order are provided in . An explicit basis has been computed in .
In , we have introduced the following symmetric second-order covariants $${\mathbf{d}}_{2} : = \operatorname{tr}_{13} {\mathbf{H}}^{2}, \qquad {\mathbf{d}}_{3} : = \operatorname{tr}_{13} {\mathbf{H}}^{3}, \qquad {\mathbf{c}}_{k} : = {\mathbf{H}}^{k-2}\operatorname{:}{\mathbf{d}}_{2}, \quad k \ge 3.$$ where ${\mathbf{H}}^{n}:={\mathbf{H}}:{\mathbf{H}}^{n-1}$ for $n\geq 2$ and $\operatorname{tr}_{13}{{{\mathbf{A}}}}$ of a fourth order tensor ${{{\mathbf{A}}}}$ is defined as $(\operatorname{tr}_{13}{{{\mathbf{A}}}})_{ij}:={{{\mathbf{A}}}}_{kikj}$ (in any orthonormal basis). We have also used the simplified notation ${\mathbf{a}}{\mathbf{b}}:={\mathbf{a}}\cdot{\mathbf{b}}$, when ${\mathbf{a}}$ and ${\mathbf{b}}$ are second order tensors.
\[rem:d3c3\] Note the following relation $${\mathbf{c}}_{3} = 2\,{\mathbf{d}}_{3}',$$ which can be checked by a direct calculation.
In addition to ${\mathbf{d}}_{2} $ and ${\mathbf{d}}_{3}$, the following second-order covariants were introduced in [@BKO1994]: $$\label{eq:Boehler-covariants}
\begin{aligned}
{\mathbf{d}}_{4} & : = {{\mathbf{d}}_{2}}^{2}, & {\mathbf{d}}_{5} & : = {\mathbf{d}}_{2} ({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}), & {\mathbf{d}}_{6} & : = {{\mathbf{d}}_{2}}^{3}, \\
{\mathbf{d}}_{7} & : = {{\mathbf{d}}_{2}}^{2} ({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}), & {\mathbf{d}}_{8} & : = {{\mathbf{d}}_{2}}^{2} ({\mathbf{H}}^{2}\operatorname{:}{\mathbf{d}}_{2}), & {\mathbf{d}}_{9} & : = {{\mathbf{d}}_{2}}^{2} ({\mathbf{H}}\operatorname{:}{{\mathbf{d}}_{2}}^{2}), \\
{\mathbf{d}}_{10} & : = {{\mathbf{d}}_{2}}^{2} ({\mathbf{H}}^{2}\operatorname{:}{{\mathbf{d}}_{2}}^{2}).
\end{aligned}$$ For $k = 2,3,4,6$, the ${\mathbf{d}}_{k}$ are *symmetric*, while they are not for $k = 5,7,8,9,10$. None of them are harmonic. These covariants were used to define the following invariants: $$\label{eq:Boehler-invariants}
J_{k} : = \operatorname{tr}{\mathbf{d}}_{k} , \qquad k = 2, \dotsc ,10,$$ which constitute a *minimal integrity basis* for ${\mathbb{H}}^{4}$ (see [@BKO1994]). In , we did not use the invariants $J_{k}$ but an alternative set of generators $I_k$. The nine invariants $J_{k}$ are not algebraically independent (neither are the nine invariants $I_k$); they are subject to some algebraic relations, which have been calculated first by Shioda [@Shi1967] (with some minor errors).
Symmetry classes {#sec:symmetry-classes}
================
Symmetry plays a fundamental role in the study of tensor representations. In this section, we recall the definitions of *symmetry groups* and *symmetry classes* of a vector ${\pmb{v}}$ in a finite dimensional representation ${\mathbb{V}}$ of a compact group $G$.
The *symmetry group* of a vector ${\pmb{v}}\in {\mathbb{V}}$ is defined as the subgroup $$G_{{\pmb{v}}} : = {\left\{g\in G,\quad g\star {\pmb{v}}= {\pmb{v}}\right\}}.$$
The *symmetry class* (or isotropy class) of a vector ${\pmb{v}}$ is the conjugacy class of its symmetry group, where the conjugacy class $[H]$ of a subgroup $H$ is defined as $$[H] : = {\left\{gHg^{-1},\quad g\in G\right\}}.$$
There is, of course, no obstruction to extend the concept of symmetry classes to a finite or infinite family of vectors belonging to different (or same) representations of $G$.
Let $\mathcal{F}$ be a finite or infinite family of vectors belonging to different (or same) representations of $G$. We define the isotropy group of $\mathcal{F}$ as the subgroup $$G_{\mathcal{F}} : = \bigcap_{{\pmb{v}}\in \mathcal{F}} G_{{\pmb{v}}}.$$ The *symmetry class* of $\mathcal{F}$ is the conjugacy class of $G_{\mathcal{F}}$ in $G$.
Note that if $\mathcal{F}$ is a vector space and $({\pmb{v}}_{i})_{i\in I}$ is any generating set of $\mathcal{F}$, then $$G_{\mathcal{F}} = \bigcap_{i\in I} G_{{\pmb{v}}_{i}}.$$ In particular, if $({\pmb{v}}_{1}, \dotsc ,{\pmb{v}}_{p})$ is a basis of $\mathcal{F}$, then $$G_{\mathcal{F}} = \bigcap_{j = 1}^{p} G_{{\pmb{v}}_{j}}.$$
Since every symmetry group of a vector ${\pmb{v}}$ in ${\mathbb{V}}$ is a closed subgroup of $G$, we are mainly interested in the closed subgroups of $G$ up to conjugacy. Now we have the following result which can be deduced from [@Bre1972 Proposition 1.9].
The set of conjugacy classes of a compact group $G$ is a *partially ordered set* (poset) induced by inclusion, which is defined as follows: $$[H_{1}] \preceq [H_{2}] \quad \text{if $H_{1}$ is conjugate to a subgroup of $H_{2}$ in $G$}.$$
\[def:at-least-at-most\] Since the symmetry classes of a given representation ${\mathbb{V}}$ form a poset, we will say that a vector ${\pmb{v}}\in {\mathbb{V}}$ (resp. a family $\mathcal{F}$) is *at least* in a given symmetry class $[H]$, if $[H] \preceq [G_{{\pmb{v}}}]$ (resp. $ [H] \preceq[G_{\mathcal{F}}]$). Similarly, we will say that it is *at most* in the symmetry class $[H]$, if $[G_{{\pmb{v}}}] \preceq [H]$ (resp. $[G_{\mathcal{F}}]\preceq [H]$).
Since we are interested in representations of the rotation group ${\mathrm{SO}}(3)$, we will recall the following result [@GSS1988].
\[lem:SO3-closed-subgroups\] Every closed subgroup of ${\mathrm{SO}}(3)$ is conjugate to one of the following list: $${\mathrm{SO}}(3),\, {\mathrm{O}}(2),\, {\mathrm{SO}}(2),\, {\mathbb{D}}_{n} (n \ge 2),\, {\mathbb{Z}}_{n} (n \ge 2),\, {\mathbb{T}},\, {\mathbb{O}},\, {\mathbb{I}},\, \text{and}\, {\mathds{1}}$$ where:
- ${\mathrm{O}}(2)$ is the subgroup generated by all the rotations around the $z$-axis and the order 2 rotation $\sigma : (x,y,z)\mapsto (x,-y,-z)$ around the $x$-axis;
- ${\mathrm{SO}}(2)$ is the subgroup of all the rotations around the $z$-axis;
- for $n \ge 2$, ${\mathbb{Z}}_{n}$ is the unique cyclic subgroup of order $n$ of ${\mathrm{SO}}(2)$, the subgroup of rotations around the $z$-axis;
- for $n \ge 2$, ${\mathbb{D}}_{n}$ is the *dihedral* group, of order $2n$. It is generated by ${\mathbb{Z}}_{n}$ and $\sigma :(x,y,z)\mapsto (x,-y,-z)$;
- ${\mathbb{T}}$ is the *tetrahedral* group, the orientation-preserving symmetry group of a given tetrahedron, which has order 12;
- ${\mathbb{O}}$ is the *octahedral* group, the orientation-preserving symmetry group of a given cube, which has order 24;
- ${\mathbb{I}}$ is the *icosahedral* group, the orientation-preserving symmetry group of a given dodecahedron, which has order 60;
- ${\mathds{1}}$ is the trivial subgroup, containing only the unit element.
\[rem:octahedral-group\] The octahedral group ${\mathbb{O}}$ is defined as the orientation-preserving symmetry group of a cube whose edges are parallel to the axes of a the canonical basis $({\pmb{e}}_{1}, {\pmb{e}}_{2}, {\pmb{e}}_{3})$ of ${\mathbb{R}}^{3}$. It corresponds to the subgroup $${\left\{g \in {\mathrm{SO}}(3);\; g\star {\pmb{e}}_{i} = \pm {\pmb{e}}_{j}\right\}}$$ of ${\mathrm{SO}}(3)$ which contains 24 elements:
- the identity $I$;
- 3 order 2 rotations around the axes ${\pmb{e}}_{1}$, ${\pmb{e}}_{2}$, ${\pmb{e}}_{3}$;
- 6 order 4 rotations around the axes ${\pmb{e}}_{1}$, ${\pmb{e}}_{2}$, ${\pmb{e}}_{3}$;
- 6 order 2 rotations around the axes ${\pmb{e}}_{1} \pm {\pmb{e}}_{2}$, ${\pmb{e}}_{1} \pm {\pmb{e}}_{3}$, ${\pmb{e}}_{2} \pm {\pmb{e}}_{3}$;
- 8 order 3 rotations around the axis ${\pmb{e}}_{1} \pm {\pmb{e}}_{2} \pm {\pmb{e}}_{3}$.
It is a classical fact, that for any representation ${\mathbb{V}}$ of a Lie group $G$, there exists only a finite number of symmetry classes [@Mos1957; @Man1962]. These classes have been detailed by Ihrig-Golubistky [@IG1984] (see also [@Oli2017]) for irreducible representations of ${\mathrm{SO}}(3)$. We get, in particular, the following posets:
1. For ${\mathbb{H}}^{1}$: $[{\mathrm{SO}}(2)] \preceq [{\mathrm{SO}}(3)]$.
2. For ${\mathbb{H}}^{2}$: $[{\mathbb{D}}_{2}] \preceq [{\mathrm{O}}(2)] \preceq [{\mathrm{SO}}(3)]$.
3. For ${\mathbb{H}}^{3}$: see .
4. For ${\mathbb{H}}^{4}$: see (same as for the Elasticity tensor [@FV1996]).
5. For ${\mathbb{H}}^{5}$: see .
The determination of symmetry classes for *reducible representations* of ${\mathrm{SO}}(3)$ has been achieved by Olive [@Oli2017], who formulated an algorithm to compute theses classes, provided a decomposition into irreducible representations is known. Using these results and the fact that $${\mathbb{S}}^{n}({\mathbb{R}}^{3})\simeq {\mathbb{H}}^{n} \oplus {\mathbb{H}}^{n-2} \oplus \dotsb \oplus {\mathbb{H}}^{n-2r},\quad r=[n/2],$$ by , we deduce the following proposition.
\[prop:symmetry-classes\] We have the following results.
1. The symmetry classes for $n$ ($n \ge 2$) first-order tensors are $${\left\{[{\mathds{1}}], [{\mathrm{SO}}(2)], [{\mathrm{SO}}(3)]\right\}}.$$
2. The symmetry classes for $n$ ($n \ge 2$) second-order symmetric tensors are $${\left\{[{\mathds{1}}], [{\mathbb{Z}}_{2}], [{\mathbb{D}}_{2}], [{\mathrm{O}}(2)], [{\mathrm{SO}}(3)]\right\}}.$$
3. The symmetry classes for one third-order totally symmetric tensor are $${\left\{[{\mathds{1}}], [{\mathbb{Z}}_{2}], [{\mathbb{Z}}_{3}], [{\mathbb{D}}_{2}], [{\mathbb{D}}_{3}], [{\mathbb{T}}], [{\mathrm{SO}}(2)], [{\mathrm{SO}}(3)]\right\}}.$$
4. The symmetry classes for one fourth-order totally symmetric tensor are (like for the Elasticity tensor) $${\left\{[{\mathds{1}}], [{\mathbb{Z}}_{2}], [{\mathbb{D}}_{2}], [{\mathbb{D}}_{3}], [{\mathbb{D}}_{4}], [{\mathbb{O}}], [{\mathrm{O}}(2)], [{\mathrm{SO}}(3)]\right\}}.$$
5. The symmetry classes for one fifth-order totally symmetric tensor are $${\left\{[{\mathds{1}}], [{\mathbb{Z}}_{2}], [{\mathbb{Z}}_{3}], [{\mathbb{Z}}_{4}], [{\mathbb{Z}}_{5}], [{\mathbb{D}}_{2}], [{\mathbb{D}}_{3}], [{\mathbb{D}}_{4}], [{\mathbb{D}}_{5}], [{\mathbb{T}}], [{\mathrm{SO}}(2)], [{\mathrm{SO}}(3)]\right\}}.$$
\[rem:vanishing-isotropic-symmetric-odd-order-tensors\] The harmonic decomposition of a totally symmetric tensor of odd order contains only factors ${\mathbb{H}}^{k}$ with $k$ odd. Moreover, an isotropic tensor in ${\mathbb{H}}^{^k}$ vanishes necessarily if $k \ge 1$ odd. Thus any totally symmetric isotropic tensor of *odd order* vanishes. This is however not true for an even order totally symmetric isotropic tensor.
Dimension of covariant spaces and symmetry {#sec:covariants-symmetry-dimension}
==========================================
Given a linear representation ${\mathbb{V}}$ of ${\mathrm{SO}}(3)$ and ${\pmb{v}}\in {\mathbb{V}}$, we define ${\mathbf{Cov}}_{k}({\pmb{v}})$ as the set of all $k$-order polynomial covariants of ${\pmb{v}}$ (see ). Note that whereas ${\mathbf{Cov}}({\mathbb{V}})$ is a polynomial algebra, and ${\mathbf{Cov}}_{k}({\mathbb{V}})$ is an infinite dimensional vector space, ${\mathbf{Cov}}_{k}({\pmb{v}})$ is the set of all *evaluations* of these covariants on the vector ${\pmb{v}}$. As such, it is a subspace of the finite dimensional real vector space ${\mathcal{P}_{k}}({\mathbb{R}}^{3})$ of homogeneous polynomials of degree $k$ on ${\mathbb{R}}^{3}$, or equivalently of the space ${\mathbb{S}}^{k}({\mathbb{R}}^{3})$ of totally symmetric tensors of order $k$. In this section, we will focus on polynomial covariants of order one and two of a vector ${\pmb{v}}\in {\mathbb{V}}$ and relate the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ and ${\mathbf{Cov}}_{2}({\pmb{v}})$ with their respective dimension.
Recall that, thanks to proposition \[prop:symmetry-classes\], the possible symmetry classes for the space ${\mathbf{Cov}}_{1}({\pmb{v}})$ are $$[{\mathds{1}}], \quad [{\mathrm{SO}}(2)], \quad [{\mathrm{SO}}(3)],$$ whereas, for ${\mathbf{Cov}}_{2}({\pmb{v}})$, they are $$[{\mathds{1}}], \quad [{\mathbb{Z}}_{2}], \quad [{\mathbb{D}}_{2}], \quad [{\mathrm{O}}(2)], \quad [{\mathrm{SO}}(3)].$$
\[prop:cov1-symmetry-classes\] Given ${\pmb{v}}\in {\mathbb{V}}$, $\dim {\mathbf{Cov}}_{1}({\pmb{v}})$ is either $0$, $1$ or $3$. Moreover, the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is:
1. $[{\mathrm{SO}}(3)]$ if and only if ${\mathbf{Cov}}_{1}({\pmb{v}}) = {\left\{0\right\}}$;
2. $[{\mathrm{SO}}(2)]$ if and only if $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 1$;
3. $[{\mathds{1}}]$ if and only if $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 3$.
\(1) If the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is $[{\mathrm{SO}}(3)]$, then, every first-order covariant vanishes and thus ${\mathbf{Cov}}_{1}({\pmb{v}}) = {\left\{0\right\}}$. Conversely, if ${\mathbf{Cov}}_{1}({\pmb{v}}) = 0$ then its symmetry class is $[{\mathrm{SO}}(3)]$.
\(2) Suppose now that the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is $[{\mathrm{SO}}(2)]$. Without loss of generality, we can suppose that the isotropy group of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is exactly ${\mathrm{SO}}(2)$. Then, $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) \ge 1$ but all first-order covariant are colinear to ${\pmb{e}}_{3}$ and thus $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 1$. Conversely, suppose that $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 1$ and let ${\pmb{u}}\ne 0$ be a basis of ${\mathbf{Cov}}_{1}({\pmb{v}})$. Then the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is just $[G_{{\pmb{u}}}] = [{\mathrm{SO}}(2)]$.
\(3) Finally, suppose that the symmetry class of ${\mathbf{Cov}}_{1}({\pmb{v}})$ is $[{\mathds{1}}]$. Then $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) \ge 2$. But if ${\pmb{u}}, {\pmb{w}}$ are two independent first-order covariants then ${\pmb{u}}\times {\pmb{w}}$ is also a first-order covariant, so that $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 3$. Conversely, if $\dim {\mathbf{Cov}}_{1}({\pmb{v}}) = 3$, we can find two independent covariants ${\pmb{u}}, {\pmb{w}}$ and thus $$G_{{\pmb{u}}} \cap G_{{\pmb{w}}} = {\mathds{1}}.$$
The case of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is more involving. Note first that the Euclidean second-order tensor ${\mathbf{q}}$ is always in ${\mathbf{Cov}}_{2}({\pmb{v}})$, thus $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \ge 1$ for every ${\pmb{v}}\in {\mathbb{V}}$. Moreover, given two covariants ${\mathbf{a}},{\mathbf{b}}$ in ${\mathbf{Cov}}_{2}({\pmb{v}})$, then $$({\mathbf{a}}{\mathbf{b}})^{s} : = \frac{1}{2}({\mathbf{a}}{\mathbf{b}}+ {\mathbf{b}}{\mathbf{a}})$$ belongs to ${\mathbf{Cov}}_{2}({\pmb{v}})$, where ${\mathbf{a}}{\mathbf{b}}$ is the standard matrix product.
\[lem:q-a-a2\] Let ${\mathbf{a}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$. Then,
1. ${\mathbf{a}}$ is orthotropic if and only if $\dim \langle {\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2} \rangle = 3$;
2. ${\mathbf{a}}$ is transversely isotropic if and only if $\dim \langle {\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2} \rangle = 2$.
Without loss of generality, we can suppose that ${\mathbf{a}}= \mathrm{diag}(\lambda_{1}, \lambda_{2}, \lambda_{3})$. Then, it is easy to check that ${\mathbf{q}}$, ${\mathbf{a}}$, ${\mathbf{a}}^{2}$ are linearly independent if and only if $$(\lambda_{2}-\lambda_{1})(\lambda_{3}-\lambda_{1})(\lambda_{3}-\lambda_{2}) \ne 0.$$ Thus, we get (1). Moreover, if ${\mathbf{a}}$ is transversely isotropic and has thus a double eigenvalue then $\dim \langle {\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2} \rangle \le 2$ but it cannot be one, otherwise, ${\mathbf{a}}$ would be isotropic. Conversely if $\dim \langle {\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2} \rangle = 2$, then ${\mathbf{a}}$ has a double eigenvalue and is hence at least transversely isotropic but it cannot be isotropic (otherwise $\dim \langle {\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2} \rangle = 1$). This achieves the proof.
Recall that a pair $({\mathbf{a}},{\mathbf{b}})$ of symmetric second-order tensors is either isotropic, transversely isotropic, orthotropic, monoclinic or triclinic by proposition \[prop:symmetry-classes\]. We have, moreover, the following result.
\[lem:orthotropic-linear-combination\] Let $({\mathbf{a}},{\mathbf{b}})$ be a pair of symmetric second-order tensors, which is either orthotropic, monoclinic or triclinic. Then, there exists a linear combination of ${\mathbf{a}}$ and ${\mathbf{b}}$ which is orthotropic.
If either ${\mathbf{a}}$ or ${\mathbf{b}}$ is orthotropic, we are done. Otherwise, both ${\mathbf{a}}$ and ${\mathbf{b}}$ are transversely isotropic, neither being isotropic. Let ${\mathbf{a}}^{\prime}$ and ${\mathbf{b}}^{\prime}$ be the deviatoric parts of ${\mathbf{a}}$ and ${\mathbf{b}}$ respectively. Note that ${\mathbf{a}}^{\prime}$, ${\mathbf{b}}^{\prime}$ are linearly independent and both transversely isotropic. If we can show that there exists a linear combination $\alpha {\mathbf{a}}^{\prime} + \beta {\mathbf{b}}^{\prime}$ which is orthotropic, then, we are done, because $$\alpha {\mathbf{a}}+ \beta {\mathbf{b}}= \alpha {\mathbf{a}}^{\prime} + \beta {\mathbf{b}}^{\prime} + \frac{1}{3}\left(\alpha \operatorname{tr}{\mathbf{a}}+ \beta \operatorname{tr}{\mathbf{b}}\right){\mathbf{q}}$$ is orthotropic. Let $$\tilde{{\mathbf{a}}} = {\mathbf{a}}^{\prime} - \frac{\operatorname{tr}({\mathbf{a}}^{\prime}{\mathbf{b}}^{\prime})}{\operatorname{tr}({{\mathbf{b}}^{\prime}}^{2})}{\mathbf{b}}^{\prime}, \qquad \tilde{{\mathbf{b}}} = {\mathbf{b}}^{\prime}$$ If $\tilde{{\mathbf{a}}}$ is orthotropic, we are done. Otherwise $\tilde{{\mathbf{a}}}$, $\tilde{{\mathbf{b}}}$ are two linearly independent, transversely isotropic deviators such that $\operatorname{tr}(\tilde{{\mathbf{a}}}\tilde{{\mathbf{b}}}) = 0$. Now, the discriminant of the characteristic polynomial of a deviatoric tensor ${\mathbf{d}}$ writes as $$(\operatorname{tr}({\mathbf{d}}^{2}))^{3}/2 - 3(\operatorname{tr}({\mathbf{d}}^{3}))^{2}.$$ Hence a deviatoric tensor ${\mathbf{d}}$ is orthotropic if and only if $$(\operatorname{tr}({\mathbf{d}}^{2}))^{3} - 6(\operatorname{tr}({\mathbf{d}}^{3}))^{2} \ne 0.$$ Let ${\mathbf{d}}(t): = t \tilde{{\mathbf{a}}} + (1-t)\tilde{{\mathbf{b}}}$. Then ${\mathbf{d}}(t)$ is orthotropic if and only if $$p(t): = (\operatorname{tr}({\mathbf{d}}(t)^{2}))^{3}-6(\operatorname{tr}({\mathbf{d}}(t)^{3}))^{2} \ne 0.$$ Moreover, a direct computation shows that the coefficient of $t^{2}$ in the polynomial $p(t)$ is $$3\operatorname{tr}({\tilde{{\mathbf{a}}}}^{2})\operatorname{tr}({\tilde{{\mathbf{b}}}}^{2})^{2}\ne 0.$$ Hence, there exists $t \in {\mathbb{R}}$ such that $p(t)\ne 0$ and for this value, ${\mathbf{d}}(t)$ is orthotropic. We have thus found a linear combination of $\tilde{{\mathbf{a}}}$, $\tilde{{\mathbf{b}}}$, and therefore of ${\mathbf{a}}^{\prime}$, ${\mathbf{b}}^{\prime}$ which is orthotropic. This achieves the proof.
\[cor:dimension-trans-iso-subspaces\] Let $F$ be a sub-vector space of ${\mathbb{S}}^{2}({\mathbb{R}}^{3})$ with $\dim F \ge 3$. Then, $F$ contains an orthotropic element.
Suppose that each element in $F$ is at least transversely isotropic. Then, by lemma \[lem:orthotropic-linear-combination\], each pair $({\mathbf{a}},{\mathbf{b}})$ of elements in $F$ is at least transversely isotropic. If each element in $F$ is isotropic, then $F$ is of dimension 0 or 1. If $F$ contains a transversely isotropic element ${\mathbf{t}}$, then for every ${\mathbf{a}}\in F$, the pair $({\mathbf{t}},{\mathbf{a}})$ is transversely isotropic and thus ${\mathbf{a}}= \alpha {\mathbf{t}}+ \beta {\mathbf{q}}$. Thus $\dim F \le 2$. This achieves the proof.
Given an orthonormal basis $({\pmb{e}}_{1},{\pmb{e}}_{2},{\pmb{e}}_{3})$ of ${\mathbb{R}}^{3}$, we will consider the following natural basis of ${\mathbb{S}}^{2}({\mathbb{R}}^{3})$ $${\mathbf{e}}_{ii}: = {\pmb{e}}_{i}\otimes {\pmb{e}}_{i} \quad \text{(no sum)},\quad {\mathbf{e}}_{ij}: = {\pmb{e}}_{i}\otimes {\pmb{e}}_{j} + {\pmb{e}}_{j}\otimes {\pmb{e}}_{i},\quad (i<j),$$ which is orthogonal but not orthonormal.
\[prop:cov2-symmetry-classes\] Given ${\pmb{v}}\in {\mathbb{V}}$, $\dim {\mathbf{Cov}}_{2}({\pmb{v}})$ is either $1$, $2$, $3$, $4$ or $6$. Moreover, the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is:
1. $[{\mathrm{SO}}(3)]$ if and only if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 1$;
2. $[{\mathrm{O}}(2)]$ if and only if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 2$;
3. $[{\mathbb{D}}_{2}]$ if and only if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 3$;
4. $[{\mathbb{Z}}_{2}]$ if and only if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 4$;
5. $[{\mathds{1}}]$ if and only if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 6$.
\(1) If the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathrm{SO}}(3)]$, then, every symmetric second-order covariant is proportional to ${\mathbf{q}}$ and hence $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 1$. Conversely, if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 1$ then ${\mathbf{q}}$ generates ${\mathbf{Cov}}_{2}({\pmb{v}})$, and its symmetry class is $[{\mathrm{SO}}(3)]$.
\(2) Suppose that the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathrm{O}}(2)]$. Then, without loss of generality, we can suppose that each symmetric second-order covariant writes as $\mathrm{diag}(\lambda,\lambda,\mu)$ and hence that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \le 2$. Since it cannot be $1$, otherwise ${\mathbf{Cov}}_{2}({\pmb{v}})$ would be reduced the one-dimensional space generated by ${\mathbf{q}}$, it must be $2$. Conversely, if $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 2$, then, there exists some non–isotropic second-order covariant ${\mathbf{a}}$ such that $({\mathbf{q}},{\mathbf{a}})$ is basis of ${\mathbf{Cov}}_{2}({\pmb{v}})$. Since ${\mathbf{a}}$ cannot be orthotropic, otherwise $({\mathbf{q}},{\mathbf{a}},{\mathbf{a}}^{2})$ would be linearly independent, by lemma \[lem:q-a-a2\], ${\mathbf{a}}$ is necessarily transversely isotropic and so is ${\mathbf{Cov}}_{2}({\pmb{v}})$.
\(3) Suppose that the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathbb{D}}_{2}]$. Then without loss of generality we can assume that each symmetric second-order covariant writes as $\mathrm{diag}(\lambda_{1},\lambda_{2},\lambda_{3})$ and hence that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \le 3$. Since this dimension cannot be $1$, neither $2$ due to points (1) and (2), its must be $3$. Conversely, suppose that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 3$. Then, by corollary \[cor:dimension-trans-iso-subspaces\], ${\mathbf{Cov}}_{2}({\pmb{v}})$ contains an orthotropic tensor ${\mathbf{c}}$, and we are done by lemma \[lem:q-a-a2\] because the orthotropic triplet $({\mathbf{q}},{\mathbf{c}},{\mathbf{c}}^{2})$ is a basis of ${\mathbf{Cov}}_{2}({\pmb{v}})$.
\(4) Suppose that the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathbb{Z}}_{2}]$. Then without loss of generality we can suppose that each symmetric second-order covariant writes as $$\left(
\begin{array}{ccc}
a_{11} & a_{12} & 0 \\
a_{12} & a_{22} & 0 \\
0 & 0 & a_{33} \\
\end{array}
\right)$$ and hence that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \le 4$. Since this dimension is necessarily $>3$ by (1), (2) and (3), it is $4$. Conversely, suppose that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 4$. Then by corollary \[cor:dimension-trans-iso-subspaces\], there exists an orthotropic covariant ${\mathbf{c}}$ in ${\mathbf{Cov}}_{2}({\pmb{v}})$ and without loss of generality, we can suppose that this covariant is diagonal. Then, by lemma \[lem:q-a-a2\], $\langle {\mathbf{q}}, {\mathbf{c}}, {\mathbf{c}}^{2} \rangle$ is a vector basis of the space of diagonal tensors, so that $\langle {\mathbf{q}}, {\mathbf{c}}, {\mathbf{c}}^{2} \rangle=\langle {\mathbf{e}}_{11}, {\mathbf{e}}_{22}, {\mathbf{e}}_{33} \rangle$, and thus each ${\mathbf{e}}_{ii}$ belongs to ${\mathbf{Cov}}_{2}({\pmb{v}})$. Let ${\mathbf{a}}$ be a second-order covariant such that $({\mathbf{e}}_{11},{\mathbf{e}}_{22},{\mathbf{e}}_{33},{\mathbf{a}})$ is a basis of ${\mathbf{Cov}}_{2}({\pmb{v}})$. Without loss of generality, we can assume that $${\mathbf{a}}= a_{12}{\mathbf{e}}_{12} + a_{13}{\mathbf{e}}_{13} + a_{23}{\mathbf{e}}_{23},$$ where the $a_{ij}$ do not vanish altogether, for instance $a_{12}\neq 0$. Then $$({\mathbf{e}}_{11}{\mathbf{a}})^{s} + ({\mathbf{e}}_{22}{\mathbf{a}})^{s} - ({\mathbf{e}}_{33}{\mathbf{a}})^{s} = a_{12}{\mathbf{e}}_{12}$$ belongs to ${\mathbf{Cov}}_{2}({\pmb{v}})$ and so does ${\mathbf{e}}_{12}$. Hence $$({\mathbf{e}}_{11},{\mathbf{e}}_{22},{\mathbf{e}}_{33},{\mathbf{e}}_{12})$$ is a basis of ${\mathbf{Cov}}_{2}({\pmb{v}})$ which has therefore the symmetry $[{\mathbb{Z}}_{2}]$.
\(5) Suppose that the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathds{1}}]$. Then $$\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \ge 5,$$ by (1), (2), (3) and (4). By corollary \[cor:dimension-trans-iso-subspaces\], there exists an orthotropic covariant ${\mathbf{c}}$ in ${\mathbf{Cov}}_{2}({\pmb{v}})$, and like in the proof of (4), we can assume that $${\mathbf{e}}_{11},{\mathbf{e}}_{22},{\mathbf{e}}_{33} \in{\mathbf{Cov}}_{2}({\pmb{v}}).$$ Since, $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) \ge 5$, the space ${\mathbf{Cov}}_{2}({\pmb{v}})$ contains two linearly independent covariants, which write $$\begin{aligned}
{\mathbf{a}}& = & a_{12}{\mathbf{e}}_{12} + a_{13}{\mathbf{e}}_{13} + a_{23}{\mathbf{e}}_{23}, \\
{\mathbf{b}}& = & b_{12}{\mathbf{e}}_{12} + b_{13}{\mathbf{e}}_{13} + b_{23}{\mathbf{e}}_{23},
\end{aligned}$$ and we can assume (without loss of generality) that the minor $$a_{12}b_{13} - a_{13}b_{12} \ne 0.$$ As in the proof of (4), we conclude then that both ${\mathbf{e}}_{12}$ and ${\mathbf{e}}_{13}$ belong to ${\mathbf{Cov}}_{2}({\pmb{v}})$. But then $$({\mathbf{e}}_{12}{\mathbf{e}}_{13})^{s} = \frac{1}{2}{\mathbf{e}}_{23}$$ belongs to ${\mathbf{Cov}}_{2}({\pmb{v}})$ and thus $\dim {\mathbf{Cov}}_{2}({\pmb{v}})=6$. Conversely, suppose that $\dim {\mathbf{Cov}}_{2}({\pmb{v}}) = 6$. Then the only possibility is that the symmetry class of ${\mathbf{Cov}}_{2}({\pmb{v}})$ is $[{\mathds{1}}]$ by (1), (2), (3) and (4). This achieves the proof.
Covariant criteria for tensor’s symmetry {#sec:covariant-criteria}
========================================
In this section, we formulate covariant criteria which restrict the symmetry class of second and fourth order tensors, using the vanishing of some of their covariants.
Second order tensors {#subsec:second-order-tensors}
--------------------
\[lem:axa2=0\] Let ${\mathbf{a}}$ be a symmetric second-order tensor. Then, ${\mathbf{a}}$ is at least transversely isotropic if and only if ${\mathbf{a}}\times {\mathbf{a}}^{2} = 0$.
Without loss of generality, we can assume that $${\mathbf{a}}= \mathrm{diag}(\lambda_{1},\lambda_{2},\lambda_{3}).$$ Then, the polynomial form of ${\mathbf{a}}\times {\mathbf{a}}^{2}$, writes $$\left( \lambda_{2} - \lambda_{1} \right) \left( \lambda_{3} - \lambda_{1}\right) \left( \lambda_{3} - \lambda_{2} \right) xyz.$$ Thus, it vanishes if and only if ${\mathbf{a}}$ is at least transversely isotropic. This achieves the proof.
\[rem:tetrahedral-symmetry\] If ${\mathbf{a}}$ is not orthotropic, then ${\mathbf{a}}\times {\mathbf{a}}^{2}$ vanishes and is thus isotropic. Otherwise, the polynomial form of ${\mathbf{a}}\times {\mathbf{a}}^{2}$ writes $k\, xyz$, with $k= \left( \lambda_{2} - \lambda_{1} \right) \left( \lambda_{3} - \lambda_{1}\right) \left( \lambda_{3} - \lambda_{2} \right) \ne 0$. This form is, of course, invariant by ${\mathbb{D}}_{2}$ but it has more symmetries. Indeed, it is invariant by the *fourth-order* rotations around $Ox$, $Oy$ and $Oz$. Considering the symmetry classes of ${\mathbb{S}}^{3}({\mathbb{R}}^{3})$ (proposition \[prop:symmetry-classes\]), and , we conclude that ${\mathbf{a}}\times {\mathbf{a}}^{2}$ has *tetrahedral* symmetry $[{\mathbb{T}}]$. An immediate corollary of this result is that $\operatorname{tr}({\mathbf{a}}\times {\mathbf{a}}^{2}) = 0$ and thus that ${\mathbf{a}}\times {\mathbf{a}}^{2}$ is harmonic, independently of the symmetry of ${\mathbf{a}}$.
\[lem:axb=0\] Let ${\mathbf{a}},{\mathbf{b}}$ be symmetric second-order tensors and suppose that ${\mathbf{a}}$ is transversely isotropic. Then, $({\mathbf{a}},{\mathbf{b}})$ is transversely isotropic if and only if ${\mathbf{a}}\times {\mathbf{b}}= 0$.
Suppose first that $({\mathbf{a}},{\mathbf{b}})$ is transversely isotropic then ${\mathbf{a}}\times {\mathbf{b}}$ is at least transversely isotropic and since it is a third-order totally symmetric tensor, it must be isotropic by proposition \[prop:symmetry-classes\] and thus vanishes by Remark \[rem:vanishing-isotropic-symmetric-odd-order-tensors\]. To prove the converse, we will use the polynomial representative ${\mathrm{a}},{\mathrm{b}}$ of ${\mathbf{a}},{\mathbf{b}}$ (see Section \[sec:sym-harm-tensors\]). The linear equation ${\mathbf{a}}\times {\mathbf{b}}= 0$ reads then $\det ({\pmb{x}}, \nabla {\mathrm{a}}, \nabla {\mathrm{b}}) = 0$. Without loss of generality we can assume that $G_{{\mathbf{a}}} = {\mathrm{O}}(2)$ and thus that $${\mathrm{a}}= \lambda (x^{2} + y^{2}) + \mu z^{2}, \qquad \lambda \ne \mu .$$ The solution is then $${\mathrm{b}}= k_{1} (x^{^{2}} + y^{2}) + k_{2} z^{2},$$ which is invariant by ${\mathrm{O}}(2)$. This achieves the proof.
Given two symmetric second order tensors ${\mathbf{a}},{\mathbf{b}}$ on the euclidean space ${\mathbb{R}}^{3}$, their commutator, a second-order skew-symmetric tensor $$[{\mathbf{a}},{\mathbf{b}}]:={\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}{\mathbf{a}}$$ can be recast as the first-order covariant $$\operatorname{tr}({\mathbf{a}}\times {\mathbf{b}}) = \frac{1}{3} \, \pmb \varepsilon:({\mathbf{a}}{\mathbf{b}}).$$
We have thus
\[lem:orthotropic-pair\] The three conditions are equivalent :
1. the pair $({\mathbf{a}}, {\mathbf{b}})$ is *at least orthotropic*.
2. $\operatorname{tr}({\mathbf{a}}\times {\mathbf{b}}) = 0$.
3. ${\mathbf{a}}, {\mathbf{b}}$ commute.
\[cor:orthotropic-pair\] Let ${\mathbf{a}},{\mathbf{b}}$ be symmetric second-order tensors. Then, $({\mathbf{a}},{\mathbf{b}})$ is orthotropic if and only if $\operatorname{tr}({\mathbf{a}}\times {\mathbf{b}}) = 0$ and $${\mathbf{a}}\times {\mathbf{a}}^{2} \ne 0, \quad \text{or} \quad {\mathbf{b}}\times {\mathbf{b}}^{2} \ne 0, \quad \text{or} \quad {\mathbf{a}}\times {\mathbf{b}}\ne 0.$$
If $({\mathbf{a}},{\mathbf{b}})$ is orthotropic, then the first-order covariant $\operatorname{tr}({\mathbf{a}}\times {\mathbf{b}})$ is necessarily isotropic by proposition \[prop:symmetry-classes\] and thus vanishes by Remark \[rem:vanishing-isotropic-symmetric-odd-order-tensors\]. Moreover, either ${\mathbf{a}}$ or ${\mathbf{b}}$ is orthotropic and thus $${\mathbf{a}}\times {\mathbf{a}}^{2} \ne 0, \qquad \text{or} \qquad {\mathbf{b}}\times {\mathbf{b}}^{2} \ne 0,$$ or both of them are transversely isotropic. In that case we necessarily have ${\mathbf{a}}\times {\mathbf{b}}\ne 0$ by lemma \[lem:axb=0\]. Conversely, if $\operatorname{tr}({\mathbf{a}}\times {\mathbf{b}}) = 0$, then the pair $({\mathbf{a}},{\mathbf{b}})$ is at least orthotropic by lemma \[lem:orthotropic-pair\]. If either ${\mathbf{a}}$ or ${\mathbf{b}}$ is orthotropic, then so is $({\mathbf{a}},{\mathbf{b}})$. Otherwise, both ${\mathbf{a}}$ and ${\mathbf{b}}$ are at least transversely isotropic, but then the condition ${\mathbf{a}}\times {\mathbf{b}}\ne 0$ forbids the pair $({\mathbf{a}},{\mathbf{b}})$ to be at least transversely isotropic. It is thus orthotropic.
We will now formulate coordinate-free conditions to classify the symmetry class of an $n$-tuple of symmetric second-order tensors.
\[thm:n-quadratic-forms\] Let $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ be an $n$-tuple of second-order symmetric tensors. Then:
1. $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *isotropic* if and only if $${\mathbf{a}}_{k}^{\prime} = 0, \quad 1 \le k \le n ,$$ where ${\mathbf{a}}_{k}^{\prime}$ is the deviatoric part of ${\mathbf{a}}_{k}$.
2. $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *transversely isotropic* if and only if there exists ${\mathbf{a}}_{j}$ such that $${\mathbf{a}}_{j}^{\prime} \ne 0, \qquad {\mathbf{a}}_{j} \times {\mathbf{a}}_{j}^{2} = 0,$$ and $${\mathbf{a}}_{j} \times {\mathbf{a}}_{k} = 0, \quad 1 \le k \le n .$$
3. $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *orthotropic* if and only if $$\operatorname{tr}({\mathbf{a}}_{k} \times {\mathbf{a}}_{l}) = 0, \quad 1 \le k,l \le n ,$$ and there exists ${\mathbf{a}}_{j}$ such that ${\mathbf{a}}_{j} \times {\mathbf{a}}_{j}^{2} \ne 0$ or there exists a pair $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ such that ${\mathbf{a}}_{i} \times {\mathbf{a}}_{j} \ne 0$.
4. $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *monoclinic* if and only if there exists a pair $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ such that $\pmb{\omega} := \operatorname{tr}({\mathbf{a}}_{i} \times {\mathbf{a}}_{j}) \ne 0$ and $$({\mathbf{a}}_{k}\pmb{\omega}) \times \pmb{\omega} = 0, \quad 1 \le k \le n .$$
\(1) $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *isotropic* if and only if ${\mathbf{a}}_{k} =\lambda_{k}{\mathbf{q}}$ for $1 \le k \le n$, which is equivalent to the condition that ${\mathbf{a}}_{k}^{\prime} = 0$ for $1 \le k \le n$.
\(2) If $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *transversely isotropic*, then, each ${\mathbf{a}}_{k}$ is at least transversely isotropic and one of them, say ${\mathbf{a}}_{j}$, is transversely isotropic. Thus ${\mathbf{a}}_{j}^{\prime} \ne 0$ and ${\mathbf{a}}_{j} \times {\mathbf{a}}_{j}^{2} = 0$ by lemma \[lem:axa2=0\]. Moreover, each pair $({\mathbf{a}}_{j}, {\mathbf{a}}_{k})$ is at least transversely isotropic and thus ${\mathbf{a}}_{j} \times {\mathbf{a}}_{k} = 0$ by lemma \[lem:axb=0\]. Conversely, if conditions in $(2)$ are satisfied, then ${\mathbf{a}}_{j}$ is transversely isotropic and each pair $({\mathbf{a}}_{j}, {\mathbf{a}}_{k})$ is transversely isotropic by lemma \[lem:axb=0\]. Thus $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is transversely isotropic.
\(3) If $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *orthotropic*, then, the ${\mathbf{a}}_{k}$ commute with each other and thus $\operatorname{tr}({\mathbf{a}}_{k} \times {\mathbf{a}}_{l}) = 0$ ($1 \le k,l \le n$) by lemma \[lem:orthotropic-pair\]. Moreover, either there exists $j \in {\left\{1,\dotsc ,n\right\}}$ such that ${\mathbf{a}}_{j}$ is orthotropic and thus ${\mathbf{a}}_{j} \times {\mathbf{a}}_{j}^{2} \ne 0$ or all the ${\mathbf{a}}_{k}$ are at least transversely isotropic. In that case, a pair of them, say $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ is orthotropic and thus ${\mathbf{a}}_{i} \times {\mathbf{a}}_{j} \ne 0$. Conversely, if $\operatorname{tr}({\mathbf{a}}_{k} \times {\mathbf{a}}_{l}) = 0$ for all $k,l$, then, we can find a basis in which there are all diagonal and the symmetry class of $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is thus at least $[{\mathbb{D}}_{2}]$. If there exists ${\mathbf{a}}_{j}$ such that ${\mathbf{a}}_{j} \times {\mathbf{a}}_{j}^{2} \ne 0$, we are done. Otherwise, all the ${\mathbf{a}}_{k}$ are at least transversely isotropic, but there exists a pair $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ such that ${\mathbf{a}}_{i} \times {\mathbf{a}}_{j} \ne 0$. Hence, both ${\mathbf{a}}_{i},{\mathbf{a}}_{j}$ are tran sversely isotropic and the pair $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ is orthotropic by lemma \[lem:axb=0\].
\(4) If $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is *monoclinic*, then, its elements have a common eigenvector, $\pmb{\omega}$, so that $({\mathbf{a}}_{k}\pmb{\omega}) \times \pmb{\omega} = 0$ ($1 \le k \le n$). Moreover, there exists a pair $({\mathbf{a}}_{i},{\mathbf{a}}_{j})$ such that $\operatorname{tr}({\mathbf{a}}_{i} \times {\mathbf{a}}_{j}) \ne 0$ and thus $\operatorname{tr}({\mathbf{a}}_{i} \times {\mathbf{a}}_{j}) = \lambda \pmb{\omega}$ with $\lambda \ne 0$. Conversely, if $\pmb{\omega} := \operatorname{tr}({\mathbf{a}}_{i} \times {\mathbf{a}}_{j}) \ne 0$, then $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ is at most monoclinic. But the condition $({\mathbf{a}}_{k}\pmb{\omega}) \times \pmb{\omega} = 0$ for all $k$ means that $\pmb{\omega}$ is a common eigenvector of ${\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n}$ and thus the symmetry group of $({\mathbf{a}}_{1}, \dotsc ,{\mathbf{a}}_{n})$ contains the second-order rotation around $\pmb{\omega}$.
Fourth order tensors {#subsec:fourth-order-tensors}
--------------------
\[lem:Sxd=0\] Let ${\mathbf{t}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ be transversely isotropic and ${\mathbf{S}}\in {\mathbb{S}}^{4}({\mathbb{R}}^{3})$. Then, $({\mathbf{S}},{\mathbf{t}})$ is transversely isotropic if and only if ${\mathbf{S}}\times {\mathbf{t}}= 0$.
Suppose first that $({\mathbf{S}},{\mathbf{t}})$ is transversely isotropic, then ${\mathbf{S}}\times {\mathbf{t}}$ is at least transversely isotropic and since it is a fifth-order symmetric tensor, it must be isotropic by proposition \[prop:symmetry-classes\] and thus vanishes by Remark \[rem:vanishing-isotropic-symmetric-odd-order-tensors\]. To prove the converse, let ${\mathrm{p}}, {\mathrm{t}}$ be the polynomial representatives of ${\mathbf{S}},{\mathbf{t}}$. Then, the linear equation ${\mathbf{S}}\times {\mathbf{t}}= 0$ reads $\det ({\pmb{x}}, \nabla {\mathrm{p}}, \nabla {\mathrm{t}}) = 0$. Without loss of generality we can assume that $G_{{\mathbf{t}}} = {\mathrm{O}}(2)$ and thus that $${\mathrm{t}}= \lambda (x^{2} + y^{2}) + \mu z^{2}, \qquad \lambda \ne \mu$$ and the solution is $${\mathrm{p}}= k_{1}z^{4} + k_{2} (x^{^{2}} + y^{2})z^{2} + k_{3}(x^{2}+y^{2})^{2},$$ which is invariant by ${\mathrm{O}}(2)$. This achieves the proof.
\[lem:tr(Hxd)=0\] Let ${\mathbf{t}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ be transversely isotropic and ${\mathbf{H}}\in {\mathbb{H}}^{4}$. Then, $({\mathbf{H}},{\mathbf{t}})$ is at least tetragonal if and only if $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0$.
Suppose first that $({\mathbf{H}},{\mathbf{t}})$ is at least tetragonal, then $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}})$ is at least tetragonal and since it is a third-order symmetric tensor, it must be isotropic by proposition \[prop:symmetry-classes\] and thus vanishes by Remark \[rem:vanishing-isotropic-symmetric-odd-order-tensors\]. To prove the converse, let ${\mathrm{p}}, {\mathrm{t}}$ be the polynomial representatives of ${\mathbf{H}},{\mathbf{t}}$. Then, the linear equation $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0$ reads $\triangle (\det ({\pmb{x}}, \nabla {\mathrm{p}}, \nabla {\mathrm{t}})) = 0$, where $\triangle$ is the Laplacian. Without loss of generality we can assume that $G_{{\mathbf{t}}} = {\mathrm{O}}(2)$ and thus that $${\mathrm{t}}= \lambda (x^{2} + y^{2}) + \mu z^{2}, \qquad \lambda \ne \mu$$ and the solution is $$\begin{gathered}
{\mathrm{p}}= k_{1} \left(6z^{2}(x^{2} + y^{2}) - (x^{4} + y^{4}) + 2z^{4}\right) \\
+ k_{2} \left(6x^{2}y^{2} - (x^{4} + y^{4})\right) + k_{3} xy\left(x^{2} - y^{2}\right),
\end{gathered}$$ which is invariant by ${\mathbb{Z}}_{4}$ and has thus at least the symmetry $[{\mathbb{D}}_{4}]$. Hence, $({\mathbf{H}},{\mathbf{t}})$ is at least tetragonal. This achieves the proof.
The cubic symmetry appears, in practice, as the more subtle to deal with. We will formulate, in the next lemma, more precise statements which allow to detect the symmetry class of a pair $({\mathbf{H}},{\mathbf{t}})$ when ${\mathbf{H}}$ is cubic and ${\mathbf{t}}$ is transversely isotropic. In that case, we know from [@Oli2017] that the symmetry class of a pair $({\mathbf{H}},{\mathbf{t}})$ is one of the following : triclinic, monoclinic, orthotropic, trigonal or tetragonal.
\[lem:cube-orientation\] Let ${\mathbf{H}}$ be a cubic fourth-order harmonic tensor and ${\mathbf{t}}\in{\mathbb{S}}^{2}({\mathbb{R}}^{2})$ be transversely isotropic. Then
1. $({\mathbf{H}},{\mathbf{t}})$ is tetragonal if and only if $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0;$$
2. $({\mathbf{H}},{\mathbf{t}})$ is trigonal if and only if $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0, \quad \text{and} \quad {\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})=0;$$
3. $({\mathbf{H}},{\mathbf{t}})$ is orthotropic if and only if $${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) \ne 0, \quad \text{and} \quad \operatorname{tr}\left({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})\right)=0;$$
4. $({\mathbf{H}},{\mathbf{t}})$ is monoclinic if and only if $$\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})) \ne 0, \quad \text{and} \quad \operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})) \times \operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})^{2}) =0.$$
\[rem:cub-orientation\] The conditions in $(3)$ are equivalent to the fact that the pair $({\mathbf{t}},{\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is orthotropic, whereas in $(4)$ they are equivalent to the fact that the pair $({\mathbf{t}},{\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is monoclinic. The cases $(1)$ and $(2)$ cover all the cases where the pair $({\mathbf{t}},{\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is transversely isotropic.
We will first investigate the four equations in lemma \[lem:cube-orientation\]. Without loss of generality, we can assume that $G_{{\mathbf{H}}}={\mathbb{O}}$ and thus that the polynomial form of ${\mathbf{H}}$ is given (up to a scaling factor) by $${\mathrm{p}}(x,y,z) = x^{4} + y^{4} + z^{4} - 3x^{2}y^{2} - 3x^{2}z^{2} - 3y^{2}z^{2}.$$ Now, every transversely isotropic second-order homogeneous polynomial ${\mathrm{t}}$ writes as $${\mathrm{t}}(x,y,z) =(\mu - \lambda)({\pmb{n}}\cdot {\pmb{x}})^{2} + \lambda {\mathrm{q}},$$ where $\lambda \ne \mu$, ${\pmb{n}}= (n_{1},n_{2},n_{3})$ is a unit vector and ${\mathrm{q}}= x^{2}+y^{2}+z^{2}$. We get thus:
- $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) =0$ if and only if $$\label{eq:sol1}
n_{1}n_{2} = n_{1}n_{3} = n_{2}n_{3} = 0;$$
- ${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) =0$ if and only if $$\label{eq:sol2}
\begin{split}
n_{1} n_{2} \left(n_{1}^{2}-n_{3}^{2}\right) & = n_{1} n_{3} \left(n_{1}^{2}-n_{2}^{2}\right) = n_{2} n_{3} \left(n_{1}^{2}-n_{2}^{2}\right) =0,
\\
n_{1} n_{2} \left(n_{2}^{2}-n_{3}^{2}\right) & = n_{1} n_{3} \left(n_{2}^{2}-n_{3}^{2}\right) = n_{2} n_{3} \left(n_{3}^{2}-n_{1}^{2}\right) =0.
\end{split}$$
- $\operatorname{tr}\left({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})\right) =0$ if and only if $$\label{eq:sol3}
n_{1} n_{2} \left(n_{1}^{2}-n_{2}^{2}\right) = n_{1} n_{3} \left(n_{1}^{2}-n_{3}^{2}\right) = n_{2} n_{3} \left(n_{2}^{2}-n_{3}^{2}\right) = 0;$$
- $\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})) \times \operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})^{2}) =0$ if and only if $$\label{eq:sol4}
\begin{split}
n_{1}^{2}n_{2}n_{3}(n_{2}^{2}-n_{3}^{2})(n_{1}^{2}-n_{3}^{2})(n_{1}^{2}-n_{2}^{2}) = 0, \\
n_{1}n_{2}^{2}n_{3}(n_{2}^{2}-n_{3}^{2})(n_{1}^{2}-n_{3}^{2})(n_{1}^{2}-n_{2}^{2}) = 0, \\
n_{1}n_{2}n_{3}^{2}(n_{2}^{2}-n_{3}^{2})(n_{1}^{2}-n_{3}^{2})(n_{1}^{2}-n_{2}^{2}) = 0.
\end{split}$$
Note also that $\eqref{eq:sol1} \implies \eqref{eq:sol2} \implies \eqref{eq:sol3} \implies \eqref{eq:sol4}$. We will now prove each statement of lemma \[lem:cube-orientation\].
\(1) Suppose first that the pair $({\mathbf{H}},{\mathbf{t}})$ is tetragonal, then $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0$ by lemma \[lem:tr(Hxd)=0\]. Conversely, if $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0$ holds, then we get and ${\pmb{n}}$ is collinear to either $${\pmb{e}}_{1}, \qquad {\pmb{e}}_{2}, \qquad {\pmb{e}}_{3}.$$ Then, both ${\mathbf{H}}$ and ${\mathbf{t}}$ are invariant by the rotation by $\pi/2$ around ${\pmb{n}}$ and the pair $({\mathbf{H}},{\mathbf{t}})$ is tetragonal.
\(2) Suppose now that the pair $({\mathbf{H}},{\mathbf{t}})$ is trigonal, then the pair of second-order covariants $({\mathbf{t}}, {\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is at least trigonal an thus transversely isotropic by proposition \[prop:symmetry-classes\]. Thus ${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})=0$ by lemma \[lem:axb=0\]. Moreover, $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0$ by point (1). Conversely, if ${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})=0$ and $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0$, then we get with at least $n_{i}n_{j} \ne 0$ for a pair $(i,j)$ ($i \ne j$). In that case, ${\pmb{n}}$ is collinear to either $${\pmb{e}}_{1}+{\pmb{e}}_{2}+{\pmb{e}}_{3}, \quad {\pmb{e}}_{1}-{\pmb{e}}_{2}+{\pmb{e}}_{3}, \quad {\pmb{e}}_{1}+{\pmb{e}}_{2}-{\pmb{e}}_{3}, \quad {\pmb{e}}_{1}-{\pmb{e}}_{2}-{\pmb{e}}_{3}.$$ Then, both ${\mathbf{H}}$ and ${\mathbf{t}}$ are invariant by the rotation by angle $2\pi/3$ around ${\pmb{n}}$ and the pair $({\mathbf{H}},{\mathbf{t}})$ is trigonal.
\(3) Suppose now that the pair $({\mathbf{H}},{\mathbf{t}})$ is orthotropic, then the first order covariant $\operatorname{tr}\left({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})\right)$ is at least orthotropic and thus vanishes. Moreover ${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) \ne 0$ by points (1) and (2). Conversely, if $\operatorname{tr}\left({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})\right)=0$ and ${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) \ne 0$, then we get with at least $n_{i} = 0$ for some $i \in{\left\{1,2,3\right\}}$. In that case ${\pmb{n}}$ is collinear to either $${\pmb{e}}_{1}+{\pmb{e}}_{2},\quad {\pmb{e}}_{1}-{\pmb{e}}_{2},\quad {\pmb{e}}_{1}+{\pmb{e}}_{3},\quad {\pmb{e}}_{1}-{\pmb{e}}_{3},\quad {\pmb{e}}_{2}+{\pmb{e}}_{3},\quad {\pmb{e}}_{2}-{\pmb{e}}_{3}.$$ Then, both ${\mathbf{H}}$ and ${\mathbf{t}}$ are invariant by the rotation by angle $\pi$ (a second-order rotation) around a pair of axes ${\pmb{e}}_{i}\pm{\pmb{e}}_{j}$ and the pair $({\mathbf{H}},{\mathbf{t}})$ is orthotropic.
\(4) Finally, suppose that the pair $({\mathbf{H}},{\mathbf{t}})$ is monoclinic, then the pair of first-order covariants $(\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})),\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})^{2}))$ is at least monoclinic and thus collinear. Moreover, $\operatorname{tr}({\mathbf{t}}\times {\mathbf{H}}\operatorname{:}{\mathbf{t}}) \ne 0$ by $(1)$, $(2)$ and $(3)$. Conversely, if $$\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})) \times \operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})^{2}) =0,$$ then we get . Since $\operatorname{tr}({\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}})) \ne 0$ cases $(1)$, $(2)$ and $(3)$ are excluded and thus the pair $({\mathbf{H}},{\mathbf{t}})$ is either monoclinic or triclinic, so we are reduced to show that it is monoclinic. If $n_{i}=0$ for some $i$, both ${\mathbf{H}}$ and ${\mathbf{t}}$ are invariant by the second-order rotation around ${\pmb{e}}_{i}$. Otherwise, we get $n_{i}=\pm n_{j}$ for a pair $(i,j)$. In that case, both ${\mathbf{H}}$ and ${\mathbf{t}}$ are invariant by the second-order rotation around $n_{i}{\pmb{e}}_{i}-n_{j}{\pmb{e}}_{j}$. This achieves the proof.
We will end this section by formulating criteria for detecting orthotropic and monoclinic symmetry for a general harmonic tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}({\mathbb{R}}^{3})$, using second-order covariants.
\[lem:orthotropic-criteria\] Let ${\mathbf{c}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ be orthotropic and ${\mathbf{H}}\in {\mathbb{H}}^{4}({\mathbb{R}}^{3})$. Then, $$G_{({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})} = G_{({\mathbf{H}},{\mathbf{c}})}.$$ In particular, $({\mathbf{H}},{\mathbf{c}})$ is orthotropic (resp. monoclinic) if and only if $$({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})$$ is orthotropic (resp. monoclinic).
Note that the inclusion $G_{({\mathbf{H}},{\mathbf{c}})} \subset G_{({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})}$ is obvious, since $({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})$ are covariants of the pair $({\mathbf{H}},{\mathbf{c}})$. To prove the reverse inclusion, we can assume, without loss of generality, that $G_{{\mathbf{c}}} = {\mathbb{D}}_{2}$ (an thus that ${\mathbf{c}}$ is diagonal). Let $g \in G_{({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})} \subset G_{{\mathbf{c}}}$. Then, $g$ is either the identity or a second-order rotation $r$ around either ${\pmb{e}}_{1}$, ${\pmb{e}}_{2}$, or ${\pmb{e}}_{3}$. Without loss of generality, we can suppose that $r$ is the rotation around ${\pmb{e}}_{3}$. Then, ${\pmb{e}}_{3}$ is a common eigenvector of ${\mathbf{c}}$, ${\mathbf{H}}\operatorname{:}{\mathbf{c}}$ and ${\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2}$. Moreover, since ${\mathbf{H}}$ is harmonic, we have ${\mathbf{H}}\operatorname{:}{\mathbf{q}}= 0$ and thus $$[({\mathbf{H}}\operatorname{:}{\mathbf{d}}){\pmb{e}}_{3}] \times {\pmb{e}}_{3} = 0, \quad \text{for} \quad {\mathbf{d}}= {\mathbf{q}},{\mathbf{c}},{\mathbf{c}}^{2}.$$ Since $({\mathbf{q}},{\mathbf{c}},{\mathbf{c}}^{2})$ and $({\mathbf{e}}_{11},{\mathbf{e}}_{22},{\mathbf{e}}_{33})$ generate the same three-dimensional vector space of diagonal matrices, this last condition can be recast as $$[({\mathbf{H}}\operatorname{:}{\mathbf{e}}_{ii}){\pmb{e}}_{3}] \times {\pmb{e}}_{3} = 0, \quad \text{for} \quad i=1,2,3.$$ and thus $$H_{1113} = H_{1123} = H_{1223} = H_{1333} = H_{2223} = H_{2333} = 0,$$ which means that ${\mathbf{H}}$ is invariant under $r$, and so $r \in G_{({\mathbf{H}}, {\mathbf{c}})}$.
Note that if ${\mathbf{a}}$, ${\mathbf{b}}$ are transversely isotropic, second-order symmetric tensors, then the pair $({\mathbf{a}},{\mathbf{b}})$ is either monoclinic, orthotropic or transversely isotropic (see [@Oli2017]), and we get the following corollary.
\[cor:transversely-isotropic-pair-criteria\] Let ${\mathbf{a}}$, ${\mathbf{b}}$ be transversely isotropic second-order symmetric tensors and ${\mathbf{H}}\in {\mathbb{H}}^{4}({\mathbb{R}}^{3})$.
1. If $({\mathbf{a}},{\mathbf{b}})$ is orthotropic, then $$G_{({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}})} = G_{({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})}.$$ In particular, $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$ is orthotropic (resp. monoclinic) if and only if $({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}})$ is orthotropic (resp. monoclinic).
2. If $({\mathbf{a}},{\mathbf{b}})$ is monoclinic, then $$G_{({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{b}})^{s})} = G_{({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})}.$$ In particular, $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$ is monoclinic if and only if $$({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{b}})^{s})$$ is monoclinic.
\(1) Suppose that $({\mathbf{a}},{\mathbf{b}})$ is orthotropic. Then there exists a basis $({\pmb{e}}_{i})$ in which both ${\mathbf{a}}$ and ${\mathbf{b}}$ are diagonal. Moreover, $({\mathbf{q}}, {\mathbf{a}}, {\mathbf{b}})$ and $({\mathbf{e}}_{11},{\mathbf{e}}_{22},{\mathbf{e}}_{33})$ generate the same three-dimensional vector space and the proof is similar to that of lemma \[lem:orthotropic-criteria\].
\(2) Suppose that $({\mathbf{a}},{\mathbf{b}})$ is monoclinic. Then, by lemma \[lem:orthotropic-linear-combination\], there exists a linear combination ${\mathbf{c}}$ of ${\mathbf{a}}$ and ${\mathbf{b}}$ which is orthotropic. But then, ${\mathbf{c}}^{2}$ is a linear combination of ${\mathbf{q}}$, ${\mathbf{a}}$, ${\mathbf{b}}$ and $({\mathbf{a}}{\mathbf{b}})^{s}$ and thus $$G_{({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{b}})^{s})} \subset G_{({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})} = G_{({\mathbf{H}},{\mathbf{c}})},$$ by lemma \[lem:orthotropic-criteria\]. Therefore $$G_{({\mathbf{a}}, {\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{b}})^{s})} \subset G_{({\mathbf{H}},{\mathbf{c}})} \cap G_{({\mathbf{a}},{\mathbf{b}})} \subset G_{({\mathbf{H}}, {\mathbf{a}},{\mathbf{b}})}.$$ The reverse inclusion being obvious, this achieves the proof.
Characterization of the Symmetry Classes of H4 {#sec:H4-symmetry-classes}
==============================================
In this section, we formulate coordinate-free conditions using covariants up to order 5 that identify the symmetry class of a given tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}$ and we prove that these conditions are both necessary and sufficient. The partially ordered set of symmetry classes for ${\mathbb{H}}^{4}$ is the same as the one for the Elasticity tensor, pictured in . The notations used in this section are those introduced in section \[sec:H4-covariant-algebra\]. We will start by connecting ${\mathbf{Cov}}_{1}({\mathbf{H}})$ and ${\mathbf{Cov}}_{2}({\mathbf{H}})$ by the following lemma.
\[lem:cov1-cov2\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. Then
1. ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$ if and only if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is at least orthotropic;
2. $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 1$ if and only if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is monoclinic;
3. $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 3$ if and only if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is triclinic.
In the proof of lemma \[lem:cov1-cov2\], some arguments are general and relies on Section \[sec:covariants-symmetry-dimension\], others depend on the very special case that, ${\mathbf{Cov}}_{1}({\mathbb{H}}^{4})$ is generated by commutators of elements in ${\mathbf{Cov}}_{2}({\mathbb{H}}^{4})$ (as can be checked in ).
\(1) If ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$, then each commutator of a pair of elements in ${\mathbf{Cov}}_{2}({\mathbf{H}})$ vanishes. Thus all the elements of ${\mathbf{Cov}}_{2}({\mathbf{H}})$ commute together and they can be represented by diagonal matrices in a common basis. All these second-order covariants are thus invariant by ${\mathbb{D}}_{2}$ and ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is thus at least orthotropic. Conversely, if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is at least orthotropic, then, since ${\mathbf{Cov}}_{1}({\mathbf{H}})$ is generated by the commutators of ${\mathbf{Cov}}_{2}({\mathbf{H}})$, it vanishes.
\(2) If $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 1$, then, by (1) ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is either monoclinic or triclinic (see proposition \[prop:cov2-symmetry-classes\]). However, if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ was triclinic and thus of dimension 6 by proposition \[prop:cov2-symmetry-classes\], then ${\mathbf{Cov}}_{2}({\mathbf{H}})={\mathbb{S}}^{2}({\mathbb{R}}^{3})$ and we could build two linearly independent commutators which belong to ${\mathbf{Cov}}_{1}({\mathbf{H}})$, which would lead to a contradiction. Therefore, ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is monoclinic. Conversely, if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is monoclinic, then, ${\mathbf{Cov}}_{1}({\mathbf{H}})$ (which is generated by commutators of ${\mathbf{Cov}}_{2}({\mathbf{H}})$) is at least monoclinic and thus monoclinic by (1).
\(3) If $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 3$, then ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is necessarily triclinic by (1) and (2). Conversely, if ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is triclinic, then $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) \ge 2$ by (1) and (2) and thus $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 3$ by proposition \[prop:cov1-symmetry-classes\].
The harmonic tensor ${\mathbf{H}}$, being a particular Elasticity tensor, can be represented by a symmetric endomorphism of the space ${\mathbb{S}}^{2}({\mathbb{R}}^{3})$, the so-called *Kelvin representation*, and thus by the matrix [@AKP2014]: $$\label{eq:harmonic-generic-matrix-form}
[{\mathbf{H}}] = \begin{pmatrix}
A & \sqrt{2}\,B \\
\sqrt{2}\,B^{t} & 2\,C \\
\end{pmatrix}$$ where $$A : = \begin{pmatrix}
\Lambda_{2}+ \Lambda_{3} & -\Lambda_{3} & -\Lambda_{2} \\
-\Lambda_{3} & \Lambda_{3} + \Lambda_{1} & -\Lambda_{1} \\
-\Lambda_{2} & -\Lambda_{1} & \Lambda_{2} + \Lambda_{1} \\
\end{pmatrix},$$ $$B : = \begin{pmatrix}
-X_{1} & Y_{1}+Y_{2} & -Z_{2} \\
- X_{2} & -Y_{1} & Z_{1}+Z_{2} \\
X_{1}+X_{2} & -Y_{2} & -Z_{1} \\
\end{pmatrix}, \quad
C : = \begin{pmatrix}
-\,\Lambda_{1} & -Z_{1} & - Y_{1} \\
-Z_{1} & -\,\Lambda_{2} & -X_{1} \\
-Y_{1} & -X_{1} & -\,\Lambda_{3}
\end{pmatrix},$$ and $B^{t}$ is the transpose matrix of $B$.
Case I: Cov2(H) is isotropic
----------------------------
\[thm:cov2-isotropic\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. The following propositions are equivalent.
1. ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is isotropic;
2. ${\mathbf{H}}$ is either *cubic* (${\mathbf{d}}_{2} \ne 0$) or *isotropic* (${\mathbf{d}}_{2} = 0$);
3. ${\mathbf{d}}_{2}$ is isotropic.
We will show that $(1) \implies (2) \implies (3) \implies (1)$.
Suppose first that $(1)$ is true, so that we have ${\mathbf{d}}_{2} = \frac{1}{3} J_{2} {\mathbf{q}}$ and ${\mathbf{d}}_{3} = \frac{1}{3} J_{3} {\mathbf{q}}$, since $\operatorname{tr}{\mathbf{d}}_{2} = J_{2}$ and $\operatorname{tr}{\mathbf{d}}_{3} = J_{3}$. Then, the covariants ${\mathbf{d}}_{k}$ defined in write as $$\label{eq:isotropic-Boehler-covariant}
\begin{array} {lll}
{\mathbf{d}}_{2} = \frac{1}{3} J_{2} {\mathbf{q}}, & {\mathbf{d}}_{3} = \frac{1}{3} J_{3} {\mathbf{q}}, & {\mathbf{d}}_{4} = \frac{1}{9} {J_{2}}^{2} {\mathbf{q}},
\\
{\mathbf{d}}_{5} = 0, & {\mathbf{d}}_{6} = \frac{1}{27} {J_{2}}^{3} {\mathbf{q}}, & {\mathbf{d}}_{7} = 0,
\\
{\mathbf{d}}_{8} = 0, & {\mathbf{d}}_{9} = 0, & {\mathbf{d}}_{10} = 0,
\end{array}$$ and we get $$3J_{4} = {J_{2}}^{2}, \quad 9J_{6} = {J_{2}}^{3}, \quad J_{5} = J_{7} = J_{8} = J_{9} = J_{10} = 0.$$ Now, $\operatorname{tr}({{\mathbf{d}}_{3}}^{2})$ is an invariant of degree 6 and should be expressible as a linear combination of the invariants $${J_{2}}^{3}, \quad {J_{3}}^{2}, \quad J_{2}J_{4}, \quad J_{6}.$$ In fact, the following relation, satisfied by any harmonic tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}$ can be checked directly by computation: $$240 \, J_{6} + 39 \, {J_{2}}^{3} + 190 \, {J_{3}}^{2} - 198 \, J_{2}J_{4} - 540 \, \operatorname{tr}({{\mathbf{d}}_{3}}^{2}) = 0.$$ When are satisfied, this leads to the relation $$30\, {J_{3}}^{2} - {J_{2}}^{3} = 0.$$ If $J_{2} = 0$, then ${\lVert{\mathbf{H}}\rVert}^{2} = J_{2} = 0$, so that ${\mathbf{H}}= 0$ is isotropic. Otherwise, we get $$\label{eq:cubic-syzigies}
\begin{aligned}
3\,J_{4} & = {J_{2}}^{2}, & J_{5} & = 0, & 30\,{J_{3}}^{2} & = {J_{2}}^{3}, & 9\,J_{6} & = {J_{2}}^{3}, \\
J_{7} & = 0, & J_{8} & = 0, & J_{9} & = 0, & J_{10} & = 0,
\end{aligned}$$ and $J_{2} \ne 0$, which are, according to [@AKP2014 Proposition 5.3], necessary and sufficient conditions for a tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}$ to be cubic.
The assertion $(2) \implies (3)$ is trivial because if ${\mathbf{H}}$ is either cubic or isotropic, then ${\mathbf{d}}_{2}$ as a covariant of ${\mathbf{H}}$ inherits its symmetry and is thus necessarily isotropic.
Suppose that $(3)$ is true, so that ${\mathbf{d}}_{2} = \alpha {\mathbf{q}}$, for some scalar $\alpha$. Then, using the fact that ${\mathbf{H}}\operatorname{:}{\mathbf{q}}= \operatorname{tr}_{34} {\mathbf{H}}= 0$ we deduce first that $${\mathbf{c}}_{k} = 0, \qquad k \ge 3.$$ Now, using remark \[rem:d3c3\], we have ${\mathbf{c}}_{3} = 2{\mathbf{d}}_{3}'$ and thus ${\mathbf{d}}_{3} = \beta {\mathbf{q}}$ for some scalar $\beta$. Since all first and second-order covariants in are build from ${\mathbf{d}}_{2}$, ${\mathbf{d}}_{3}$ and the ${\mathbf{c}}_{k}$, we deduce that they are all isotropic. But every symmetric second-order covariant is obtained as a linear combination of either a product of an invariant with a second-order covariant from or the symmetric product of two first-order covariants from . Therefore, ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is isotropic. This achieves the proof.
According to [@AKP2014 Proposition 5.3], an harmonic tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}$ is either cubic or isotropic if and only if the relations are satisfied. Surprisingly, a consequence of theorem \[thm:cov2-isotropic\] (which uses this result) is that an harmonic tensor ${\mathbf{H}}\in {\mathbb{H}}^{4}$ is either cubic or isotropic if and only if $3\,J_{4} = {J_{2}}^{2}$ (one only needs to check the first relation of ). Indeed, by theorem \[thm:cov2-isotropic\], ${\mathbf{H}}$ is at least cubic if and only if ${\mathbf{d}}_{2}$ is isotropic, which is equivalent to ${\mathbf{d}}_{2}^{\prime} = 0$. But $${\lVert{\mathbf{d}}_{2}^{\prime}\rVert}^{2} = \operatorname{tr}\left({\mathbf{d}}_{2} - \frac{1}{3}J_{2}{\mathbf{q}}\right)^{2}
= \operatorname{tr}\left({\mathbf{d}}_{2}^{2} - \frac{2}{3}J_{2}{\mathbf{d}}_{2} + \frac{1}{9}J_{2}^{2}{\mathbf{q}}\right)
= \frac{1}{3}\left(3\,J_{4} - {J_{2}}^{2}\right).$$ The magic here is that in the proof of theorem \[thm:cov2-isotropic\], the knowledge of an integrity basis of ${\mathbf{Cov}}({\mathbb{H}}^{4})$ (given in ) has been used (it was not known when [@AKP2014] was written). This remark is important because it has always been pointed out by the elders that the knowledge of covariants, rather than invariants, is the cornerstone to understand the geometry of a representation. This statement is thus illustrated here.
Case II: Cov2(H) is transversely isotropic
------------------------------------------
\[thm:cov2-transversely-isotropic\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. The following propositions are equivalent.
1. ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is transversely isotropic;
2. ${\mathbf{H}}$ is *tetragonal*, *trigonal* or *transversely isotropic*;
3. the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is transversely isotropic.
By virtue of lemma \[lem:axa2=0\], lemma \[lem:axb=0\] and theorem \[thm:cov2-transversely-isotropic\], condition $(3)$ in theorem \[thm:cov2-transversely-isotropic\] can be recast as $${{\mathbf{d}}_{2}}^{\prime} \ne 0, \qquad {\mathbf{d}}_{2} \times ({\mathbf{d}}_{2})^{2} = 0, \qquad {\mathbf{c}}_{3} \times {\mathbf{d}}_{2} = 0.$$
We will show that $(1) \implies (2) \implies (3) \implies (1)$.
Suppose first that $(1)$ is true, then, without loss of generality, we can assume that all symmetric second-order covariants are invariant by the subgroup ${\mathrm{O}}(2)$ and that at least one of them, say ${\mathbf{a}}$ writes as $${\mathbf{a}}= \lambda{\mathbf{q}}+ \mu \pmb{\tau}, \qquad \mu \ne 0.$$ Thus, $\pmb{\tau} = \mathrm{diag}(1,1,-2)$ is an eigenvector of ${\mathbf{H}}$ (because ${\mathbf{H}}\operatorname{:}{\mathbf{q}}=0$ and ${\mathbf{H}}\operatorname{:}{\mathbf{a}}$ is traceless) and we have $$Y_{2} = 0, \qquad X_{1} + X_{2} = 0, \qquad Z_{1} = 0, \qquad \Lambda_{1} = \Lambda_{2}.$$ Now we compute ${\mathbf{d}}_{2}$ from and write that it must be invariant by ${\mathrm{O}}(2)$ and we get $$\begin{aligned}
4Z_{2}X_{1} + (\Lambda_{1} + 4\Lambda_{3})Y_{1}& = &0,\\
(\Lambda_{1} + 4\Lambda_{3})X_{1}-4Z_{2}Y_{1}& = &0.
\end{aligned}$$ The solutions of this system break into two alternatives:
1. either $Z_{2} = 0$ and $\Lambda_{1} = -4\Lambda_{3}$,
2. or, $X_{1} = Y_{1} = 0$.
In the first case, we get $$[{\mathbf{H}}] = \begin{pmatrix}
-3\Lambda_{3} & -\Lambda_{3} & 4\Lambda_{3} & -\sqrt{2}X_{1} & \sqrt{2}Y_{1} & 0 \\
-\Lambda_{3} & -3\Lambda_{3} & -4\Lambda_{3} & -\sqrt{2} X_{1} & -\sqrt{2}Y_{1} & 0 \\
4\Lambda_{3} & 4\Lambda_{3} & -8\Lambda_{3} & 0 & 0 & 0 \\
- \sqrt{2}X_{1} & +\sqrt{2}X_{1} & 0 & 8\Lambda_{3} & 0 & -2 Y_{1} \\
\sqrt{2}Y_{1} & -\sqrt{2}Y_{1} & 0 & 0 & 8\Lambda_{3} & -2 X_{1} \\
0 & 0 & 0 & -2 Y_{1} & -2 X_{1} & -2\Lambda_{3}
\end{pmatrix}$$ which is at least trigonal since $g\star {\mathbf{H}}= {\mathbf{H}}$ for all $g\in {\mathbb{Z}}_{3}$. In the second case, we get $$[{\mathbf{H}}] = \begin{pmatrix}
\Lambda_{1} + \Lambda_{3} & -\Lambda_{3} & -\Lambda_{1} & 0 & 0 & -\sqrt{2}Z_{2} \\
-\Lambda_{3} & \Lambda_{1} + \Lambda_{3} & -\Lambda_{1} & 0 & 0 & \sqrt{2}Z_{2} \\
-\Lambda_{1} & -\Lambda_{1} & 2\Lambda_{1} & 0 & 0 & 0 \\
0 & 0 & 0 & -2\Lambda_{1} & 0 & 0 \\
0 & 0 & 0 & 0 & -2\Lambda_{1} & 0 \\
-\sqrt{2}Z_{2} & \sqrt{2}Z_{2} & 0 & 0 & 0 & -2\Lambda_{3}
\end{pmatrix}$$ which is at least tetragonal since $g\star {\mathbf{H}}= {\mathbf{H}}$ for all $g\in {\mathbb{Z}}_{4}$.
The assertion $(2) \implies (3)$ is trivial because if $(2)$ is true then the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is either transversely isotropic or isotropic, but it cannot be isotropic by virtue of theorem \[thm:cov2-isotropic\].
Finally, suppose that $(3)$ is true. Note first that ${\mathbf{d}}_{2}$ cannot be isotropic, because of theorem \[thm:cov2-isotropic\]. Without loss of generality, we can assume, therefore, that $${\mathbf{d}}_{2}^{\prime} = \mu_{2} \pmb{\tau}, \qquad {\mathbf{c}}_{3} = \mu_{3}\pmb{\tau},$$ where $\mu_{2} \ne 0$ and $\pmb{\tau} = \mathrm{diag}(1,1,-2)$. We get thus $${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}^{\prime} = \mu_{2} {\mathbf{H}}\operatorname{:}\pmb{\tau} = \mu_{3}\pmb{\tau}$$ leading to $${\mathbf{H}}\operatorname{:}\pmb{\tau} = \frac{\mu_{3}}{\mu_{2}} \pmb{\tau},$$ which means that $\pmb{\tau}$ is an eigenvector of ${\mathbf{H}}$. But then $${\mathbf{c}}_{4} = {\mathbf{H}}\operatorname{:}{\mathbf{c}}_{3} = \frac{\mu_{3}^{2}}{\mu_{2}} \pmb{\tau}, \qquad {\mathbf{c}}_{5} = {\mathbf{H}}\operatorname{:}{\mathbf{c}}_{4} = \frac{\mu_{3}^{3}}{\mu_{2}^{2}} \pmb{\tau},$$ and thus, the triple $({\mathbf{d}}_{2},{\mathbf{c}}_{3},{\mathbf{c}}_{4}, {\mathbf{c}}_{5})$ is transversely isotropic. We deduce then from , that ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is transversely isotropic.
We will now formulate conditions which allow to distinguish between the three remaining cases: transversely isotropic, trigonal and tetragonal.
\[cor:transiso-trigo-tetra-criteria\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. Then
1. ${\mathbf{H}}$ is *transversely isotropic* if and only if ${\mathbf{d}}_{2}$ is transversely isotropic and $${\mathbf{H}}\times {\mathbf{d}}_{2} = 0;$$
2. ${\mathbf{H}}$ is *tetragonal* if and only if ${\mathbf{d}}_{2}$ is transversely isotropic, $${\mathbf{H}}\times {\mathbf{d}}_{2} \ne 0, \quad \text{and} \quad \operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) = 0;$$
3. ${\mathbf{H}}$ is *trigonal* if and only if ${\mathbf{d}}_{2}$ is transversely isotropic, $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) \ne 0, \quad \text{and} \quad ({\mathbf{H}}: {\mathbf{d}}_{2}) \times {\mathbf{d}}_{2} = 0.$$
Note first that if ${\mathbf{H}}$ is either transversely isotropic, tetragonal or trigonal then, ${\mathbf{d}}_{2}$ is necessarily transversely isotropic, by theorem \[thm:cov2-isotropic\].
\(1) If ${\mathbf{H}}$ is *transversely isotropic*, then, ${\mathbf{H}}\times {\mathbf{d}}_{2} = 0$ by lemma \[lem:Sxd=0\]. Conversely, if ${\mathbf{d}}_{2}$ is transversely isotropic and ${\mathbf{H}}\times {\mathbf{d}}_{2} = 0$, then, $({\mathbf{H}},{\mathbf{d}}_{2})$ is transversely isotropic by lemma \[lem:Sxd=0\] and so is ${\mathbf{H}}$.
\(2) If ${\mathbf{H}}$ is *tetragonal*, then, $\operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) = 0$ by lemma \[lem:tr(Hxd)=0\] and ${\mathbf{H}}\times {\mathbf{d}}_{2} \ne 0$ by (1). Conversely, if the conditions in (2) are satisfied, then, $({\mathbf{H}}, {\mathbf{d}}_{2})$ is at least tetragonal by lemma \[lem:tr(Hxd)=0\], and so is ${\mathbf{H}}$. Since ${\mathbf{H}}$ cannot be isotropic or cubic by theorem \[thm:cov2-isotropic\] (because ${\mathbf{d}}_{2}$ is assumed to be transversely isotropic), it is either tetragonal or transversely isotropic, the later case being excluded by the condition ${\mathbf{H}}\times {\mathbf{d}}_{2} \ne 0$.
\(3) If ${\mathbf{H}}$ is *trigonal*, then, the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is transversely isotropic and thus $$({\mathbf{H}}: {\mathbf{d}}_{2}) \times {\mathbf{d}}_{2} = {\mathbf{c}}_{3} \times {\mathbf{d}}_{2} = 0,$$ by lemma \[lem:axb=0\]. Moreover, $\operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) \ne 0$ by lemma \[lem:tr(Hxd)=0\]. Conversely, if the conditions in $(3)$ holds, then, the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is transversely isotropic by lemma \[lem:axb=0\] and ${\mathbf{H}}$ is either *tetragonal*, *trigonal* or *transversely isotropic* by theorem \[thm:cov2-transversely-isotropic\]. Since ${\mathbf{H}}$ cannot be transversely isotropic by lemma \[lem:Sxd=0\], nor tetragonal by lemma \[lem:tr(Hxd)=0\], it is necessarily trigonal.
We will end this subsection with two lemmas which characterise the symmetry class of a pair $({\mathbf{H}}, {\mathbf{t}})$ where ${\mathbf{H}}$ is a fourth-order harmonic tensor and ${\mathbf{t}}$ is a transversely isotropic second-order symmetric tensor. This completes the results of Section \[sec:covariant-criteria\] and will be very useful to prove our main theorem in Section \[sec:Ela-symmetry-classes\].
\[lem:trigonal-pair\] Let ${\mathbf{t}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ be transversely isotropic and ${\mathbf{H}}\in {\mathbb{H}}^{4}({\mathbb{R}}^{3})$ be an harmonic fourth-order tensor. Then, the pair $({\mathbf{H}}, {\mathbf{t}})$ is trigonal if and only if $$\label{eq:trigonal-pair}
({\mathbf{H}}\operatorname{:}{\mathbf{t}})\times {\mathbf{t}}=0,\quad {\mathbf{d}}_{2}\times {\mathbf{t}}=0, \quad \text{and} \quad \operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0.$$
Suppose first that $({\mathbf{H}}, {\mathbf{t}})$ is trigonal. Then the triplet of second-order covariants $({\mathbf{H}}\operatorname{:}{\mathbf{t}},{\mathbf{d}}_{2},{\mathbf{t}})$ is at least trigonal and thus transversely isotropic by proposition \[prop:symmetry-classes\]. We have thus $({\mathbf{H}}\operatorname{:}{\mathbf{t}})\times {\mathbf{t}}=0$ and ${\mathbf{d}}_{2} \times {\mathbf{t}}=0$ by lemma \[lem:axb=0\]. Moreover, $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0$, by lemma \[lem:tr(Hxd)=0\]. Conversely, suppose that conditions are satisfied. Then, ${\mathbf{d}}_{2}$ is at least transversely isotropic by lemma \[lem:axb=0\].
1. If ${\mathbf{d}}_{2}$ is isotropic, then ${\mathbf{H}}$ is cubic by theorem \[thm:cov2-isotropic\] (it cannot vanish because we assume $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0$). But then, $({\mathbf{H}}, {\mathbf{t}})$ is trigonal by lemma \[lem:cube-orientation\].
2. If ${\mathbf{d}}_{2}$ is transversely isotropic, then ${\mathbf{d}}_{2}' = \lambda {\mathbf{t}}'$ with $\lambda \ne 0$ and thus $$({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}) \times {\mathbf{d}}_{2} =0, \quad \text{and} \quad \operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) \ne 0.$$ Therefore ${\mathbf{H}}$ is trigonal by corollary \[cor:transiso-trigo-tetra-criteria\] and so is the pair $({\mathbf{H}}, {\mathbf{t}})$.
\[cor:transversely-isotropic-triplet\] Let ${\mathbf{t}}\in {\mathbb{S}}^{2}({\mathbb{R}}^{3})$ be a transversely isotropic and ${\mathbf{H}}$ be an harmonic fourth-order tensor. Then, the pair $({\mathbf{H}}, {\mathbf{t}})$ is either trigonal, tetragonal or transversely isotropic if and only if the triplet $({\mathbf{d}}_{2}, {\mathbf{t}}, {\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is transversely isotropic.
Suppose first that $({\mathbf{H}}, {\mathbf{t}})$ is either trigonal, tetragonal or transversely isotropic. Then the triplet of second-order covariants $({\mathbf{d}}_{2}, {\mathbf{t}}, {\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is at least transversely isotropic and thus transversely isotropic. Conversely, suppose that $({\mathbf{d}}_{2}, {\mathbf{t}}, {\mathbf{H}}\operatorname{:}{\mathbf{t}})$ is transversely isotropic. Then we have $${\mathbf{d}}_{2} \times {\mathbf{t}}= 0, \quad \text{and} \quad ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) \times {\mathbf{t}}= 0,$$ by lemma \[lem:axb=0\]. If $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0$, then, the pair $({\mathbf{H}}, {\mathbf{t}})$ is either tetragonal or transversely isotropic by lemma \[lem:tr(Hxd)=0\]. If $\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) \ne 0$, then, the pair $({\mathbf{H}}, {\mathbf{t}})$ is trigonal by lemma \[lem:trigonal-pair\]. This achieves the proof.
Case III: Cov2(H) is orthotropic
--------------------------------
\[lem:v5-V6\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. Then $${\pmb{v}}_{5} = {\pmb{v}}_{6} = 0 \quad \implies \quad {\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}},$$ where ${\pmb{v}}_{5}: = \pmb{\varepsilon} \operatorname{:}({\mathbf{d}}_{2}{\mathbf{c}}_{3})$ and ${\pmb{v}}_{6}: = \pmb{\varepsilon} \operatorname{:}({\mathbf{d}}_{2}{\mathbf{c}}_{4})$.
If ${\pmb{v}}_{5} = {\pmb{v}}_{6} = 0$, then the commutators $[{\mathbf{d}}_{2},{\mathbf{c}}_{3}]$ and $[{\mathbf{d}}_{2},{\mathbf{c}}_{4}]$ vanish. Without loss of generality, we can assume that ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ are diagonal matrices. We will now show that $$[{\mathbf{c}}_{3},{\mathbf{c}}_{4}] = 0.$$
1. If ${\mathbf{d}}_{2}$ is orthotropic, then ${\mathbf{c}}_{4}$ is also diagonal (since $[{\mathbf{d}}_{2},{\mathbf{c}}_{4}]=0$) and thus $[{\mathbf{c}}_{3},{\mathbf{c}}_{4}] = 0$.
2. If ${\mathbf{d}}_{2}$ is transversely isotropic, we can assume, without loss of generality, that ${\mathbf{d}}_{2} = \mathrm{diag}(\lambda,\lambda,\mu)$ where $\lambda \ne \mu$. Then, since ${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}$ is also diagonal, we get $$X_{1} + X_{2} = Y_{2} = Z_{1} = 0.$$ Expressing now that $({{\mathbf{d}}}_{2})_{11} = ({{\mathbf{d}}}_{2})_{22}$ and $({{\mathbf{d}}}_{2})_{12} = 0$, we have $$Z_{2}(\Lambda_{1}-\Lambda_{2}) = (\Lambda_{3} + 2\Lambda_{1} + 2\Lambda_{2})(\Lambda_{1}-\Lambda_{2}) = 0.$$ But, since $[{\mathbf{d}}_{2},{\mathbf{c}}_{4}] = 0$ where ${\mathbf{c}}_{4} = {\mathbf{H}}^{2}\operatorname{:}{\mathbf{d}}_{2}$, we get $$({{\mathbf{c}}}_{4})_{13} = ({{\mathbf{c}}}_{4})_{23} = 0,$$ and thus $$(\Lambda_{1}-\Lambda_{2})Y_{1} = (\Lambda_{1}-\Lambda_{2})X_{2}=0.$$
- If $\Lambda_{1}=\Lambda_{2}$, then, $({{\mathbf{c}}}_{4})_{12} = (\mu-\lambda)(\Lambda_{2}-\Lambda_{1})Z_{2}=0$. Thus ${{\mathbf{c}}}_{4}$ is diagonal and $[{\mathbf{c}}_{3},{\mathbf{c}}_{4}] = 0$.
- If $\Lambda_{1}\ne \Lambda_{2}$, then $$X_{2}=Y_{1}=Z_{2} = 0, \qquad \Lambda_{3} + 2\Lambda_{1} + 2\Lambda_{2} =0$$ and, once again, ${{\mathbf{c}}}_{4}$ is diagonal and thus $[{\mathbf{c}}_{3},{\mathbf{c}}_{4}] = 0$.
3. if ${\mathbf{d}}_{2}$ is isotropic, then all second order covariants vanish (by theorem \[thm:cov2-isotropic\]).
In each case, ${\mathbf{d}}_{2}$, ${\mathbf{c}}_{3}$, ${\mathbf{c}}_{4}$ commute with each other and thus all the first-order covariants in vanish, leading to ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$.
\[thm:cov2-orthotropic\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. The following propositions are equivalent.
1. ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is orthotropic;
2. ${\mathbf{H}}$ is *orthotropic*;
3. ${\pmb{v}}_{5} = {\pmb{v}}_{6} = 0$ and the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is orthotropic.
In that case, $G_{{\mathbf{H}}} = G_{({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})}$.
\[rem:orthotropic-triplet\] Condition $(3)$ implies that the triplet $({\mathbf{d}}_{2},{\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is orthotropic by lemma \[lem:v5-V6\]. Conversely, if the triplet $({\mathbf{d}}_{2},{\mathbf{c}}_{3},{\mathbf{c}}_{4})$ is orthotropic, then $(3)$ holds because if $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ was at least transversely isotropic, then so would be ${\mathbf{Cov}}_{2}({\mathbf{H}})$ by theorem \[thm:cov2-transversely-isotropic\] and theorem \[thm:cov2-isotropic\], which would lead to a contradiction. Thus, these two conditions are equivalent. However, checking condition $(3)$ requires less computations than checking that $({\mathbf{d}}_{2},{\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is orthotropic using theorem \[thm:n-quadratic-forms\].
We will show that $(1) \implies (2) \implies (3) \implies (1)$.
Suppose first that ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is orthotropic and thus of dimension 3 by proposition \[prop:cov2-symmetry-classes\]. Then, by corollary \[cor:dimension-trans-iso-subspaces\], there exists ${\mathbf{c}}\in {\mathbf{Cov}}_{2}({\mathbf{H}})$ which is orthotropic. By lemma \[lem:q-a-a2\], we deduce that ${\mathbf{Cov}}_{2}({\mathbf{H}}) = \langle {\mathbf{q}}, {\mathbf{c}}, {\mathbf{c}}^{2} \rangle$, and thus, without loss of generality, we can assume that ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is the space of all diagonal tensors. Now, since ${\mathbf{H}}\operatorname{:}{\mathbf{q}}= 0$, ${\mathbf{H}}\operatorname{:}{\mathbf{c}}$ and ${\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2}$ are second-order symmetric covariants, we deduce moreover that the space of diagonal matrices is invariant under ${\mathbf{H}}$, which has thus the matrix representation $$\label{eq:orthotropic-matrix-form}
[{\mathbf{H}}] = \begin{pmatrix}
\Lambda_{2} + \Lambda_{3} & -\Lambda_{3} & -\Lambda_{2} & 0 & 0 & 0 \\
-\Lambda_{3} & \Lambda_{3}+\Lambda_{1} & -\Lambda_{1} & 0 & 0 & 0 \\
-\Lambda_{2} & -\Lambda_{1} & \Lambda_{1}+\Lambda_{2} & 0 & 0 & 0 \\
0 & 0 & 0 & -2 \Lambda_{1} & 0 & 0 \\
0 & 0 & 0 & 0 & -2\Lambda_{2} & 0 \\
0 & 0 & 0 & 0 & 0 & -2\Lambda_{3}
\end{pmatrix}$$ which is the normal form of an harmonic tensor which is *at least* orthotropic. Since it cannot be of lower symmetry by theorem \[thm:cov2-transversely-isotropic\] and theorem \[thm:cov2-isotropic\], we conclude that ${\mathbf{H}}$ is orthotropic.
Suppose now that ${\mathbf{H}}$ is orthotropic. Then, ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$ by proposition \[prop:cov1-symmetry-classes\] and thus ${\pmb{v}}_{5} = {\pmb{v}}_{6} = 0$. Moreover, the pair $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is at least orthotropic and thus orthotropic by theorem \[thm:cov2-transversely-isotropic\] and theorem \[thm:cov2-isotropic\]. Thus we get $(3)$.
Finally, suppose that $(3)$ holds. Then by lemma \[lem:v5-V6\], ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$ and thus ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is at least orthotropic by lemma \[lem:cov1-cov2\] and thus orthotropic since $({\mathbf{d}}_{2},{\mathbf{d}}_{3})$ is orthotropic.
Case IV: Cov2(H) is monoclinic
------------------------------
\[lem:v5\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. Then $${\pmb{v}}_{5} = 0 \implies \dim {\mathbf{Cov}}_{1}({\mathbf{H}}) \le 1,$$ where ${\pmb{v}}_{5} : = \pmb{\varepsilon} \operatorname{:}({\mathbf{d}}_{2}{\mathbf{c}}_{3})$.
If ${\pmb{v}}_{5} = 0$, then ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ commute. We will distinguish 2 cases whether ${\mathbf{d}}_{2}$ is orthotropic or transversely isotropic (if ${\mathbf{d}}_{2}$ is isotropic, the result already holds by theorem \[thm:cov2-isotropic\]).
\(1) Suppose that ${\mathbf{d}}_{2}$ is transversely isotropic. Without loss of generality we can assume that ${\mathbf{d}}_{2} = \mathrm{diag}(\lambda,\lambda,\mu)$, $\lambda \ne \mu$, and that ${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}$ is diagonal. We get then $$X_{1}+X_{2} = 0, \qquad Y_{2} = 0, \qquad Z_{1} = 0.$$ Using these substitutions, we have $$\begin{aligned}
({\mathbf{d}}_{2})_{11} & = 4\Lambda_{3}^{2} + 2 \Lambda_{2}\Lambda_{3} + 4\Lambda_{2}^{2} + 4Z_{2}^{2} + 6Y_{1}^{2} + 6 X_{1}^{2} \\
({\mathbf{d}}_{2})_{22} & = 4\Lambda_{3}^{2} + 2\Lambda_{1}\Lambda_{3} + 4\Lambda_{1}^{2} + 4Z_{2}^{2} + 6Y_{1}^{2} + 6X_{1}^{2} \\
({\mathbf{d}}_{2})_{12} & = (\Lambda_{1}-\Lambda_{2})Z_{2} \\
({\mathbf{d}}_{2})_{13} & = (4\Lambda_{3} - 2\Lambda_{2} + 3\Lambda_{1})Y_{1} + 4X_{1}Z_{2} \\
({\mathbf{d}}_{2})_{23} & = (4\Lambda_{3} - 2\Lambda_{1} + 3\Lambda_{2})X_{1} - 4Y_{1}Z_{2} \\
\end{aligned}$$ and thus $$\begin{aligned}
& (\Lambda_{1} - \Lambda_{2})Z_{2} = 0, \\
& (4\Lambda_{3} - 2\Lambda_{2} + 3\Lambda_{1})Y_{1} + 4X_{1}Z_{2} = 0, \\
& (4\Lambda_{3} - 2\Lambda_{1} + 3\Lambda_{2})X_{1} - 4Y_{1}Z_{2} = 0, \\
& (\Lambda_{1} - \Lambda_{2})(2\Lambda_{1} + 2\Lambda_{2} + \Lambda_{3}) = 0.
\end{aligned}$$ (a) If $\Lambda_{1} = \Lambda_{2}$, we get $$\begin{aligned}
(4\Lambda_{3} + \Lambda_{1})Y_{1} + 4X_{1}Z_{2} & = 0, \\
-(4\Lambda_{3} + \Lambda_{1})X_{1} + 4Y_{1}Z_{2} & = 0.
\end{aligned}$$ Then either the determinant of the system $16Z_{2}^{2}+(4\Lambda_{3} + \Lambda_{1})^{2}$ does not vanish, and thus $ X_{1} = Y_{1} = 0$. In this case we get $$X_{1} = X_{2} = Y_{1} = Y_{2} = 0$$ and ${\mathbf{H}}$ is invariant under the rotation by angle $\pi$ around ${\pmb{e}}_{3}$. Otherwise, we have $Z_{2} = 0$ and $4\Lambda_{3} + \Lambda_{1} = 0$. In that case, ${\mathbf{c}}_{4}$ is also diagonal and commutes thus with both ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ and we are done by lemma \[lem:v5-V6\].
\(b) If $\Lambda_{1} \ne \Lambda_{2}$ , then $Z_{2} = 0$ and $$\begin{aligned}
Y_{1}(4\Lambda_{3} - 2\Lambda_{2} + 3\Lambda_{1}) & = 0, \\
X_{1}(4\Lambda_{3} - 2\Lambda_{1} + 3\Lambda_{2}) & = 0, \\
\Lambda_{3} + 2\Lambda_{2} + 2\Lambda_{1} & = 0. \\
\end{aligned}$$ If $X_{1}= 0$ or $Y_{1} = 0$ then, we are done since ${\mathbf{H}}$ is at least monoclinic in either cases. Thus we can assume that $$\begin{aligned}
4\Lambda_{3} - 2\Lambda_{2} +3\Lambda_{1} & = 0, \\
4\Lambda_{3} + 3\Lambda_{2} - 2\Lambda_{1} & = 0, \\
\Lambda_{3} + 2\Lambda_{2} + 2\Lambda_{1} & = 0,
\end{aligned}$$ but the unique solution of this linear system is $\Lambda_{1} = \Lambda_{2} = \Lambda_{3} = 0$, and then ${\mathbf{c}}_{4} = 0$. Again, we are done by lemma \[lem:v5-V6\].
\(2) Suppose that ${\mathbf{d}}_{2}$ is orthotropic. Our strategy will be to show that ${\pmb{v}}_{6} = \pmb\varepsilon\operatorname{:}({\mathbf{d}}_{2}{\mathbf{c}}_{4})$ is a common eigenvector of both ${\mathbf{d}}_{2}$, ${\mathbf{c}}_{3}$ and ${\mathbf{c}}_{4}$, in which case $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 1$ (if ${\pmb{v}}_{6} = 0$, then ${\mathbf{Cov}}_{1}({\mathbf{H}}) = {\left\{0\right\}}$ by lemma \[lem:v5-V6\]). Note that, if we can prove that ${\pmb{v}}_{6}$ is an eigenvector of ${\mathbf{d}}_{2}$, then, we are done because $${\mathbf{d}}_{2}({\mathbf{c}}_{3}{\pmb{v}}_{6}) = {\mathbf{c}}_{3}({\mathbf{d}}_{2}{\pmb{v}}_{6}), \qquad {\mathbf{d}}_{2}({\mathbf{c}}_{4}{\pmb{v}}_{6}) = {\mathbf{c}}_{4}({\mathbf{d}}_{2}{\pmb{v}}_{6})$$ and ${\mathbf{d}}_{2}$ (which is orthotropic) has only simple eigenvalues. Now ${\mathbf{d}}_{2} {\pmb{v}}_{6}$ can be recast as a product of covariants in . Indeed, we have $$\label{eq:d2v6}
{\mathbf{d}}_{2} {\pmb{v}}_{6} = J_{2}{\pmb{v}}_{6}-{\pmb{v}}_{8b},$$ where $${\pmb{v}}_{6} = \pmb\varepsilon\operatorname{:}({\mathbf{d}}_{2}{\mathbf{c}}_{4}), \quad \text{and} \quad {\pmb{v}}_{8b} = \pmb\varepsilon\operatorname{:}({\mathbf{d}}_{2}^{2}{\mathbf{c}}_{4}).$$
Without loss of generality, we can assume that ${\mathbf{d}}_{2}$ is diagonal, and hence that $({{{\mathbf{q}}}},{\mathbf{d}}_{2},{\mathbf{d}}_{2}^{2})$ is a basis of the space of diagonal matrices. Therefore $$\label{eq:c3diag}
{\mathbf{c}}_{3} = \alpha{{\mathbf{q}}} + \beta{\mathbf{d}}_{2} + \gamma{\mathbf{d}}_{2}^{2}.$$ If $\gamma = 0$, then $${\mathbf{c}}_{4} = {\mathbf{H}}\operatorname{:}{\mathbf{c}}_{3} = \beta ({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}) = \beta{\mathbf{c}}_{3}$$ and we are done by lemma \[lem:v5-V6\]. Therefore, we can suppose that $\gamma \ne 0$. Contracting with ${\mathbf{c}}_{4}$ both sides of , we get $${\mathbf{c}}_{4}{\mathbf{c}}_{3} = \alpha{\mathbf{c}}_{4} + \beta{\mathbf{c}}_{4}{\mathbf{d}}_{2} + \gamma{\mathbf{c}}_{4}{\mathbf{d}}_{2}^{2},$$ and contracting with $\pmb\varepsilon$ leads to $$\label{eq:v7beq1}
{\pmb{v}}_{7b} = \pmb\varepsilon\operatorname{:}({\mathbf{c}}_{4}{\mathbf{c}}_{3}) = -\beta{\pmb{v}}_{6}-\gamma{\pmb{v}}_{8b}.$$ Now, contracting ${\mathbf{H}}$ with both sides of , we get $${\mathbf{c}}_{4} = {\mathbf{H}}\operatorname{:}{\mathbf{c}}_{3} = \beta{\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2} + \gamma{\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}^{2} = \beta{\mathbf{c}}_{3} + \gamma{\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}^{2}.$$ But ${\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}^{2}$ can be recast as a product of covariants in . Indeed $$8{\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}^{2} = (-2J_{2}J_{3} + 8J_{5}){{\mathbf{q}}}-2J_{3}{\mathbf{d}}_{2} + 7J_{2}{\mathbf{c}}_{3} + 10{\mathbf{c}}_{5}-12({\mathbf{d}}_{2}{\mathbf{c}}_{3})^{s}.$$ Therefore (remember that ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ commute), we have $${\mathbf{c}}_{4} = \left(\beta + \frac{7\gamma}{8}J_{2}\right){\mathbf{c}}_{3} + \gamma\left(J_{5}-\frac{J_{2}J_{3}}{4}\right){\mathbf{q}}- \frac{\gamma}{4}J_{3}{\mathbf{d}}_{2} + \frac{5\gamma}{4}{\mathbf{c}}_{5} - \frac{3\gamma}{2}{\mathbf{d}}_{2}{\mathbf{c}}_{3}$$ and thus $${\mathbf{c}}_{4}{\mathbf{c}}_{3} = \left(\beta + \frac{7\gamma}{8}J_{2}\right) {\mathbf{c}}_{3}^{2} + \gamma \left(J_{5} - \frac{J_{2}J_{3}}{4}\right) {\mathbf{c}}_{3} - \frac{\gamma}{4}J_{3}{\mathbf{d}}_{2}{\mathbf{c}}_{3} + \frac{5\gamma}{4}{\mathbf{c}}_{5}{\mathbf{c}}_{3} - \frac{3\gamma}{2}{\mathbf{d}}_{2}{\mathbf{c}}_{3}^{2}.$$ Hence $${\pmb{v}}_{7b} = \pmb\varepsilon\operatorname{:}({\mathbf{c}}_{4}{\mathbf{c}}_{3}) = \frac{5\gamma}{3}\pmb\varepsilon\operatorname{:}({\mathbf{c}}_{5}{\mathbf{c}}_{3}).$$ But $\pmb\varepsilon\operatorname{:}({\mathbf{c}}_{5}{\mathbf{c}}_{3})$ can be recast as a product of covariants in . Indeed $$\label{eq:c5c3}
15\, \pmb\varepsilon \operatorname{:}({\mathbf{c}}_{5}{{\mathbf{c_{3}}}}) = 4J_{3}{\pmb{v}}_{5} + 15J_{2}{\pmb{v}}_{6} + 18{\pmb{v}}_{8a} - 24{\pmb{v}}_{8b},$$ where $${\pmb{v}}_{5} = \pmb\varepsilon \operatorname{:}({\mathbf{d}}_{2}{{\mathbf{c_{3}}}}) = 0, \quad \text{and} \quad {\pmb{v}}_{8a} = \pmb\varepsilon \operatorname{:}({\mathbf{d}}_{2}{{\mathbf{{c_{3}}^{2}}}}) = 0.$$ We have thus $${\pmb{v}}_{7b} = \frac{5\gamma}{3}J_{2}{\pmb{v}}_{6}-\frac{8\gamma}{3}{\pmb{v}}_{8b}.$$ Using , we deduce that $$\frac{5\gamma}{3}J_{2}{\pmb{v}}_{6} - \frac{8\gamma}{3}{\pmb{v}}_{8b} = {\pmb{v}}_{7b} = -\beta{\pmb{v}}_{6}-\gamma{\pmb{v}}_{8b}$$ and hence that $${\pmb{v}}_{8b} = \left(J_{2} + \frac{3\beta}{5\gamma}\right){\pmb{v}}_{6}.$$ Therefore $${\mathbf{d}}_{2}{\pmb{v}}_{6} = J_{2}{\pmb{v}}_{6}-{\pmb{v}}_{8b} = \frac{3\beta}{5\gamma}{\pmb{v}}_{6}$$ and ${\pmb{v}}_{6}$ is an eigenvector of ${\mathbf{d}}_{2}$, which achieves the proof.
\[cor:cov1-v5\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. Then, $$\label{eq:v5-v5}
{\pmb{v}}_{5} \times \left[({\pmb{v}}_{5}\cdot {\mathbf{H}}\cdot {\pmb{v}}_{5}) {\pmb{v}}_{5}\right] = 0 \quad \text{and} \quad {\pmb{v}}_{5} \times \left[({\pmb{v}}_{5} \cdot {\mathbf{H}}^{2} \cdot {\pmb{v}}_{5}) {\pmb{v}}_{5}\right] = 0,$$ if and only if ${\mathbf{H}}$ is at least monoclinic.
Suppose first that is satisfied. If ${\pmb{v}}_{5} = 0$, we are done by lemma \[lem:v5\]. Otherwise, we can suppose, without loss of generality, that ${\pmb{v}}_{5} = k {\pmb{e}}_{1}$ with $k \ne 0$. But then we get $$H_{1112} = H_{1113} = 0, \qquad ({\mathbf{H}}^{2})_{1112} = ({\mathbf{H}}^{2})_{1113} = 0,$$ and thus $$\begin{aligned}
Y_{1} + Y_{2} & = 0, \\
Z_{2} & = 0, \\
2X_{1}Y_{2} -(\Lambda_{2}-\Lambda_{3})Z_{1} & = 0, \\
(\Lambda_{2}-\Lambda_{3})Y_{2} + 2X_{1} Z_{1} & = 0.
\end{aligned}$$ If $4X_{1}^{2} +(\Lambda_{2}-\Lambda_{3})^{2} \ne 0$, then $Y_{2} = Z_{1} = 0$ and we are done (since then, ${\mathbf{H}}$ is a normal form of a monoclinic tensor). Otherwise, we get $\Lambda_{3} = \Lambda_{2}$ and $X_{1} = 0$. Then ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ commute so that ${\pmb{v}}_{5} = 0$, which leads to a contradiction. Conversely, if ${\mathbf{H}}$ is at least monoclinic, then $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) \le 1$ by proposition \[prop:cov1-symmetry-classes\], and thus we get . This achieves the proof.
\[thm:cov2-monoclinic\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. The following propositions are equivalent.
1. ${\mathbf{H}}$ is monoclinic;
2. ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is monoclinic;
3. the triplet $({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is monoclinic.
In that case, $G_{{\mathbf{H}}} = G_{({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})}$.
We will prove that $(1) \implies (2) \implies (3) \implies (1)$.
Suppose first that $(1)$ holds. Then, ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is at least monoclinic. But ${\mathbf{Cov}}_{2}({\mathbf{H}})$ cannot be orthotropic, transversely isotropic, nor isotropic by Theorems \[thm:cov2-orthotropic\], \[thm:cov2-transversely-isotropic\], and \[thm:cov2-isotropic\]. Thus ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is monoclinic.
Suppose now that $(2)$ holds. Then, the triplet $({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is at least monoclinic. Since it cannot be at least orthotropic by lemma \[lem:v5-V6\] and lemma \[lem:cov1-cov2\], it is thus monoclinic.
Finally, suppose that $(3)$ holds. Then, ${\mathbf{H}}$ is either monoclinic or triclinic. Moreover, there exists a basis where each element of the triplet $({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ can be written as $$\left(
\begin{array}{ccc}
* & * & 0 \\
* & * & 0 \\
0 & 0 & * \\
\end{array}
\right)$$ and using the results of , we can conclude that $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) = 1$. But then, ${\mathbf{H}}$ is at least monoclinic by corollary \[cor:cov1-v5\] and thus monoclinic.
Case V: Cov2(H) is triclinic
----------------------------
\[thm:cov1-triclinic\] Let ${\mathbf{H}}\in {\mathbb{H}}^{4}$ be a fourth order harmonic tensor. The following propositions are equivalent.
1. ${{{\mathbf{H}}}}$ is triclinic;
2. ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is triclinic;
3. the triplet $({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is triclinic.
We will prove that $(1) \implies (2) \implies (3) \implies (1)$. If $(1)$ holds, then $(2)$ holds by Theorems \[thm:cov2-isotropic\], \[thm:cov2-transversely-isotropic\], \[thm:cov2-orthotropic\] and \[thm:cov2-monoclinic\]. If $(2)$ holds, then $(3)$ holds because if $({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})$ is at least monoclinic, then $\dim {\mathbf{Cov}}_{1}({\mathbf{H}}) \le 1$ and thus ${\mathbf{Cov}}_{2}({\mathbf{H}})$ is at least monoclinic by lemma \[lem:cov1-cov2\]. Finally, if $(3)$ holds, then ${\mathbf{H}}$ is necessarily triclinic.
Characterization of the Symmetry Class of an Elasticity tensor {#sec:Ela-symmetry-classes}
==============================================================
The harmonic decomposition of the Elasticity tensor was first obtained by Backus [@Bac1970] (see also [@Cow1989; @Bae1993]) and is given by $${\mathbb{E}\mathrm{la}}\simeq {\mathbb{H}}^{0} \oplus {\mathbb{H}}^{0} \oplus {\mathbb{H}}^{2} \oplus {\mathbb{H}}^{2} \oplus {\mathbb{H}}^{4}.$$ More precisely (see [@BKO1994] for instance), given an orthonormal frame $({\pmb{e}}_{1}, {\pmb{e}}_{2}, {\pmb{e}}_{3})$, each Elasticity tensor ${\mathbf{E}}$ can be written as $$\label{eq:boehler-decomposition}
\begin{split}
E_{ijkl} & = \lambda \delta_{ij} \delta_{kl} + \mu(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}) \\
& \quad + \delta_{ij} a_{kl} + \delta_{kl} a_{ij} \\
& \quad + \delta_{ik} b_{jl} + \delta_{jl} b_{ik} + \delta_{il} b_{jk} + \delta_{jk} b_{il} \\
& \quad + H_{ijkl}.
\end{split}$$ In this decomposition, $\lambda,\mu$ (the generalized Lamé coefficients) and the deviators ${\mathbf{a}},{\mathbf{b}}$ are related to the *dilatation tensor* ${{\mathbf{d}}}:=\operatorname{tr}_{12} {{{\mathbf{E}}}}$ and the *Voigt tensor* ${{\mathbf{v}}}:=\operatorname{tr}_{13}{{{\mathbf{E}}}}$ by the following process [@Cow1989]. Starting with , we get $${\mathbf{d}}= (3\lambda + 2 \mu){\mathbf{q}}+ 3{\mathbf{a}}+ 4{\mathbf{b}}, \quad
{\mathbf{v}}= (\lambda + 4 \mu){\mathbf{q}}+ 2{\mathbf{a}}+ 5{\mathbf{b}}.$$ Taking the traces of each equation, one obtains $$\operatorname{tr}({\mathbf{d}}) = 9\lambda + 6 \mu , \quad
\operatorname{tr}({\mathbf{v}}) = 3\lambda + 12 \mu,$$ and, finally: $$\begin{aligned}
\lambda & = \frac{1}{15}( 2 \operatorname{tr}({\mathbf{d}}) - \operatorname{tr}({\mathbf{v}})), & \mu & = \frac{1}{30}(- \operatorname{tr}({\mathbf{d}}) + 3 \operatorname{tr}({\mathbf{v}})), \\
{\mathbf{a}}& = \frac{1}{7}( 5{\mathbf{d}}^{\prime} - 4 {\mathbf{v}}^{\prime}) , & {\mathbf{b}}& = \frac{1}{7} ( -2{\mathbf{d}}^{\prime} + 3 {\mathbf{v}}^{\prime}),
\end{aligned}$$ where ${\mathbf{d}}^{\prime}:={{\mathbf{d}}}-\frac{1}{3}\operatorname{tr}( {{\mathbf{d}}})\,{\mathbf{q}}$ and ${\mathbf{v}}^{\prime}:={{\mathbf{v}}}-\frac{1}{3}\operatorname{tr}({{\mathbf{v}}})\,{\mathbf{q}}$ are the deviatoric parts of ${\mathbf{d}}$ and ${\mathbf{v}}$ respectively.
The fourth-order harmonic component ${\mathbf{H}}$ is obtained using ${\mathbf{S}}:= ({\mathbf{E}})^{s}$, the total symmetrization of ${\mathbf{E}}$, given by $$S_{ijkl} = \frac{1}{3} (E_{ijkl}+E_{ikjl}+E_{iljk}).$$ The traceless part of ${\mathbf{S}}$, ${\mathbf{H}}$, is then given by $${\mathbf{H}}= {\mathbf{S}}- \frac{2}{7} {\mathbf{q}}\odot \left({\mathbf{d}}' + 2 {\mathbf{v}}'\right) - \frac{1}{15}\left(\operatorname{tr}{\mathbf{d}}+ 2 \operatorname{tr}{\mathbf{v}}\right) {\mathbf{q}}\odot {\mathbf{q}}$$ where $$({\mathbf{a}}\odot {\mathbf{b}})_{ijkl} := \frac{1}{6} \big( a_{ij} b_{kl} + b_{ij} a_{kl}+a_{ik} b_{jl} +b_{ik} a_{jl} +a_{il} b_{jk} + b_{il} a_{jk})$$ if ${\mathbf{a}}$ and ${\mathbf{b}}$ are two symmetric second order tensors.
An Elasticity tensor ${\mathbf{E}}$ can thus be written as $${\mathbf{E}}= ({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}}, \lambda, \mu),$$ where $\lambda, \mu$ are scalars, ${\mathbf{a}}, {\mathbf{b}}\in {\mathbb{H}}^{2}$ and ${\mathbf{H}}\in {\mathbb{H}}^{4}$. This decomposition is however *not unique*. Indeed, substituting for $({\mathbf{a}}, {\mathbf{b}})$ any *invertible linear combination* of them would lead to a similar decomposition and the same is true for the pair of scalars $(\lambda,\mu)$. In particular, in the following theorem, one can use $({\mathbf{d}}^{\prime}, {\mathbf{v}}^{\prime})$ instead of $({\mathbf{a}}, {\mathbf{b}})$, for instance.
We will now state our main theorem, which characterizes, using polynomial covariants, the symmetry class of an Elasticity tensor.
\[thm:main\] Let ${\mathbf{E}}= ({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}}, \lambda, \mu) \in {\mathbb{E}\mathrm{la}}$ be an harmonic decomposition of an Elasticity tensor ${\mathbf{E}}$, where ${\mathbf{H}}\in {\mathbb{H}}^{4}$, ${\mathbf{a}}, {\mathbf{b}}\in {\mathbb{H}}^{2}$ and $\lambda, \mu$ are scalars. Then
1. ${\mathbf{E}}$ is isotropic if and only if ${\mathbf{a}}= {\mathbf{b}}= {\mathbf{d}}_{2} = 0$.
2. ${\mathbf{E}}$ is cubic if and only if ${\mathbf{a}}= {\mathbf{b}}= {\mathbf{d}}_{2}^{\prime} = 0$ and ${\mathbf{d}}_{2} \ne 0$.
3. ${\mathbf{E}}$ is transversely isotropic if and only if $({\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}})$ is transversely isotropic and $${\mathbf{H}}\times {\mathbf{d}}_{2} = {\mathbf{H}}\times {\mathbf{a}}= {\mathbf{H}}\times {\mathbf{b}}= 0.$$
4. ${\mathbf{E}}$ is tetragonal if and only if $({\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}})$ is transversely isotropic, $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) = \operatorname{tr}({\mathbf{H}}\times {\mathbf{a}}) = \operatorname{tr}({\mathbf{H}}\times {\mathbf{b}}) = 0,$$ and $${\mathbf{H}}\times {\mathbf{d}}_{2} \ne 0, \quad \text{or} \quad {\mathbf{H}}\times {\mathbf{a}}\ne 0, \quad \text{or} \quad {\mathbf{H}}\times {\mathbf{b}}\ne 0.$$
5. ${\mathbf{E}}$ is trigonal if and only if $({\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}})$ is transversely isotropic, $${\mathbf{d}}_{2} \times ({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}) = {\mathbf{a}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{a}}) = {\mathbf{b}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{b}}) = 0,$$ and $$\operatorname{tr}( {\mathbf{H}}\times {\mathbf{d}}_{2}) \ne 0, \quad \text{or} \quad \operatorname{tr}({\mathbf{H}}\times {\mathbf{a}}) \ne 0, \quad \text{or} \quad \operatorname{tr}({\mathbf{H}}\times {\mathbf{b}}) \ne 0.$$
6. ${\mathbf{E}}$ is orthotropic if and only if the family of second-order tensors $$\mathcal{F}_{o} := {\left\{{\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}^{2}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}^{2}\right\}}$$ is orthotropic.
7. ${\mathbf{E}}$ is monoclinic if and only if the family of second-order tensors $$\mathcal{F}_{m} := \left\{{\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}^{2}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}^{2}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{b}})^{s}, {\mathbf{H}}\operatorname{:}({\mathbf{a}}{\mathbf{d}}_{2})^{s}, {\mathbf{H}}\operatorname{:}({\mathbf{b}}{\mathbf{d}}_{2})^{s}\right\}$$ is monoclinic.
8. ${\mathbf{E}}$ is triclinic if and only if none of the preceding conditions holds.
Explicit covariant relations on a finite family $\mathcal{F}$ of second-order tensors which characterize its symmetry class are provided by theorem \[thm:n-quadratic-forms\].
Note that if the family $\mathcal{F}_{o}$ is transversely isotropic, then, the triplet $({\mathbf{d}}_{2},{\mathbf{a}},{\mathbf{b}})$ is transversely isotropic. Otherwise, it would be isotropic but then, ${\mathbf{E}}$ would be either isotropic or cubic by points (1) and (2), and $\mathcal{F}_{o}$ would be isotropic itself, because covariants of ${\mathbf{E}}$ cannot have less symmetry than ${\mathbf{E}}$ itself. This would lead to a contradiction. Hence, if the family $\mathcal{F}_{o}$ is transversely isotropic, then, $({\mathbf{d}}_{2},{\mathbf{a}},{\mathbf{b}})$ is transversely isotropic and thus ${\mathbf{E}}$ is either transversely isotropic (3), tetragonal (4), or trigonal (5).
Note first that the symmetry class of ${\mathbf{E}}$ is the same as the symmetry class of the triplet $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ (see Section \[sec:symmetry-classes\]).
\(1) If ${\mathbf{E}}$ is isotropic, then, ${\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}}$ are all isotropic and thus vanish, since they all belong to irreducible representations. Conversely, if ${\mathbf{a}}= {\mathbf{b}}= {\mathbf{d}}_{2} = 0$, then, ${\mathbf{H}}$ vanishes because ${\lVert{\mathbf{H}}\rVert}^{2} = \operatorname{tr}{\mathbf{d}}_{2}$. Thus, $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ is isotropic.
\(2) If ${\mathbf{E}}$ is cubic, then, all second-order symmetric covariant are isotropic. Thus, ${\mathbf{a}}= {\mathbf{b}}= {\mathbf{d}}_{2}^{\prime} = 0$ but ${\mathbf{d}}_{2} \ne 0$ (otherwise, $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ would be isotropic, by point (1)). Conversely, if ${\mathbf{a}}= {\mathbf{b}}= {\mathbf{d}}_{2}^{\prime} = 0$ and ${\mathbf{d}}_{2} \ne 0$, then, ${\mathbf{H}}$ is cubic according to theorem \[thm:cov2-isotropic\] and so is $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$.
For the sequel of the proof, note that, so far, that we have proved that ${\mathbf{E}}$ is either isotropic, or cubic if and only if the family of second-order covariants $$\mathcal{F}_{i} := {\left\{{\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}}\right\}}$$ is isotropic.
\(3) If ${\mathbf{E}}$ is transversely isotropic, then, the triplet $({\mathbf{a}}, {\mathbf{b}}, {\mathbf{d}}_{2})$ is thus transversely isotropic. Moreover, each pair $({\mathbf{H}},{\mathbf{d}}_{2})$, $({\mathbf{H}},{\mathbf{a}})$, $({\mathbf{H}},{\mathbf{b}})$ is at least transversely isotropic and thus $${\mathbf{H}}\times {\mathbf{d}}_{2} = {\mathbf{H}}\times {\mathbf{a}}= {\mathbf{H}}\times {\mathbf{b}}= 0,$$ by lemma \[lem:Sxd=0\] and Remark \[rem:Sxq=0\]. Conversely, if the conditions in (3) are satisfied, then, at least one of the covariants ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{d}}_{2}$ (call it ${\mathbf{t}}$) is transversely isotropic and $${\mathbf{H}}\times {\mathbf{t}}= 0.$$ Therefore, the pair $({\mathbf{H}},{\mathbf{t}})$ is transversely isotropic according to lemma \[lem:Sxd=0\] and so is the triplet $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$.
\(4) If ${\mathbf{E}}$ is tetragonal, then, each pair of covariants $({\mathbf{H}},{\mathbf{d}}_{2})$, $({\mathbf{H}},{\mathbf{a}})$, $({\mathbf{H}},{\mathbf{b}})$ is at least tetragonal and thus $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{d}}_{2}) = \operatorname{tr}({\mathbf{H}}\times {\mathbf{a}}) = \operatorname{tr}({\mathbf{H}}\times {\mathbf{b}}) = 0,$$ by lemma \[lem:tr(Hxd)=0\] and Remark \[rem:Sxq=0\]. Now, since $({\mathbf{a}}, {\mathbf{b}}, {\mathbf{d}}_{2})$ is transversely isotropic, at least one of the covariants ${\mathbf{d}}_{2}$, ${\mathbf{a}}$, ${\mathbf{b}}$ is transversely isotropic and thus $${\mathbf{H}}\times {\mathbf{d}}_{2} \ne 0, \quad \text{or} \quad {\mathbf{H}}\times {\mathbf{a}}\ne 0, \quad \text{or} \quad {\mathbf{H}}\times {\mathbf{b}}\ne 0,$$ by lemma \[lem:Sxd=0\] (otherwise, one of the pairs $({\mathbf{H}},{\mathbf{d}}_{2})$, $({\mathbf{H}},{\mathbf{a}})$, $({\mathbf{H}},{\mathbf{b}})$ would be at least transversely isotropic and so would be $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$). Conversely, if conditions in (4) are satisfied, we can find a covariant ${\mathbf{t}}$ among ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{d}}_{2}$ such that $$\operatorname{tr}({\mathbf{H}}\times {\mathbf{t}}) = 0, \quad \text{and} \quad {\mathbf{H}}\times {\mathbf{t}}\ne 0.$$ Then, ${\mathbf{t}}$ is necessarily transversely isotropic by Remark \[rem:Sxq=0\] and thus $({\mathbf{H}},{\mathbf{t}})$ is at least tetragonal by lemma \[lem:tr(Hxd)=0\]. Moreover, $({\mathbf{H}},{\mathbf{t}})$ cannot be transversely isotropic, nor isotropic by lemma \[lem:Sxd=0\]. Since it cannot be either cubic (since ${\mathbf{t}}$ is transversely isotropic), it is in fact tetragonal, and so is the triplet $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$.
\(5) If ${\mathbf{E}}$ is trigonal, then, ${\mathbf{Cov}}_{2}({\mathbf{E}})$ is at least transversely isotropic and we get, in particular, $${\mathbf{d}}_{2} \times ({\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}) = {\mathbf{a}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{a}}) = {\mathbf{b}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{b}}) = 0,$$ by lemma \[lem:axb=0\] and Remark \[rem:Sxq=0\]. Moreover, $({\mathbf{a}}, {\mathbf{b}}, {\mathbf{d}}_{2})$ is transversely isotropic and thus at least one of the covariants ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{d}}_{2}$ (call it ${\mathbf{t}}$) is transversely isotropic. But then, $$G_{{\mathbf{E}}} = G_{({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})} = G_{{\mathbf{H}}} \cap G_{({\mathbf{d}}_{2},{\mathbf{a}},{\mathbf{b}})} = G_{{\mathbf{H}}} \cap G_{{\mathbf{t}}}.$$ Thus, the pair $({\mathbf{H}}, {\mathbf{t}})$ is trigonal and $\operatorname{tr}( {\mathbf{H}}\times {\mathbf{t}}) \ne 0$ by lemma \[lem:trigonal-pair\]. Conversely, if conditions in (5) are satisfied, we can find a covariant ${\mathbf{t}}$ among ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{d}}_{2}$ (and thus at least transversely isotropic) such that $${\mathbf{t}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{t}}) = 0, \quad {\mathbf{d}}_{2} \times {\mathbf{t}}= 0, \quad \text{and} \quad \operatorname{tr}( {\mathbf{H}}\times {\mathbf{t}}) \ne 0.$$ Then, ${\mathbf{t}}$ is necessarily transversely isotropic by Remark \[rem:Sxq=0\] and thus $({\mathbf{H}},{\mathbf{t}})$ is trigonal by lemma \[lem:trigonal-pair\], and so is the triplet $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$.
For the sequel of the proof, note that, so far, that we have proved that ${\mathbf{E}}$ is either transversely isotropic, tetragonal or trigonal if and only if the family of second-order covariants $$\mathcal{F}_{ti} := {\left\{{\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}}, {\mathbf{c}}_{3}, {\mathbf{H}}\operatorname{:}{\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{b}}\right\}}$$ is transversely isotropic.
\(6) If ${\mathbf{E}}$ is orthotropic, then, the family of second-order covariants $\mathcal{F}_{o}$ is at least orthotropic. Since, moreover, $$\mathcal{F}_{i} \subset \mathcal{F}_{ti} \subset \mathcal{F}_{o},$$ $\mathcal{F}_{o}$ cannot be isotropic by points (1) and (2), neither transversely isotropic by points (3), (4) and (5). It is thus orthotropic. Conversely, if $\mathcal{F}_{o}$ is orthotropic, then $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ is either orthotropic, monoclinic or triclinic because $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ cannot have higher symmetry than its covariants. If either ${\mathbf{a}}$ or ${\mathbf{b}}$ is orthotropic, then, $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ is orthotropic by lemma \[lem:orthotropic-criteria\]. The same conclusion holds if either ${\mathbf{d}}_{2}$ or ${\mathbf{c}}_{3}$ is orthotropic by theorem \[thm:cov2-orthotropic\]. Otherwise, ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{d}}_{2}$ and ${\mathbf{c}}_{3}$ are each at least transversely isotropic. In that case, if either $({\mathbf{a}},{\mathbf{b}})$, $({\mathbf{a}},{\mathbf{d}}_{2})$, $({\mathbf{a}},{\mathbf{c}}_{3})$, $({\mathbf{b}},{\mathbf{d}}_{2})$, $({\mathbf{b}},{\mathbf{c}}_{3})$ or $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is orthotropic, then, $({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}})$ is orthotropic by corollary \[cor:transversely-isotropic-pair-criteria\] and the fact that ${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}$ and ${\mathbf{c}}_{4} = {\mathbf{H}}\operatorname{:}{\mathbf{c}}_{3}$ . Thus, we can assume that the quadruplet $({\mathbf{a}},{\mathbf{b}},{\mathbf{d}}_{2},{\mathbf{c}}_{3})$ is transversely isotropic (it cannot be isotropic, otherwise, so would be $\mathcal{F}_{o}$, because ${\mathbf{H}}\operatorname{:}{\mathbf{q}}= 0$). Note that, in this case, the alternative ${\mathbf{d}}_{2}$ transversely isotropic is excluded, otherwise, $({\mathbf{d}}_{2},{\mathbf{c}}_{3})$ would be transversely isotropic and so would be $\mathcal{F}_{o}$ by theorem \[thm:cov2-transversely-isotropic\]. Therefore, ${\mathbf{d}}_{2}^{\prime} =0$ and ${\mathbf{H}}$ is either isotropic or cubic by theorem \[thm:cov2-isotropic\]. The case where ${\mathbf{H}}$ is isotropic (and thus vanishes) is excluded because then, $\mathcal{F}_{o}$ would be at least transversely isotropic. We can thus finally assume that ${\mathbf{H}}$ is cubic. Then, either ${\mathbf{a}}$ or ${\mathbf{b}}$ is transversely isotropic. Let suppose it is ${\mathbf{a}}$. Then ${\mathbf{b}}$ is collinear to ${\mathbf{a}}$ (since ${\mathbf{a}}$ and ${\mathbf{b}}$ are deviators) and the pair $({\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}})$ has the same symmetry group as $\mathcal{F}_{o}$ and is thus orthotropic. Th erefore, $$\operatorname{tr}({\mathbf{a}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{a}})) = 0, \qquad {\mathbf{a}}\times ({\mathbf{H}}\operatorname{:}{\mathbf{a}}) \ne 0,$$ and $({\mathbf{H}}, {\mathbf{a}})$ is orthotropic by lemma \[lem:cube-orientation\], and so is $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$.
\(7) If ${\mathbf{E}}$ is monoclinic then the family of covariants $\mathcal{F}_{m}$ is at least monoclinic and thus monoclinic, by points (1)–(6) and because $$\mathcal{F}_{i} \subset \mathcal{F}_{ti} \subset \mathcal{F}_{o} \subset \mathcal{F}_{m}.$$ Conversely, suppose that $$G_{\mathcal{F}_{m}} = {\left\{id,r\right\}},$$ where $r$ is a second-order rotation. Then, $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$ is at most monoclinic, because it cannot have higher symmetry than its covariants. Besides, $$r \in G_{\mathcal{F}_{m}} \subset G_{({\mathbf{a}},{\mathbf{b}})},$$ so we have only to check that $r\in G_{{\mathbf{H}}}$, to prove that $$r \in G_{{\mathbf{E}}} = G_{{\mathbf{H}}} \cap G_{({\mathbf{a}},{\mathbf{b}})}.$$ Now, since $$r \in G_{\mathcal{F}_{m}} \subset G_{({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})},$$ ${\mathbf{H}}$ is at least monoclinic by theorem \[thm:cov1-triclinic\]. If ${\mathbf{H}}$ is either monoclinic or orthotropic, then, we are done by theorem \[thm:cov2-monoclinic\] and theorem \[thm:cov2-orthotropic\], because, in these cases we have $$G_{{\mathbf{H}}} = G_{({\mathbf{d}}_{2}, {\mathbf{c}}_{3}, {\mathbf{c}}_{4})}.$$ If ${\mathbf{H}}$ is either transversely isotropic, tetragonal or trigonal, then $({\mathbf{d}}_{2}, {\mathbf{c}}_{3})$ is transversely isotropic by theorem \[thm:cov2-transversely-isotropic\]. Thus ${\mathbf{d}}_{2}$ is transversely isotropic and ${\mathbf{d}}_{2} \times {\mathbf{c}}_{3} = 0$ with ${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}$. But then, the triplet $({\mathbf{d}}_{2}, {\mathbf{a}}, {\mathbf{b}})$ is at most orthotropic, otherwise the family $\mathcal{F}_{m}$ would have the same symmetry group as $({\mathbf{d}}_{2}, {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2})$ and would be transversely isotropic. Therefore, either ${\mathbf{a}}$ or ${\mathbf{b}}$ (let call it ${\mathbf{c}}$) is orthotropic, and we are done by lemma \[lem:orthotropic-criteria\], because $$r \in G_{\mathcal{F}_{m}} \subset G_{({\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}, {\mathbf{H}}\operatorname{:}{\mathbf{c}}^{2})} = G_{({\mathbf{H}},{\mathbf{c}})} \subset G_{{\mathbf{H}}},$$ or ${\mathbf{a}}$ and ${\mathbf{b}}$ are both transversely isotropic but one of the three pair $({\mathbf{a}},{\mathbf{b}})$, $({\mathbf{a}}, {\mathbf{d}}_{2})$ or $({\mathbf{b}}, {\mathbf{d}}_{2})$ (let call it $({\mathbf{t}}_{1},{\mathbf{t}}_{2})$) is either orthotropic or monoclinic, and we are done by corollary \[cor:transversely-isotropic-pair-criteria\] (since ${\mathbf{c}}_{3} = {\mathbf{H}}\operatorname{:}{\mathbf{d}}_{2}$), because then $$r \in G_{\mathcal{F}_{m}} \subset G_{({\mathbf{H}}, {\mathbf{t}}_{1},{\mathbf{t}}_{2})} \subset G_{{\mathbf{H}}}.$$ Suppose now that ${\mathbf{H}}$ is cubic. If either ${\mathbf{a}}$ or ${\mathbf{b}}$ is orthotropic, then, we are done by lemma \[lem:orthotropic-criteria\] and the same conclusion holds, by corollary \[cor:transversely-isotropic-pair-criteria\], if ${\mathbf{a}}$ and ${\mathbf{b}}$ are transversely isotropic but the pair $({\mathbf{a}},{\mathbf{b}})$ is orthotropic or monoclinic. We can thus assume that the pair of deviators $({\mathbf{a}},{\mathbf{b}})$ is transversely isotropic (it cannot be isotropic otherwise, so would be $\mathcal{F}_{m}$). In that case, either ${\mathbf{a}}$ or ${\mathbf{b}}$ does not vanish and is thus transversely isotropic. Suppose, for instance, that ${\mathbf{a}}\ne 0$. Then, ${\mathbf{b}}$ is collinear to ${\mathbf{a}}$ and $$G_{\mathcal{F}_{m}} = G_{({\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}})}.$$ Thus, $({\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}})$ is monoclinic and $$r \in G_{({\mathbf{a}}, {\mathbf{H}}\operatorname{:}{\mathbf{a}})} = G_{({\mathbf{a}}, {\mathbf{H}})},$$ by lemma \[lem:cube-orientation\] and Remark \[rem:cub-orientation\]. Finally, if ${\mathbf{H}}$ is isotropic, ${\mathbf{H}}= 0$ and we are done. This achieves the proof.
Covariants of binary forms {#sec:binary-forms-covariants}
==========================
A binary form ${\mathbf{f}}$ of degree $n$ is a homogeneous complex polynomial in two variables $u,v$ of degree $n$: $${\mathbf{f}}({\pmb{\xi}}) = a_{0}u^{n} + a_{1}u^{n-1}v + \dotsb + a_{n-1}uv^{n-1} + a_{n}v^{n},$$ where ${\pmb{\xi}}= (u,v)\in {\mathbb{C}}^{2}$ and $a_{k}\in {\mathbb{C}}$. The set of all binary forms of degree $n$ is a complex vector space of dimension $n + 1$ which will be denoted by ${\mathrm{S}_{n}}$. The special linear group $${\mathrm{SL}}(2,{\mathbb{C}}) : = {\left\{\gamma: =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
,\quad ad-bc = 1\right\}}$$ acts naturally on ${\mathbb{C}}^{2}$ and induces a left action on ${\mathrm{S}_{n}}$, given by $$(\gamma \star {\mathbf{f}})({\pmb{\xi}}): = {\mathbf{f}}(\gamma^{-1} {\pmb{\xi}}),$$ where $\gamma\in {\mathrm{SL}}(2,{\mathbb{C}})$. The spaces ${\mathrm{S}_{n}}$ are irreducible representations of ${\mathrm{SL}}(2,{\mathbb{C}})$ (see [@Ste1994] for instance) and every complex algebraic linear representation $V$ of ${\mathrm{SL}}(2,{\mathbb{C}})$ can be decomposed into a direct sum $$V \simeq {\mathrm{S}_{n_{1}}}\oplus \dotsc \oplus {\mathrm{S}_{n_{p}}}.$$
The *transvectant* of index $r$ of two binary forms ${\mathbf{f}}\in {\mathrm{S}_{n}}$ and ${\mathbf{g}}\in {\mathrm{S}_{p}}$ is defined as $$\label{eq:transvectant}
{\lbrace {\mathbf{f}},{\mathbf{g}}\rbrace_{r}} = \frac{(n-r)!}{n!}\frac{(p-r)!}{p!} \sum_{i = 0}^{r}(-1)^{i} \binom{r}{i} \frac{\partial^{r} {\mathbf{f}}}{\partial^{r-i}u \partial^{i} v}
\frac{\partial^{r} {\mathbf{g}}}{\partial^{i}u \partial^{r-i}v},$$ which is a binary form of degree $n + p-2r$ (which vanishes if $r > \min(n,p)$).
For two $n$-th powers binary forms $$\label{eq:n-powers-transvectant}
({\mathbf{a}}{\pmb{\xi}})^{n} : = (a_{1}u + a_{2}v)^{n},\quad ({\mathbf{b}}{\pmb{\xi}})^{p} : = (b_{1}u + b_{2}v)^{p},$$ we get the particularly simple form $${\lbrace ({\mathbf{a}}{\pmb{\xi}})^{n},({\mathbf{b}}{\pmb{\xi}})^{p}\rbrace_{r}} = ({\mathbf{a}}{\mathbf{b}})^{r}({\mathbf{a}}{\pmb{\xi}})^{n-r}({\mathbf{b}}{\pmb{\xi}})^{p-r},$$ where by definition $({\mathbf{a}}{\mathbf{b}}): = a_{1}b_{2}-a_{2}b_{1}$.
The covariant algebra of $V$ is defined as $${\mathbf{Cov}}(V) : = {\mathbb{C}}[V\oplus {\mathbb{C}}^{2}]^{{\mathrm{SL}}(2,{\mathbb{C}})}.$$ The **degree** of a covariant ${\mathbf{h}}\in {\mathbf{Cov}}(V)$ is the total degree $d$ of ${\mathbf{h}}$ in ${\mathbf{f}}\in V$, whereas the total degree $k$ of ${\mathbf{h}}$ in $\xi \in {\mathbb{C}}^{2}$ is called the **order** of ${\mathbf{h}}$.
The key point is that the transvectant of two binary forms is ${\mathrm{SL}}(2,{\mathbb{C}})$-equivariant and that ${\mathbf{Cov}}(V)$ is generated by the infinite set of *iterated transvectants* [@GY2010; @Olv1999; @Oli2017]: $${\mathbf{f}}_{1}, \dotsc ,{\mathbf{f}}_{p} \quad {\lbrace {\mathbf{f}}_{i},{\mathbf{f}}_{j}\rbrace_{r}}, \quad {\lbrace {\mathbf{f}}_{i},{\lbrace {\mathbf{f}}_{j},{\mathbf{f}}_{k}\rbrace_{r}}\rbrace_{s}}, \quad \dotsc$$
\[rem:even-order-covariants\] A consequence of this observation is that for every integer $n\geq 1$, the covariant algebra ${\mathbf{Cov}}({\mathrm{S}_{2n}})$ is generated by *even order* covariants.
The remarkable achievement of Gordan is that he was able to provide a constructive (and extremely efficient) way to obtain a *finite* generating set of transvectants for the covariant algebra of finite dimensional representation of ${\mathrm{SL}}(2,{\mathbb{C}})$. This algorithm is now known as *Gordan’s algorithm* (see [@Oli2017]). There are in fact two versions of this algorithm; one of them produces a basis for ${\mathbf{Cov}}({\mathrm{S}_{n}})$, provided we know bases for ${\mathbf{Cov}}({\mathrm{S}_{k}})$, for each $k < n$. The other one produces a basis for ${\mathbf{Cov}}(V_{1}\oplus V_{2})$, if we know bases for ${\mathbf{Cov}}(V_{1})$ and ${\mathbf{Cov}}(V_{2})$. More precisely, if ${\left\{{\mathbf{f}}_{1}, \dotsc , {\mathbf{f}}_{p}\right\}}$ and ${\left\{{\mathbf{g}}_{1}, \dotsc , {\mathbf{g}}_{q}\right\}}$ generate respectively ${\mathbf{Cov}}(V_{1})$ and ${\mathbf{Cov}}(V_{2})$, then the covariant algebra ${\mathbf{Cov}}(V_{1} \oplus V_{2})$ is generated by the finite family of transvectants $${\lbrace {\mathbf{f}}_{1}^{\alpha_{1}}\dotsb {\mathbf{f}}_{p}^{\alpha_{p}},{\mathbf{g}}_{1}^{\beta_{1}}\dotsb {\mathbf{g}}_{q}^{\beta_{q}}\rbrace_{r}},$$ where the integers $(\alpha_{i},\beta_{i},u,v,r)$ are the *irreducible solutions* of the *Diophantine equation* $$\sum_{i = 1}^{p} a_{i} \alpha_{i} = u + r, \qquad \sum_{j = 1}^{p} b_{j} \beta_{j} = v + r$$ and $a_{i},b_{j}$ are the orders of ${\mathbf{f}}_{i},{\mathbf{g}}_{j}$.
Using this algorithm, we will formulate a theorem which connects generating sets for ${\mathbf{Cov}}({\mathrm{S}_{2n}})$ and ${\mathbf{Inv}}({\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}})$. First, observe that there is a natural covariant mapping $$\psi : {\mathbb{C}}^{2} \to {\mathrm{S}_{2}}, \qquad \eta \mapsto {\mathbf{w}}_{\eta},$$ where $${\mathbf{w}}_{\eta}(\pmb{{\pmb{\xi}}}) : = (\eta_{1}v - \eta_{2}u)^{2}, \qquad {\pmb{\xi}}= (u,v).$$ By pullback, this mapping induces an algebra homomorphism $$\psi^{*} : {\mathbb{C}}[{\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}}]^{{\mathrm{SL}}(2,{\mathbb{C}})} \to {\mathbb{C}}[{\mathrm{S}_{2n}}\oplus {\mathbb{C}}^{2}]^{{\mathrm{SL}}(2,{\mathbb{C}})} = {\mathbf{Cov}}({\mathrm{S}_{2n}})$$ given by $$\psi^{*}({\mathrm{p}})({\mathbf{f}}, {\pmb{\xi}}) = {\mathrm{p}}({\mathbf{f}}, {\mathbf{w}}_{\eta}),\qquad {\mathrm{p}}\in {\mathbb{C}}[{\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}}]^{{\mathrm{SL}}(2,{\mathbb{C}})}.$$ Consider now the covariant linear mapping $$\label{eq:psi-section-definition}
\varsigma: {\mathbf{Cov}}({\mathrm{S}_{2n}}) \to {\mathbb{C}}[{\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}}]^{{\mathrm{SL}}(2,{\mathbb{C}})}, \qquad {\mathbf{h}}\mapsto {\mathrm{p}}({\mathbf{f}},{\mathbf{w}}) := \sum_{k=0}^{r} {\lbrace {\mathbf{h}}_{2k},{\mathbf{w}}^{k}\rbrace_{2k}}$$ where ${\mathbf{h}}({\mathbf{f}},{\pmb{\xi}})=\sum_{k=0}^{r}{\mathbf{h}}_{2k}({\mathbf{f}},{\pmb{\xi}})$ is the decomposition of ${\mathbf{h}}$ into homogeneous covariants of order $2k$ (see remark \[rem:even-order-covariants\]). We have the following result.
The algebra homomorphism $\psi^{*}$ is surjective and $\varsigma$ is a linear equivariant section of $\psi^{*}$. In other words $$\psi^{*} \circ \varsigma = \mathrm{Id}.$$
Note first that *$\varsigma$ is linear but is not an algebra homomorphism*. We will show that $\varsigma$ is a section of $\psi^{*}$ (as a linear mapping) and the surjectivity will follow. If ${\mathbf{h}}$ is homogeneous of order $2r$, we have $$\varsigma({\mathbf{h}}) = {\lbrace {\mathbf{h}},{\mathbf{w}}^{r}\rbrace_{2r}},$$ and hence $$[(\psi^{*} \circ \varsigma)({\mathbf{h}})]({\mathbf{f}},\eta) = {\lbrace {\mathbf{h}},{\mathbf{w}}_{\eta}^{r}\rbrace_{2r}}.$$ Thus, if ${\mathbf{h}}({\mathbf{f}},{\pmb{\xi}}) = ({\mathbf{a}}\pmb{{\pmb{\xi}}})^{2r} = (a_{1}u + a_{2}v)^{2r}$ is a $2r$-th power binary form, we get $$(\psi^{*} \circ \varsigma)({\mathbf{h}})({\mathbf{f}},\eta) = {\lbrace (a_{1}u + a_{2}v)^{2r},(\eta_{1}v - \eta_{2}u)^{2r}\rbrace_{2r}} = (a_{1}\eta_{1} + a_{2}\eta_{2})^{2r} = {\mathbf{h}}({\mathbf{f}},\eta),$$ by virtue of . Since every binary form of degree $2r$ is a linear combination of $2r$-th power binary forms, this achieves the proof.
\[thm:basis-for-S2n-oplus-S2\] Let ${\left\{{\mathbf{h}}_{1},\dotsc ,{\mathbf{h}}_{N}\right\}}$ be a minimal basis for ${\mathbf{Cov}}({\mathrm{S}_{2n}})$. Then a minimal basis for the joint invariant algebra $${\mathbb{C}}[{\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}}]^{{\mathrm{SL}}(2,{\mathbb{C}})}$$ is given by $${\left\{\varsigma({\mathbf{h}}_{1}),\dotsc ,\varsigma({\mathbf{h}}_{N}), \Delta\right\}}$$ where $\Delta({\mathbf{w}}) : = b_{1}^{2} - b_{0}b_{2}$, if ${\mathbf{w}}({\pmb{\xi}}) := b_{0} u^{2} + b_{1}uv + b_{2}v^{2} \in {\mathrm{S}_{2}}$.
The result is still true if we replace, in the theorem, ${\mathrm{S}_{2n}}$ by a direct sum of binary forms of even degree ${\mathrm{S}_{2n_{1}}} \oplus \dotsb \oplus {\mathrm{S}_{2n_{k}}}$.
Applying Gordan’s algorithm to obtain a basis for ${\mathbf{Inv}}({\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}})$, and since ${\mathbf{Cov}}({\mathrm{S}_{2}})$ is generated by the binary form ${\mathbf{w}}$ itself and the invariant $\Delta$, we deduce that a generating set for ${\mathbf{Inv}}({\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}})$ is given by $\Delta$ and transvectants $${\lbrace {\mathbf{h}}_{1}^{\alpha_{1}}\cdots {\mathbf{h}}_{N}^{\alpha_{N}},{\mathbf{w}}^{r}\rbrace_{2r}},$$ where $(\alpha_{i},2r)$ is an irreducible solution of $$\label{eq:Diophantine-equation}
\sum_{i = 1}^{N} \alpha_{i} a_{i} = 2r$$ and ${\mathbf{h}}_{1}, \dotsc ,{\mathbf{h}}_{N}$ are generators for ${\mathbf{Cov}}({\mathrm{S}_{2n}})$, all of them being of even order.
Now observe that, if a product ${\mathbf{h}}_{1}^{\alpha_{1}}\cdots {\mathbf{h}}_{N}^{\alpha_{N}}$ contains more than two factors, then $(\alpha_{i},2r)$ is reducible. Indeed it can be written as a sum of two non-trivial solutions $(\alpha^{1}_{i},2r_{1})$ and $(\alpha^{2}_{i},2r_{2})$ of , where $2r_{1} + 2r_{2} = 2r$. Thus, a finite set of generators for ${\mathbf{Inv}}({\mathrm{S}_{2n}}\oplus {\mathrm{S}_{2}})$ is given by $\Delta$ and $${\lbrace {\mathbf{h}}_{i},{\mathbf{w}}^{r_{i}}\rbrace_{2r_{i}}}, \qquad i = 1, \dotsc , N$$ where $2r_{i}$ (see remark \[rem:even-order-covariants\]) is the order of ${\mathbf{h}}_{i}$. To achieve the proof, it remains to show that if ${\left\{{\mathbf{h}}_{1},\dotsc ,{\mathbf{h}}_{N}\right\}}$ is minimal the same is true for ${\left\{\varsigma({\mathbf{h}}_{1}),\dotsc ,\varsigma({\mathbf{h}}_{N}), \Delta\right\}}$. To do so, observe that if for some $i \in {\left\{1, \dotsc , N\right\}}$, there exists a polynomial $P$ such that $$\varsigma({\mathbf{h}}_{i}) = P(\Delta,\varsigma({\mathbf{h}}_{j})), \qquad j \ne i,$$ then using the fact that $\psi^{*}$ is an algebra homomorphism, we get $${\mathbf{h}}_{i} = \psi^{*}(\varsigma({\mathbf{h}}_{i})) = \psi^{*}(P(\Delta,\varsigma({\mathbf{h}}_{j}))) = P(\psi^{*}(\Delta), \psi^{*}(\varsigma({\mathbf{h}}_{j}))) = P(0,{\mathbf{h}}_{j}),$$ because $\psi^{*}(\Delta) = \Delta ({\mathbf{w}}_{\eta}) = \Delta ((\eta_{1}v - \eta_{2}u)^{2}) = 0$, which leads to a contradiction.
Covariants of harmonic tensors {#sec:harmonic-tensors-covariants}
==============================
There is a closed relation between covariant/invariant algebras of harmonic polynomials of three variables and those of binary forms which is recalled in this section (see also [@OKA2017 Appendix B]).
The complexification of the ${\mathrm{SO}}(3)$-representation on the real space of harmonic polynomials ${\mathcal{H}_{n}}({\mathbb{R}}^{3})$ extends to a representation of the complex algebraic group $${\mathrm{SO}}(3,{\mathbb{C}}) : = {\left\{P \in \mathrm{M}_{3}({\mathbb{C}}); \; P^{t}P = {\mathrm{I}},\, \det P = 1\right\}}$$ on the space of complex harmonic polynomials ${\mathcal{H}_{n}}({\mathbb{C}}^{3})$, which remains irreducible. There is, moreover, a group homomorphism [@OKA2017 Appendix B] $$\pi : {\mathrm{SL}}(2,{\mathbb{C}}) \to {\mathrm{SO}}(3,{\mathbb{C}}), \qquad \gamma \mapsto \operatorname{Ad}_{\gamma},$$ where $$\operatorname{Ad}_{\gamma} : M \mapsto \gamma M \gamma^{-1}, \qquad \gamma \in {\mathrm{SL}}(2,{\mathbb{C}}), \quad M \in {\mathfrak{sl}}(2,{\mathbb{C}})$$ is the adjoint action of ${\mathrm{SL}}(2,{\mathbb{C}})$ on its Lie algebra ${\mathfrak{sl}}(2,{\mathbb{C}})$. When restricted to the real Lie group $${\mathrm{SU}}(2,{\mathbb{C}}) : = {\left\{\gamma \in {\mathrm{SL}}(2,{\mathbb{C}});\; \bar{\gamma}^{t} \gamma = {\mathrm{I}}\right\}},$$ it induces the well-known two-fold covering $$\pi : {\mathrm{SU}}(2,{\mathbb{C}}) \to {\mathrm{SO}}(3,{\mathbb{R}}), \qquad \gamma \mapsto \operatorname{Ad}_{\gamma}.$$ Using these constructions, ${\mathcal{H}_{n}}({\mathbb{C}}^{3})$ becomes an ${\mathrm{SL}}(2,{\mathbb{C}})$-representation if we set $$\label{eq:actiongamma}
\gamma \star {\mathrm{h}}: = \pi(\gamma) \star {\mathrm{h}}, \qquad {\mathrm{h}}\in{\mathcal{H}_{n}}({\mathbb{C}}^{3}),\quad \gamma \in {\mathrm{SL}}(2,{\mathbb{C}}),$$ and ${\mathcal{H}_{n}}({\mathbb{R}}^{3})$ becomes an ${\mathrm{SU}}(2,{\mathbb{C}})$-representation if we set $$\gamma \star {\mathrm{h}}: = \pi(\gamma) \star {\mathrm{h}}, \qquad {\mathrm{h}}\in{\mathcal{H}_{n}}({\mathbb{R}}^{3}),\quad \gamma \in {\mathrm{SU}}(2,{\mathbb{C}}),$$ both of them remaining irreducible.
Now, there is an equivariant isomorphism between the space ${\mathcal{H}_{n}}({\mathbb{C}}^{3})$ of complex harmonic polynomials of degree $n$ and binary forms of degree $2n$. This isomorphism derives from an equivariant mapping introduced first in Cartan’s theory of spinors going back to 1913 (see [@Car1981 Chapter 3]) and rediscovered later by Backus [@Bac1970]. More precisely, let us introduce the *Cartan map* $$\label{eq:Cartan-map}
\phi : {\mathbb{C}}^{2} \to {\mathfrak{sl}}(2,{\mathbb{C}}), \qquad {\pmb{\xi}}\mapsto {\pmb{\xi}}\, {\pmb{\xi}}^{\omega}=
\begin{pmatrix}
-uv & u^{2} \\
-v^{2} & uv
\end{pmatrix},$$ where $${\pmb{\xi}}=
\begin{pmatrix}
u \\
v
\end{pmatrix}
, \qquad {\pmb{\xi}}^{\omega} =
\begin{pmatrix}
-v & u
\end{pmatrix},$$ and ${\pmb{\xi}}^{\omega}$ means the covariant version of the vector ${\pmb{\xi}}$, defined using the determinant $\omega$ on ${\mathbb{C}}^{2}$ (a nondegenerate bilinear form). The main property of this mapping is that it is ${\mathrm{SL}}(2,{\mathbb{C}})$-equivariant, meaning that $$\phi(\gamma {\pmb{\xi}}) = \operatorname{Ad}_{\gamma} \phi({\pmb{\xi}}), \qquad \forall \gamma \in {\mathfrak{sl}}(2,{\mathbb{C}}).$$ Choosing the following basis $$\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix},
\qquad
\begin{pmatrix}
0 & i \\
i & 0
\end{pmatrix},
\qquad
\begin{pmatrix}
i & 0 \\
0 & -i
\end{pmatrix},$$ of the Lie algebra ${\mathfrak{sl}}(2,{\mathbb{C}})$ (corresponding to multiplication by $i$ of Pauli matrices), allows us to identify ${\mathfrak{sl}}(2,{\mathbb{C}})$ with ${\mathbb{C}}^{3}$, using the parametrization $$\begin{pmatrix}
iz & x+iy \\
-x+iy & -iz
\end{pmatrix}.$$ In this basis, the Cartan map writes $$\label{eq:explicit-Cartan-map}
\phi : {\mathbb{C}}^{2} \to {\mathbb{C}}^{3}, \qquad (u,v) \mapsto \left( x= \frac{u^{2} + v^{2}}{2}, y= \frac{u^{2} - v^{2}}{2i}, z= iuv \right).$$ By pullback, the Cartan map $\phi$ induces an equivariant isomorphism $$\phi^{*} : {\mathcal{H}_{n}}({\mathbb{C}}^{3}) \to {\mathrm{S}_{2n}}, \qquad {\mathrm{h}}\mapsto {\mathrm{h}}\circ \phi ,$$ which is equivariant in the following sense (using ) $$\phi^{*}( \operatorname{Ad}_{\gamma} \star {\mathrm{h}}) = \gamma \star \phi^{*}({\mathrm{h}}), \qquad {\mathrm{h}}\in {\mathcal{H}_{n}}({\mathbb{C}}^{3}), \, \gamma \in {\mathrm{SL}}(2, {\mathbb{C}}).$$
\[thm:Hn-S2n-isomorphism\] The linear mapping $\phi^{*}: {\mathcal{H}_{n}}({\mathbb{C}}^{3}) \to {\mathrm{S}_{2n}}$ defined by $$(\phi^{*}({\mathrm{h}}))(u,v) : = {\mathrm{h}}\left( \frac{u^{2} + v^{2}}{2}, \frac{u^{2} - v^{2}}{2i}, iuv \right).$$ is an ${\mathrm{SL}}(2,{\mathbb{C}})$-equivariant *isomorphism*. The invariant algebras ${\mathbb{C}}[{\mathcal{H}_{n}}({\mathbb{C}}^{3})]^{{\mathrm{SO}}(3,{\mathbb{C}})}$ and ${\mathbb{C}}[{\mathrm{S}_{2n}}]^{{\mathrm{SL}}(2, {\mathbb{C}})}$ are thus isomorphic.
The equivariant isomorphism $\phi^{*}: {\mathcal{H}_{n}}({\mathbb{C}}^{3}) \to {\mathrm{S}_{2n}}$ is unique, up to a scaling factor, thanks to Schur’s lemma. A different basis and thus a different representation has been considered by Backus in its study of the Elasticity tensor, as [@Bac1970 Eq. 50] $$(u,v) \mapsto \left( u^{2} - v^{2}, -i(u^{2} + v^{2}), 2uv \right).$$ Note also that a different representation was given in [@OKA2017 Theorem 5.1], as $$(u,v) \mapsto \left( \frac{u^{2} - v^{2}}{2}, \frac{u^{2} + v^{2}}{2i}, uv \right).$$ However, expression seems to be finally more convenient, especially when one works with transvectants.
Let ${{\mathrm{S}_{2n}}^{{\mathbb{R}}}} : = \phi^{*}({\mathcal{H}_{n}}({\mathbb{R}}^{3}))$ be the space of binary forms which correspond to real harmonic polynomials. This space is characterized as follows $${{\mathrm{S}_{2n}}^{{\mathbb{R}}}} = {\left\{{\mathbf{f}}\in {\mathrm{S}_{2n}}; \; S{\mathbf{f}}= {\mathbf{f}}\right\}},$$ where $S$ is the linear involution of ${\mathrm{S}_{2n}}$ defined by $$(S{\mathbf{f}})(u,v) = \bar{{\mathbf{f}}}(-v,u),$$ and where $\bar{{\mathbf{f}}}(u,v) := \overline{{\mathbf{f}}(\bar{u},\bar{v})}$. This means that if $${\mathbf{f}}= \sum_{k=0}^{2n} a_{k}u^{k}v^{2n-k},$$ then, $${\mathbf{f}}\in {{\mathrm{S}_{2n}}^{{\mathbb{R}}}} \iff a_{2n-k} = (-1)^{k}\overline{a_{k}}, \qquad k = 0, \dotsc, 2n.$$
Note that ${{\mathrm{S}_{2n}}^{{\mathbb{R}}}}$ is invariant under the action of ${\mathrm{SU}}(2,{\mathbb{C}})$ and that the decomposition of the space ${\mathrm{S}_{2n}}$ into irreducible components of ${\mathrm{SU}}(2,{\mathbb{C}})$ writes $${\mathrm{S}_{2n}} = {{\mathrm{S}_{2n}}^{{\mathbb{R}}}} \oplus i{{\mathrm{S}_{2n}}^{{\mathbb{R}}}},$$ where $i{{\mathrm{S}_{2n}}^{{\mathbb{R}}}}$ is characterized by the functional equation $S{\mathbf{f}}= -{\mathbf{f}}$. Moreover, since we have the following commuting relations $$\partial_{u} \circ S = S \circ \partial_{v}, \qquad \partial_{v} \circ S = -S \circ \partial_{u},$$ we deduce that $${\lbrace S{\mathbf{f}},S{\mathbf{g}}\rbrace_{r}} = S{\lbrace {\mathbf{f}},{\mathbf{g}}\rbrace_{r}},$$ by . Therefore, we have the following result.
Let ${\mathbf{f}}\in {{\mathrm{S}_{2n}}^{{\mathbb{R}}}}$ and ${\mathbf{g}}\in {{\mathrm{S}_{2p}}^{{\mathbb{R}}}}$. Then ${\lbrace {\mathbf{f}},{\mathbf{g}}\rbrace_{2r}} \in {{\mathrm{S}_{2n+2p-2r}}^{{\mathbb{R}}}}$.
In particular, an iterated transvectant of order 0 (*i.e.* an invariant) is necessary real when evaluated on binary forms in ${{\mathrm{S}_{2n}}^{{\mathbb{R}}}}$, because ${{\mathrm{S}_{0}}^{{\mathbb{R}}}} = {\mathbb{R}}$. Therefore, if $I_{1}, \dotsc, I_{N}$ are invariants of binary forms in ${\mathrm{S}_{2n_{1}}}\oplus \dotsb \oplus {\mathrm{S}_{2n_{p}}}$ obtained by such a transvectant process, they become real polynomials when evaluated on ${{\mathrm{S}_{2n_{1}}}^{{\mathbb{R}}}}\oplus \dotsb \oplus {{\mathrm{S}_{2n_{p}}}^{{\mathbb{R}}}}$.
Consider now the covariant algebra $${\mathbf{Cov}}({\mathbb{V}}) = {\mathbb{R}}[{\mathbb{V}}\oplus {\mathbb{R}}^{3}]^{{\mathrm{SO}}(3)}$$ where $${\mathbb{V}}: = {\mathcal{H}_{n_{1}}}({\mathbb{R}}^{3})\oplus \dotsb \oplus {\mathcal{H}_{n_{p}}}({\mathbb{R}}^{3}).$$ Then, using the group morphism $\pi : {\mathrm{SU}}(2,{\mathbb{C}}) \to {\mathrm{SO}}(3,{\mathbb{R}})$ and the isomorphism introduced in theorem \[thm:Hn-S2n-isomorphism\], we deduce an explicit real algebra isomorphism $${\mathbf{Cov}}({\mathbb{V}}) \simeq {\mathbb{R}}[{{\mathrm{S}_{2n_{1}}}^{{\mathbb{R}}}}\oplus \dotsb \oplus {{\mathrm{S}_{2n_{p}}}^{{\mathbb{R}}}} \oplus {{\mathrm{S}_{2}}^{{\mathbb{R}}}}]^{{\mathrm{SU}}(2,{\mathbb{C}})},$$ where we have made the trivial identification ${\mathbb{R}}^{3} = {\mathcal{H}_{1}}({\mathbb{R}}^{3})$ and we have the following result.
\[thm:decomplexification\] Let ${\left\{{\mathbf{g}}_{1},\dotsc,{\mathbf{g}}_{N}\right\}}$ be a *minimal* generating set of the complex covariant algebra $${\mathbb{C}}[{\mathrm{S}_{2n_{1}}}\oplus \dotsb \oplus {\mathrm{S}_{2n_{p}}} \oplus {\mathrm{S}_{2}}]^{{\mathrm{SL}}(2,{\mathbb{C}})}$$ obtained by iterated transvectants. Then, by restriction, the set ${\left\{{\mathbf{g}}_{1},\dotsc,{\mathbf{g}}_{N}\right\}}$ defines a *minimal* generating set of the real invariant algebra $${\mathbb{R}}[{{\mathrm{S}_{2n_{1}}}^{{\mathbb{R}}}}\oplus \dotsb \oplus {{\mathrm{S}_{2n_{p}}}^{{\mathbb{R}}}} \oplus {{\mathrm{S}_{2}}^{{\mathbb{R}}}}]^{{\mathrm{SU}}(2,{\mathbb{C}})}.$$
In practice, to obtain an explicit basis of the covariant algebra $${\mathbf{Cov}}({\mathcal{H}_{n_{1}}}({\mathbb{R}}^{3})\oplus \dotsb \oplus {\mathcal{H}_{n_{p}}}({\mathbb{R}}^{3}))$$ starting from the knowledge of a basis $${\left\{{\mathbf{h}}_{1},\dotsc,{\mathbf{h}}_{N}\right\}}$$ of the covariant algebra $${\mathbf{Cov}}({\mathrm{S}_{2n_{1}}}\oplus \dotsb \oplus {\mathrm{S}_{2n_{p}}}),$$ obtained by iterated transvectants, one can use lemma \[lem:trad-transvectants\] to translate these iterated transvectants and obtained tensorial expressions (or their polynomial counterparts, using the results of Section \[sec:covariant-tensor-operations\]) for the generators of $${\mathbf{Cov}}({\mathcal{H}_{n_{1}}}({\mathbb{R}}^{3})\oplus \dotsb \oplus {\mathcal{H}_{n_{p}}}({\mathbb{R}}^{3})),$$ don’t omitting to add ${\mathrm{q}}: = x^{2} + y^{2} + z^{2}$ to this list.
\[lem:trad-transvectants\] Let ${\mathbf{H}}_{1} \in {\mathbb{H}}^{n}({\mathbb{R}}^{3})$ and ${\mathbf{H}}_{2} \in {\mathbb{H}}^{p}({\mathbb{R}}^{3})$ be two harmonic tensors. Then we have $$\label{eq:even-order-transvectant}
{\lbrace \phi^{*}{\mathbf{H}}_{1},\phi^{*}{\mathbf{H}}_{2}\rbrace_{2r}} = 2^{-r}\phi^\ast(({\mathbf{H}}_{1} \overset{(r)}{\cdot}{\mathbf{H}}_{2})^{s}_0)$$ and $$\label{eq:odd-order-transvectant}
{\lbrace \phi^{*}{\mathbf{H}}_{1},\phi^{*}{\mathbf{H}}_{2}\rbrace_{2r+1}} = \kappa(n,p,r) \phi^\ast((\operatorname{tr}^r({\mathbf{H}}_{1} \times {\mathbf{H}}_{2}))_0)$$ where $$\kappa(n,p,r) = \frac{1}{2^{2r+1}} \frac{(n+p-1)! (n-r-1)!(p-r-1)!}{(n+p-1-2r)! (n-1)! (p-1)! }.$$
Invariants of the Elasticity tensor {#sec:elasticity-invariants}
===================================
A minimal integrity basis for the Elasticity tensor ${\mathbf{E}}= ({\mathbf{H}}, {\mathbf{a}}, {\mathbf{b}}, \lambda,\mu)$ was produced for the first time in [@OKA2017], using Gordan’s algorithm [@Oli2017]. In this appendix, we provide an alternative minimal integrity basis, using the covariants of ${\mathbf{H}}$ given in .
This basis has been checked to be correct using the Hilbert series of ${\mathbf{Inv}}({\mathbb{E}\mathrm{la}})$ and using the method which was outlined in section \[sec:H4-covariant-algebra\]. This basis consists in:
1. 15 simple invariants:
- $\lambda$, $\mu$;
- the simple invariants of ${\mathbf{a}}$ and ${\mathbf{b}}$: $\operatorname{tr}{\mathbf{a}}^{2}$, $\operatorname{tr}{\mathbf{a}}^{3}$, $\operatorname{tr}{\mathbf{b}}^{2}$, $\operatorname{tr}{\mathbf{b}}^{3}$:
- and the nine simple invariants of ${\mathbf{H}}$, computed first in [@BKO1994]: $$\begin{gathered}
\operatorname{tr}{{\mathbf{d}}}_{2},\quad \operatorname{tr}{{\mathbf{d}}}_{3},\quad \operatorname{tr}{{\mathbf{d}}}_{2}^{2},\quad \operatorname{tr}\left({{\mathbf{d}}}_{2}{{\mathbf{d}}}_{3}\right),\quad \operatorname{tr}{{\mathbf{d}}}_{2}^{3}, \\
\quad \operatorname{tr}\left({{\mathbf{d}}}_{2}^{2}{{\mathbf{d}}}_{3}\right), \quad \operatorname{tr}\left({{\mathbf{d}}}_{2}{{\mathbf{d}}}_{3}^{2}\right), \quad \operatorname{tr}{{\mathbf{d}}}_{3}^{3}, \quad \operatorname{tr}\left({{\mathbf{d}}}_{2}^{2}{{\mathbf{d}}}_{3}^{2}\right);
\end{gathered}$$
2. 4 joint invariants of $({\mathbf{a}},{\mathbf{b}})$: $$\operatorname{tr}\left({\mathbf{a}}{\mathbf{b}}\right),\quad \operatorname{tr}\left({\mathbf{a}}^{2}{\mathbf{b}}\right),\quad \operatorname{tr}\left({\mathbf{a}}{\mathbf{b}}^{2}\right),\quad \operatorname{tr}\left({\mathbf{a}}^{2}{\mathbf{b}}^{2}\right)$$
3. 52 joint invariants of $({\mathbf{H}},{\mathbf{a}})$, and similarly 52 joint invariants of $({\mathbf{H}},{\mathbf{b}})$ given in , where $${{{{\vphantom{\pmb{C}}}^{\textrm{3,3}}\pmb{C}}}} = \operatorname{tr}\big({\mathbf{H}}\times {{\mathbf{d}}}_{2}\big) ,\quad {{{{\vphantom{\pmb{C}}}^{\textrm{4b,5}}\pmb{C}}}}=\big({\mathbf{H}}^{2}\big)^s\times{{\mathbf{d}}}_{2},\quad {\pmb{v}}_{5} = \pmb\varepsilon\operatorname{:}\big({{\mathbf{d}}}_{2}{{\mathbf{c}}}_{3}\big).$$
4. 174 joint invariant of $({\mathbf{H}},{\mathbf{a}},{\mathbf{b}})$ given in , and , where $$\begin{gathered}
{{{{\vphantom{\pmb{C}}}^{\textrm{3,7}}\pmb{C}}}} = {\mathbf{H}}\times\big({\mathbf{H}}^{2}\big)^s, \quad {{{{\vphantom{\pmb{C}}}^{\textrm{3,9}}\pmb{C}}}} = \big( \big({\mathbf{H}}\cdot{\mathbf{H}}\big)^s\times {\mathbf{H}}\big), \\ {{{{\vphantom{\pmb{C}}}^{\textrm{3,5}}\pmb{C}}}} = {\mathbf{H}}\times {{\mathbf{d}}}_{2}, \quad {{{{\vphantom{\pmb{C}}}^{\textrm{4,7}}\pmb{C}}}}=\big( {\mathbf{H}}\times \big({\mathbf{H}}^{3}\big)^s\big).
\end{gathered}$$
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---
address: 'California Institute of Technology, Pasadena, CA 91125, USA'
author:
- 'JOHN H. SCHWARZ'
title: 'SOME PROPERTIES OF TYPE I${}^{\prime}$ STRING THEORY'
---
\#1\#2\#3\#4[[\#1]{}[**\#2**]{}, \#3 (\#4)]{}
Introduction
============
I am pleased to contribute to this volume in memory of Yuri Golfand. His name will be remembered by future generations of physicists for his 1971 paper with Likhtman,[@golfand71] which introduced the four-dimensional super-Poincaré algebra for the first time. Recognizing that such a symmetry algebra is a consistent mathematical possibility was certainly a remarkable achievement. It is a curious coincidence that this paper appeared within a few days of Pierre Ramond’s paper on fermionic strings.[@ramond71] Communications were not so good in those days, and the Golfand–Likhtman work was not generally known (at least in the West) for several years. As a result, its influence in driving the development of supersymmetry was not as great as it should have been. In fact, supersymmetric theories in two dimensions were developed to describe the world-sheet theory of RNS strings,[@gervais71] and this motivated Wess and Zumino to seek four-dimensional analogs.[@zumino74] Only years later did we understand that RNS strings, properly interpreted, have local 10-dimensional spacetime supersymmetry.[@gliozzi77]
The version of the theory that received the most attention prior to 1985 was the one containing both open and closed strings, which Mike Green and I called the type I theory, since it has one ten-dimensional supersymmetry. In 1984 we showed that this theory is inconsistent (due to gauge anomalies) unless the gauge group is chosen to be SO(32).[@green84] Then the anomalies cancel, and consistency is achieved. In this manuscript, I propose to review some of the interesting features that appear when one of the spatial dimensions is chosen to be a circle. In this case an alternative $T$ dual description, known as type I${}^{\prime}$, is available. This description gives a different viewpoint for understanding various phenomena, such as gauge symmetry enhancement. The material presented here is not new, though it may be organized somewhat differently than has been done before.
T Duality
=========
Let $X^\mu (\sigma,\tau)$ denote the embedding functions of a closed string world-sheet in ten-dimensional spacetime. In the case of a trivial flat geometry, the world sheet field equations are simple two-dimensional wave equations. Suppose that one of the nine spatial dimensions, $X^9$ say, is circular with radius $R$. Denoting $X^9$ by $X$ for simplicity, the general solution of the wave equation is $$\label{Xform}
X = mR\sigma + \frac{n}{R} \tau + {\rm ~periodic~terms}.$$ The parameter $\sigma$ labels points along the string and is chosen to have periodicity $2\pi$. Thus $m$ is an integer, called the winding number, which is the number of times the string wraps the spatial circle. The parameter $\tau$ is world-sheet time, and correspondingly $p = n/R$ is the momentum along the circle. Single-valuedness of $e^{ipX}$ requires that $n$ is an integer, called the Kaluza–Klein excitation number.
The general solution of the $2d$ wave equation consists of arbitrary left-moving and right-moving pieces $$X(\sigma, \tau) = X_L (\sigma + \tau) + X_R (\sigma-\tau).$$ In the particular case described above we have $$\begin{aligned}
X_L &=& \frac{1}{2} \left(mR + \frac{n}{R}\right) (\sigma + \tau) + \ldots
\nonumber\\
X_R &=& \frac{1}{2} \left(mR - \frac{n}{R}\right) (\sigma - \tau) + \ldots .\end{aligned}$$ T duality is the world-sheet field transformation $X_R \rightarrow - X_R, X_L
\rightarrow X_L$ (or vice versa) together with corresponding transformations of world-sheet fermi fields. There are two issues to consider: the transformation of the world-sheet action and the transformation of the space-time geometry. The world-sheet action may or may not be invariant under T duality, depending on the theory, but the classical description of the spacetime geometry is always radically changed. Let us examine that first: $$X = X_L + X_R \rightarrow X_L - X_R = \frac{n}{R} \sigma + mR\tau + \ldots .$$ Comparing with eq.(\[Xform\]), we see that this describes a closed string on a circle of radius $1/R$ with winding number $n$ and Kaluza–Klein excitation number $m$. Thus we learn the rule that under T duality $R
\rightarrow 1/R$ and $m \leftrightarrow n$.[@giveon94] In the case of type I or type II superstrings, world-sheet supersymmetry requires that $\psi_R^9 \rightarrow -
\psi_R^9$ at the same time. This has the consequence for type II theories of interchanging the IIA theory (for which space-time spinors associated with left-movers and right-movers have opposite chirality) and the IIB theory (for which they have the same chirality). Thus T duality is not a symmetry in this case — rather it amounts to the equivalence of the IIA theory compactified on a circle with radius $R$ and the IIB theory on a circle with radius $1/R$. If we compactified on a torus instead, and performed T duality transformations along two of the cycles, then this would take IIA to IIA or IIB to IIB and would therefore be a symmetry.
In recent years, D-branes have played a central role in our developing understanding of string theory.[@polchinski95] These are dynamical objects, which can be regarded as nonperturbative excitations of the theory. They have the property that open strings can end on them. When they have $p$ spatial dimensions they are called D$p$-branes. If a D$p$-brane is a flat hypersurface, the coordinates can be chosen so that it fills the directions $X^m, \ m = 0,1, \ldots, p$ and has a specified position in the remaining “transverse” dimensions $X^i = d^i$ where $i = p
+ 1, \ldots, 9$. An open string ending on such a D-brane is required to satisfy Neumann boundary conditions in tangential directions $$\partial_\sigma X^M|_{\sigma = 0} = 0 \qquad m = 0, 1, \ldots , p,$$ and Dirichlet boundary conditions in the transverse directions $$X^i = d^i \qquad i = p + 1, \ldots, 9.$$
A remarkable fact, which is easy to verify, is that the T duality transformation $X_R \rightarrow - X_R$ interchanges Dirichlet and Neumann boundary conditions. This implies that an “unwrapped” D$p$-brane, which is localized on the circle, is mapped by T duality into a D$(p + 1)$-brane that is wrapped on the dual circle. This rule meshes nicely with the fact that the IIA theory has stable (BPS) D$p$-branes for even values of $p$ and the IIB theory has stable D$p$-branes for odd values of $p$. An obvious question that arises is how the wrapped D-brane encodes the position along the circle of the original unwrapped D-brane. The answer is that a type II D-brane has a U(1) gauge field $A$ in its world volume, and as a result a wrapped D-brane has an associated Wilson line $e^{i\oint A}$. This gives the dual description of position on the circle.
Type I Superstrings
===================
Type IIB superstrings have a world-sheet parity symmetry, denoted $\Omega$. This $Z_2$ symmetry amounts to interchanging the left- and right-moving modes on the world sheet: $X_L^\mu \leftrightarrow X_R^\mu, \, \psi_L^\mu
\leftrightarrow \psi_R^\mu$. This is a symmetry of IIB and not of IIA, because only in the IIB case do the left and right-moving fermions carry the same space-time chirality. When one gauges this $Z_2$ symmetry, the type I theory results.[@sagnotti87] The projection operator $\frac{1}{2} (1 + \Omega)$ retains the left-right symmetric parts of physical states, which implies that the resulting type I closed strings are unoriented. In addition, it is necessary to add a twisted sector — the type I open strings. These are strings whose ends are associated to the fixed points of $\sigma \rightarrow 2\pi - \sigma$, which are at $\sigma = 0$ and $\sigma = \pi$. These strings must also respect the $\Omega$ symmetry, so they are also unoriented. The type I theory has half as much supersymmetry as type IIB (16 conserved supercharges instead of 32 — corresponding to a single Majorana–Weyl spinor). This supersymmetry corresponds to the diagonal sum of the $L$ and $R$ supersymmetries of the IIB theory.
This “orientifold” construction of the type I theory has the entire 10d spacetime as a fixed point set, since $\Omega$ does not act on $x^\mu$. Correspondingly a spacetime-filling orientifold plane (an O9-plane) results. This orientifold plane turns out to carry $-32$ units of $RR$ charge, which must be cancelled by adding 32 D9-planes. Rather than proving this, we can make it plausible by recalling that $n$ type I D9-planes carry an SO($n$) gauge group. Moreover, we know that the total charge must be cancelled and that SO(32) is the only orthogonal group allowed by anomaly cancellation requirements. Correspondingly, these are the unique choices allowed by tadpole cancellation. As a remark on notation, let me point out that instead of speaking of 32 D$9$-branes, we could equivalently speak of 16 D$9$-branes and their mirror images. This distinction is simply one of conventions. The important point is that when $n$ type I D9-branes and their $n$ mirror images coincide with an O9-plane, the resulting system has an unbroken SO(2n) gauge symmetry.
The Type I$^\prime$ Theory
==========================
We now wish to examine the T-dual description of the type I theory on a spacetime of the form $R^9 \times S^1$, where the circle has radius $R$. We have seen that IIB is T dual to IIA and that type I is an orientifold projection of IIB. Therefore, one should not be surprised to learn that the result is a certain orientifold projection of type IIA compactified on the dual circle $\tilde{S}^1$ of radius $R' = 1/R$. The resulting T dual version of type I has been named type IA and Type I$^\prime$ by various authors. We shall adopt the latter usage here.
We saw that T duality for a type II theory compactified on a circle corresponds to the world-sheet symmetry $X_R \rightarrow - X_R, \psi_R \rightarrow -
\psi_R$, for the component of $X$ and $\psi$ along the circle. This implies that $X = X_L + X_R \rightarrow X' = X_L - X_R$. In the case of type II theories, we saw that $X'$ describes a dual circle $\tilde{S}^1$ of radius $R' =
1/R$. In the type I theory we gauge world-sheet parity $\Omega$, which corresponds to $X_L \leftrightarrow X_R$. Evidently, in the T dual formulation this corresponds to $X' \rightarrow - X'$. Therefore this gauging gives an orbifold projection of the dual circle: $\tilde{S}^1 /Z_2$. More precisely the $Z_2$ action is an orientifold projection that combines $X' \rightarrow - X'$ with $\Omega$. This makes sense because $\Omega$ above is not a symmetry of the IIA theory, since left-moving and right-moving fermions have opposite chirality. However, the simultaneous spatial reflection $X' \rightarrow - X'$ compensates for this mismatch.
The orbifold $\tilde{S}^1 /Z_2$ describes half of a circle. In other words, it is the interval $0 \leq X' \leq \pi R'$. The other half of the circle should be regarded as also present, however, as a mirror image that is also $\Omega$ reflected. Altogether the statement of T duality is the equivalence of the compactified IIB orientifold $(R^9 \times S^1)/\Omega$ with the type IIA orientifold $(R^9 \times S^1)/ \Omega \cdot {\mathcal I}_1$. The symbol ${\mathcal I}_1$ represents the reflection $X' \rightarrow - X'$.
The fixed-point set in the type I${}^{\prime}$ construction consists of a pair of orientifold 8-planes located at $X' = 0$ and $X' = \pi R'$. Each of these carries $-16$ units of $RR$ charge. Consistency of the type I${}^{\prime}$ theory requires adding 32 D$8$-branes. Of these, 16 reside in the interval $0 \leq X' \leq \pi R'$ and 16 are their mirror images located in the interval $\pi R' \leq X' \leq 2\pi
R'$. Clearly, these D$8$-branes are the T duals of the D$9$-branes of the type I description.
The positions of the D$8$-branes along the interval are determined in the type I description by Wilson lines in the Cartan subalgebra of SO(32). Since this group has rank 16, its Cartan subalgebra has 16 generators. Let $A^I$ denote the component of the corresponding 16 gauge fields along the circular direction. These correspond to compact U(1)’s, so their values are characterized by angles $\theta_I$. These determine the dual positions of the D$8$-branes to be $$X'_I = \theta_I R', \quad I = 1,2, \ldots, 16.$$
The SO(32) symmetry group is broken by the Wilson lines to the subgroup that commutes with the Wilson line matrix. In terms of the type I${}^{\prime}$ description this gives the following rules:
- When $n$ D$8$-branes coincide in the interior of the interval, this corresponds to an unbroken U($n$) gauge group.
- When $n$ D$8$-branes coincide with an O8-plane they give an unbroken SO($2n$) gauge group.
In both cases the gauge bosons arise as zero modes of $8-8$ open strings. In the second case the mirror-image D$8$-branes also contribute. As we will explain later, this is not the whole story. Further symmetry enhancement can arise in other ways.
The case of trivial Wilson line (all $A^I = 0$) corresponds to having all 16 D$8$-branes (and their mirror images) coincide with one of the D$8$-branes. This gives SO(32) gauge symmetry, of course. In addition there are two U(1) factors. The corresponding gauge fields arise as components of the 10d metric and B field: $g_{\mu 9}$ and $B_{\mu 9}$. One combination of these belongs to the 9d supergravity multiplet, whereas the other combination belongs to a 9d vector supermultiplet.
Somewhat more generally, consider the Wilson line $$\left(\begin{array}{cc}
I_{16 + 2N} & 0\\
0 & I_{16 - 2N} \end{array} \right).$$ This corresponds to having $8 + N$ D$8$-branes coincide with the O8-plane at $X' = 0$ and $8-N$ D$8$-branes with the O8-plane at $X' = \pi R'$. Generically this gives rise to the gauge symmetry $$SO(16 + 2N) \times SO(16 - 2N) \times U(1)^2.$$ However, from the S-dual heterotic description of the type I theory, one knows that for a particular value of the radius further symmetry enhancement is possible. Specifically, for heterotic radius $R_H^2 = N/8$ one finds the gauge symmetry enhancement $$SO(16 - 2N) \times U(1) \rightarrow E_{9 - N}.$$ This radius, converted to type I metric, corresponds to $R^2 = gN/8$. This symmetry enhancement will be explained from a type I$^\prime$ viewpoint later. There are other interesting extended symmetries such as SU(18) and SO(34), which might also be understood from a type I$^\prime$ viewpoint, but will not be considered here.
D0-Branes
=========
The type I$^\prime$ theory is constructed as a type IIA orientifold. As such, its bulk physics — away from the orientifold planes — is essentially that of the type IIA theory. More precisely, there are number of distinct type IIA vacua distinguished by the difference in the number of D$8$-branes to the left and the right. When these numbers match, one has the ordinary IIA vacuum. When they don’t one has a “massive” IIA vacuum of the kind first considered by Romans.[@romans86] In any case, the ordinary IIA vacuum admits various even-dimensional D-branes. Here I wish to focus on D$0$-branes. Later we will discuss what happens to them when they cross a D$8$-brane and enter a region with a different IIA vacuum.
D$0$-branes of the type I$^\prime$ theory correspond to type I D-strings that wrap the compactification circle. The Wilson line on the D-strings controls the positions of the dual D$0$-branes. A collection of $n$ coincident type 1 D-strings has an O($n$) world-volume gauge symmetry. Unlike the case of D$9$-branes the reflection element is included, so that the group really is O($n$) and not SO($n$). This means that in the case of a single D string it is $O(1) = Z_2$. Thus in this case there are two possible values for the Wilson line $(\pm 1)$. The dual type I$^\prime$ description is a single D$0$-brane stuck to one of the orientifold planes, with the value of the Wilson line controlling which one it is.
A single D$0$-brane of type I$^\prime$ stuck to an orientifold-plane cannot move off the plane into the bulk. However, a pair of them can do so. To understand this, let us consider a pair of wrapped D strings of type I, coincident in the other dimensions, which carries an O(2) gauge symmetry. Again, this is T dual to a pair of type I$^\prime$ D0-branes with positions controlled by the choice of O(2) Wilson line. The inequivalent choices of Wilson line are classified by conjugacy classes of the O(2) gauge group. So we should recall what they are. It is important that O(2), unlike its SO(2) subgroup, is non-Abelian. Correspondingly, there are conjugacy classes of two types:
- The SO(2) subgroup has classes labeled by an angle $\theta$. Including the effect of the reflection, inequivalent classes correspond to range $0 \leq
\theta \leq \pi$. Such a conjugacy class describes a D$0$-brane at $X' =
\theta R'$ in the bulk, together with the mirror image at $X' = (2\pi - \theta)
R'$. We see that to move into the bulk a second (mirror image) D0-brane had to be provided.
- The reflection elements of O(2) all belong to the same conjugacy class. A representative is the matrix $\left(\begin{array}{cc} 1 & 0\\ 0 &
-1\end{array}\right)$. This class corresponds to one stuck D$0$-brane on each O8-plane.
Brane Creation
==============
The solutions of massive type IIA supergravity were investigated by Polchinski and Witten,[@polchinski95a] who showed that they involve a metric and dilaton that vary in one direction. In the context of the type I$^\prime$ theory this means they vary in all regions for which the number of D$8$-branes to the left and to the right are unequal. Thus the only case for which this effect does not occur is the SO(16) $\times$ SO(16) configuration with D$8$-branes attached to each of the O8-planes. (This case is closely related to the M theory description of the $E_8\times E_8$ theory.[@horava95])
We can avoid describing the $X'$ dependence of the metric explicitly by using proper distance $s$ as a coordinate along the interval. (This requires holding the other coordinates fixed.) Then one has $0 \leq s \leq \pi R', R' = 1/R$. We didn’t address the issue earlier, but when we said the interval has length $\pi R'$ we really did mean its proper distance. In terms of this coordinate there is a varying dilaton field, and hence a varying string coupling constant $g_A (s)$. Only in regions with half of the D$8$-branes to the left and half to the right is it constant.
The function $g_A (s)$ was obtained by Polchinski and Witten by solving the field equations. A more instructive way of obtaining and understanding the result uses the brane creation process. Consider an isolated D$0$-brane in a region where $g_A(s)$ is constant. Now suppose the D$0$-brane crosses a D$8$-brane to enter a region where $g_A(s)$ is varying. What happens is that the D$0$-brane emerges on the other side with a fundamental string stretched between it and the D$8$-brane. This phenomenon, called Hanany–Witten effect,[@hanany96] has been derived by a variety of means.[@bachas97a] It occurs in many different settings that are related by various duality transformations. (For example, two suitably oriented M5-branes can cross to give rise to a stretched M2-brane.[@dealwis97]) The intuitive reason that string creation is required can be understood as follows. The original D$0$-brane configuration preserved half the supersymmetry and was BPS. Therefore a delicate balance of focus ensured that it was stable at rest. When it crosses the D$8$-brane (adiabatically) the amount of supersymmetry remains unchanged and so it should still be stable at rest. To be specific, let us consider the D$8$-brane configuration discussed earlier with $8 + N$ D$8$-branes on the $X' = 0$ O8-plane and $8 - N$ D$8$-branes on the $X' = \pi R'$ O8-plane. In this case $N$ fundamental strings should connect the D$0$-brane to the $X' = 0$ O8-plane. The BPS condition implies that the mass of the D$0$-brane should be independent of its position in the interval. Recalling that the mass of a type IIA D$0$-brane is $1/g_A$, we therefore conclude that for this configuration $$\label{Dmass}
M_{D0} = \frac{1}{g_A (0)} = \frac{1}{g_A(s)} + N T_{F1} s.$$ Here $T_{F1} = \frac{1}{2\pi}$ is the tension of a fundamental type IIA string (in string units). We therefore see that $g_A(s)$ is the reciprocal of a linear function whenever $N\not= 0$. Thus, for $N\not= 0$ it necessarily develops a pole if $R'$ is too large.
The mass $M_{D0}$ can also be computed in the type I picture in terms of a pair of wrapped D strings with Wilson line. The mass is independent of the O(2) Wilson line, since it is independent of the $X'$ coordinate. However, it does depend on the SO(32) Wilson line. Altogether the mass is a sum of two contributions: $$M_{D0} = M_{\rm winding} + M_{\rm Wilson} .$$ The winding term contribution is given by simple classical considerations: $$M_{\rm winding} = 2 \cdot 2\pi R \cdot T_{D1} = \frac{2R}{g}.$$ A more careful analysis is required to obtain the Wilson line contribution $$M_{\rm Wilson} = \frac{N}{4R} .$$ Note that this contribution vanishes for large $R$.
We now come to the main point. There is a special value of $R'$, the one for which the coupling diverges at the $X' = \pi R'$ orientifold plane. In this case $$\label{vanish}
\frac{1}{g_A (\pi R')} = 0,$$ which implies, using eq. (\[Dmass\]), that $$M_{D0} = \frac{N}{2R} = \frac{2R}{g} + \frac{N}{4R},$$ and hence that $$R^2 = gN/8.$$ This is precisely the value that we previously asserted gives the symmetry enhancement SO($16 - 2N$) $\times$ U(1) $\rightarrow E_{9-N}$. The reason that there is symmetry enhancement is that there are additional massless vectors with appropriate quantum numbers. They arise as the ground states of open strings connecting the D$8$-branes to a stuck D$0$-brane.[@bergman97; @bachas97] This works because the stuck D$0$-brane is massless in this case, as a consequence of eq. (\[vanish\]). This accounts for all the extra gauge bosons when $N>2$. In the $E_7$ and $E_8$ cases, there are additional states attributable to a single bulk D0-brane near $X' = \pi R'$.
Conclusion
==========
The study of supersymmetric theories has come a long way since Golfand’s pioneering work. I presume that he would be pleased.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to O. Bergman for very helpful discussions. This work was supported in part by the U.S. Dept. of Energy under Grant No. DE-FG03-92-ER40701.
References {#references .unnumbered}
==========
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|
---
abstract: 'We compare existing high spectral resolution (R = $\lambda/\Delta\lambda \sim$ 40,000) Ca[ii]{} K observations ($\lambda_{\rm air}$=3933.66Å) towards 88 mainly B-type stars, and new observations taken using ISIS on the William Herschel Telescope at R $\sim$ 10,000 towards 3 stars taken from the Palomar-Green Survey, with 21-cm H[i]{} emission-line profiles, in order to search for optical absorption towards known intermediate and high velocity cloud complexes. Given certain assumptions, limits to the gas phase abundance of Ca[ii]{} are estimated for the cloud components. We use the data to derive the following distances from the Galactic plane ($z$); 1) Tentative lower $z$-height limits of 2800 pc and 4100 pc towards Complex C using lack of absorption in the spectra of HD341617 and PG0855+294, respectively. 2) A weak lower $z$-height of 1400 pc towards Complex WA-WB using lack of absorption in EC09470–1433 and a weak lower limit of 2470 pc using lack of absorption in EC09452–1403. 3) An upper [$z$-height]{} of 2470 pc towards a southern intermediate velocity cloud (IVC) with $v_{\rm LSR}$=–55 kms$^{-1}$ using PG2351+198. 4) Detection of a possible IVC in Ca[ii]{} absorption at $v_{\rm LSR}$=+52 kms$^{-1}$ using EC20104–2944. No associated H[i]{} in emission is detected. At this position, normal Galactic rotation predicts velocities of up to $\sim$ +25 kms$^{-1}$. The detection puts an upper $z$-height of 1860 pc to the cloud. 5) Tentative H[i]{} and Ca[ii]{} K detections towards an IVC at $\sim$ +70 kms$^{-1}$ in the direction of HVC Complex WE, sightline EC06387–8045, indicating that the IVC may be at a $z$-height lower than 1770 pc. 6) Detection of Ca[ii]{} K absorption in the spectrum of PG0855+294 in the direction of IV20, indicating that this IVC has a $z$-height smaller than 4100 pc. 7) A weak lower $z$-height of 4300 pc towards a small HVC with $v_{\rm LSR}$=+115 kms$^{-1}$ at $l,b$=200$^{\circ}$,+52$^{\circ}$, using lack of absorption in the Ca[ii]{} K spectrum of PG0955+291.'
date: Accepted Received in original form
title: 'Ca[ii k]{} interstellar observations towards early disc and halo stars - distances to intermediate and high-velocity clouds'
---
\[firstpage\]
ISM: general – ISM: clouds – ISM: structure – stars: early-type
Introduction
============
This paper is the second of a pair that uses a sample of mainly B-type stars to probe the interstellar medium of the disc and halo of the Milky Way in Ca[ii]{} K ($\lambda_{\rm air}$=3933.663 Å). In the first (Smoker et al. 2003; hereafter Paper 1), we considered the abundance of Ca[ii]{} K, variations in the element over degree scales, and the distribution of this species as a function of distance from the Galactic plane ($z$). In the current work, we use the 88 sightlines in our sample, plus new observations towards three other stars, to search for Ca[ii]{} K absorption in gas in intermediate and high-velocity clouds, the purpose being to try and improve the distance limits to these still enigmatic objects.
Intermediate and high velocity clouds (hereafter IHVCs) are objects with absolute values of their velocities in the local standard of rest (LSR) of between $\sim$40–100 and $>$ $\sim$ 100 kms$^{-1}$, respectively. These velocities are not explicable by simple rotation of gas around the Galactic centre, and hence it has been postulated that the clouds are either material within the Galactic halo at distances of $\sim$1–5 kpc (e.g. review by Wakker & van Woerden 1997; Putman et al. 2003), or objects left over from the formation of the Milky Way, with distances of several hundreds of kpc (Blitz et al. 1999, Braun & Burton 1999). Both types of object have been extensively studied in H[i]{}, although until recently, the ionised component of the clouds remained uncertain. This situation has been rectified by results from the Wisconsin H$\alpha$ mapper (WHAM) that indicate that IHVCs contain ionised gas (e.g. Haffner, Reynolds & Tufte 2001; Smoker et al. 2002; Tufte et al. 2002), the ionisation being caused by either collisional ionisation, photoionisation by the extragalactic ionising field, and/or the escape of photons from the disc of the Galaxy. To attempt to determine which is the most likely source of ionisation, the distance to IHVCs would be of great help, as to the present-day, there is still a dearth of distance measurements, particularly towards HVCs (Wakker 2001). The current paper once more attempts to address this issue, by searching for IHVC components in the Ca[ii]{} line of a sample of mainly early-type stars, located in the Galactic disc and halo. The description of the reduction and analysis of the majority (88) of these stars was described in Paper 1, with new observations towards a further 3 objects being described in this paper.
Section \[observations\] describes new William Herschel Telescope observations towards three stars within IVC Complex K, Section \[results\] presents the results of the WHT observations, plus a Table comparing the 91 stars in the current sample with IVC and HVC emission-line features found in either the Leiden-Dwingeloo Northern H[i]{} survey (Hartmann & Burton 1997), or the Villa-Elisa Southern H[i]{} survey (Arnal et. al 2000).
In Section \[disc\] we discuss the Ca[ii]{}K to H[i]{} ratio, or upper limit, for sightlines with a IHVC H[i]{} detection. In Section \[distance\] we use these limits to derive new upper or lower distance estimates towards IHVCs. Finally, Section \[concl\] contains the summary.
Observations and data reduction {#observations}
===============================
\[newobsdist\]
The new observations described in this paper were taken using the Intermediate dispersion Spectrograph and Imaging System (ISIS), located on the WHT, during 3–4 Aug. 2001. The blue-arm was used, with the H2400B grating and a 1.0 arcsec slit, giving an instrumental FWHM resolution of $\sim$ 30 kms$^{-1}$ and wavelength coverage from $\sim$ 3800$-$4160 Å. Three stars towards IVC Complex K were observed, PG numbers 1718+519, 1725+252 and 1738+505, which were reduced using standard methods to obtain the equivalent widths (EWs) and velocity centroids of the Ca[ii]{} K components. The signal to noise ratio obtained towards the three objects listed above was $\sim$ 70, 130 and 110, respectively.
Results
=======
New WHT results
---------------
Fig. \[fig1\] shows both the complete wavelength coverage, and the Ca[ii]{} K line only, towards the three PG stars. Table \[tab1\] shows the quantities derived from these spectra. The Ca[ii]{} K interstellar reduced equivalent width of low-velocity material towards the three sample stars (=$EW\times$sin($b$)) is 90, 96 and 119 mÅ. Given a Ca[ii]{} K scaleheight of $\sim$ 800 pc and REW at infinity of $\sim$115 mÅ (Paper 1), this indicates that the stars are quite distant. Of the three objects, PG1718+519 turned out to be a binary, whose spectrum is contaminated by a late-type companion star. PG1725+252 is an early-type star with a stellar velocity of $-62\pm$6 kms$^{-1}$. The ‘intermediate velocity’ line at $-63$ kms$^{-1}$ is hence likely to be stellar. This was checked by running a model atmosphere code using a solar Calcium abundance with T$_{\rm eff}$=26,000 K, log$(g)$=5.0 (Theissen et al. 1993) and microturbulance velocity of 0 and 5 kms$^{-1}$. These produced equivalent width estimates of 44 and 65 mÅ, similar to the measured value of 40$\pm$5 mÅ. Note that, at this high gravity a microturbulance velocity of 0 kms$^{-1}$ may well be the best choice. Finally, PG1738+505 has a stellar velocity of 27$\pm$3 kms$^{-1}$ and displays a relatively broad interstellar profile.
----------------------------------- ----------------------- --------------------- --------------
Star PG1718+519 PG1725+252 PG1738+505
$l,b$ (deg) 79.00, 34.94 48.21, 28.74 77.54,31.84
$z$ (pc) 2260 320 510
$v_{*}^{\rm LSR}$ (kms$^{-1}$) $-$41$\pm$8 $-62\pm6$ +27$\pm$3
EW(Ca[ii]{} K) LV (mÅ) 157$\pm$6 199$\pm$5 225$\pm$5
EW(Ca[ii]{} K)$\times$sin($b$) LV 90$\pm$3 96$\pm$3 119$\pm$3
$v$(Ca[ii]{} K) LV (kms$^{-1}$) +1$\pm$1 +15$\pm$1 +6.5$\pm$1.0
FWHM(Ca[ii]{} K) LV (kms$^{-1}$) 37$\pm$2 34$\pm$1 61$\pm$2
$v$(Ca[ii]{} K) IV (kms$^{-1}$) $-90\pm 10$ (stellar) $-63\pm3$ (stellar) –
FWHM(Ca[ii]{} K) IV (kms$^{-1}$) 37$\pm$10 64$\pm$10 –
EW(Ca[ii]{} K) IV (mÅ) 20$\pm$7 40$\pm$5 $<$10
$N$(H[i]{}) IV (cm$^{-2}$) 10$^{19}$ 10$^{19}$ 10$^{19}$
----------------------------------- ----------------------- --------------------- --------------
![WHT spectra of three Palomar-Green stars in the direction of Complex K. The left-hand panels show the entire $\lambda$-range observed, with the right-hand panels showing the spectra at the wavelength of Ca[ii]{} K, in velocity space and in the Local Standard of Rest.[]{data-label="fig1"}](./md1099fig1.eps)
Comparison of all Ca[ii]{} sample spectra with H[i]{} survey data to search for IHVCs {#comp1}
-------------------------------------------------------------------------------------
Table \[tab2\] lists the sightlines from the current sample where an IHVC was detected either in H[i]{} or Ca[ii]{} K, and compares these sightlines with known intermediate and high-velocity cloud complexes, taken from Wakker (2001). This table should be used in conjunction with the notes on individual sightlines given in Sect. \[individual\]. The H[i]{} data are from either Leiden-Dwingeloo Northern H[i]{} survey (Hartmann & Burton 1997), or the Villa-Elisa Southern H[i]{} survey (Arnal et al. 2000) for IHVCs. Columns 1 to 4 give the stellar name, LSR velocity in kms$^{-1}$, $z$-height in pc and signal-to-noise ratio in the stellar continuum at $\sim$ 3933 Å, respectively. The references for 71 of the distances are taken from Paper 1, while three distance estimates for the three PG stars towards the newly-observed Complex K are taken from Theissen et al. (1993) and de Boer et al (1997). Three other distance estimates are taken from Lynn et al. (2004). The remaining objects do not yet have distances available.
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Star $v_{\rm LSR}^{*}$ $|z^{*}|$ SNR IHVC $v_{\rm LSR}^{\rm IHVC}$ FWHM log$N$ $T_{B}^{\rm peak}$ $v_{\rm LSR}^{\rm IHVC}$ log$N$ log($N_{\rm pred}$)
name (H[i]{}) (H[i]{}) (H[i]{}) (H[i]{}) (Ca[ii]{}) (Ca[ii]{}) (Ca[ii]{})
PG0009+036 +160 9090 35 PPA –48$\pm$5.0 20.0$\pm$5.0 19.32$\pm$0.02 0.59$\pm$0.02 –43.3 11.96 11.67
EC00179–6503 +40 3150 40 MS +141.1$\pm$1.2 70.0$\pm$3.0 19.32$\pm$0.03 0.16$\pm$0.02 – $<$11.16 11.68
EC00237–2317 +86 – 60 MS –121.3$\pm$1.7 29.4$\pm$3.5 19.11$\pm$0.06 0.22$\pm$0.02 – $<$10.98 11.60
EC00321–6320 –30 664 120 MS +78.4$\pm$1.8 25.8$\pm$2.4 19.16$\pm$0.04 0.29$\pm$0.02 – $<$10.68 11.62
“ & “ & “ & “ & MS &+109.8$\pm$2.4 & 27.6$\pm$4.2 & 19.10$\pm$0.04 & 0.24$\pm$0.02 & – & $<$10.68 & 11.60\
“ & “ & “ & “ & MS &+171.4$\pm$0.4 & 7.4$\pm$1.0 & 18.90$\pm$0.05 & 0.59$\pm$0.05 & – & $<$10.68 & 11.54\
“ & “ & “ & “ & MS &+180.6$\pm$0.6 & 7.5$\pm$1.1 & 18.78$\pm$0.07 & 0.42$\pm$0.05 & – & $<$10.68 & 11.50\
HD38666 & +93 & 181 & 300 & Other & 41.0$\pm$2.0 & 11.1$\pm$2.2 & 18.71$\pm$0.02 & 0.24$\pm$0.07 & +42.6 & 10.77 & 11.48\
“ & +93 & “ & “ & Other & 52.1$\pm$0.4 & 20.7$\pm$2.8 & 19.26$\pm$0.02 & 0.44$\pm$0.03 & – & $<$10.28 & 11.76\
EC05490–4510 & +16 & 800 & 50 & Other & +50 to +80 & – & – & – & – & $<$11.06 & –\
EC06012–7810 & +30 & 1500 & 30 & MB &+207.9$\pm$1.0 & 26.9$\pm$1.7 & 18.81$\pm$0.03 & 0.13$\pm$0.02 & – & $<$11.28 & –\
“ & “ & “ & & MB &+288.0$\pm$0.3 & 30.8$\pm$0.6 & 19.45$\pm$0.03 & 0.43$\pm$0.07 & – & $<$11.28 & 11.70\
EC06387–8045 & +49 & 1770 & 12 & Other &–39.4$\pm$3.5 & 56.3$\pm$5.3 & 19.43$\pm$0.04 & 0.18$\pm$0.03 & – & $<$11.68 & 11.70\
“ & “ & “ & & Other & +70.8$\pm$1.0 & 19.6$\pm$2.0 & 18.82$\pm$0.10 & 0.17$\pm$0.02 & +75.0 & 11.50 & 11.52\
PG0833+699 & +23 & 1980 & 30 & LLIV Ar &–69.4$\pm$0.6 & 15.7$\pm$1.3 & 19.34$\pm$0.04 & 0.71$\pm$0.04 & –67.0 & 11.46 & 11.67\
“ & & “ & & LLIV Ar &–40.1$\pm$0.5 & 26.9$\pm$1.5 & 19.78$\pm$0.03 & 1.17$\pm$0.04 & –45.0 & 11.28 & 11.80\
PG0855+294 & +58 & 4100 & 40 & C &–168.7$\pm$1.3 & 22.8$\pm$3.5 & 19.00$\pm$0.04 & 0.23$\pm$0.03 & – & $<$11.16 & 11.57\
“ & “ & “ & “ & IV Arch & –23.1$\pm$1.0 & 4.8$\pm$0.5 & 19.44$\pm$0.10 & 3.20$\pm$0.05 & – & – & –\
“ & “ & “ & “ & IV Arch & –27.0$\pm$3.0 & 22.0$\pm$2.0 & 20.07$\pm$0.10 & 2.47$\pm$0.10 & –29.0 & 11.90 & 11.89\
EC09452–1403 & +226 & 2470 & 25 & WB & +120.0$\pm$10 & 24.6$\pm$10.2 & 18.65$\pm$0.16 & 0.10$\pm$0.04 & – & $<$11.36 & 11.46\
EC09470–1433 & – & 1400 & 60 & Other & +55.0$\pm$5.0 & 20.0$\pm$5.0 & 18.95$\pm$0.09 & 0.25$\pm$0.03 & – & $<$10.98 & 11.35\
“ & “ & “ & “ & WB & +114.0$\pm$8.0 & 25.0$\pm$5.0 & 18.97$\pm$0.05 & 0.25$\pm$0.03 & – & $<$10.98 & 11.35\
PG0955+291 & +72 & 4300 & 35 & IV Spur & –29.3$\pm$1.2 & 59.2$\pm$2.0 & 19.81$\pm$0.02 & 0.58$\pm$0.02 & – & $<$11.22 & 11.81\
“ & “ & “ & “ & “ & – & – & – & – & –45.0 & 11.74 & –\
“ & “ & “ & “ & “ & – & – & – & – & –64.0 & 11.26 & –\
“ & “ & “ & “ & Other & +115.5$\pm$2.3 & 20.9$\pm$4.0 & 18.60$\pm$0.10 & 0.09$\pm$0.03 & – & $<$11.22 & 11.45\
PG1008+689 & –11 & 950 & 45 & IV Arch & –44.3$\pm$0.3 & 21.9$\pm$0.8 & 19.62$\pm$0.03 & 1.30$\pm$0.06 & – & $<$11.11 & –\
“ & “ & “ & “ & IV Arch & – & – & – & – & –39.0 & 11.48 & –\
“ & “ & “ & “ & IV Arch & – & – & – & – & –49.0 & 11.62 & –\
EC10087–1411 & +96 & 620 & 40 & Other & +63.0$\pm$5 & 32.2$\pm$4.6 & 19.30$\pm$0.04 & 0.32$\pm$0.04 & – & $<$11.16 & 11.66\
EC11074–2912 & – & 950 & 25 & Other &–48.6$\pm$3.3 & 36.5$\pm$10.5 & 19.36$\pm$0.05 & 0.32$\pm$0.04 & – & $<$11.36 & –\
EC11507–2253 & +210 & – & 45 & Other & +45.0$\pm$10 & 30.0$\pm$10 & 19.11$\pm$0.06 & 0.22$\pm$0.03 & – & $<$11.11 & –\
PG1213+456 & –15 & 2700 & 20 & IV Arch &–57.8$\pm$0.3 & 22.8$\pm$0.9 & 19.55$\pm$0.02 & 0.82$\pm$0.02 & –56.9 & 11.83 & 11.73\
PG1243+275 & +107 & 6200 & 25 &IV Ar/Sp &–23.7$\pm$0.4 & 39.4$\pm$0.8 & 19.91$\pm$0.01 & 1.08$\pm$0.02 & – & $<$11.36 & –\
“ & “ & “ & “ &IV Ar/Sp & – & – & – & – & –42.8 & 11.28 & –\
LS3510 & 0 & – & 200 & Other & +38.1$\pm$5.0 & 17.5$\pm$4.0 & 20.52$\pm$0.10 & – & – & $<$10.46 & 12.03\
“ & “ & – & “ & Other & +53.4$\pm$5.0 & 11.7$\pm$4.0 & 20.24$\pm$0.10 & – & – & $<$10.46 & 11.94\
LS3604 & –30 & – & 160 & Other & +38.8$\pm$1.0 & 37.8$\pm$2.5 & 21.13$\pm$0.03 & – & – & $<$10.56 & 12.21\
“ & “ & “ & “ & Other & +41.8$\pm$0.6 & 7.6$\pm$0.5 & 20.31$\pm$0.03 & – & – & $<$10.56 & 11.96\
“ & “ & “ & “ & Other & +51.0$\pm$0.5 & 4.0$\pm$0.5 & 19.63$\pm$0.03 & – & – & $<$10.56 & 11.76\
LS3694 & –29 & – & 150 & Other & +40.9$\pm$5.0 & 44.1$\pm$5.2 & 20.86$\pm$0.04 & – & – & $<$10.58 & 12.13\
“ & “ & – & “ & Other & +55.6$\pm$1.0 & 7.8$\pm$1.0 & 20.06$\pm$0.04 & – & – & $<$10.58 & 11.89\
LS3751 & –24 & – & 170 & Other & – & – & – & – & +38.6 & 11.33 & –\
“ & “ & – & “ & Other & +56.9$\pm$0.5 & 5.8$\pm$0.8 & 19.86$\pm$0.03 & – & – & $<$10.52 & 11.83\
“ & “ & – & “ & Other & +59.7$\pm$0.5 & 18.5$\pm$2.0 & 20.16$\pm$0.03 & – & – & $<$10.52 & 11.92\
PG1725+252 & –64 & 320 & 120 & C &–154.0$\pm$1.7 & 29.8$\pm$3.6 & 19.47$\pm$0.03 & 0.36$\pm$0.04 & – & $<$11.28 & 11.71\
PG1738+505 & +24 & 510 & 100 & K & –98.2$\pm$0.7 & 26.4$\pm$2.1 & 19.47$\pm$0.03 & 0.59$\pm$0.04 & – & $<$11.36 & 11.71\
HD341617 & +63 & 2800 & 110 & C &–118.0$\pm$10 & 35.0$\pm$10.0 & 19.18$\pm$0.04 & 0.31$\pm$0.02 & – & $<$10.62 & 11.62\
“ & “ & “ & “ & Other & – & – & & – & –45.0 & 10.80 & –\
NGC6712 ZNG-1 & –102 & 500 & 50 & MS & – & – & – & – & – & $<$11.06 & –\
LS5112 & –120 & – & 150 & GCN?, MS & – & – & – & – & –137.3 & 11.33 & –\
EC19071–7643 & –17 & 970 & 110 & Other? &–40 to –60 & – & – & – & – & $<$10.72 & –\
EC19489–5641 & – & – & 60 & Other &+36.0$\pm$3.0 & 15.0$\pm$5.0 & 19.35$\pm$0.05 & 0.68$\pm$0.03 & – & $<$10.98 & 11.68\
EC19490–7708 & – & 800 & 70 & Other &+43.0$\pm$1.4 & 23.8$\pm$2.8 & 19.08$\pm$0.06 & 0.30$\pm$0.04 & – & $<$10.91 & 11.59\
EC19596–5356 & +200 &20000 & 55 & Other &+38.7$\pm$0.4 & 12.1$\pm$0.9 & 19.15$\pm$0.02 & 0.58$\pm$0.04 & +43.4 & 11.45 & 11.61\
EC20089–5659 & –17 & 409 & 140 & Other & +50.0$\pm$1.0 & 28.0$\pm$1.7 & 19.50$\pm$0.03 & 0.57$\pm$0.03 & – & $<$10.61 & 11.72\
EC20104–2944 & +145 & 1860 & 70 & Other & +44$\pm$5 & 34.4$\pm$2.6 & 19.30$\pm$0.04 & 0.30$\pm$0.02 & +52.7 & 11.72 & 11.66\
M15ZNG–1 & –100 & 4600 & 80 & gp & +69.6$\pm$0.6 & 18.2$\pm$1.4 & 19.34$\pm$0.04 & 0.61$\pm$0.04 & +65.0 & 12.38 & 11.67\
EC23169–2235 & +79 & 2220 & 30 & Other & –30 to –50 & – & – & – & – & $<$11.28 & –\
PG2351+198 & –275 & 2470 & 50 & IVS &–55.0$\pm$5.0 & 20.0$\pm$5 & 18.70$\pm$0.14 & 0.14$\pm$0.05 & –55.80 & 11.12 & 11.48\
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Star $v_{\rm LSR}^{*}$ $|z^{*}|$ SNR log$N$
(Ca[ii]{})
EC00358–1516 +84 4294 120 $<$10.68
EC00468–5622 +4 1971 140 $<$10.61
EC01483–6806 +61 2100 100 $<$10.76
EC03240–6229 –17 1600 30 $<$11.28
EC03342–5243 +84 1210 200 $<$10.46
EC03462–5813 +24 650 50 $<$11.06
EC04420–1908 +192 1124 35 $<$11.22
EC04460–3215 –17 880 20 $<$11.46
EC05229–6058 +17 1060 30 $<$11.28
EC05438–4741 +37 1800 35 $<$11.22
EC05515–6107 +75 2280 35 $<$11.22
EC05515–6231 –21 335 100 $<$10.76
EC05582–5816 +66 670 45 $<$11.11
PG0823+499 +12 1000 45 $<$11.11
PG0914+001 +80 8440 35 $<$11.22
PG0934+145 +105 5820 30 $<$11.28
EC09414–1325 +60 1540 40 $<$11.16
PG0954+049 +90 2400 30 $<$11.28
EC10500–1358 +92 3330 30 $<$11.28
EC10549–2953 –15 800 40 $<$11.16
HD97917 +11 – 120 $<$10.68
PG1205+228 +156 2340 60 $<$10.98
PG1212+369 –32 2600 30 $<$11.28
PG1310+316 –55 8100 20 $<$11.45
EC13139–1851 +18 1060 30 $<$11.28
PG1323–086 –41 12600 140 $<$10.61
PG1351+393 –24 6400 25 $<$11.36
EC14102–1337 –20 – 40 $<$11.16
HD137569 –24 500 400 $<$10.16
EC15374–1552 –56 – 50 $<$11.06
LSIV–0401 +105 4700 80 $<$10.86
M10ZNG–1 +90 2970 140 $<$10.61
PG1704+222 –22 3700 80 $<$10.85
PG1708+142 +180 10000 30 $<$11.28
PG1718+519 –41 2260 100 $<$10.76
M22ZNG-5 –130 330 70 $<$10.92
EC19304–5337 +165 – 60 $<$10.98
EC19337–6743 –8 390 150 $<$10.58
EC19476–4109 –5 842 110 $<$10.72
EC19563–7205 –10 – 15 $<$11.58
EC19579–4259 +16 180 120 $<$10.68
EC19586–3823 –96 1513 70 $<$10.92
EC20011–5005 –168 3927 40 $<$11.16
EC20068–7324 +71 – 30 $<$11.28
EC20252–3137 +23 1642 100 $<$10.76
EC20292–2414 +8 1000 35 $<$11.22
EC20411–2704 +18 200 30 $<$11.28
EC20485–2420 –40 2100 110 $<$10.72
PG2120+062 –56 2500 50 $<$11.06
PG2146+087 +19 1250 45 $<$11.11
PG2219+094 –17 4190 60 $<$10.98
PG2229+099 –10 5220 12 $<$11.68
PG2345+241 +82 2920 60 $<$10.98
PG2356+167 +3 2030 30 $<$11.28
-------------- ------------------- ----------- ----- ------------
: Sightlines with neither a H[i]{} nor Ca[ii]{} IHVC detection. The meaning of the columns is explained in Sect. \[comp1\]. []{data-label="tab3"}
If the sightline is towards a known IVC or HVC, Columns 5–9 give the IHVC name (mostly taken following Wakker 2001) LSR velocity in kms$^{-1}$ of H[i]{} gas at this position on the sky, full width half maximum value (FWHM) of the H[i]{} profile in kms$^{-1}$, log(H[i]{} column density in cm$^{-2}$) and peak brightness temperature in K, respectively. These values were obtained via Gaussian profile fitting using [elf]{} within [dipso]{} (Howarth et al. 1996). In the Northern hemisphere, the H[i]{} values are taken from the nearest stray-radiation corrected Leiden-Dwingeloo H[i]{} survey pointing (Hartmann & Burton 1997), which sampled the sky north of declination=$-$30 degrees at a resolution of 0.5 degrees. In the Southern hemisphere, the corresponding values are from the Instituto Argentino de Radioastronomía Villa-Elisa Southern Sky survey (Arnal et al. 2000), which surveyed the sky south of declination=$-$25 deg. The version of the Villa-Elisa survey that we have used has been corrected for the effects of stray radiation. In this case, linear interpolation of the four nearest profiles was performed. In the region of overlap between the two surveys we used the Leiden-Dwingeloo data.
In the case where optical absorption is detected, Column 10 gives the LSR velocity assuming that the feature is due to Ca[ii]{} K, with Column 11 giving either the log(Ca[ii]{} column density in cm$^{-2}$) of the IHVC, or (more frequently) its upper limit from the current dataset. Given the instrumental full width half maximum resolution, $\Delta\lambda_{\rm instr}$, the observed SNR in the continuum, $\sigma_{\rm cont}$, was used to calculate a limiting equivalent width in Å, $EW_{\rm lim}$(Ca[ii]{}), thus;
$$EW_{\rm lim}({\rm Ca \,II})=5\sigma_{\rm cont}^{-1} \Delta\lambda_{\rm instr}.
\label{eqlim}$$
Typically, the spectra have a instrumental resolution of 0.1Å and SNR=30, which means that the calculated limiting equivalent width is on the order of 20 mÅ, which is compatible with visual inspection of the spectra. Once this limiting equivalent width has been estimated, it can be used to determine the limiting column density $N_{\rm lim}$, assuming that we are on the linear part of the curve of growth, viz;
$$N_{\rm lim}=1.13 \times 10^{20} \frac{EW_{\rm lim}}{\lambda^{2}f},
\label{nlim}$$
where $\lambda$ is the wavelength in Å and $f$ is the oscillator strength of Ca[ii]{} K, taken to be 0.634 (Morton 1991). Finally, Column 12 gives the predicted value of the Ca[ii]{} column density that would be expected from the observed H[i]{} column, viz;
$${\rm log}(N_{\rm p}^{\rm WM00})({\rm CaII}) = (0.30 \times ({\rm log}(N_{H})) + 5.87,
\label{wm00CaII}$$
which is taken from Wakker & Mathis (2000, henceforth WM00). Note that using this relationship only gives an approximate estimation of the real Ca[ii]{} column density because; 1) The H[i]{} column density is derived from a large beam and does not take fine-scale structure into account, 2) The H-to-Ca ratio varies somewhat from cloud-to-cloud, 3) the H-to-Ca ratio may vary within clouds and 4) We are using $N_{\rm HI}$ and not $N_{\rm Htot}$. Savage et al. (2000) find that $N$(H[i]{})(Ly-alpha)=0.6–1.0$\times N$(H[i]{})(21-cm).
Finally, Table \[tab3\] lists the sightlines where no IHVC was detected in either the H[i]{} or optical Ca[ii]{} K spectra.
Discussion {#disc}
==========
In this section, we give notes on individual sightlines, then collate the information in Sect. \[distance\] to try to improve the distance estimates to IHVC complexes.
Notes on individual sightlines {#individual}
------------------------------
We first provide notes on individual sightlines, comparing H[i]{} data from either the Leiden-Dwingeloo or Villa-Elisa 21-cm H[i]{} surveys with our optical Ca[ii]{} K absorption line profiles. Where there is either H[i]{} emission or Ca[ii]{} K absorption seen with $|v_{\rm LSR}| >\sim$ 40 kms$^{-1}$, the sightline is discussed below. The remaining sightlines are not considered in the current paper. Where previous authors have determined distances to IVCs or HVCs using the same stars as the current dataset, the references are given. Lack of references hence implies that these sightlines have not previously been searched for IVCs/HVCs. For each sightline, we also give an estimate of the Ca[ii]{} column density predicted from the H[i]{} column density and estimated using equation \[wm00CaII\]. The 1$\sigma$ scatter on this relation is 0.42 dex (WM00). The number of standard deviations that the upper limit is away from the predicted value is also given. If the observed upper limit to Ca[ii]{} column density is significantly lower than that predicted by equation \[wm00CaII\], this implies that, [*under the assumptions given in Sect. \[comp1\]*]{}, then a non-detection is likely to be caused by the star being closer than the IHVC. Note that for the current sample, we consider only the H[i]{} component, and neglect any ionised hydrogen present. Presence of H[ii]{} would increase the estimated value of log($N_{\rm CaII}^{\rm WM00}$) and make it more likely that a non-detection could be used to derive a lower distance limit. In some IVCs at least, depending on the filling factor, the ionised component of H could be as much as that contained in neutral material (Smoker et al. 2002).
Individual sightlines, towards which there are possible or likely IHVC detections, are now discussed. The H[i]{} and Ca[ii]{} K spectra towards these sightlines being displayed in Fig. \[fig2\]. Where there exist higher-resolution 21-cm H[i]{} spectra towards the sample stars, this is noted in the comments for the individual sightlines.
[*PG0009+036*]{}: The Dwingeloo H[i]{} spectrum shows an IVC at –48 kms$^{-1}$. This material lies in the direction of the Pegasus-Pisces Arch (PPA; Wakker 2001). This star was previously observed by de Boer et al. (1994) with IUE, although the spectrum was of low quality and no limit to the IVC distance could be set. De Boer et al. (1994) also presented a Effelsberg H[i]{} profile in this direction, which is towards Magellanic Stream Cloud V. They find a log($N_{\rm HI}$) value of 19.26 for the IVC, close to the LDS value of 19.32. Our Ca[ii]{} spectrum shows [*tentative*]{} evidence for absorption at –43.4 kms$^{-1}$, although the signal to noise is low. If the feature is real, then it has a log($N_{\rm CaII}$) value of 11.96, compared to the WM00 estimated value of 11.67. In this case, the $z$-height of the cloud would be less than $\sim$ 9000 pc.
[*EC00179–6503*]{}: This sightline lies in the direction of the Magellanic Stream. The Villa-Elisa H[i]{} spectrum shows HV material at $\sim$ +145 kms$^{-1}$, possibly made up of two separate components with central velocities of +135 and +177 kms$^{-1}$ and FWHM velocity widths of 48 and 21 kms$^{-1}$. A single-component fit has a FWHM of $\sim$ 70 kms$^{-1}$. There is no corresponding Ca[ii]{} detection to a limit of log($N_{\rm CaII}$)=11.16. At this H[i]{} column density WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.66, hence at the 1.5$\sigma$ level this HVC is likely to be at a $z$-height exceeding $\sim$ 3150 pc.
[*EC00237–2317*]{}: This sightline lies in the direction of the Magellanic Stream. The Dwingeloo H[i]{} spectrum shows a HVC at –120 kms$^{-1}$, although the column density is low ($N_{\rm HI}$=1.3$\times$10$^{19}$ cm$^{-2}$). There is no corresponding detection in Ca[ii]{} K with log($N_{\rm CaII}$)$<$ 10.98. For this $N_{\rm H}$, WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.60, hence at a 1.5$\sigma$ level, the lack of Ca[ii]{} absorption implies that the HVC is further away than the star. The distance towards EC00237–2317 is currently unknown, no Stromgren photometry exists and only the Si[ii]{} line is observed, so no temperature has been derived for the object (Lynn, unpublished result). Note that the optical absorption feature at –190 kms$^{-1}$ is likely to be S[ii]{} at $\lambda_{\rm air}$=3931.91Å.
[*EC00321–6320*]{}: The Villa-Elisa H[i]{} spectrum shows an IVC and two HVCs at +78.4, +109.8 and +175 kms$^{-1}$ with log($N_{\rm HI}$) values of 19.16, 19.10 and 19.14 respectively. The Parkes spectrum shown in Wakker et al. (2001) (sightline HD003175) has components at +73, +104 and +168 kms$^{-1}$, with log($N_{\rm HI}$) values of 19.00, 18.91 and 19.41 respectively. Optical absorption at +155 kms$^{-1}$ is present that merges in with the 175 kms$^{-1}$ H[i]{} feature, but is likely to be He[i]{} at $\lambda_{\rm air}$=3935.95Å. The IVC/HVC at +78.4 and +109.8 kms$^{-1}$ are not detected in Ca[ii]{} to a limit of log($N_{\rm CaII}$)=10.68, compared with the WH00 prediction of 11.62 and 11.60, respectively. Thus at the 2.5$\sigma$ level, these HI features are likely to be further away than the stellar $z$-height of 664 pc. Using the H[i]{} results compiled in Wakker et al. (2001) results in the same conclusion.
[*HD38666 ($\mu$ Col.)*]{}: The Villa-Elisa H[i]{} spectrum shows blended IVC components at $\sim$ +41 and $\sim$ +53 kms$^{-1}$. This star has been observed in the UV with the Goddard echelle spectrograph, and the results discussed by both Howk, Savage & Fabian (1999) and Brandt et al. (1999). The latter found absorption features at +3, +21, +33 and +42 kms$^{-1}$. The +42 kms$^{-1}$ component was also detected in Ca[ii]{} K in the current spectrum. It thus seems likely that there are two IVC components, one at +42 kms$^{-1}$ and one at +52 kms$^{-1}$. The latter is not detected in Ca[ii]{} K to a limit of log($N_{\rm CaII}$)=10.28, compared to the WM00 prediction of 11.76. This indicates that this component is likely to be further away than HD38666, which has a $z$-height of $\sim$ 181 pc.
[*EC05490–4510*]{}: The Villa-Elisa H[i]{} spectrum shows a weak extended wing in H[i]{} from +50 to +80 kms$^{-1}$ that merges with LV gas. No such wing is present in the Ca[ii]{} data although this is too faint to be detected in any case with the current data.
[*EC06012–7810*]{}: This sightline lies in the direction of the Magellanic Bridge. The Villa-Elisa H[i]{} spectrum shows possible emission at +207 kms$^{-1}$, plus a separate peak at +288 kms$^{-1}$. WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.70 for the latter feature, compared to the observational limit of log($N_{\rm CaII}$)$<$11.28. Hence at the $\sim$ 3$\sigma$ level, the +294 kms$^{-1}$ feature is at a $z$-height exceeding $\sim$ 1500 pc. Finally, note that the optical absorption feature at +216 kms$^{-1}$ is likely to be He[i]{} at $\lambda_{\rm air}$=3935.95Å.
[*EC06387–8045*]{}: The Villa-Elisa H[i]{} spectrum shows a marginal H[i]{} detection at –39 kms$^{-1}$ that merges in with LV gas and also at +71 kms$^{-1}$. The latter feature shows a tentative detection in Ca[ii]{} K at +75 kms$^{-1}$, although the SNR is low. [*If*]{} both features are real and are the same parcel of gas, then this IVC has an upper $z$-height limit of 1770 pc and log($N_{\rm CaII}$) value of 11.50, compared with the predicted value from WM00 of 11.52.
[*PG0833+699*]{}: This sightline has been discussed previously by Ryans et al. (1997b) who used the current data to determine an upper limit to the distance of LLIV1. The Dwingeloo H[i]{} spectrum shows two strong components at –69.3 and –40.1 kms$^{-1}$ with log($N_{\rm HI}$) values of 19.34 and 19.78, merged with LV gas. The Lovell-telescope H[i]{} profile from Ryans et a. (1997b) shows the same components at –70.5 and –43.0 kms$^{-1}$ with log($N_{\rm HI}$) values of 19.36 and 20.01. As noted by Ryans (1997b), both of these components are detected in Ca[ii]{} in absorption, giving an upper limit of z=1980 pc towards these IVCs.
[*PG0855+294*]{}: The Villa-Elisa H[i]{} spectrum shows a weak HVC at about –169 kms$^{-1}$. There is no corresponding Ca[ii]{} detection. WM00 predicts log($N_{\rm CaII}$)=11.57, compared with our limit of 11.16. The data imply that this HVC is further away than the stellar $z$-height of 4100 pc at a 1$\sigma$ level only. Fig. \[lower\_limits\] shows the result of a model fit using log$N$(Ca[ii]{})=11.57, $b$=9.7 kms$^{-1}$ (estimated from the H[i]{} profile), and $v_{\rm LSR}$=–168.7 kms$^{-1}$, superimposed on the observed spectrum. Part of the IV Arch at –27 kms$^{-1}$ is also detected in H[i]{}. This has a corresponding Ca[ii]{} K detection at –29 kms$^{-1}$, giving an upper $z$-height to this IVC (IV20 from Kuntz & Danly 1996) of 4100 pc.
![Solid lines: observed Ca[ii]{} K spectra towards four IHVCs towards which H[i]{} is detected. Dashed lines: Ca[ii]{} K model-fit spectra calculated using the $v_{\rm LSR}^{\rm IHVC}$, FWHM(H[i]{}) and log($N_{\rm pred}$) values given in Table 2. []{data-label="lower_limits"}](./md1099fig3.eps)
[*EC09452–1403*]{}: The Dwingeloo H[i]{} spectrum shows a possible HVC at +120 kms$^{-1}$, but its brightness temperature is only 0.10 K. There is no corresponding Ca[ii]{} detection. WM00 predicts log($N_{\rm CaII}^{\rm WM00}$)=11.46, compared with our upper limit of 11.36. Hence the current data does not constrain the distance to this HVC, even assuming it is real and not baseline ripple. Fig. \[lower\_limits\] shows the result of a model fit using log$N$(Ca[ii]{})=11.46, $b$=10.5 kms$^{-1}$ (estimated from the H[i]{} profile), and $v_{\rm LSR}$=+120.0 kms$^{-1}$, superimposed on the observed spectrum.
[*EC09470–1433*]{}: The Dwingeloo H[i]{} spectrum shows a marginal IVC detection at +55 kms$^{-1}$ ($T_{B}^{\rm peak}$=0.25 K). The HVC detection at +114 kms$^{-1}$ is more secure. The optical absorption at +56 kms$^{-1}$ is likely to be stellar S[ii]{} at $\lambda_{\rm air}$=3933.264Å. If this feature were interstellar in nature, it would also be present in the CaH spectrum shown in Fig. \[ec09470spectrum\]. However, this is not the case. The detection at +86 kms$^{-1}$ is likely to be stellar Ca[ii]{} K due to the velocity and velocity width being consistent with a stellar feature, plus the lack of obvious H[i]{} at this position. Concerning the H[i]{} feature at +114 kms$^{-1}$, there is no obvious associated Ca[ii]{} K. WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.35, compared with our upper limit of 10.98. Hence at the $\sim$ 1$\sigma$ level only, this HVC is likely to lie at a $z$-height exceeding 1400 pc. Fig. \[lower\_limits\] shows the result of a model fit using log$N$(Ca[ii]{})=11.35, $b$=10.6 kms$^{-1}$ (estimated from the H[i]{} profile), and $v_{\rm LSR}$=+114.0 kms$^{-1}$, superimposed on the observed spectrum.
![CaH and K spectra towards EC09470-1433.[]{data-label="ec09470spectrum"}](./md1099fig4.eps)
[*PG0955+291*]{}: The Dwingeloo H[i]{} spectrum shows an IV feature at –29 kms$^{-1}$ extending to –70 kms$^{-1}$ with log($N_{\rm HI}$)=19.81. An additional H[i]{} detection at +115.5 kms$^{-1}$ is weak although appears real. This small HVC has no corresponding Ca[ii]{} absorption detection. Fig. \[lower\_limits\] shows the result of a model fit to the Ca[ii]{} spectrum using log$N$(Ca[ii]{})=11.45, $b$=7.5 kms$^{-1}$ (estimated from the H[i]{} profile), and $v_{\rm LSR}$=+115 kms$^{-1}$, superimposed on the observed spectrum.
The Ca[ii]{} K data (previously analysed by Ryans et al. 1997a) also show Ca[ii]{} absorption at –45 and –64 kms$^{-1}$ which were used to set an upper $z$-height limit of 4300 pc to the IVC IV18.
[*PG1008+689*]{}: The Dwingeloo H[i]{} spectrum shows H[i]{} associated with the LLIV arch at –44 kms$^{-1}$ with log($N_{\rm HI}$)=19.62$\pm$0.03. Ryans et al (1997a) also detected this feature in their Lovell telescope data, with a velocity of –45.4 kms$^{-1}$ and log($N_{\rm HI}$)=19.55. They used these data to place an upper distance limit to this IVC. In the H[i]{} spectrum there is only one component, whereas in the Ca[ii]{} data there are components at $\sim$–39 and –49 kms$^{-1}$. The total Ca[ii]{} column density contained in these two features is log($N_{\rm CaII}$)=11.86.
[*EC10087–1411*]{}: The Dwingeloo H[i]{} spectrum shows a clear IVC at +63 kms$^{-1}$. The cloud is not seen in Ca[ii]{} K absorption to a limit of log($N_{\rm CaII}$)=11.16 compared to the value of 11.66 predicted by WM00. Hence the current data put a lower $z$-height limit of 620 pc to the current cloud, at a 1$\sigma$ limit only.
[*EC11074–2912*]{}: The intermediate-velocity feature at –48.6 kms$^{-1}$ merges in with LV gas. No Ca[ii]{} absorption is seen below a velocity of –30 kms$^{-1}$.
[*EC11507–2253*]{}: The Dwingeloo H[i]{} spectrum shows a possible IVC at +45 kms$^{-1}$ that merges into the LV gas. This feature is also weakly detected in optical absorption in the Ca[ii]{} K and Ca[ii]{} H spectra. If the two features probe the same material, the IVC is closer than the (unknown) stellar distance.
[*PG1213+456*]{}: The Dwingeloo H[i]{} spectrum shows an IVC at –58 kms$^{-1}$ with log($N_{\rm HI}$)=19.55. The Lovell-telescope H[i]{} data of Ryans et al. (1997a) shows a feature at –60 kms$^{-1}$ and log($N_{\rm HI}$)=19.40 which they used to determine the distance to IV17.
[*PG1243+275*]{}: The Dwingeloo H[i]{} spectrum shows an LV feature at –23 kms$^{-1}$ that extends to –60 kms$^{-1}$. The Ca[ii]{} K spectrum shows a corresponding absorption feature at –40 kms$^{-1}$, blended in with lower-velocity gas. [*If*]{} these are the same feature, then the IV gas must lie at a $z$-height of less than 6200 pc.
[*LS 3510*]{}: The Villa-Elisa H[i]{} spectrum shows strong IV emission at +38 and +53 kms$^{-1}$. There are no corresponding Ca[ii]{} K detections which imply that the distance to these features is greater than that of the (unknown) stellar distance.
[*LS 3694*]{}: The Villa-Elissa H[i]{} spectrum shows strong IV emission at +41 and +55 kms$^{-1}$. There are no corresponding Ca[ii]{} K detections which imply that the distance to these features is greater than that of the (unknown) stellar distance.
[*LS 3751*]{}: The Villa-Elissa H[i]{} spectrum shows strong IV emission at $\sim$ +58 kms$^{-1}$. There are no corresponding Ca[ii]{} K detections which imply that the distance to these features is greater than that of the (unknown) stellar distance. A IVC is, however, detected in Ca[ii]{} K absorption at +39 kms$^{-1}$, placing an upper limit to this feature at equal to the (unknown) stellar distance.
[*PG1725+252*]{}: The Dwingeloo H[i]{} spectrum shows an HVC at –154 kms$^{-1}$. There is no corresponding feature seen on the low-resolution WHT spectrum. WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.71, compared with our upper limit of 11.28. Hence at the 1$\sigma$ level the HVC is at a $z$-height larger than 320 pc.
[*PG1738+505*]{}: The Dwingeloo H[i]{} spectrum shows an IVC at –98 kms$^{-1}$. There is no associated Ca[ii]{} K absorption feature to a limit of $\sim$ 11.36, compared to the predicted value of 11.71. Hence these data say imply that the cloud is at a $z$-height greater than the star of 510 pc at $<$ 1$\sigma$ level only.
[*HD341617*]{}: The Dwingeloo H[i]{} spectrum shows an HVC at –118 kms$^{-1}$, connected to LV material. No corresponding feature is seen in the Keck Ca[ii]{} K spectrum. Fig. \[lower\_limits\] shows the result of a model fit using log$N$(Ca[ii]{})=11.62, $b$=14.9 kms$^{-1}$ (estimated from the H[i]{} profile) and $v_{\rm LSR}$=–118 kms$^{-1}$, superimposed on the observed spectrum. A [*possible*]{} IVC is detected in Ca[ii]{} K at –58 kms$^{-1}$. This could be associated with the wing of gas extending from LV to HV material, but this is not certain. If so, then this gas lies at a distance closer than the $z$-height of 2800 pc estimated by Mooney et al. (2002).
[*LS5112*]{}: The Ca[ii]{} K spectrum shows an absorption feature at $\sim$ –137 kms$^{-1}$ with FWHM of 8.0 kms$^{-1}$. As the instrumental resolution is $\sim$7.0 kms$^{-1}$, the feature is essentially unresolved. This narrow-width feature is superimposed on top of the Ca[ii]{} K stellar line at $v_{\rm LSR}$=–120 kms$^{-1}$, which has a FWHM of 58 kms$^{-1}$. The narrow absorption feature is also present in the CaH spectrum as presented in Fig. \[ls5112spectrum\]. If this is interstellar nature, it places an upper distance as equal to the (unknown) stellar distance. However, no H[i]{} is detected at this velocity on the Leiden-Dwingeloo survey. There is of course the possibility that the feature is circumstellar. The positive-velocity features at high velocity are associated with normal differential rotation.
![Raw extracted CaH and K spectra towards LS5112. The narrow-velocity feature at -138 kms$^{-1}$ marked “HVC?” is present in both spectra. []{data-label="ls5112spectrum"}](./md1099fig5.eps)
[*EC19071–7643*]{}: The Villa-Elisa survey shows an H[i]{} wing to the LV gas that extends to $\sim$ –60 kms$^{-1}$. Due to the stellar velocity being $\sim$ –17 kms$^{-1}$ it is difficult to determine whether the wing also seen in Ca[ii]{} K is stellar or interstellar in nature. Hence the current data are of no use in determining the distance to this IVC.
[*EC19489–5641*]{}: The Villa-Elisa survey H[i]{} spectrum shows a IV feature at +36 kms$^{-1}$ that merges in with LV gas. There is no corresponding feature on the Ca[ii]{} K spectrum and the stellar distance is unknown. There are narrow and weak features in the Ca[ii]{} K spectrum at –128 kms$^{-1}$ ($\lambda_{\rm air}$=3931.91Å), –88 kms$^{-1}$ ($\lambda_{\rm air}$=3932.41Å) and –78 kms$^{-1}$ ($\lambda_{\rm air}$=3932.54). The –128 kms$^{-1}$ feature is likely to be stellar S[ii]{} ($\lambda_{\rm air}$=3931.91Å). For the other two features the nearest stellar lines could be S[ii]{} at $\lambda_{\rm air}$=3932.30Å, Ar[ii]{} at 3932.55Å or Sc[i]{} at $\lambda_{\rm air}$=3932.60Å, the latter of which would not be seen in a B-type spectrum (e.g. Hambly et al. 1997). There thus remains the possibility that these lines are interstellar although there is lack of associated H[i]{}.
[*EC19490–7708*]{}: The Villa-Elisa survey shows a possible IVC at +43 kms$^{-1}$. The feature is not visible on the Ca[ii]{} K spectrum. WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.59, compared with our upper limit of 10.91. Hence at the 1.5$\sigma$ level, the current observations put a lower $z$-height limit of $\sim$ 800 pc towards this IVC. The feature at $\sim$ +175 kms$^{-1}$ is likely He[i]{} at $\lambda_{\rm air}$=3935.9Å.
[*EC19596–5356*]{}: The Villa-Elisa survey shows a possible IV feature at $\sim$ +38.7 kms$^{-1}$ that merges in with LV gas. The feature is also detected on the Ca[ii]{} spectrum, although it again is blended with LV gas. The log($N_{\rm CaII}$) value of 11.45 compares to that predicted by WM00 of 11.61. If the features are coincident, this IVC must be closer than that of the star. Assuming a He abundance of 10.96 with the estimated $T_{\rm eff}$=16500 K, log($g$)=4.0 leads to a (large!) stellar distance of 48.1 kpc or a $z$-height of 20 kpc (Lynn, unpublished results), hence the IVC must be closer than this.
[*EC20089–5659*]{}: The Villa-Elisa H[i]{} spectrum shows strong IVC emission at +50 kms$^{-1}$. There is no obvious Ca[ii]{} K absorption at this velocity on-top of the broad stellar line. WM00 predict log($N_{\rm CaII}^{\rm WM00}$)=11.72 compared with our limit of $<$ 10.61. Hence it seems likely that this IVC is at a $z$-height exceeding $\sim$ 400 pc.
[*EC20104–2944*]{}: The optical spectrum shows Ca[ii]{} absorption at +52 kms$^{-1}$ with FWHM $\sim$ 10 kms$^{-1}$ with no obvious associated H[i]{} emission on the LDS. At this position, normal Galactic rotation predicts velocities of up to $\sim$ +25 kms$^{-1}$. The absorption is not stellar as EC20104–2944 has a projected rotational velocity of 50 kms$^{-1}$. The presence of optical absorption implies that the gas lies at a $z$-height of less than $\sim$ 1860 pc. Finally, the line at $\sim$ 140 kms$^{-1}$ is probably stellar He[i]{} at $\lambda_{\rm air}$=3935.95Å.
[*EC20411–2704*]{}: The Dwingeloo survey shows low-level ($<$0.2 K) H[i]{} from $\sim$ –30 kms$^{-1}$ to –150 kms$^{-1}$. This is not visible in Ca[ii]{} K, although the expected absorption would in any case be too weak to measure.
[*M15 ZNG–1*]{}: The Dwingeloo survey shows a previously-identified IVC at +70 kms$^{-1}$ whose distance is less than 3 kpc. At a resolution in H[i]{} of $\sim$ 1 arcmin, Smoker et al. (2002) find that this IVC has a large log($N_{\rm CaII}$) value of 11.67 and log($N_{\rm HI}$) value towards this sightline of 18.70. The current paper adds nothing to the information concerning this IVC.
[*EC23169–2235*]{}: The Dwingeloo survey shows tentative evidence for H[i]{} from $\sim$ –30 kms$^{-1}$ to –100 kms$^{-1}$, although this could be baseline ripple. The feature in any case would be too weak to detect via Ca[ii]{} absorption hence the current data say nothing about the distance towards this IVC.
[*PG2351+198*]{}: The Dwingeloo survey shows a possible IVC at –55 kms$^{-1}$, although blended with LV gas. The feature in the Ca[ii]{} K spectrum at –100 kms$^{-1}$ is likely He[i]{} at $\lambda$=3935.95Å. The feature at –55 kms$^{-1}$ on the optical spectrum is at a rest wavelength of $\sim$ 3936.55Å. This could either be interstellar Ca[ii]{} K in absorption, or much less likely, stellar Mn[i]{} at $\lambda_{\rm air}$=3936.76Åwhich is normally not seen in B-type stars.
Distance limits for individual cloud complexes {#distance}
==============================================
In the following section we collate the measurements given in the previous section in order to attempt to use them to provide distance limits to known IVC and HVC complexes.
Complex C
---------
Complex C is a huge H[i]{} feature visible in the Northern Hemisphere. Its distance is not well constrained; Wakker (2001) gives a firm lower $z$-height limit of 800 pc and a weak lower $z$-height limit of 4300 pc. We note that previous observations towards the QSO PG1351+640 in Ca[ii]{} have found log($N_{\rm CaII}$)=11.91 at log($N_{\rm HI}$)=18.86. Hence, our upper limits of log($N_{\rm CaII}$)$<$11.28 at log($N_{\rm HI}$)=19.47 for PG1725+252 ($v_{\rm LSR}$=–154 kms$^{-1}$), log($N_{\rm CaII}$)$<$10.62 at log($N_{\rm HI}$)=19.18 for HD341617 ($v_{\rm LSR}$=–118 kms$^{-1}$) and log($N_{\rm CaII}$)$<$11.16 at log($N_{\rm HI}$)=19.00 ($v_{\rm LSR}$=–169 kms$^{-1}$) for PG0855+294 would appear to give weak lower $z$-height limits of 320, 2800 and 4100 pc, respectively towards the corresponding parts of this complex.
Complex K and an IVC towards it
-------------------------------
Complex K is a northern cloud which exhibits weak H$\alpha$ emission (Haffner et al. 2001) and has LSR velocities ranging from $\sim$–65 to $\sim$ –95 kms$^{-1}$ and an existing upper $z$-height limit of 4500 pc. Its deviation velocity of $\sim$ –80 kms$^{-1}$ (Wakker 2001) puts it on the borderline between the normal IVC/HVC demarcation. Although a total of 8 of our stars intersect regions of this cloud, in only one of these sightlines is H[i]{} seen at velocities associated with Complex K; at $v_{\rm LSR}$=–98 kms$^{-1}$ towards PG1738+505 at a $z$-height of 510 pc. Towards this sightline, it is unclear as to whether or not absorption in Ca[ii]{} K was detected (Fig. \[fig1\]).
In the same region of the sky, we detect marginal Ca[ii]{} K absorption with $v_{\rm LSR}$=–45 kms$^{-1}$ towards HD341617 at $z$=2800 pc. The absolute value of the velocity is probably too low for this IVC to be considered part of Complex K. In any case, H[i]{} in emission is only marginally detected towards this sightline at this velocity. Concluding, the current data do not say anything definitive about the distance to Complex K or the possible IVC towards HD341617.
Complex gp
----------
Complex gp is a southern positive-velocity IVC with an LSR velocity of $\sim$ +70 kms$^{-1}$ and an existing distance bracket of 300–2000 pc. Parts of it are in the same area of the sky as the Magellanic Stream, although the velocities of the features are different. The cloud is a strong H$\alpha$ emitter and has tentatively been detected using [iras]{} (Smoker et al. 2002). A total of 5 of our sample intersect regions of this cloud, however, only in the previously-known direction of M15 is H[i]{} detected. Hence the current observations add nothing to the distance estimate of this cloud.
Complex GCN
-----------
Complex CGN contains a number of small clouds in the region of the Galactic Centre, which have velocities in the range of $\sim$ –340 to –170 kms$^{-1}$ (Wakker & van Woerden 1991 and refs. therein). One of our stars of unknown distance, LS5112, intersects the general vicinity of the cloud. However, although we see absorption in the spectrum at $\sim$ –138 kms$^{-1}$ (close to the stellar velocity of –120 kms$^{-1}$), that may be interstellar or circumstellar Ca[ii]{} H and K, there is no associated H[i]{} emission at this velocity.
Complex WA–WB
-------------
A few of our sightlines are in the same part of the sky as complexes WA–WB (Wannier, Wrixon & Wilson 1972). Towards EC09452–1403 (complex WB) and EC09470–1433 H[i]{} HV gas is detected at $v_{\rm LSR}$=+120 and $v_{\rm LSR}$=+114 kms$^{-1}$, respectively, with log($N_{\rm HI}$)=18.65 and 18.97. Neither of these sightlines is detected in Ca[ii]{} absorption. Due to its relative faintness, EC09452–1403 only imposes a weak lower $z$-height limit of 2470 pc, although the lack of Caii to a limit of log($N_{\rm CaII}$)$<$10.98 towards EC09470–1433 means it is likely that this complex lies at a $z$-height exceeding 1400 pc. Finally, a possible IVC is seen in the H[i]{} spectrum in the vicinity of EC10087–1411. This is not seen in absorption at the 1$\sigma$ level, hence if it exists, a weak lower $z$-height limit of 620 pc is present for this object.
Complex WE and an IVC towards it
--------------------------------
Complex WE is a HVC at low-intermediate Galactic latitude with velocities of $\sim$+110 kms$^{-1}$ compared to an expected differential galactic rotation of between 0 and –100 kms$^{-1}$ in this direction (Wakker 2001). The $z$-height was previously thought to be $<$ 3200 pc. Towards one of our sightlines, EC06387–8045, we find a weak H[i]{} detection at +73 kms$^{-1}$, with a correspondingly tentative Ca[ii]{} K absorption at +75 kms$^{-1}$. However, it is not clear that this cloud is associated with complex WE, which has a velocity on this part of the sky of $\sim$+120 kms$^{-1}$; it is likely that this object is just an unrelated IVC. If both H[i]{} and Ca[ii]{} features are real, and there is doubt about this, then the upper $z$-height for this cloud would be 1770 pc. Note that additionally, in the H[i]{} spectrum of EC06012–7810, there is a wing extending up to 150 kms$^{-1}$ that could be associated with Complex WE. This is not visible in the Ca[ii]{} spectrum, although given its broad nature it would be very difficult to detect. Other IVCs are also seen in H[i]{} emission towards stars LS3510, LS3604, LS3694 and LS3751, with velocities of $\sim$ +40 to +60 kms$^{-1}$. None of these are seen in Ca[ii]{} K absorption, indicating that these IVCs are further than the associated (unknown) stellar distances. The only IVC seen in absorption is at a velocity of $\sim$ +39 kms$^{-1}$ towards LS3751.
Southern IVCs including the Pegasus-Pisces Arch
-----------------------------------------------
Of the current sample, in only PG 0009+036 and PG2351+198 do we see H[i]{} at intermediate velocity towards the IV South map of Wakker (2001, Fig. 17). The former sightline lies towards the Pegasus-Pisces Arch (PPA; Wakker 2001). Our marginal Ca[ii]{} detection at $v$=–43 kms$^{-1}$ towards this sightline would place this part of the IV gas at a $z$-height of $<$ 9000 pc. A higher SNR spectrum towards this star would be useful. Towards PG2351+198 at $z$=2470 pc Ca[ii]{} K absorption is seen coincident with H[i]{} emission at –55 kms$^{-1}$, thus providing an upper distance limit to this part of the Southern IVC complex, assuming the feature is not Mn[i]{} in absorption.
The IV Arch and Spur
--------------------
The IV arch and Spur covers a large part of the Northern sky, and consists of H[i]{} with velocities around +60 kms$^{-1}$ (Kuntz & Danly 1996) and $z$-height of between 800 and 1500 pc. Many of our sightlines have previously been analysed by Ryans et al. (1997a,b). The only new observations that we have are detections of Ca[ii]{} K towards PG1243+275 at –43 kms$^{-1}$, with an implied upper $z$-height limit of 6200 pc and detection of Ca[ii]{} K towards PG0855+294 towards IV20, giving an upper $z$-height limit of 4100 pc towards this cloud. These observations do not improve the previously-existing distance bracket towards the IV Arch or Spur.
The Magellanic Stream and IV clouds towards it
----------------------------------------------
The Magellanic Stream is an enormous H[i]{} feature, spanning more than 100 degrees of both the northern and southern sky. It is thought to be tidal in nature, formed by the interaction of the Magellanic Clouds with the Milky Way, and velocities across it of $\approx$ –300 to +300 kms$^{-1}$ (Wakker 2001; Putman et al. 2003 and refs. therein). Although a few of our sightlines intersect the Stream and are detected in the H[i]{} spectra, no Ca[ii]{} components are seen, due to the fact that the head of stream is at a distance of $\sim$ 55 kpc, compared with the stellar distances of $<$ 10 kpc.
A few IVCs have been tentatively detected towards stream sightlines in the current dataset, although of course they need not be associated with it. EC19596–5356 shows H[i]{} emission at +39 kms$^{-1}$ and possible Ca[ii]{} absorption at +43 kms$^{-1}$, leading to a tentative upper $z$-height limit of $\sim$ 20 kpc. Similarly, EC20104–2944 shows Ca[ii]{} absorption at 53 kms$^{-1}$ with no obvious associated H[i]{} emission on the LDS. At this position, normal Galactic rotation predicts velocities of up to $\sim$ +25 kms$^{-1}$.
Previously uncatalogued HVCs
----------------------------
In one of our sightlines, towards PG0955+291, there is a small, previously-uncatalogued HVC at $l,b$=200$^{\circ}$,+52$^{\circ}$ with $v_{\rm LSR}$=+115 kms$^{-1}$. Due to its relative faintness in H[i]{} ($\sim$4$\times$10$^{18}$ cm$^{-2}$), the current observations only put a weak lower $z$-height limit of 4300 pc towards this cloud (see Fig. \[lower\_limits\]).
Summary and Conclusions {#concl}
=======================
We have searched for interstellar absorption in the Ca[ii]{} K line for traces of intermediate and high velocity clouds. A number of clouds were detected in this species, although, as normal in this kind of work, many showed no evidence for Ca[ii]{} absorption in the spectra. Under assumptions concerning the abundance variation of the gas and changes in column density over the IHVC in question, the current data were used to estimate lower distance limits towards a number of IHVCs. Future work should follow up a number of the tentative distance estimates, by obtaining higher signal to noise spectra than is present in a number of the current stars and higher-resolution H[i]{} data.
acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the staffs of Isaac Newton Group of telescopes, La Palma, Spain, the Anglo-Australian Observatory, Coonababraran, Australia, the European Southern Observatory, Cerro Paranal, Chile (programme ID 67.D-0010A) and the W.M. Keck observatory, Hawaii, U.S.A., for help in taking these observations. HRMK, WRJR and RSIR thank [pparc]{} for financial support for some of this work. BBL and CJM would like to thank the Department for Employment and Learning, Northern Irelend for funding. JVS would like to thank the APS division of Queen’s University Belfast for hospitality as part of the visiting fellows programme, the European Southern Observatory for travel funds and D. E. Faria for useful comments. FPK is grateful to AWE Aldermaston for the award of a William Penney Fellowship. This research has made use of the [simbad]{} database, operated at CDS, Strasbourg, France. Finally, we would like to thank the anonymous referee for many useful suggestions and corrections to the text.
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abstract: 'It has been suggested that the Z-mode instability driven by energetic electrons with a loss-cone type velocity distribution is one candidate process behind the continuum and zebra pattern of solar type-IV radio bursts. Both the temperature of background plasma ($T_0$) and the energy of energetic electrons ($v_e$) are considered to be important to the variation of the maximum growth rate ($\gamma_{max}$). Here we present a detailed parameter study on the effect of $T_0$ and $v_e$, within a regime of the frequency ratio ($10 \leq \frac{\omega_{pe}}{\Omega_{ce}} \leq 30$). In addition to $\gamma_{max}$, we also analyze the effect on the corresponding wave frequency ($\omega^r_{max}$) and propagation angle ($\theta_{max}$). We find that (1) $\gamma_{max}$ in-general decreases with increasing $v_e$, while its variation with $T_0$ is more complex depending on the exact value of $v_e$; (2) with increasing $T_0$ and $v_e$, $\omega^r_{max}$ presents step-wise profiles with jumps separated by gradual or very-weak variations, and due to the warm-plasma effect on the wave dispersion relation $\omega^r_{max}$ can vary within the hybrid band (the harmonic band containing the upper hybrid frequency) and the band higher; (3) the propagation is either perpendicular or quasi-perpendicular, and $\theta_{max}$ presents variations in line with those of $\omega^r_{max}$, as constrained by the resonance condition. We also examine the profiles of $\gamma_{max}$ with $\frac{\omega_{pe}}{\Omega_{ce}}$ for different combinations of $T_0$ and $v_e$ to clarify some earlier calculations which show inconsistent results.'
author:
- Chuanyang Li
- Yao Chen
- Xiangliang Kong
- 'M. Hosseinpour'
- Bing Wang
title: 'Effect of the temperature of background plasma and the energy of energetic electrons on Z-mode excitation'
---
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Introduction
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Recent studies on moving type-IV solar radio bursts (t-IVms, slowly-drifting wide band continuum observed at metric-decimetric wavelengths) reveal that the t-IVm sources are associated with an eruptive high-temperature dense structure ([@Vasanth16; @Vasanth19]). This is possible since the events of study are recorded at metric wavelengths by both the *Nançay Radioheliograh* (NRH: [@Kerdraon97]) and at Extreme Ultraviolet (EUV) by the Atmospheric Imaging Assembly on board the *Solar Dynamics Observatory* (AIA/SDO: [@Lemen12; @Pesnell12]). Further differential emission analysis of AIA data shows that the source temperature is around several MK and the density is at the level of $10^8$ cm$^{-3}$ at a heliocentric distance of $\sim$1.2–1.5 Solar Radii (R$_\odot$) at frequencies around 200-300 MHz ([@Vasanth19]). At this height of the solar atmosphere, the magnetic field strength is in general around or less than several Gauss (see, e.g., [@Dulk78; @Cho07; @Ramesh10; @Chen11; @Feng11]). Thus, the metric t-IVm bursts are generated within a plasma regime with the plasma-electron-cyclotron frequency ratio ($\frac{\omega_{pe}}{\Omega_{ce}}$) much larger than unity (mostly larger than 10). Based on the observations and some related earlier theoretical studies (e.g., [@Winglee86; @Benacek17]), [@Vasanth16; @Vasanth19] suggested that the t-IV continuum belongs to coherent plasma emission generated by energetic electrons trapped within the eruptive magnetic structure.
Energetic electrons trapped by a magnetic structure can develop a loss-cone type distribution with an inversion of population along the perpendicular direction in the velocity space, i.e., $\frac{\partial f}{\partial v_\perp} > 0$, where $f$ represents the velocity distribution function of energetic electrons. They can drive kinetic instabilities and excite plasma waves (e.g., [@Freund76; @Freund77; @Wu79; @Wu85; @Winglee86]). In the parameter regime of $\frac{\omega_{pe}}{\Omega_{ce}} \gg 1$, such distribution can result in the Z-mode instability and excites enhanced Z-mode waves, which are the slow branch of the extraordinary (X) mode and corresponding to obliquely (or perpendicularly) propagating Langmuir waves. Under certain conditions, such as propagating in inhomogeneous magnetic field and nonuniform plasmas, Z-mode waves may transform into escaping electromagnetic mode and be observed as radio bursts such as t-IV bursts (e.g., [@Winglee86]).
Many earlier studies have applied the Z-mode instability to explain the origin of the intriguing embedding zebra structure of t-IV bursts (e.g., [@Winglee86; @Yasnov04; @Zlotnik13; @Benacek17]). Zebras refer to the numerous emission stripes that are almost parallel to each other superposed on the t-IV continuum, as manifested on the solar radio dynamic spectra ([@Kundu65; @Slottje72; @Kruger79; @Chernov01; @Chernov10; @Chernov12; @Tan14]). Note that the Z-mode instability is also called as the double-plasma resonance (DPR) in many references ([@Yasnov04; @Benacek17; @Benacek18]), since for cold plasmas the instability reaches the maximum growth rate when the upper hybrid frequency ($\omega_{UH}=\sqrt{\omega_{pe}^2 + \Omega_{ce}^2}$) equals a harmonic of $\Omega_{ce}$. It is found that the most important parameter relevant to solar radio bursts is the frequency ratio ($\frac{\omega_{pe}}{\Omega_{ce}}$) which strongly modulates the values of the maximum of the Z-mode growth rate ($\gamma_{max}$). With increasing $\frac{\omega_{pe}}{\Omega_{ce}}$, the profile of $\gamma_{max}$ manifests peaks at frequencies close to harmonics of $\Omega_{ce}$, i.e., $\gamma_{max}$ reaches maximum when Z-mode frequency is close to $n\Omega_{ce}$ where $n$ is an integer. Note that the exact mechanism(s) accounting for the t-IVm continuum and zebras are yet to be determined. For the continuum, both incoherent gyro-synchrotron and coherent plasma emission have been suggested, while more scenarios exist for zebras, such as the Bernstain wave mode, whistler wave mode, the DPR or the Z-mode instability (see the review in [@Chernov10] and the lastest statistical study by [@Tan14]). In this study, only the last possibility, i.e., the mechanism involving the Z-mode instability, will be explored.
[@Winglee86] is one of the first to investigate the effect of $\frac{\omega_{pe}}{\Omega_{ce}}$ on $\gamma_{max}$ of Z-mode. They proposed that the Z-mode instability driven by energetic electrons with loss-cone type distribution might be able to explain both features. The zebra pattern is due to the peak of the growth rate at certain frequencies that are separated by electron gyro-frequency. Between these peaks, the Z-mode can still be excited though at a lower growth rate. The continuum is explained with inhomogeneity of the magnetic field and density within a large-scale source region. Continuous change of background parameters can result in a continuous variation of plasma characteristic frequencies, leading to the Z-mode growth within the corresponding frequency range. According to [@Winglee86], this may serve as a unified scenario for both type-IV continuum and zebras. Using the standard loss-cone distribution and the Dory, Guest, and Harris (DGH, [@Dory65]) distribution for energetic electrons as inputs, they concluded that the latter is likely more relevant to t-IV bursts with significant zebra structures.
Following [@Winglee86], with the DGH distribution [@Yasnov04] and [@Benacek17] further investigated the effect of background plasma temperature and energy of energetic electrons on the $\frac{\omega_{pe}}{\Omega_{ce}}$ dependence of Z-mode growth rate. Yet, the two studies drew inconsistent conclusions. [@Yasnov04] concluded that the zebra pattern with more significant peaks of $\gamma_{max}$ forms for higher energy of energetic electrons, and $\gamma_{max}$ increases with increasing $T_0$ (from 2 MK to 20 MK), while [@Benacek17] concluded that the profile of $\gamma_{max}$ versus $\frac{\omega_{pe}}{\Omega_{ce}}$ expresses distinct peaks with relatively low energy of energetic electrons and does not change significantly with $T_0$. Thus, a clarification is required for a consistent understanding of the variation of the growth rate with $\frac{\omega_{pe}}{\Omega_{ce}}$.
In addition, most earlier studies focused on the variation of the growth rate with the plasma-cyclotron frequency ratio, and did not pay much attention to other critical parameters such as the propagation angles ($\theta_{max}$) and the frequencies ($\omega^r_{max}$) at which the instability develops at the maximum growth rate. [@Lee13] conducted a parameter study on the electron cyclotron maser instability which directly excites the escaping X and ordinary (O) modes ([@Wu79]) and checked the variation of both parameters for the regime of $\frac{\omega_{pe}}{\Omega_{ce}} \leq 5$. [@Yi13] extends the study to investigate the variations of the two parameters of Z-mode instability. Both studies are based on cold plasma approximation. The latter concluded that for $\frac{\omega_{pe}}{\Omega_{ce}} \leq 6$ two Z-mode bands appear for a specific set of parameters with one very narrow band and one relatively wide band. The narrow band is associated with a very short wavelength and may suffer from strong thermal cyclotron damping effect. Thus, it may not survive in warm plasmas, and the wide band may be more relevant to the radio burst. However, the growth rate of the narrow band is often larger than its wide band counterpart, and may be mistakenly picked out when the wave spectrum is not carefully analyzed. This, together with the observational indications from t-IVms ([@Vasanth16; @Vasanth19]), points to the significance of including warm-plasma effect when calculating the Z-mode growth rate.
Only a few studies considered such effect on Z-mode growth [@Winglee86; @Yasnov04; @Benacek17], with various combinations of the background plasma temperature (represented by $T_0$ or the corresponding thermal speed $v_0=\sqrt{\frac{k_BT_0}{m}}$) and energy of energetic electrons (represented by $\emph{v}_e$) of the DGH distribution. Unfortunately, only a few discrete values of the two parameters have been taken into account. For example, [@Benacek17] investigated the Z-mode instability for $v_0$ = 0, 0.009, and 0.018 c, and $v_e$ = 0.1, 0.2, and 0.3 c, while [@Yasnov04] only considered two different values of $T_0$ (2 and 20 MK). This means that present parameter studies on the effect of $T_0$ and $v_e$ are rather incomplete and more detailed parameter study may be necessary. Indeed, according to the calculations presented below, such study leads to important novel results that have not been reported.
In summary of this section, a detailed parameter study on the effect of the temperature of background plasma and the energy of energetic electrons is required to better understand the Z-mode instability driven by trapped electrons. This serves as the major motivation of the present study. The following section introduces basic assumptions, the wave dispersion relation (see also the Appendix), and parameters used in the calculations, and in Section 3 results of the parameter study are presented. A summary and discussion are given in the last section.
Basic assumptions, dispersion relation, and parameters
======================================================
The present study is based on the general kinetic dispersion relation for small-amplitude waves propagating in uniform magnetized warm plasmas (see, e.g., [@Baldwin69; @Wu85]), which is a linear wave solution to the collisionless Vlasov-Maxwell system. The plasma consists of two components of electrons, one is the background warm plasma with the Maxwellian distribution ($f_0$), the other is the energetic electrons with the DGH distribution ($f_e$), given by $$f(u_{\perp},u_{\parallel})=\frac{n_e}{n_0}f_e+\left(1-\frac{n_e}{n_0}\right)f_0,$$ $$f_0=\frac{1}{(2\pi)^{3/2}v_0^{3}}\exp\left(-\frac{u^{2}}{2v_0^{2}}\right),$$ $$f_e=\frac{u_\perp^{2j}}{2^j(2\pi)^{3/2}v_e^{3+2j}j!}\exp\left(-\frac{u_\perp^{2}+u_\parallel^{2}}{2v_e^{2}}\right),$$ where $f$ is the total electron distribution function, $n_0$ and $n_e$ are number density of thermal and energetic electrons, respectively. *j* is the order of the DGH distribution function ($f_e$) and is set to be 1, $v_e$ is the mean velocity of energetic electrons, $u_\perp \ (u_\parallel)$ represents the averaged perpendicular (parallel) momentum per unit mass of electrons. The ions are assumed to be static since only modes with frequencies much higher than ion characteristic frequencies are considered, thus only electrons contribute to the general dispersion relation.
In Figure 1, we demonstrate the total electron distribution ($f$) with white contours, superposed by maps of $\lg(\partial f/\partial u_\perp +1)$ which represents the $u_\perp$ gradient of $f$. The corresponding temperatures of panels a, b and c are $T_0$ = 0, 2, and 4 MK respectively, and the related electron velocity is fixed at 0.2 c. With increasing $T_0$, $f_0$ occupies a larger area of the velocity space, and gets closer to the phase space occupied by energetic electrons, as expected. This of course affects the growth rate of Z-mode instability which is determined by the integral (see Equation A.5) along the resonance curve as defined in the velocity space by the resonance condition $$\gamma_L\omega_r-n\Omega_{ce}-k_\parallel u_\parallel=0,$$ where $\gamma_L$ is the Lorentz factor, $k_\parallel$ is the parallel wave number, and $\Omega_{ce}$ is the electron cyclotron frequency.
We assume that $n_e \ll n_0$. Then, the wave modes are determined by thermal electrons while the instability is energized by energetic electrons. This allows us to utilize fluid equations of warm plasmas to derive the dispersion relation of Z-mode from which the real part of the wave frequency ($\omega_r=\omega_r(\vec k)$) is deduced, where $\vec k \ (= k \cos\theta \hat{e}_z + k \sin\theta \hat{e}_x)$ represents the wave vector, and $\theta$ is the angle between the background magnetic field $\vec {B}_0 \ (= B_0 \hat{e}_z)$ and $\vec k$. The solution of the growth rate (i.e., the imaginary part of the wave frequency, $\gamma$) can be greatly simplified with this assumption (see Equation A.4). Please check the Appendix for details of $\gamma$, the general kinetic, and fluid dispersion relations.
Both dispersion relations are solved numerically. Major parameters are $T_0$, $v_e$, and $\frac{\omega_{pe}}{\Omega_{ce}}$. The density ratio $\frac{n_e}{n_0}$ is included in the growth rate, therefore its exact value is not important as long as it remains small enough. For $T_0$, considering the t-IVm observations introduced earlier and usual warm-plasma parameters of the solar corona, we vary $T_0$ in a range of \[0, 8\] MK or $\sim$ \[0, 1\] keV; for $v_e$ we vary it in a range of \[0.15, 0.4\] c or $\sim$ \[5, 50\] keV, where c is the speed of light. Thus, the weakly-relativistic approximation can be applied.
Regarding the range of $\frac{\omega_{pe}}{\Omega_{ce}}$, as mentioned in the introduction this parameter can be much larger than unity for plasmas within the t-IVm sources, we therefore set its range to be \[10, 30\]. According to observations on t-IV bursts with zebra patterns, the number of stripes and the corresponding estimated harmonics are often larger than 10, sometimes reaching 30 (e.g., [@Kuijpers75; @Aurass03; @Zlotnik03; @Chernov05; @Kuznetsov07]). This means the adopted range of $\frac{\omega_{pe}}{\Omega_{ce}}$ is relevant to the study of t-IV radio bursts.
Another reason of using this range of $\frac{\omega_{pe}}{\Omega_{ce}}$ stems from the limitation of the fluid dispersion relation of Z-mode. For cold plasmas, the frequency of Z-mode is determined by the upper hybrid frequency ($\omega_r=\omega_{UH}$). For warm plasmas, its kinetic dispersion relation is greatly affected by the cyclotron resonance effect, being split into many branches (usually called electron Bernstein modes). The branch within the hybrid band (i.e., the band contains $\omega_{UH}$, given by $(s-1) \Omega_{ce}\leq \omega_r \leq s \Omega_{ce}$, $s$ is a positive integer) with normal dispersion corresponds to the Z-mode, the part with abnormal dispersion corresponds to the electron cyclotron mode. Only the normal-dispersion part is of interest here. Dispersion relations given by the fluid equations and the plasma kinetic theory for Z-mode in warm plasmas of Maxwellian distribution are plotted in Figure 2, for $\frac{\omega_{pe}}{\Omega_{ce}}=$ 5, 10, 15, and 20. It can be seen that for $\frac{\omega_{pe}}{\Omega_{ce}}=$ 5, the fluid dispersion relation deviates away from the kinetic one at frequencies higher than $5.7 \ \Omega_{ce}$, while for larger values ($\geq$ 10) the fluid dispersion relation represents a good approximation to the kinetic one, at least for the normal dispersion parts within the hybrid band and one band higher (i.e., the band of $s \Omega_{ce}\leq \omega_r \leq (s+1) \Omega_{ce}$, see also ). We therefore use the fluid dispersion relation of Z-mode and limit our discussion to the regime of $10 \leq \frac{\omega_{pe}}{\Omega_{ce}} \leq 30$. As seen from our results, all the obtained values of $\omega_r$ are within these two bands, justifying the usage of the fluid dispersion relation.
The general resonance condition (Equation 4) can be simplified under weak relativistic approximation as $$u_\perp^{2}/c^2+\left(u_\parallel/c-u_0/c\right)^2=r^2, r^2=N^2\cos^2\theta+2\left(\frac{n\Omega_{ce}}{\omega}-1\right)$$ where $u_0/c=N\cos\theta$ and $N=kc/\omega_r$. Thus, the resonance curve in the $u$ space is a circle with the radius given by $r$ and the location of the center given by $u_0$. The growth rate is determined by both the details of the distribution function and the resonance curve. The instability grows when the resonance curve passes through regions with large and positive $\partial f/\partial u_\perp$. Such examples are shown in Figure 1 as black arcs or half circles. On the other hand, if the resonance curve passes through regions of small and/or negative values of $\partial f/\partial u_\perp$, the growth rate is either small or negative corresponding to wave damping. Such examples are plotted as yellow and green half circles in Figure 1. In our calculations, the absorption or damping effect of thermal electrons is taken into account.
Parameter study on the Z-mode instability
=========================================
In this section, we present the parameter study on Z-mode excitation, focusing on the effect of [$T_0$ ]{}and $v_e$. It is done by numerically solving the dispersion relation using the fluid approximation of warm plasmas to determine the mode frequency ($\omega_r$, normalized by $\Omega_{ce}$, Equation A.1), and integrating the kinetic dispersion relation to get the growth rate ($\gamma$, normalized by $\Omega_{ce}n_e/n_0$, Equations A.4). As mentioned, we limit our study in the parameter regime of $10 \leq \omega_{pe}/\Omega_{ce} \leq 30$, $0\leq T_0 \leq 8$ MK, and 0.15 c $\leq v_e \leq$ 0.4 c, with the weakly relativistic approximation. For each set of parameters, we calculate $\gamma$ within an appropriate range of ($\omega_r, \theta$). This yields a map of $\gamma$ over ($\omega_r, \theta$), through which we find the maximum growth rate $\gamma_{max}$ and the corresponding wave frequencies $\omega_r^{max}$ and $\theta_{max}$ for further analysis.
To demonstrate the method, we first present the study with a fixed $\omega_{pe}/\Omega_{ce}$ (= 15). This allows us to look into the individual contribution from various harmonics ($n$) and reach some general conclusions regarding the effect of $T_0$ and $v_e$. Then, we examine more details of their effect on the Z-mode instability over the given regime of $\omega_{pe}/\Omega_{ce}$.
Effect of [$T_0$ ]{}and $v_e$ on Z-mode growth with [$\omega_{pe} / \Omega_{ce}$ ]{}= 15
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In Figures 3–5, we plot the map of the growth rate at various individual harmonic ($n$) (panels a–e) as well as their sum (panel f), for [$\omega_{pe} / \Omega_{ce}$ ]{}= 15, $v_e = 0.15 \ c$ and $T_0 = $ 0, 2, 4 MK. From these maps, it is easy to tell the maximum growth rate for each harmonic. As expected, for different combination of parameters there always exists a specific harmonic $n_d$ at which the growth rate dominates over other harmonics, and only the two nearby harmonics with $n=n_d\pm1$ contribute significantly to the wave growth rate (given by the sum over harmonics). In addition, for all cases considered in this study, the Z-mode achieves the maximum growth rate always at the perpendicular or quasi-perpendicular direction. These features are consistent with earlier studies (e.g., [@Winglee86]) that have used the DGH distribution functions for energetic electrons.
For $T_0 = 0$, i.e., the cold-plasma case, we have $n_d$ = 15 with $\gamma^n_{max} = 4.372$, $\omega^n_r = 15.032$, and $\theta^n = 88.8^\circ$, and similar values are obtained for the summed growth rate, indicating the dominance of this harmonic. For $T_0 = 2$ MK, we have $n_d$ = 16 with $\gamma^n_{max} = 4.303$, $\omega^n_r = 15.892$, and $\theta^n = 90^\circ$, and the summed growth rate also has similar or identical values. For $T_0 = 4$ MK, we have $n_d$ = 16 with $\gamma^n_{max} = 5.527$, $\omega^n_r = 15.761$, and $\theta^n = 90^\circ$, again the summed grow rate has similar or identical values. These results indicate that for warm plasmas the number of the dominant harmonic can deviate away from the value given by [$\omega_{pe} / \Omega_{ce}$ ]{}(= 15).
This presents one important difference between the cold- and warm- plasma situations. For cold plasmas the frequency is fixed to the upper hybrid frequency $\omega_{UH}$, while for warm plasmas the Z-mode can grow in a much broader range covering the whole hybrid band and the bands higher (see Figure 2).
As seen from Figure 4, we see that the wave growth rates at $n=17$ and $n=15$ have smaller yet comparable maximum growth rates with that at $n_d=16$. The contributions from these three harmonics have been labeled in the map of the total wave growth rate (Figure 4f). From Figure 5, a very similar situation is observed for the three harmonics (15, 16 ($n_d$), and 17), yet the contribution of $n=15$ cannot be recognized from the summed map (Figure 5f). This is simply due to the result that the strong damping effect at $n_d = 16$ cancels the wave growth at $n=15$ within the corresponding regime of ($\omega_r, \theta$).
Another interesting observation is that, for both $T_0 = 2$ and 4 MK, the wave growth pattern with $\gamma > 0$ at the dominant harmonic splits into two parts by a strong absorption (or damping) regime, due to the presence of warm plasmas. In general, the thermal damping extends to a larger parameter space of ($\omega_r, \theta$) according to Figures 4 and 5.
The above results demonstrate the significance of including warm-plasma effect into the calculation of the Z-mode dispersion relation and their growth rate.
To further understand the resonance condition which decides the instability, we select three points in the ($\omega_r, \theta$) space, including the maximum growth rate at the dominant harmonic $n_d$, a weaker growth rate located nearby (at $\theta=88^\circ$), and a point in the strong wave damping region (see vertical arrows in panels c of Figures 3–5). The resonance curves given by corresponding parameters have been plotted onto the relevant velocity distribution functions, as shown in Figure 1.
Note that for cold plasmas, the absorption effect is due to the negative gradient of the distribution of energetic electrons $f_e$, and only two points, corresponding to the maximum and a weaker growth rate have been selected. As seen from Figure 1a, both curves pass through a significant part of the region with a positive gradient of $f_e$.
For warm plasmas, the resonance circle at the maximum growth rate is given by a zero of a very small value of $u_0 \ (\approx 0)$, corresponding to a nearly perpendicular propagation ($\theta \approx 90$). This makes the curve to sample the positive gradient region of $f$ most efficiently and thus leads to the maximum growth rate at the perpendicular propagation. The resonance curve corresponding to the weaker growth rate has a much larger radius and samples a part of the Maxwellian region with a significant negative gradient due to large number of electrons there, this makes the growth less efficient. On the other hand, the curve corresponding to strong damping passes through a significant section, including the central part, of the Maxwellian. This makes the growth rate negative and the Z-mode wave can not grow.
In Figure 6, we show the maps of the summed growth rate for $v_e = 0.15$ c (upper), 0.2 c (middle), and 0.3 c (lower) and $T_0 = 0$ (left), 2 (middle), 4 (right) MK. For ease of comparison, we re-present the results for $v_e = 0.15$ c. The variation trend is very clear from cold to warm plasmas with increasing $T_0$ and $v_e$, which can be summarized as: (1) the frequency range of wave growth increases significantly due to the warm-plasma effect on the wave dispersion relation, as already mentioned; (2) appearance of strong wave absorption or damping due to the inclusion of thermal electrons, in particular, the growth region at the dominant harmonic ($n_d$) splits into two parts by the damping of Maxwellian; (3) the maximum wave growth always appears at perpendicular or quasi-perpendicular direction, as required by the resonance curve to sample the most positive gradient of $f$, as elaborated above; (4) with increasing $v_e$, a clear trend with non-negligible contributions from more harmonics and a slight decrease of $\gamma_{max}$ are observed.
The above studies are based on a few discrete values of $T_0$ and $v_e$, as done in most earlier studies. Here in Figure 7 we show variation profiles of the three parameters of Z-mode instability ($\gamma_{max}, \omega^r_{max}, \theta_{max}$) with $T_0$ and $v_e$. This allows us to explore more details of the parameter dependence.
As seen from the left panels of Figure 7, for $v_c \leq 0.25$ c the maximum growth rate manifests an obvious oscillation with increasing $T_0$. For instance, for $v_e = 0.15$ c, $\gamma_{max}$ reaches its peak of 5.349 at 3 MK; for $v_e = 0.2$ c, $\gamma_{max}$ reaches its peak of 2.926 at 4.5 MK. On the other hand, for $v_e > 0.25$ c, the oscillation pattern is not significant, in other words, $\gamma_{max}$ is almost independent of $T_0$. The wave frequencies $\omega^r_{max}$ vary within the hybrid band for lower $T_0$ and may jump into the band higher for larger $T_0$. The jumping point is different for different $v_e$. For $v_e = 0.15$ c, $\omega^r_{max}$ jumps from 15.270 to 15.932 around 1.125 MK, and for $v_e = 0.25$ c, $\omega^r_{max}$ jumps from 15.383 to 15.776 around 3.375 MK. Before the jumps, $\omega^r_{max}$ increases gradually while after the jumps $\omega^r_{max}$ presents a slow declining trend. This gives the interesting behavior of the stepwise variation of $\omega^r_{max}$. The stepwise jumping point happens at different values of $T_0$ for different $v_e$. This behavior is mainly due to the increase of the number of the dominant harmonic ($n_d$) by unity, in response to the continuous increase of $T_0$. Note that the values of $n_d$ have been written onto the upper panel. Further discussion will be presented in the last section.
For large $T_0$ and low $v_e$, it can be seen that $\omega^r_{max}$ is in the band higher than the hybrid band. We highlight that $\omega^r_{max}$ can appear in both bands, its exact values depend on the values of $T_0$ and $v_e$. These results are significant to studies on t-IV solar radio bursts with or without zebra patterns since this basically determines the frequencies of emission. It also affects any further studies to infer the magnitude of the magnetic field strength and plasma density on the basis of radio spectral data. More discussion is presented in the last section.
The propagation angle at the maximum growth rate ($\theta_{max}$) is or very close to 90$^\circ$, consistent with earlier studies on Z-mode instability using the DGH distribution of energetic electrons. Here, we show that $\theta_{max}$ varies in accordance with $\omega^r_{max}$. This is because that the two parameters must vary coherently to meet the resonance condition (Equation 4).
From the right panels of Figure 7, we see that both $\gamma_{max}$ and $\omega^r_{max}$ decline in-general with increasing $v_e$. This trend becomes not significant when $v_e$ is large enough, say, $v_e > 0.25$ c. In other words, when $v_e$ is large enough, both wave parameters only weakly depend on $v_e$. Again, a stepwise variation of $\omega^r_{max}$ exists when $v_e$ increases continuously. This happens at $v_e$ = 0.189 c for $T_0 = 2$ MK, and at $v_e$ = 0.255 c for $T_0 = 4$ MK, also due to the jump of the dominant harmonic number by unity. Note that the number $n_d$ has been written onto the upper panel. The wave grows and propagates mainly along the perpendicular or quasi-perpendicular direction, and the angle $\theta_{max}$ presents a variation pattern in accordance with that of the $\omega^r_{max}$.
Effect of [$T_0$ ]{}and [$v_e$ ]{}on Z-mode growth with 10 $\le$ [$\omega_{pe} / \Omega_{ce}$ ]{}$\le$ 30
---------------------------------------------------------------------------------------------------------
In this subsection, we investigate the effect of [$\omega_{pe} / \Omega_{ce}$ ]{}on $\gamma_{max}$ of the instability, as done in many earlier studies. Varying [$\omega_{pe} / \Omega_{ce}$ ]{}can be understood as varying the magnitude of the background magnetic field and the plasma density. Thus, this makes a preliminary study of wave excitation in inhomogeneous media if assuming the spatial scale of the inhomogeneity is much larger than the wavelength.
The ratio [$\omega_{pe} / \Omega_{ce}$ ]{}varies in a range of \[10, 30\]. For any specific value of $\omega_{pe}/\Omega_{ce}$, we conduct the parameter study as those described above and find the corresponding $\gamma_{max}$, then plot the profiles of $\gamma_{max}$ versus $\omega_{pe}/\Omega_{ce}$. A major purpose of this subsection is to clarify some inconsistent results given by earlier publications, as stated earlier.
From Figures 8a–8b, we plot such profiles for different values of $T_0$ while fixing $v_e$ at 0.2 c. The most prominent feature of the profiles is their oscillations, with quasi-periodic peaks and valleys. The distance between neighboring peaks is about one $\Omega_{ce}$. This pattern is well known and has been applied to explain the presence of zebra patterns of solar t-IV radio bursts (e.g., [@Winglee86; @Yasnov04; @Kuznetsov07; @Benacek17]). Another important feature is the overall increasing trend of $\gamma_{max}$ with increasing $\omega_{pe}/\Omega_{ce}$ and the very-weak-yet-discernible decreasing trend with increasing $T_0$. This result is different from that presented by [@Benacek17] who show that $\gamma_{max}$ decreases in-general with increasing $\omega_{pe}/\Omega_{ce}$. The difference might be due to the specific simplifications used in their model. For example, they have neglected the term containing the parallel gradient of $f_e$ (i.e., $\partial f_e/\partial u_\parallel$) when calculating the growth rate. In addition, the specific peaks shift towards lower values of [$\omega_{pe} / \Omega_{ce}$ ]{}with increasing $T_0$. This is basically consistent with those given by [@Benacek17] (see Figure 6 of this reference).
The ratio between values of $\gamma_{max}$ at neighboring peaks and bottoms can be used to characterize the flatness of the profile, as plotted in Figure 8c. It can be seen that the ratio declines with increasing $\omega_{pe}/\Omega_{ce}$. For instance, for [$\omega_{pe} / \Omega_{ce}$ ]{}= 15, the ratio is about 2.1 at $T_0 = 1$ MK and $\sim$ 1.9 at $T_0 = 3$ MK, while for [$\omega_{pe} / \Omega_{ce}$ ]{}= 25, the ratio is $\sim$ 1.6 at $T_0 = 1$ MK and $\sim$ 1.3 at $T_0 = 3$ MK. The flatness variation trend of $\gamma_{max}$ versus $\omega_{pe}/\Omega_{ce}$ is basically consistent with the result of [@Benacek17]. For $\omega_{pe}/\Omega_{ce} < 15$, the peak-bottom ratio decreases with increasing $T_0$, while the variation of flatness of $\gamma_{max}$ with [$T_0$ ]{}is not very regular for larger values of $\omega_{pe}/\Omega_{ce} \ (> 15)$. The latter might be caused by inregular oscillations and steep changes of $\gamma_{max}$ at the bottom of the profiles.
In Figure 9, we show similar profiles of $\gamma_{max}$ with [$\omega_{pe} / \Omega_{ce}$ ]{}for different $v_e$ while fixing $T_0$ to be 2 MK. A very weak overall increasing trend of $\gamma_{max}$ with increasing [$\omega_{pe} / \Omega_{ce}$ ]{}can be identified from the upper panel. From the profiles of peak-bottom ratio (Figure 9b), it can be seen that the ratio declines with increasing [$\omega_{pe} / \Omega_{ce}$ ]{}and also with increasing $v_e$. For $v_e \geq 0.3$ c and [$\omega_{pe} / \Omega_{ce}$ ]{}$\geq 15$, the ratio is around or less than 1.2, indicating that the zebra pattern may not be recognizable under these conditions. The result presented in Figure 9 is basically consistent with those presented by [@Winglee86] and [@Benacek17]. Here we extend the calculations to a larger parameter regime of [$\omega_{pe} / \Omega_{ce}$ ]{}.
Summary and discussion
======================
Many earlier studies have studied the Z-mode instability to understand the origin of zebras of t-IV bursts. Among various parameters, the ratio of $\omega_{pe}$ and $\Omega_{ce}$ (decided by the magnetic field strength and plasma density) plays a major role on the maximum of the instability growth rate ($\gamma_{max}$). In addition to this ratio, the temperature of background plasma ($T_0$) and the energy of energetic electrons ($v_e$) are also important. Earlier studies only considered a few discrete values of the two parameters and revealed inconsistent results. Here we revisit the problem with a more complete parameter study on the effect of these parameters. The parameter regimes of interest, relevant to latest observations on t-IV sources, are taken to be $10 \leq \omega_{pe}/\Omega_{ce} \leq 30$, $0\leq T_0 \leq 8$ MK, and 0.15 c $\leq v_e \leq $ 0.4 c.
For a specific value of [$\omega_{pe} / \Omega_{ce}$ ]{}(= 15), it was found that $\gamma_{max}$ presents a general decreasing trend with increasing $v_e$, and an obvious oscillation with increasing $T_0$ for $v_e = 0.15 - 0.25$ c while it is almost independent of $T_0$ for larger $v_e$. In addition, with increasing $T_0$ and $v_e$, the frequency at $\gamma_{max}$ presents step-wise profiles with jumps separated by gradual or weak variations in the range covering the hybrid band and the band higher. The propagation angle $\theta_{max}$ varies accordingly as constrained by the resonance condition. The cause of these jumps, as mentioned in Section 3, is due to the change of the dominant harmonic number ($n_d$) by unity. Further explanation is given as follows. To achieve the maximum growth rate, the resonance curve must sample the appropriate region of the velocity distribution ($f$), i.e., with a large positive velocity gradient of $f$ and enough number of particles along the curve. This puts strong constraints on $\omega_r \ (k)$ and $\theta$, at which the maximum growth rate is obtained. These parameters, along with $n$, decide the resonance curve. In Figure 7, we increase $T_0$ and $v_e$ gradually. The two parameters change the velocity gradient of $f$ and thus the resonance condition at $\gamma_{max}$. Generally speaking, a larger $T_0$, i.e., a more-expanded Maxwellian distribution, corresponds to a more-expanded resonance curve at $\gamma_{max}$, while $f$ with a larger $v_e$ will result in the opposite trend of the resonance curve at $\gamma_{max}$. In addition, the dispersion relation of Z-mode changes with $T_0$. This further affects the growth rate.
As seen from our results, below certain thresholds, the maximum rate can still be obtained for fixed harmonic number $n$ with gradually-changing values of $\omega_r$, $\gamma_L$, and $\theta$. Yet, above the thresholds, the maximum growth rate moves to the nearby harmonic, $n+1$ for increasing $T_0$ and $n-1$ for increasing $v_e$, due to the above-mentioned opposite trend of $T_0$ and $v_e$ on the velocity distribution function $f$. This change of harmonic number leads to the jumps of various parameters observed in Figure 7.
One explanation of zebras is that each stripe is given by a peak of growth rate which appear for continuous variation of $\omega_{pe}/\Omega_{ce}$. For cold plasmas the peak is reached when $\omega_{UH} = s \Omega_{ce}$, i.e., when the upper hybrid frequency equals to a harmonic of electron cyclotron frequency. This is not correct for warm plasmas. Taking the warm-plasma effect into account, within a range of $10 \le $ [$\omega_{pe} / \Omega_{ce}$ ]{}$\le 30$ we studied the influences of $T_0$ and $v_e$ on the variation of the peaks and relevant maximum-minimum ratios of growth rate. It was found that the ratios always decline and the location of peaks shift towards lower values of $\omega_{pe}/\Omega_{ce}$, with increasing $\omega_{pe}/\Omega_{ce}$. The ratios do not present a simple variation trend with increasing $T_0$, and it in-general declines with increasing $v_e$ for $v_e \leq 0.3$ c. For larger $v_e$, the ratio remains around or less than 1.2. Such small values of ratio may lead to continuum without recognizable zebras.
During solar flares, both heating and particle acceleration take place. This leads to change of the plasma temperature ($T_0$) and energy of energetic electrons ($v_e$). Thus, it is natural to suggest that the frequency variations of zebra stripes may be partially due to the on-going heating and acceleration processes (e.g., [@Yasnov04]). For example, according to our calculations, with continuous plasma heating and particle acceleration, the wave frequency may change either gradually or suddenly. And the maximum growth rate also changes accordingly. This may affect the morphology of the stripes and their emission intensity. Thus, the calculations presented here are significant to explanations of zebras and further studies to infer coronal parameters, such as the magnetic field strength in the source region (e.g., [@Tan12]). In addition, it should be highlighted that both magnetic field and plasma density change rapidly during solar flares, and this may also have an important effect on the wave growth rate as well as the presence of zebra stripes and their spectral morphology. Further studies to explore origins of various types of zebras should take these factors into account.
Acknowledgements {#acknowledgements .unnumbered}
================
This study is supported by the National Natural Science Foundation of China (11790303 (11790300), 11750110424 and 11873036). X.K. also acknowledges the support from the Young Elite Scientists Sponsorship Program by China Association for Science and Technology, and the Young Scholars Program of Shandong University, Weihai. The authors are grateful to the anonymous referee for valuable comments.
The growth rate and dispersion tensor of Z-mode instability for warm plasmas
============================================================================
The wave frequency in a collisionless Vlasov-Maxwell system is written as $\omega = \omega_r + i\gamma$, where $\omega_r$ is determined by the dispersion relation using the following fluid approximation of warm plasmas for X (Z) mode $$\text{Re}\overleftrightarrow{\Lambda}(\vec{k},\omega_r)=
\left( \begin{array}{ccc}
-N^2\cos^2\theta & 0 & N^2\sin\cos\theta \\
0 & -N^2 & 0 \\
N^2\sin\cos\theta & 0 & -N^2\sin^2\theta
\end{array}
\right )+\overleftrightarrow{\epsilon}=0,$$ $$\overleftrightarrow{\epsilon}=\overleftrightarrow{I}-\frac{\omega_{pe}^2}{\omega_r^2}\frac{\overleftrightarrow{C}_e}{\Delta_e}, \Delta_e=(1-\frac{\Omega_{ce}^2}{\omega_r})(1-3N^2v_0^2\cos^2\theta)-3N^2v_0^2\sin^2\theta,$$ $$\overleftrightarrow{C}_e=
\left( \begin{array}{ccc}
1-3N^2v_0^2\cos^2\theta & -i\frac{\Omega_{ce}}{\omega_r}(1-3N^2v_0^2\cos^2\theta) & 3N^2v_0^2\sin\theta\cos\theta \\
i\frac{\Omega_{ce}}{\omega_r}(1-3N^2v_0^2\cos^2\theta) & 1-3N^2v_0^2 & i\frac{\Omega_{ce}}{\omega_r}3N^2v_0^2\sin\theta\cos\theta \\
3N^2v_0^2\sin\theta\cos\theta & -i\frac{\Omega_{ce}}{\omega_r}3N^2v_0^2\sin\theta\cos\theta & 1-\frac{\Omega_{ce}^2}{\omega_r^2}-3N^2v_0^2\sin^2\theta
\end{array}
\right ),$$ where $N=kc/\omega_r$ and $k$ are the refractive index and the wave number, respectively, and $\theta$ is the angle of propagation (i.e., the angle between $\vec{k}$ and $\vec{B}$), and $v_0=\sqrt{k_BT_0/m_e}$ is the thermal velocity of background electrons.
Under the assumption of $\omega_r \gg \left|\gamma\right|$, the growth rate $\gamma$ is given by $$\gamma=-\frac{\text{Im}\Lambda(\vec{k},\omega_r)}{\frac{\partial}{\partial\omega_r}\text{Re}\Lambda(\vec{k},\omega_r)}.$$ $\text{Im}\Lambda(\vec{k},\omega_r)$ is the imaginary part of the kinetic dispersion relation given by [e.g. @Baldwin69] $$\begin{split}
\text{Im}\Lambda(\vec{k},\omega_r)=&2\pi\frac{\omega_{pe}^2}{\omega_r^2}\int_{-\infty}^{+\infty}du_{\parallel}\int_{0}^{+\infty}du_{\perp}\Bigg\{\frac{u_{\parallel}}{\gamma_L}\left(u_\perp\frac{\partial}{\partial u_\parallel}-u_\parallel\frac{\partial}{\partial u_\perp}\right)\times f(u_\perp,u_\parallel)\hat{e}_z\hat{e}_z+\\
&\omega_r\left[\frac{\partial}{\partial u_\perp}+\frac{k_\parallel}{\gamma_L\omega_r}\left(u_\perp\frac{\partial}{\partial u_\parallel}-u_\parallel\frac{\partial}{\partial u_\perp}\right)\right]\times f(u_\perp,u_\parallel)\sum_{n=-\infty}^{\infty}\frac{\overleftrightarrow{T_n}(b)}{\gamma_L\omega_r-n\Omega_{ce}-k_\parallel u_\parallel}\Bigg\}
\end{split},$$ $$\overleftrightarrow{T_n}(b)=
\left( \begin{array}{ccc}
\frac{n^2\Omega_{ce}^2}{k_\perp^2}J_n^2(b) & -i\frac{n\Omega_{ce}}{k_\perp}u_\perp J_n(b)J_n^\prime(b) & \frac{n\Omega_{ce}}{k_\perp}u_\parallel J_n^2(b)\\
i\frac{n\Omega_{ce}}{k_\perp}u_\perp J_n(b)J_n^\prime(b) & u_\perp^2J_n^{\prime2}(b) & iu_\perp u_\parallel J_n(b)J_n^\prime(b)\\
\frac{n\Omega_{ce}}{k_\perp}u_\parallel J_n^2(b) & -iu_\perp u_\parallel J_n(b)J_n^\prime(b) & u_\parallel^2J_n^2(b)
\end{array}
\right ),$$ where $u_\parallel=p_\parallel /m_e=\gamma_L v_\parallel$, $u_\perp=p_\perp /m_e=\gamma_L v_\perp$, $\omega_{pe}=\sqrt{n_ee^2 /m_e\varepsilon_0}$ is the plasma frequency, $\gamma_L=\left(1-\frac{v^2}{c^2}\right)^{-1/2}$ is the Lorentz factor, $f(u_\perp,u_\parallel)$ is the total distribution function of electrons, and $J_n(b)$ the first-kind Bessel function of the *n*th order, $J_n^\prime(b)$ is its partial derivative with respect to $b \ (=k_\perp u_\perp /\Omega_{ce})$.
Aurass, H., Klein, K.-L., Zlotnik, E. Y., & Zaitsev, V. V. 2003, , 410, 1001 Baldwin, D. E., Bernstein, I. B., & Weenink, M. P. H. 1969, Advances in Plasma Physics, 3, 1 Ben[á]{}[č]{}ek, J., & Karlick[ý]{}, M. 2018, , 611, A60 Ben[á]{}[č]{}ek, J., Karlick[ý]{}, M., & Yasnov, L. V. 2017, , 598, A106 Chen, Y., Feng, S. W., Li, B., et al. 2011, , 728, 147 Chernov, G. P., Sych, R. A., Meshalkina, N. S., Yan, Y., & Tan, C. 2012, , 538, A53 Chernov, G. P., Yan, Y. H., Fu, Q. J., & Tan, C. M. 2005, , 437, 1047 Chernov, G. P., Yasnov, L. V., Yan, Y.-H., & Fu, Q.-J. 2001, , 1, 6, 525 Chernov, G. P. 2010, Research in Astronomy and Astrophysics, 10, 821 Cho, K.-S., Lee, J., Gary, D. E., Moon, Y.-J., & Park, Y. D. 2007, , 665, 799 Dory, R. A., Guest, G. E., & Harris, E. G. 1965, Physical Review Letters, 14, 131 Dulk, G. A., & McLean, D. J. 1978, , 57, 279 Feng, S. W., Chen, Y., Li, B., et al. 2011, , 272, 119 Freund, H. P., & Wu, C. S. 1977, Physics of Fluids, 20, 619 Freund, H. P., & Wu, C. S. 1976, Physics of Fluids, 19, 299 Kerdraon, A., & Delouis, J.-M. 1997, Coronal Physics from Radio and Space Observations, 483, 192 Kr[ü]{}ger, A. 1979, Geophysics and Astrophysics Monographs Kuijpers, J. 1975, , 40, 405 Kundu, M. R. 1965, New York: Interscience Publication Kuznetsov, A. A., & Tsap, Y. T. 2007, , 241, 127 Lee, S.-Y., Yi, S., Lim, D., et al. 2013, Journal of Geophysical Research (Space Physics), 118, 7036 Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, , 275, 17 Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, , 275, 3 Ramesh, R., Kathiravan, C., & Sastry, C. V. 2010, , 711, 1029 Slottje, C. 1972, , 25, 210 Tan, B., Tan, C., Zhang, Y., M[é]{}sz[á]{}rosov[á]{}, H., & Karlick[ý]{}, M. 2014, , 780, 129 Tan, B., Yan, Y., Tan, C., Sych, R., & Gao, G. 2012, , 744, 166 Vasanth, V., Chen, Y., Feng, S., et al. 2016, , 830, L2 Vasanth, V., Chen, Y., Lv, M., et al. 2019, , 870, 30 Winglee, R. M., & Dulk, G. A. 1986, , 307, 808 Wu, C. S. 1985, , 41, 215 Wu, C. S., & Lee, L. C. 1979, , 230, 621 Yasnov, L. V., & Karlick[ý]{}, M. 2004, , 219, 289 Yi, S., Lee, S.-Y., Kim, H.-E., et al. 2013, Journal of Geophysical Research (Space Physics), 118, 7584 Zlotnik, E. Y., Zaitsev, V. V., Aurass, H., Mann, G., & Hofmann, A. 2003, , 410, 1011 Zlotnik, E. Y. 2013, , 284, 579
[^1]: E-mail: licy001@163.com
[^2]: E-mail: yaochen@sdu.edu.cn
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abstract: 'The mesonic excitations and $s$-wave $\pi$–$\pi$ scattering lengths at finite temperature are studied in the two-flavor Polyakov–Nambu–Jona-Lasinio (PNJL) model. The masses of $\pi$ meson and $\sigma$ meson, pion-decay constant, the pion-quark coupling strength, and the scattering lengths $a_{0}$ and $a_{2}$ at finite temperature are calculated in the PNJL model with two forms of Polyakov-loop effective potential. The obtained results are almost independent of the choice of the effective potentials. The calculated results in the PNJL model are also compared with those in the conventional Nambu–Jona-Lasinio model and indicate that the effect of color confinement screens the effect of temperature below the critical one in the PNJL model. Furthermore, the Goldberger-Treiman relation and the Gell-Mann–Oakes–Renner relation are extended to the case at finite temperature in the PNJL model.'
author:
- |
[Wei-jie Fu$^{a}$, and Yu-xin Liu$^{a,b,}$[^1] ]{}\
\
\
\
title: 'Mesonic excitations and $\pi$–$\pi$ scattering lengths at finite temperature in the two-flavor Polyakov–Nambu–Jona-Lasinio model'
---
[**PACS Numbers:** ]{} [12.38.Aw, 11.30.Rd, 13.75.Lb, 14.40.Aq ]{}
Introduction
============
QCD thermodynamics and phase diagram, especially about the restoration of the chiral symmetry and the deconfinement phase transition which are expected to occur in ultra-relativistic heavy-ion collisions [@Shuryak2004; @Gyulassy2005; @Shuryak2005; @Arsene2005; @Back2005; @Adams2005; @Adcox2005; @Blaizot2007] or in the interior of neutron stars [@Weber2005; @Alford2007; @Alford2008; @Fu2008b], has been a subject of intense investigation in recent years. One significant aspect to investigate the restoration of the chiral or axial symmetry and the deconfinement phase transition is to study the variation of properties of particles propagating in hot and/or dense medium [@Costa2004; @Hansen2007; @Costa2008]. In this work, we focus on the influence of a hot medium on the properties of light pseudoscalar ($\pi$) and scalar ($\sigma$) mesons, and $\pi$–$\pi$ scattering lengths. Special attentions are paid to their dramatic variations near the regime where the chiral phase transition and the deconfinement phase transition occur. We expect to extract the signals of phase transition from mesonic excitations and $\pi$–$\pi$ interactions in the hot medium.
A promising phenomenological approach to study the low-energy processes involving the pseudoscalar and scalar mesons at zero temperature and finite temperature is the Nambu–Jona-Lasinio (NJL) model [@Nambu1961; @Volkov1984; @Klevansky1992; @Hatsuda1994; @Alkofer1996; @Buballa2005]. The most important advantage of the NJL model is that it introduces a mechanism of the dynamical breaking of chiral symmetry (due to the quark-antiquark condensate). However, the NJL model has its disadvantage, which is the lack of the description of color confinement. To include some effects of color confinement, a Polyakov-loop improved Nambu–Jona-Lasinio (PNJL) model has been developed recent years [@Meisinger9602; @Pisarski2000; @Fukushima2004; @Mocsy2004; @Arriola2006; @Ratti2006a; @Ratti2006b; @Fu2008; @Ciminale2008]. In the PNJL model, the Polyakov-loop as a classical field couples to quarks and thus suppresses the contributions from wrong degrees of freedom (color non-singlet) to the thermodynamics below the critical temperature. Therefore, the introduction of the Polyakov-loop represents some aspects of the color confinement, at least on the level of statistics [@Fu2008]. The validity of the PNJL model has been confirmed in a series of works by confronting the PNJL results with the lattice QCD data [@Ratti2006a; @Ratti2006b; @Ghosh2006; @Ratti2006c; @zhang2006]. The phase structure and thermodynamics in the PNJL model have recently been explored extensively [@Sasaki2006; @Weise2007; @Fu2008; @Ciminale2008; @Ghosh2008; @Abuki2008; @Tuominen2008; @Abuki2008b; @Kashiwa2008; @Costa2008b; @Fukushima2008a; @Contrera2008; @Fukushima2008b; @Mukherjee2007; @Hiller2008], and the impact of Polyakov-loop dynamics on the chiral susceptibility or quark number susceptibility [@Sasaki2006; @Weise2007], QCD critical endpoint [@Kashiwa2008; @Costa2008b] and critical surface [@Fukushima2008a], and the color superconductivity phase transition [@Ratti2006b; @Ciminale2007; @Abuki2008c; @Dumm2008] have attracted lots of interests. Furthermore, fluctuations beyond the mean field approximation have been included in the PNJL model [@Blaschke2007; @Hell2007], and the PNJL model has also been extended to the regime of imaginary chemical potential [@Sakai2008; @Sakai2008b; @Kashiwa2008b; @Sakai2008c] and $0+1$ dimensions [@Dusling2008], and applied to analyze the flavors of quark-gluon-plasma [@Mueller2008] and the isentropic trajectories on QCD phase diagram [@Fukushima2009].
The properties of pseudoscalar and scalar mesons at finite temperature for two [@Hansen2007] and three [@Costa2008] flavor systems have also been investigated in the PNJL model. In Ref. [@Hansen2007], The mesonic correlators and spectral functions for $\pi$ and $\sigma$ mesons were obtained. It was found that the $\pi$-$\sigma$ degeneracy in the chiral symmetry restored phase was still satisfied after coupling quarks to the Polyakov-loop and the role of $\pi$ meson as Goldstone boson was also confirmed in the PNJL model. It was also found that, although the PNJL model can not cure the problem of the conventional NJL model as for the unphysical width of the $\sigma$ meson, the PNJL results on the decay width improved slightly the NJL ones [@Hansen2007]. In order to further study the broken chiral symmetry and its restoration in the mesonic sector in the PNJL model which makes the investigation of the interplay between the restoration of chiral symmetry and the deconfinement phase transition possible, it is necessary to study the Goldberger-Treiman relation [@Goldberger1958] and the Gell-Mann–Oakes–Renner relation [@Gell1968] which are direct results due to the chiral symmetry breaking. Furthermore, one of the most fundamental hadronic processes of QCD at the mesonic level, the pion-pion scattering, $\pi+\pi\rightarrow\pi+\pi$, at finite temperature, which provides a direct link between the theoretical formalism of chiral symmetry and experiment, also deserves to be investigated. In this work, we will then study the problems mentioned above in the PNJL model.
The paper is organized as follows. In Sec. II we simply review the formalism of the two flavor PNJL model. In Sec. III we discuss the mesonic excitations at finite temperature in the PNJL model. The dependence of the pion-decay constant, pion-quark coupling strength, and the relation between the mass of $\sigma$ meson and that of $\pi$ meson on the temperature are studied in the PNJL model. We also extend the Goldberger-Treiman relation and Gell-Mann–Oakes–Renner relation to a formalism which is appropriate at finite temperature. In Sec. IV we study the $s$-wave $\pi$–$\pi$ scattering lengths in the PNJL model and compare the results in the PNJL model with those in the conventional NJL. Finally, in Sec. V, we give a summary and conclusions.
The PNJL model
==============
The Lagrangian density for the two-flavor PNJL model is given as [@Ratti2006a] $$\begin{aligned}
\mathcal{L}_{PNJL}&=&\bar{\psi}\left(i\gamma_{\mu}D^{\mu}-\hat{m}_{0}\right)\psi
+G\left[\left(\bar{\psi}\psi\right)^{2}
+\left(\bar{\psi}i\gamma_{5}\vec{\tau}\psi\right)^{2}\right] \nonumber \\
&& -\mathcal{U}\left(\Phi[A],\bar{\Phi}[A] \,
,T\right),\label{lagragian}\end{aligned}$$ where $\psi=(\psi_{u},\psi_{d})^{T}$ is the quark field, $$D^{\mu}=\partial^{\mu}-iA^{\mu}\quad\textrm{with}\quad
A^{\mu}=\delta^{u}_{0}A^{0}\quad\textrm{,}\quad
A^{0}=g\mathcal{A}^{0}_{a}\frac{\lambda_{a}}{2}=-iA_4.$$ The gauge coupling $g$ is combined with the SU(3) gauge field $\mathcal{A}^{\mu}_{a}(x)$ to define $A^{\mu}(x)$ for convenience and $\lambda_{a}$ are the Gell-Mann matrices in color space. $\hat{m}_{0}=\textrm{diag}(m_{u},m_{d})$ is the current quark mass matrix. Throughout this work, we take $m_{u}=m_{d}\equiv m_{0}$, assuming the isospin symmetry is reserved on the Lagrangian level. The four-fermion interaction with an effective coupling strength $G$ for scalar and pseudoscalar channels has $\mathrm{SU_{V}}(2)\times
\mathrm{SU_{A}}(2)\times \mathrm{U_{V}}(1)$ symmetry, which is broken to $\mathrm{SU_{V}}(2)\times \mathrm{U_{V}}(1)$ when $m_{0}\neq 0$. Here $\tau^{a}(a=1,2,3)$ in the Lagrangian density (Eq. ) are Pauli matrices in flavor space.
The $\mathcal{U}\left(\Phi,\bar{\Phi},T\right)$ in the Lagrangian density is the Polyakov-loop effective potential, which controls the Polyakov-loop dynamics and can be expressed in terms of the trace of the Polyakov-loop $\Phi=(\mathrm{Tr}_{c}L)/N_{c}$ and its conjugate $\bar{\Phi}=(\mathrm{Tr}_{c}L^{\dag})/N_{c}$. Here the Polyakov-loop $L$ is a matrix in color space, which can be explicitly given as [@Ratti2006a] $$L\left(\vec{x}\right)=\mathcal{P}\exp\left[i\int_{0}^{\beta}d\tau\,
A_{4}\left(\vec{x},\tau\right)\right]
=\exp\left[i \beta A_{4} \right]\, ,$$ where $\beta=1/T$ is the inverse of the temperature. The Polyakov-loop effective potential has the $Z(3)$ center symmetry like the pure-gauge QCD Lagrangian. When the temperature is lower than a critical value ($T_{0}\simeq 270\,\mathrm{MeV}$ in pure gauge QCD [@Ratti2006a]), the value of $\Phi$ (and $\bar{\Phi}$) which minimizes the Polyakov-loop effective potential is zero, meaning that the phase is color confined and has the $Z(3)$ symmetry. However, when the temperature is above the critical temperature $T_{0}$, $\Phi$ develops a nonzero value which minimizes the effective potential and the system is transited from a $Z(3)$ symmetric, confined phase to a $Z(3)$ symmetry broken, deconfined phase. The temperature dependent Polyakov-loop effective potential is chosen to reproduce the lattice data for both the expectation value of the Polyakov-loop [@Kaczmarek2002] and some thermodynamic quantities [@Boyd1996]. In the PNJL Lagrangian in Eq. , the coupling between the Polyakov-loop and quarks is uniquely determined by the covariant derivative $D_{\mu}$.
In previous works, two possible forms for the Polyakov-loop effective potential have been well developed. Following our previous work [@Fu2008], we denote them as $\mathcal{U}_{\mathrm{pol}}(\Phi,\bar{\Phi},T)$ and $\mathcal{U}_{\mathrm{imp}}(\Phi,\bar{\Phi},T)$, respectively. The former is a polynomial in $\Phi$ and $\bar{\Phi}$ [@Ratti2006a] and the latter is an improved effective potential in which the higher order polynomial terms in $\Phi$ and $\bar{\Phi}$ are replaced by a logarithm [@Ratti2006b]. Both the effective potentials are taken in our work to investigate whether our results depend on the details of the Polyakov-loop effective potential. These two effective potentials have the following forms $$\frac{\mathcal{U}_{\mathrm{pol}}\left(\Phi,\bar{\Phi},T\right)}{T^{4}}
= -\frac{b_{2}(T)}{2}\bar{\Phi}\Phi -\frac{b_{3}}{6}
(\Phi^{3}+{\bar{\Phi}}^{3})+\frac{b_{4}}{4}(\bar{\Phi}\Phi)^{2} \, ,$$ with $$b_{2}(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}
{\left(\frac{T_{0}}{T}\right)}^{2}
+a_{3}{\left(\frac{T_{0}}{T}\right)}^{3},$$ and $$\begin{aligned}
\frac{\mathcal{U}_{\mathrm{imp}}\left(\Phi,\bar{\Phi},T\right)}{T^{4}}
& = &-\frac{1}{2}A(T)\bar{\Phi}\Phi
+B(T)\ln\left[1-6\bar{\Phi}\Phi+4({\bar{\Phi}}^{3}+\Phi^{3})
-3(\bar{\Phi}\Phi)^{2}\right] \, ,\end{aligned}$$ with $$A(T)=A_{0}+A_{1}\left(\frac{T_{0}}{T}\right)
+A_{2}{\left(\frac{T_{0}}{T}\right)}^{2},\quad
B(T)=B_{3}{\left(\frac{T_{0}}{T}\right)}^{3}.$$ A precise fit of the parameters in these two effective potentials has been performed to reproduce some pure-gauge lattice QCD data in Refs. [@Ratti2006a; @Ratti2006b]. The results are listed in Table \[pol\_para\], Table \[imp\_para\], respectively. The parameter $T_{0}$ is the critical temperature for the deconfinement phase transition to take place in the pure-gauge QCD and $T_{0}$ is chosen to be $270\,\mathrm{MeV}$ according to the lattice calculations.
$a_{0}$ $a_{1}$ $a_{2}$ $a_{3}$ $b_{3}$ $b_{4}$
--------- --------- --------- --------- --------- ---------
6.75 $-1.95$ 2.625 $-7.44$ 0.75 7.5
: Parameters for the polynomial effective potential $\mathcal{U}_{\mathrm{pol}}$[]{data-label="pol_para"}
$A_{0}$ $A_{1}$ $A_{2}$ $B_{3}$
--------- --------- --------- ---------
3.51 $-2.47$ 15.2 $-1.75$
: Parameters for the improved effective potential $\mathcal{U}_{\mathrm{imp}}$[]{data-label="imp_para"}
In the NJL sector of the model, three parameters need to be determined: the three-momentum cutoff $\Lambda$, the current quark mass $m_{0}$, and the coupling strength $G$. In our work we employ the zero-temperature values of the quark condensate, pion decay constant and the mass of pion to fix the parameters. The obtained results are given in Table \[NJL\_para\].
$\Lambda\,(\mathrm{MeV})$ $G\,({\mathrm{GeV}}^{-2})$ $m_{0}\,(\mathrm{MeV})$ $|\langle\bar{\psi}_{u}\psi_{u}\rangle|^{1/3}\,(\mathrm{MeV})$ $f_{\pi}\,(\mathrm{MeV})$ $m_{\pi}\,(\mathrm{MeV})$
--------------------------- ---------------------------- ------------------------- ---------------------------------------------------------------- --------------------------- ---------------------------
659.28 4.773 5.32 250.0 92.4 139.3
: Parameters of the NJL sector of the model and the physical quantities being fitted []{data-label="NJL_para"}
Mesonic excitations at finite temperature
=========================================
Before we study the properties of mesonic excitations at finite temperature in the PNJL model, the gap equation whose solution provides the constituent mass of the quark should be given. As presented in Ref. [@Hansen2007], such gap equation in the Hartree approximation reads $$m=m_{0}+2GT\mathrm{Tr}\sum_{n=-\infty}^{+\infty}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{-1}{p\!\!\!\slash-m+\gamma^{0}(-iA_{4})},\label{gap}$$ where the imaginary time formalism is used and the temporal component of the four-momentum is discretized, i.e. $p_{0}=i\omega_{n}$ and $\omega_{n}=(2n+1)\pi T$ is the Matsubara frequency for a fermion; $m$ is the constituent mass of the quark; $\mathrm{Tr}$ is the trace which operates over Dirac, flavor, and color spaces. Here the three-momentum cut-off is employed. After a sum of the Matsubara frequencies, Eq. can be written as $$m=m_{0}+2GN_{f}\sum_{c=1}^{N_{c}}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{2m}{E_{p}}
\left\{1-f\left[E_{p}-(-i{A_{4}}_{cc})\right]-f\left[E_{p}+(-i{A_{4}}_{cc})\right]\right\},\label{gap2}$$ where $E_{p}=(p^{2}+m^{2})^{1/2}$ and the summation over the color index can be further written as $$\begin{aligned}
&&\sum_{c=1}^{N_{c}}f\left[E_{p}-(-i{A_{4}}_{cc})\right]\nonumber \\
& =&\sum_{c=1}^{N_{c}}\frac{1}{e^{\beta E_{p}}e^{i\beta{A_{4}}_{cc}}+1}\nonumber \\
&=&\left[\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{22}}+1\right)\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{33}}+1\right)+\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{11}}+1\right)\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{33}}+1\right)\right.\nonumber \\
&&\left.+\left(e^{\beta E_{p}}e^{i\beta{A_{4}}_{11}}\! + \!
1\right)\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{22}} \! + \! 1\right)\right] \left[\left(e^{\beta E_{p}}e^{i\beta{A_{4}}_{11}} \! + \!
1\right)\left(e^{\beta E_{p}}e^{i\beta{A_{4}}_{22}} \! + \!
1\right)\left(e^{\beta
E_{p}}e^{i\beta{A_{4}}_{33}} \! + \! 1\right)\right]^{-1}\nonumber\\
&=&N_{c}\frac{\bar{\Phi}e^{-\beta E_{p}}+2\Phi e^{-2\beta
E_{p}}+e^{-3\beta E_{p}}}{1+3\bar{\Phi}e^{-\beta E_{p}}+3\Phi
e^{-2\beta E_{p}}+e^{-3\beta
E_{p}}}=N_{c}f_{\Phi}^{+}(E_{p}),\label{distribution1}\end{aligned}$$ where the distribution function $f_{\Phi}^{+}(E_{p})$ in the PNJL model has been given in Ref. [@Hansen2007] with another method and we follow their notations. We can find that, when $\Phi=\bar{\Phi}=1$, $f_{\Phi}^{+}(E_{p})$ becomes the conventional Fermi-Dirac distribution function. In the same way, the summation of the last term in Eq. is $$\begin{aligned}
&&\sum_{c=1}^{N_{c}}f\left[E_{p}+(-i{A_{4}}_{cc})\right]\nonumber \\
& =&N_{c}\frac{\Phi e^{-\beta E_{p}}+2\bar{\Phi} e^{-2\beta
E_{p}}+e^{-3\beta E_{p}}}{1+3\Phi e^{-\beta E_{p}}+3\bar{\Phi}
e^{-2\beta E_{p}}+e^{-3\beta
E_{p}}}=N_{c}f_{\Phi}^{-}(E_{p}).\label{distribution2}\end{aligned}$$ Finally, the gap equation is given by $$m=m_{0}+2GN_{f}N_{c}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{2m}{E_{p}}
\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right].\label{gap3}$$ The gap equation in the PNJL model at finite temperature can also be simply derived from the gap equation at zero temperature, which is $$m=m_{0}+8GmN_{f}N_{c}iI_{1},\label{gap4}$$ where $$I_{1}=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}-m^{2}}.\label{I1}$$ To calculate the integral $I_{1}$ at finite temperature in the PNJL model, we just need to replace the integral in $p_{0}$ with $iT\sum_{n}\frac{1}{N_{c}}\sum_{c}$ with $p_{0}=
i\omega_{n}-i{A_{4}}_{cc}$, i.e. $$\begin{aligned}
I_{1}&=&iT\sum_{n=-\infty}^{+\infty}\frac{1}{N_{c}}\sum_{c=1}^{N_{c}}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{1}{(i\omega_{n}-i{A_{4}}_{cc})^{2}-{E_{p}}^{2}}\nonumber \\
&
=&-i\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{1}{2E_{p}}
\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right].\end{aligned}$$
We have shown that calculations at finite temperatures in the PNJL model can be simply derived from calculations at zero temperature above. Therefore, in the following we investigate the mesonic excitations at finite temperature in the PNJL model starting from those at zero temperature. We follow the formalism in Ref. [@Klevansky1992] and the $\pi$ and $\sigma$ mesons correspond to the pseudoscalar isovector modes and the scalar isoscalar mode, respectively. For the pseudoscalar modes, defining the operators $$\tau^{\pm}=\frac{1}{\sqrt{2}}(\tau_{1}\pm i\tau_{2}),$$ we can reexpress the four-fermion term in the pseudoscalar channel in the Lagrangian in Eq. as $$\left(\bar{\psi}i\gamma_{5}\vec{\tau}\psi\right)^{2}=2\left(\bar{\psi}i\gamma_{5}\tau^{+}\psi\right)
\left(\bar{\psi}i\gamma_{5}\tau^{-}\psi\right)+\left(\bar{\psi}i\gamma_{5}\tau_{3}\psi\right)
\left(\bar{\psi}i\gamma_{5}\tau_{3}\psi\right).$$ The effective interaction resulting from the exchange of a $\pi$ meson can be obtained as an infinite sum of loops in the random-phase approximation (RPA) [@Klevansky1992] and the leading order terms in $N_{c}$ is shown diagrammatically in Fig. \[f1\].
![Schematic representation of the effective interaction for the pseudoscalar modes in the RPA, where the double dashed line represents the effective propagator of $\pi$ mesons and the solid lines are quark lines; the black dots denote the effective coupling between $\pi$ meson and quarks. Here, $T_{i}=T_{j}=\tau_{3}$ for $\pi^{0}$, and $T_{i}=\tau^{\pm}$, $T_{j}=\tau^{\mp}$ for $\pi^{\pm}$.[]{data-label="f1"}](RPA.eps)
Using the symbols in Ref. [@Klevansky1992], the left hand side of the equation in Fig. \[f1\] can be denoted as $iU_{ij}(k^{2})$. Summing up all the terms on the right-hand side, we obtain $$iU_{ij}(k^{2})=i\gamma_{5}T_{i}\frac{2iG}{1-2G\Pi_{ps}(k^{2})}i\gamma_{5}T_{j}.\label{Uij}$$ Comparing Eq. with the equation in Fig. \[f1\], one can find that the mass of $\pi$ mesons is related to the pole of Eq. , which is the solution of the following equation [@Klevansky1992] $$1-2G\Pi_{ps}(k^{2})=0.\label{pion_equation}$$ Furthermore, the coupling strength between $\pi$ meson and quarks $g_{\pi qq}$ can be obtained as $$g_{\pi qq}^{2}=\left[\frac{\partial \Pi_{ps}(k^{2})}{\partial
k^{2}}\right]^{-1}\bigg|_{k^{2}=m_{\pi}^{2}}.\label{pion_coupling}$$ Therefore, the information of $\pi$ mesons is included in the pseudoscalar polarization $\Pi_{ps}(k^{2})$, which reads $$-i\Pi_{ps}(k^{2})=-\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\mathrm{Tr}\left[i\gamma_{5}T_{i}iS(k+p)i\gamma_{5}T_{j}iS(p)\right],\label{pion_polarization}$$ where $iS(p)=i/(p\!\!\!\slash-m)$ is the propagator of quarks. After calculating the trace in Eq. , one has $$\begin{aligned}
-i\Pi_{ps}(k^{2})&=&4N_{c}N_{f}\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\frac{1}{p^{2}-m^{2}}-2N_{c}N_{f}k^{2}\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\frac{1}{(p^{2}-m^{2})[(k+p)^{2}-m^{2}]}\nonumber\\
&=&4N_{c}N_{f}I_{1}-2N_{c}N_{f}k^{2}I(k),\label{pion_polarization2}\end{aligned}$$ where we have used the function $I_{1}$ given in Eq. and also defined the function $I(k)$ with the same symbols as used in Refs. [@Klevansky1992; @Schulze1995; @Quack1995], i.e. $$I(k)=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\frac{1}{(p^{2}-m^{2})[(k+p)^{2}-m^{2}]}.\label{Ik}$$ Furthermore, we introduce another two functions as done in Refs. [@Schulze1995; @Quack1995], which will be used in the following: $$K(k)=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\frac{1}{(p^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]},\label{Kk}$$ $$L(k)=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\frac{1}{(p^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]^{2}}.\label{Lk}$$ Then, substituting the expression of the pseudoscalar polarization in Eq. into Eq. , we have $$1-8GN_{c}N_{f}iI_{1}+4GN_{c}N_{f}k^{2}iI(k)=0.\label{pion_equation2}$$ Upon inserting the gap equation (in Eq. ) into the above equation, one obtains [@Klevansky1992] $$\frac{m_{0}}{m}+4GN_{c}N_{f}k^{2}iI(k)=0,\label{pion_equation2}$$ whose solution gives the mass of the pseudoscalar mode. The explicit expression for the coupling between $\pi$ meson and quarks can be easily obtained upon substituting Eq. into Eq. , and, in turn, it reads [@Schulze1995] $$g_{\pi
qq}^{2}=\frac{i}{N_{c}N_{f}}\frac{1}{I(m_{\pi})+I(0)-m_{\pi}^{2}K(m_{\pi})}.\label{pion_coupling2}$$
In the case of finite temperature in the PNJL model, we need to extend the function $I(k)$ in the same way as taken for the function $I_{1}$. When the three momentum is vanishing, i.e. $k=(\omega,\,0)$, $I(\omega,\,0)$ at finite temperature in the PNJL model is $$I(\omega,\,0)=-i\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{1}{E_{p}(\omega^{2}-4E_{p}^{2})}\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right].\label{Ik2}$$ Then Eq. at finite temperature can be rewritten as $$\frac{m_{0}}{m}+4GN_{c}N_{f}m_{\pi}^{2}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{1}{E_{p}(m_{\pi}^{2}-4E_{p}^{2})}\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right]=0.\label{pion_equation3}$$ Furthermore, we have $$K(\omega,\,0)=i\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{\omega^{2}-12E_{p}^{2}}{4E_{p}^{3}(\omega^{2}-4E_{p}^{2})^{2}}
\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right],\label{Kk2}$$ $$L(\omega,\,0)=i\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}
\frac{\omega^{2}-20E_{p}^{2}}{2E_{p}^{3}(\omega^{2}-4E_{p}^{2})^{3}}
\left[1-f_{\Phi}^{+}(E_{p})-f_{\Phi}^{-}(E_{p})\right].\label{Lk2}$$
In the same way, properties of $\sigma$ mesons can be extracted from the scalar polarization $\Pi_{s}(k^{2})$, which is $$\begin{aligned}
-i\Pi_{s}(k^{2})&=&-\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\mathrm{Tr}\left[iS(k+p)iS(p)\right]\nonumber\\
&=&4N_{c}N_{f}I_{1}-2N_{c}N_{f}(k^{2}-4m^{2})I(k).\label{sigma_polarization}\end{aligned}$$ Therefore, employing the RPA approximation for the scalar channel in the same procedure as for the effective interaction in the pseudoscalar channel, one could determine the mass of $\sigma$ meson which is the pole of its corresponding effective propagator, i.e. $1-2G\Pi_{s}(m_{\sigma}^{2})=0$, explicitly given by $$\frac{m_{0}}{m}+4GN_{c}N_{f}(m_{\sigma}^{2}-4m^{2})iI(m_{\sigma})=0.\label{sigma_equation}$$ The relation between the mass of $\sigma$ meson and that of $\pi$ meson could be obtained by comparing Eq. and Eq. as $$m_{\sigma}^{2}=4m^{2}+m_{\pi}^{2}\frac{I(m_{\pi})}{I(m_{\sigma})},\label{mass_relation}$$ which returns to the relation given in Ref. [@Klevansky1992] when the difference between $I(m_{\pi})$ and $I(m_{\sigma})$ is neglected.
In the following, we would investigate the pion-decay constant $f_{\pi}$ at finite temperature in the PNJL model with a starting of the definition of $f_{\pi}$ $$\langle 0 |
J^{i}_{5\mu}(x)|\pi^{j}\rangle=ik_{\mu}f_{\pi}\delta^{ij} .
\label{fpi_definition}$$ Considering the explicit expression of the left hand side in the PNJL model $$\langle 0 |
J^{i}_{5\mu}(x)|\pi^{j}\rangle=-\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}
\mathrm{Tr}\left[i\gamma_{\mu}\gamma_{5}\frac{\tau^{i}}{2}iS(k+p)ig_{\pi
qq}\gamma_{5}\tau^{j}iS(p)\right] ,$$ we arrive at $$f_{\pi}=-4iN_{c}g_{\pi qq}mI(m_{\pi}).\label{fpi}$$ Employing the expression of the pion-quark coupling in Eq. and considering the fact $N_{f} =2$ in the present case, one has [@Schulze1995] $$f_{\pi}^{2}=-8iN_{c}m^{2}\frac{I^{2}(m_{\pi})}{I(0)+I(m_{\pi})-m_{\pi}^{2}K(m_{\pi})},\label{fpi2}$$ and $$f_{\pi}^{2}g_{\pi
qq}^{2}=4m^{2}\frac{I^{2}(m_{\pi})}{\left[I(0)+I(m_{\pi})-m_{\pi}^{2}K(m_{\pi})\right]^{2}}\equiv
m^{2}r^{2},\label{GT}$$ where we have defined a symbol $r$ as $$r\equiv
\frac{2I(m_{\pi})}{I(0)+I(m_{\pi})-m_{\pi}^{2}K(m_{\pi})}.\label{r}$$ When the temperature is approaching zero, $I(m_{\pi}) \approx I(0)$, $K(m_{\pi}) \approx 0$, $r\rightarrow 1$. Eq. returns then to the quark level version of the Goldberger-Treiman relation at zero temperature [@Goldberger1958]. As we show below, when the temperature is near some critical temperature, $r$ deviates from $1$ evidently. Furthermore, from Eq. , one has the mass of $\pi$ meson as $$m_{\pi}^{2}=-\frac{m_{0}}{m}\frac{1}{4GN_{c}N_{f}iI(m_{\pi})}.$$ One may also combine this equation with Eq. . It gives consequently $$m_{\pi}^{2}f_{\pi}^{2}=\frac{m_{0}m}{G}\frac{I(m_{\pi})}{I(0)+I(m_{\pi})-m_{\pi}^{2}K(m_{\pi})}.\label{GM}$$ And the constituent mass of the quark $m$ is related with the condensate of quark by $$\begin{aligned}
m&=&-2GN_{f}\langle \bar{u}u\rangle+m_{0}\nonumber \\
&=&-2G\langle \bar{\psi}\psi\rangle+m_{0}.\label{condensate}\end{aligned}$$ Replacing the constituent mass in Eq. with the quark condensate in Eq. , we obtain $$m_{\pi}^{2}f_{\pi}^{2}=-m_{0}\langle \bar{\psi}\psi\rangle
r+\frac{m_{0}^{2}}{2G}r = -m_{0}\langle \bar{\psi}\psi\rangle r
\Big[ 1 + \frac{m_{0}}{2G \vert \langle \bar{\psi}\psi\rangle \vert
} \Big] \, .\label{GM2}$$ As mentioned above, at zero temperature $r\rightarrow 1$. Considering the lowest-order contribution in $m_{0}$, one obtains then $m_{\pi}^{2}f_{\pi}^{2}\simeq -m_{0}\langle
\bar{\psi}\psi\rangle$, which is the lowest-order approximation to the Gell-Mann–Oakes–Renner relation [@Gell1968].
Before our numerical calculations, we need to give the equations to determine the values of the Polyakov-loop $\Phi$ and its conjugate $\bar{\Phi}$. In the mean-field approximation or equivalently the Hartree approximation, the thermodynamical potential density for the Lagrangian density in Eq. is given as [@Hansen2007] $$\begin{aligned}
\Omega(\Phi,\bar{\Phi},m,T)&=&\frac{(m_{0}-m)^{2}}{4G}+\mathcal{U}(\Phi,\bar{\Phi},T)
-2N_{f}N_{c}\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}E_{p}\nonumber \\
&&-2N_{f}T\int_{\Lambda}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\left[
\ln\left(1+N_{c}\bar{\Phi}e^{-\beta E_{p}}+N_{c}\Phi e^{-2\beta
E_{p}}+e^{-3\beta E_{p}}\right)\right.\nonumber \\
&&\left.+\ln\left(1+N_{c}\Phi e^{-\beta E_{p}}+N_{c}\bar{\Phi}
e^{-2\beta E_{p}}+e^{-3\beta E_{p}}\right)\right].\label{potential}\end{aligned}$$ Minimizing this thermodynamical potential with respect to $\Phi$ and $\bar{\Phi}$, we obtain equations $$\frac{\partial \Omega}{\partial \Phi}=0,\qquad \frac{\partial
\Omega}{\partial \bar{\Phi}}=0.\label{Eq_Phi}$$ In the absence of chemical potential, these two equations are identical, and so $\Phi=\bar{\Phi}$ [@Ratti2006a]. In the same way, Minimizing the thermodynamical potential in Eq. with respect to the value of the constituent quark mass $m$, the gap equation in Eq. can also be obtained.
In the following, we present our numerical results. First of all, we give our calculated values for several characteristic temperatures. These characteristic temperatures include the pseudo-transition temperature for chiral crossover, $T_{\chi}$, corresponding to the maximum of $-\mathrm{d}m/\mathrm{d}T$ [@Ratti2006a; @Fu2008; @Hansen2007], the pseudo-transition temperature for deconfinement crossover, $T_{P}$, corresponding to the maximum of $\mathrm{d}\Phi/\mathrm{d}T$, the Mott temperature $T_{M}$ for $\pi$ meson, defined by $$m_{\pi}(T_{M})=2m(T_{M}),$$ meaning that the pion can dissociate into a constituent quark and an antiquark above the Mott temperature, and the dissociation temperature for $\sigma$ meson $T_{d}^{\sigma}$ [@Quack1995], defined by $$m_{\sigma}(T_{d}^{\sigma})=2m_{\pi}(T_{d}^{\sigma}).$$ These characteristic temperatures except for $T_{P}$, can also serve in the conventional NJL model. Numerical results for these characteristic temperatures in the PNJL model with two Polyakov-loop effective potentials are shown in Table \[temperatues\]. Here, for comparison we also list the results in the conventional NJL model.
$T_{\chi}\,(\mathrm{MeV})$ $T_{P}\,(\mathrm{MeV})$ $T_{M}\,(\mathrm{MeV})$ $T_{d}^{\sigma}\,(\mathrm{MeV})$
------------------------------------- ---------------------------- ------------------------- ------------------------- ----------------------------------
PNJL ($\mathcal{U}_{\mathrm{pol}}$) 253.2 245.4 264.6 253.0
PNJL ($\mathcal{U}_{\mathrm{imp}}$) 245.0 232.0 259.6 246.3
NJL 184.4 — 201.2 181.9
: Several critical temperatures in the conventional NJL model and the PNJL model with two Polyakov-loop effective potentials ($T_{0}=270\,\mathrm{MeV}$ is chosen for these two effective potentials).[]{data-label="temperatues"}
In Fig. \[f2\] we illustrate our calculated results of the masses of $\pi$ and $\sigma$ mesons, the mass of constituent quark, and the Polyakov-loop as functions of the temperature. Fig. \[f2\] shows evidently that at a temperature not very high, the masses of the constituent quark, the pion and the $\sigma$ mesons maintain the same as the corresponding one at zero temperature. As the temperature is around the critical one, these masses vary abruptly. And further, if the temperature is very high, the masses of $\pi$ and $\sigma$ mesons become degenerate, which indicates that the chiral symmetry is restored at high temperature. Such a feature is consistent with that given in the framework of Bethe-Salpeter equation combining with the Dyson-Schwinger equations (see for example Ref. [@Maris2001]). We also find that two different Polyakov-loop effective potentials do not result in qualitative differences but only slightly quantitative deviations as the left panel of Fig. \[f2\] shows. Furthermore, looking through the right panel of Fig. \[f2\], we can notice that the chiral phase transition occurs at relatively lower temperature in the conventional NJL model.
![\[f2\] Left panel: calculated masses of $\pi$ meson, $\sigma$ meson, constituent quark and the Polyakov-loop as functions of the temperature. Here, thick curves and thin curves correspond to the results with the polynomial Polyakov-loop effective potential $\mathcal{U}_{\mathrm{pol}}$, the improved effective potential $\mathcal{U}_{\mathrm{imp}}$, respectively. Right panel: masses of $\pi$, $\sigma$, and $m$ as functions of the temperature in the PNJL model with $\mathcal{U}_{\mathrm{imp}}$ (thick curves) and in the conventional NJL model (thin curves).](Mass.EPS)
In order to compare the obtained results in the PNJL model with those in the conventional NJL model more conveniently, we scale the temperature in unit of Mott temperature $T_{M}$ and re-display the results in Fig. \[f3\]. One can recognize that, in the PNJL model, only when the temperature is very near the phase transition temperature, masses of mesons and constituent quark begin to deviate from their values at zero temperature obviously. While in the conventional NJL model, these masses begin to deviate from their zero-temperature values at much lower temperature, about $0.4\,T_{M}$. This phenomenon can be attributed to the fact that, in the low temperature, chiral symmetry is broken and the quark and antiquark are in the confined hadronic phase in the PNJL model, contributions from thermal excitations of one and two quarks or antiquarks are suppressed as the distribution functions in Eq. and Eq. show when the Polyakov-loop $\Phi$ approaches zero. This is a manifestation of color confinement on the level of statistics. While in the conventional NJL model, due to the lack of the appearance of color confinement, contributions from one and two quarks or antiquarks become significant even at low temperature, which results in the phenomenon mentioned above.
![\[f3\] Calculated masses of $\pi$ meson, $\sigma$ meson, and constituent quark as functions of the temperature in unit of Mott temperature $T_{M}$ in the PNJL model with $\mathcal{U}_{\mathrm{imp}}$ (thick lines) and in the conventional NJL model (thin lines).](Mass-Tscaled.EPS)
In Fig. \[f4\] we show the square of the pion-quark coupling strength $g_{\pi qq}^{2}$ and the pion-decay constant $f_{\pi}$ as functions of the temperature in unit of Mott temperature in the PNJL model with polynomial and improved effective potentials and in the conventional NJL model. As Eq. shows, when temperature approaches the Mott temperature $T_{M}$ from below, i.e. when the mass of $\pi$ meson is about twice mass of the constituent quark, $iK(m_{\pi})\rightarrow \infty$. Therefore, the pion-quark coupling strength and pion-decay constant vanish at $T_{M}$, as Eq. and Eq. show. Furthermore, One can also find in Fig. \[f4\] that, in the PNJL model, $g_{\pi
qq}^{2}$ and $f_{\pi}$ almost keep invariant with the increase of the temperature when the temperature is not high and these two quantities decrease rapidly only when the temperature is above $0.8\,T_{M}$. While in the conventional NJL model these two quantities begin to decrease at about $0.4\,T_{M}$. This behavior is also due to the lack of the color confinement in the conventional NJL model as the same as the behavior of masses of mesons and constituent quark as functions of temperature shown in Fig. \[f3\].
![\[f4\] Left panel: calculated results of the square of the pion-quark coupling strength $g_{\pi qq}^{2}$ (in Eq. ) as a function of the temperature in unit of Mott temperature $T_{M}$ in the PNJL and the conventional NJL model. Right panel: calculated results of the pion-decay constant $f_{\pi}$ (in Eq. ) as a function of the temperature in unit of Mott temperature.](Fgpi-fpi.EPS)
We have shown above that the Goldberger-Treiman relation and Gell-Mann–Oakes-Renner relation at finite temperature are different from those at zero temperature in that a factor $r$ defined in Eq. is introduced. The calculated behavior of $r$ as function of temperature is displayed in Fig. \[f5\] and one can find that, when the temperature is below $0.9\,T_{M}$, $r$ is almost a constant very near $1$, which indicates that these two important relations at vacuum still serve well in the large region of temperature $0\sim 0.9\,T_{M}$. However, when the temperature is above $0.9\,T_{M}$, $r$ decreases very rapidly and vanishes at $T=T_{M}$. In the region of the temperature $0.9\,T_{M} \sim T_{M}$, the Goldberger-Treiman relation and Gell-Mann–Oakes-Renner relation at vacuum should be extended to Eq. , Eq. , respectively.
![\[f5\] Calculated factor $r$ defined in Eq. as a function of the temperature in unit of Mott temperature in the PNJL and the conventional NJL model.](Fr-T.EPS)
$\pi$–$\pi$ scattering lengths
==============================
The formalism of $s$-wave $\pi$–$\pi$ scattering lengths at zero temperature in the conventional NJL model has been established in Refs. [@Bernard1991; @Bernard1992; @Schulze1995], and it has been extended to the case at finite temperature by Quack et al. [@Quack1995]. In this work we follow the notation and calculation given in Ref. [@Schulze1995]. The invariant amplitude of $\pi$–$\pi$ scattering has the form: $$\langle\,
cp_{c};dp_{d}|i\mathcal{M}|ap_{a};bp_{b}\,\rangle=iA(s,t,u)\delta_{ab}\delta_{cd}
+iB(s,t,u)\delta_{ac}\delta_{bd}+iC(s,t,u)\delta_{ad}\delta_{bc},\label{total_am}$$ where $a$, $b$, $c$, and $d$ are the isospin labels, and $s$, $t$ and $u$ are the Mandelstam variables, $s=(p_{a}+p_{b})^{2}$, $t=(p_{a}-p_{c})^{2}$ and $u=(p_{a}-p_{d})^{2}$. The amplitude of definite total isospin $I$, defined by $A_{I}$, can be projected out, given by Ref. [@Schulze1995] $$A_{0}=3A+B+C, \quad A_{1}=B-C,\; \mathrm{and} \quad
A_{2}=B+C.\label{A_I}$$ When the scattering is at the kinematic threshold, we obtain the scattering lengths, i.e. $$a_{I}=\frac{1}{32\pi}A_{I}(s=4m_{\pi}^{2}, t=0, u=0).\label{length}$$ For simplicity, the pion momenta can be chosen as $$p_{a}=p_{b}=p_{c}=p_{d}=p,\; \mathrm{and}\quad
p^{2}=m_{\pi}^{2},\label{momenta}$$ which can be verified to fulfill the threshold condition in Eq. . To lowest order in $1/N_{c}$, there are two types of Feynman diagrams contributing to the $s$-wave $\pi$–$\pi$ scattering [@Bernard1991; @Schulze1995], i.e.the box diagram and the $\sigma$-propagation diagram. Here we also present them in Fig. \[f6\]. The three diagrams in the first row of Fig. \[f6\] are the box diagrams and the ones in the second row are the $\sigma$-propagation diagrams.
![Feynman diagrams contributing to the $s$-wave $\pi$–$\pi$ scattering (see also Ref. [@Bernard1991; @Schulze1995]). Here the external momenta for pions are chosen to be the special case in Eq. .[]{data-label="f6"}](FDpi-pi.eps)
Following the calculation of Ref. [@Schulze1995], we obtain respective amplitude for each diagram in Fig. \[f6\] as $$\begin{aligned}
i\mathcal{M}_{a}&=&(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})
(-4N_{c}N_{f}g_{\pi qq}^{4})[I(0)+I(p)-p^{2}K(p)]\nonumber \\
&\equiv&(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})iT_{a},\label{a}\\
i\mathcal{M}_{b}&=&(\delta_{ab}\delta_{cd}-\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})
(-4N_{c}N_{f}g_{\pi qq}^{4})[I(0)+I(p)-p^{2}K(p)]\nonumber \\
&\equiv&(\delta_{ab}\delta_{cd}-\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})iT_{b},\\
i\mathcal{M}_{c}&=&(-\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})
(-8N_{c}N_{f}g_{\pi qq}^{4})\left[I(0)+\frac{p^{4}}{2}L(p)-2p^{2}K(p)\right]\nonumber \\
&\equiv&(-\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})iT_{c},\end{aligned}$$ $$\begin{aligned}
i\mathcal{M}_{d}&=&\delta_{ab}\delta_{cd}(8N_{c}N_{f}g_{\pi
qq}^{4})\frac{I^{2}(p)}{\left(1-\frac{p^{2}}{m^{2}}\right)I(2p)+\frac{m_{\pi}^{2}}{4m^{2}}I(m_{\pi})}\nonumber \\
&\equiv&\delta_{ab}\delta_{cd}iT_{d},\\
i\mathcal{M}_{e}&=&\delta_{ac}\delta_{bd}(8N_{c}N_{f}g_{\pi
qq}^{4})\frac{[I(0)-p^{2}K(p)]^{2}}{I(0)+\frac{m_{\pi}^{2}}{4m^{2}}I(m_{\pi})}\nonumber \\
&\equiv&\delta_{ac}\delta_{bd}iT_{e},\\
i\mathcal{M}_{f}&=&\delta_{ad}\delta_{bc}(8N_{c}N_{f}g_{\pi
qq}^{4})\frac{[I(0)-p^{2}K(p)]^{2}}{I(0)+\frac{m_{\pi}^{2}}{4m^{2}}I(m_{\pi})}\nonumber \\
&\equiv&\delta_{ad}\delta_{bc}iT_{f}.\label{f}\end{aligned}$$ From above equations, one can notice $T_{b}=T_{a}$, and $T_{f}=T_{e}$. Substituting Eqs. — into Eq. , one obtains $$A=2T_{a}-T_{c}+T_{d}, \qquad B=C=T_{c}+T_{e}.$$ Therefore, employing Eq. we have $$\begin{aligned}
A_{0}&=&6T_{a}-T_{c}+3T_{d}+2T_{e},\nonumber \\
A_{1}&=&0,\nonumber \\
A_{2}&=&2(T_{c}+T_{e}).\label{A_I2}\end{aligned}$$
![Calculated scattering amplitudes $T_{a}$, $T_{c}$, $T_{d}$, and $T_{e}$ as functions of the temperature in unit of Mott temperature in the PNJL and the conventional NJL models.[]{data-label="f7"}](Length-T.EPS)
In Fig. \[f7\] we present our calculated results of the scattering amplitudes $T_{a}$, $T_{c}$, $T_{d}$, and $T_{e}$ as functions of the temperature in unit of $T_{M}$ in the PNJL model with polynomial Polyakov-loop effective potential and improved effective potential and in the conventional NJL model. The results of the conventional NJL model in our work are roughly consistent with those given in Ref. [@Quack1995]. However, there exists a difference which reads that our present calculation indicates that the scattering amplitude $T_{a}$ approaches zero at the Mott temperature $T_{M}$, the calculation in Ref. [@Quack1995] gives that $T_{a}$ is divergent at $T_{M}$. Recalling the analysis above, we would emphasized that, when the temperature approaches to $T_{M}$, $iK(m_{\pi})$ in Eq. and $iL(m_{\pi})$ in Eq. are divergent and the degree of divergence of $iL(m_{\pi})$ is higher than that of $iK(m_{\pi})$. Substituting the expression of the coupling between $\pi$ meson and quarks $g_{\pi
qq}^{2}$ in Eq. into Eq. , we find $T_{a}\propto 1/(-iK(m_{\pi}))$ when the temperature is near the Mott temperature, therefore, $T_{a}$ approaches zero at $T_{M}$. In the same way, we find that $T_{d}$ approaches zero, $T_{c}$ is divergent, and $T_{e}$ approaches a finite value at the Mott temperature. Furthermore, when the temperature is equal to the dissociation temperature of $\sigma$ meson, i.e. $T=T_{d}^{\sigma}$, we have $m_{\sigma}=2m_{\pi}$, which results in that the $\sigma$ propagator in diagram d of Fig. \[f6\] and also the amplitude $T_{d}$ (see Fig. \[f7\]) are divergent at $T_{d}^{\sigma}$. Comparing the results of the PNJL model with those of the conventional NJL model, one can also recognize the similar behavior as obtained previously, which reads that the $T$-matrix amplitudes calculated in the PNJL model deviate from their values at zero temperature only when the temperature is near the critical temperature, while the deviation occurs much earlier in the conventional NJL model. Taking amplitude $T_{d}$ for example, since the mass of $\sigma$ meson decreases with the increase of the temperature much earlier in the conventional NJL model than that in the PNJL model, as Fig. \[f3\] shows, we expect that the divergence in $T_{d}$ also occurs earlier in the conventional NJL model, which is verified in Fig. \[f7\]. In addition, we find that the value of $T_{d}^{\sigma}/T_{M}$ calculated in the conventional NJL model is about $0.90$, smaller than $0.96$ in the PNJL model with polynomial effective potential, and $0.95$ in the PNJL model with improved effective potential.
![Calculated $s$-wave $\pi$–$\pi$ scattering lengths $a_{0}$ and $a_{2}$ as functions of the temperature in unit of Mott temperature in the PNJL and the conventional NJL models.[]{data-label="f8"}](Length-a.EPS)
In Fig. \[f8\], we show the calculated results of the $s$-wave $\pi$–$\pi$ scattering lengths $a_{0}$ and $a_{2}$ as functions of the temperature in unit of $T_{M}$ in the PNJL model with two Polyakov-loop effective potentials and in the conventional NJL model. Since $a_{0}$ and $a_{2}$ contain contribution from the $T$-matrix amplitude $T_{c}$ as Eq. shows, they are both divergent at $T=T_{M}$. Furthermore, $a_{0}$ also contains $T_{d}$, so it diverges at $T=T_{d}^{\sigma}$ as well. At zero temperature, we have $a_{0}=0.173$ and $a_{2}=-0.045$, which are consistent with the Weinberg values $(a_{0})^{W}=7m_{\pi}^{2}/(32\pi
f_{\pi}^{2})=0.158$ and $(a_{2})^{W}=-2m_{\pi}^{2}/(32\pi
f_{\pi}^{2})=-0.045$ [@Quack1995; @Bernard1991]. On the experimental side, the Geneva-Saclay collaboration provided the often quoted values $a_{0}=0.26\pm 0.05$ and $a_{2}=-0.028\pm
0.012$ [@Froggatt1977; @Nagels1979], and recent years experiment E865 at Brookhaven National Laboratory, USA, has given new values $a_{0}=0.203\pm 0.033\pm 0.004$ and $a_{2}=-0.055\pm 0.023 \pm
0.003$ [@Truol2000], and also $a_{0}=0.216\pm 0.013\pm 0.004\pm
0.005$ [@Pislak2001]. The scattering lengths $a_{0}$ and $a_{2}$ are almost independent of the temperature until the temperature is increased to $0.9T_{M}$ in the PNJL model, and after that they vary drastically with temperature. While in the conventional NJL model $a_{0}$ and $a_{2}$ begin to vary with temperature at about $0.6T_{M}$ and the temperature at which $a_{0}$ diverges due to the $\sigma$-meson dissociation is also lower in the conventional NJL model. From the above analysis, one can recognize that the physical meaning of the divergence of the $s$-wave $\pi$–$\pi$ scattering lengths $a_{0}$ and $a_{2}$ at the Mott temperature $T_{M}$ for $\pi$ meson is clear and consistent with that shown in Ref. [@Quack1995]. At $T=T_{M}$, $\pi$ meson can dissociate into a constituent quark and an antiquark, the $\pi$ meson is then unbound and its radius becomes infinite. Mathematically, the geometrical size of the pion meson can be described by its charge radius $r$ which is related to $f_{\pi}$ through $f_{\pi}^{2}\propto 1/\langle
r^{2}\rangle$ [@Quack1995]. Employing the Weinberg relations cited above, we have the relation between the scattering lengths and the charge radius of $\pi$ meson as $|a|\propto \langle
r^{2}\rangle$, which clearly indicates that the divergence of $\pi$–$\pi$ scattering lengths at the pion Mott temperature is closely related with the melting of the pion meson. The relation between the divergence of the $s$-wave $\pi$–$\pi$ scattering lengths and the delocalization of the pion meson at $T_{M}$ is then confirmed not only in the conventional NJL model but also in the PNJL model. As for the divergence of the $s$-wave $\pi$–$\pi$ scattering length $a_{0}$ at the dissociation temperature for $\sigma$ meson $T_{d}^{\sigma}$, we should note that this divergence corresponds to the situation that the propagator for $\sigma$ meson displayed in the part d of Fig. \[f6\] is on shell, which means that the $\pi$–$\pi$ scattering in the s channel couples resonantly with the $\sigma$ meson field. The divergence (from positive infinite to negative infinite) of the $a_{0}$ and the mass relation $m_{\sigma} = 2 m_{\pi}$ suggest that, in general point of view, a very loosely bound state may appear at the dissociation temperature $T_{d}^{\sigma}$. However, detailed investigation is required to clarify its mechanism.
Summary
=======
In summary, we have studied the mesonic excitations at finite temperature in the two flavor PNJL model. The masses of $\pi$ meson and $\sigma$ meson, pion-decay constant, and the pion-quark coupling strength at finite temperature are calculated in the PNJL model with two forms of Polyakov-loop effective potential. Their variation behaviors with temperature, especially when the temperature takes a value near the critical one, are investigated in details. We find that the results calculated in the PNJL model are almost independent of the choice of the Polyakov-loop effective potential. We also compare our calculated results in the PNJL model with those in the conventional NJL model. We find that, since in the PNJL model, the Polyakov-loop which is coupled with quarks suppresses the unwanted degrees of freedom below the critical temperature, all quantities describing the properties of mesons deviate from their values at zero temperature only when the temperature is very near the critical temperature, for example the Mott temperature, and they vary with temperature rapidly in a very narrow region near the critical one. While in the conventional NJL model, these quantities begin to vary with temperature much earlier. Therefore, we conclude that the effect of color confinement screens the effect of temperature below the critical temperature. Furthermore, we have investigated the Goldberger-Treiman relation and the Gell-Mann–Oakes–Renner relation at finite temperature in the PNJL model, and we find that when the temperature is below about $0.9T_{M}$, where $T_{M}$ is the Mott temperature for $\pi$ meson, these two important relations are hardly changed by the effect of temperature. However, the Goldberger-Treiman relation and the Gell-Mann–Oakes–Renner relation should be corrected once the temperature is in the region of $0.9T_{M}\sim T_{M}$.
In this work, we have also investigated the $s$-wave $\pi$–$\pi$ scattering lengths at finite temperature in the PNJL model. The obtained results in the PNJL model are also compared with those in the conventional NJL model. We find that scattering length $a_{0}$ is divergent at Mott temperature for $\pi$ meson, $T_{M}$, and at dissociation temperature for $\sigma$ meson, $T_{d}^{\sigma}$, and scattering length $a_{2}$ is divergent at $T_{M}$, which are consistent with the results in the conventional NJL model. Due to the effect of color confinement, the dissociation temperature for $\sigma$ meson $T_{d}^{\sigma}$ calculated in unit of $T_{M}$ in the PNJL model is relatively larger than that given in the conventional NJL model. In the same way, the influence of the temperature on the scattering lengths $a_{0}$ and $a_{2}$ below the critical temperature is suppressed by the color confinement in the PNJL model. In addition, the characteristic of the scattering amplitude $T_{a}$ at the Mott temperature calculated in the PNJL model is different from that given previously in the conventional NJL model.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the National Natural Science Foundation of China under contract No. 10425521 and No. 10675007, and the Major State Basic Research Development Program under contract No. G2007CB815000. Helpful discussions with Dr. Craig D. Roberts at Argonne National Laboratory, USA, and Prof. Chuan Liu, Prof. Han-qing Zheng and Dr. Lei Chang are acknowledged with great thanks.
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[^1]: Corresponding author, e-mail address: yxliu@pku.edu.cn
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---
author:
- 'Guihua Gong, Chunlan Jiang, Liangqing Li and Cornel Pasnicu'
title: '**A reduction theorem for $AH$ algebras with the ideal property**'
---
[**Abstract**]{}
Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{\phi_{1,2}}A_{2}\xrightarrow{\phi_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ with $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that $\sup_{n,i}dim(X_{n,i})<+\infty$. In this article, we prove that $A$ can be written as the inductive limit of $$B_{1}\longrightarrow B_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$ where $B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ are $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$ (all of them are connected simplicial complexes of dimension at most three), $s_{n}$ and $\{n,i\}$ are positive integers and $Q_{n,i}\in M_{\{n,i\}}(C(Y_{n,i}))$ are projections. This theorem unifies and generalizes the reduction theorem for real rank zero $AH$ algebras due to Dadarlat and Gong (\[D\], \[G3\] and \[DG\]) and the reduction theorem for simple $AH$ algebras due to Gong (see \[G4\]).
*Keywords*: $C^*$-algebra, AH algebra, ideal property, Elliott intertwining, Reduction theorem\
*AMS subject classification*: Primary: 46L05, 46L35.
**§1. Introduction**
Successful classification results have been obtained for real rank zero $AH$ algebras (see \[Ell1\], \[Lin1-3\] \[EG1-2\], \[EGLP\], \[D\], \[G1-3\], \[DG\]) and simple $AH$ algebras (see \[Ell2-3\], \[Li1-3\], \[G4\], \[EGL1-2\]) in the case of no dimension growth (this condition can be relaxed to a certain slow dimension growth condition). To unify these two classification theorems, we will consider $AH$ algebras with the ideal property (see \[Ji-Jiang\] and \[GJLP\]). This article is a continuation of the paper \[GJLP\]—we obtain the reduction theorem for arbitrary $AH$ algebras $A$ (of no dimension growth) with the ideal property. That is, we remove the restriction that $K_{*}(A)$ is torsion free in the paper \[GJLP\]. Since we do not assume that $K_{*}(A)$ is torsion free, we must involve higher dimensional spaces such as $T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k}$, and $S^2$ in our reduction theorem (see below). This makes the main result of this paper much more difficult to prove than the one in \[GJLP\], as $C(X)$ is not stably generated when $\dim(X)\geq 2$.
The ideal property is a property of structural interest for a $C^*$-algebra. Many interesting and important $C^*$-algebras have the ideal property. It was proved by Cuntz-Echterhoff-Li that semigroup $C^*$-algebras of $ax + b$-semigroups over Dedekind domains have the ideal property (\[CEL\]). A generalization of this result appeared in \[L\]. Interesting examples of crossed product $C^*$-algebras with the ideal property could be found, e.g., in \[Pa-Ph1\] and \[Pa-Ph2\]. Other important results involving the ideal property have been proved, e.g., in \[Pa-R1-2\], \[Pa-Ph1\] and \[Pa-Ph2\]. Many $C^*$-algebras coming from $\mathbb{Z}$ dynamical systems on compact metric spaces are $AH$ algebras (see \[EE\], \[Phi1-2\], \[Lin4\] and \[LinP\]).
An $AH$ algebra is a nuclear $C^{*}$-algebra of the form $A=\lim\limits_{\longrightarrow}(A_{n}, \phi_{n,m})$ with $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}C(X_{n,i})P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_{n}$, $[n,i]$ are positive integers, $M_{[n,i]}(C(X_{n,i}))$ are algebras of $[n,i]\times[n,i]$ matrices with entries in $C(X_{n,i})$—the algebra of complex - valued continuos functions on $X_{n,i}$—, and finally $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections (see \[Bla\]). Let $T_{\uppercase\expandafter{\romannumeral2}, k}$ (and $T_{\uppercase\expandafter{\romannumeral3}, k}$) be a connected finite simplicial complex with $H^{1}(T_{\uppercase\expandafter{\romannumeral2}, k})=0$ and $H^{2}(T_{\uppercase\expandafter{\romannumeral2}, k})=\mathbb{Z}/k\mathbb{Z}$ (and $H^{1}(T_{\uppercase\expandafter{\romannumeral3}, k})=0=H^{2}(T_{\uppercase\expandafter{\romannumeral3}, k})$ and $H^{3}(T_{\uppercase\expandafter{\romannumeral3}, k})=\mathbb{Z}/k\mathbb{Z}$, respectively). Recall that the unit circle is denoted by $S^{1}$ and the 2-dimensional unit sphere is denoted by $S^{2}$. In this article, we will prove that an $AH$ algebra with ideal property and no dimension growth can be rewritten as an $AH$ inductive limit with the spaces $X_{n,i}$ being $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$. (For the background information, we refer the readers to \[GJLP\].) The result in this paper plays an essential role in the classification of the $AH$ algebras with the ideal property and with no dimension growth (see \[GJL\]).
**§2. Notation and prelimilary**
The following notations are quoted from \[G4\] or \[GJLP\].
**2.1.** If $A$ and $B$ are two $C^{\ast}$-algebras, we use $Map(A,B)$ to denote **the space of all linear, completely positive $\ast$-contractions** from $A$ to $B$. If both $A$ and $B$ are unital, then $Map(A,B)_{1}$ will denote the subset of $Map(A,B)$ consisting of unital maps. By word “map”, we shall mean a linear, completely positive $\ast$-contraction between $C^{\ast}$-algebras, or else we shall mean a continuous map between topological spaces, which one will be clear from the context.
**By a homomorphism between $C^{\ast}$-algebras, will be meant a $\ast$-homomorphism.** Let $Hom(A,B)$ denote the **space of all the homomorphisms** from $A$ to $B$. Similarly, if both $A$ and $B$ are unital, let $Hom(A,B)_{1}$ denote the subset of $Hom(A,B)$ consisting of all the untial homomorphisms.
**Definition 2.2.** Let $G\subset A$ be a finite set and $\delta>0$. We shall say that $\phi\in Map(A,B)$ is $G-\delta$ **multiplicative** if $$\parallel\phi(ab)-\phi(a)\phi(b)\parallel<\delta$$ for all $a,b\in G$.
**2.3.** In the notation for an inductive system $(A_{n},\phi_{n,m})$, we understand that $\phi_{n,m}=\phi_{m-1,m}\circ\phi_{m-2,m-1}\circ\cdots\circ\phi_{n,n+1},$ where all $\phi_{n,m}:A_{n}\rightarrow A_{m}$ are homomorphisms.
We shall assume that, for any summand $A^{i} _{n}$ in the direct sum $A_{n}=\bigoplus^{t_{n}}_{i=1}A^{i} _{n}$, necessarily $\phi_{n,n+1}(\textbf{1}_{A^{i} _{n}})\neq0$, since, otherwise, we could simply delete $A^{i} _{n}$ from $A_{n}$ without changing the limit algebra.
**2.4.** If $A_{n}=\bigoplus_{i}A^{i} _{n}$ and $A_{m}=\bigoplus_{j}A^{j} _{m}$, we use $\phi^{i,j}_{n,m}$ to denote the partial map of $\phi_{n,m}$ from the $i$-th block $A^{i} _{n}$ of $A_{n}$ to the $j$-th block $A^{j} _{m}$ of $A_{m}$.
**2.5.** By 2.3 of \[Bla\] and Theorem 2.1 of \[EGL2\], we know that any $AH$ algebra can be written as an inductive limit $A=lim(A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}C(X_{n,i})P_{n,i}, \phi_{n,m})$, where $X_{n,i}$ are finite simplicial complexes and $\phi_{n,m}$ are injective. In this article we will assume that $X_{n,i}$ are (path) connected finite simplicial complexes and $\phi_{n,m}$ are injective.
It is well known that, for any connected finite simplicial complex $X$, there is a metric $d$ on $X$ with the following property: for any $x\in X$ and $\eta>0$, the $\eta$-ball centered in $x$, $B_{\eta}(x)=\{x^{\prime}\in X|~ d(x^{\prime},x)<\eta\}$ is path connected. So in this article, we will always assume that the metric on a connected simplicial complex has this property.
**2.6.** In this article, we assume that the inductive limit system satisfies the no dimension growth condition: there is an $M\in \mathbb{N}$ such that for any $n,i,$ $$dim(X_{n,i})\leqslant M.$$
The condition could be relaxed to a so called very slow dimension growth for our main theorem. Since the proof of this slightly more general case is quite tedious, we will leave it to a subsequent paper.
**2.7.** By 1.3.3 of \[G4\], any $AH$ algebra $$A=lim(A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, \phi_{n,m})$$ is isomorphic to a limit corner subalgebra (see Definition 1.3.2 of \[G4\] for this concept) of $\tilde{A}=\lim\limits_{\longrightarrow}(\tilde{A}_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \tilde{\phi}_{n,m})$—an inductive limit of full matrix algebras over $X_{n,i}$. Once we prove that $\tilde{A}$ is an inductive limit of homogeneous algebras over the spaces in the following list: $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$, then $A$ itself is also an inductive limit of such kind. Therefore in this article, we will assume $A$ itself is an inductive limit of finite direct sum of full matrix algebras over $X_{n,i}$. That is $P_{n,i}=\textbf{1}_{M_{[n,i]}(C(X_{n,i}))}$.
**2.8.** Let $Y$ be a compact metrizable space. Let $P\in M_{k_{1}}(C(Y))$ be a projection with rank($P$)=$k\leqslant k_{1}$. For each $y$, there is a unitary $u_{y}\in M_{k_{1}}(\mathbb{C})$ (depending on $y$) such that $$P(y)=u_{y}\left(
\begin{array}{cccccc}
1 & \ & \ & \ & \ &\ \\
\ & \ddots & \ & \ & \ &\ \\
\ & \ & 1 & \ & \ &\ \\
\ & \ & \ & 0 & \ &\ \\
\ & \ & \ & \ & \ddots \ & \ \\
\ & \ & \ & \ & \ & 0 \ \\
\end{array}
\right)u^{*}_{y},$$ where there are $k$ $1^{'}s$ on the diagonal. If the unitary $u_{y}$ can be chosen to be continuous in $y$, then $P$ is called a **trivial projection**.
It is well known that any projection $P\in M_{k_{1}}(C(Y))$ is locally trivial. That is, for any $y_{0}\in Y$, there is an open set $U_{y_{0}}\ni y_{0}$, and there is a continuous unitary-valued function $$u: U_{y_{0}}\longrightarrow M_{k_{1}}(\mathbb{C})$$ such that the above equation holds for $u(y)$ (in place of $u_{y}$) for any $y\in U_{y_{0}}$.
If $P$ is trivial, then $P M_{k_{1}}(C(X))P\cong M_{k}(C(X))$.
**2.9.** Let $X$ be a compact metrizable space and $\psi: C(X)\longrightarrow P M_{k_{1}}(C(Y))P$ be a unital homomorphism. For any given point $y\in Y$, there are points $$x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\in X,$$ and a unitary $U_{y}\in M_{k_{1}}(\mathbb{C})$ such that $$\psi(f)(y)=P(y)U_{y}\left(
\begin{array}{cccccc}
f(x_{1}(y)) & \ & \ & \ & \ &\ \\
\ & \ddots & \ & \ & \ &\ \\
\ & \ & f(x_{k}(y)) & \ & \ &\ \\
\ & \ & \ & 0 & \ &\ \\
\ & \ & \ & \ & \ddots \ & \ \\
\ & \ & \ & \ & \ & 0 \ \\
\end{array}
\right)U^{*}_{y}P(y)\in P(y)M_{k_{1}}(\mathbb{C})P(y),$$ for all $f\in C(X)$. Equivalently, there are $k$ rank one orthogonal projections $p_{1},p_{2},\cdots,p_{k}$ with $\sum^{k}_{i=1}p_{i}(y)=P(y)$ and $x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\in X$, such that $$\psi(f)(y)=\sum\limits^{k}\limits_{i=1}f(x_{i}(y))p_{i}(y),~~~~\forall f\in C(X).$$
Let us denote the set $\{x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\}$, counting multiplicities, by SP$\psi_{y}$ (see \[Pa1\]). In other words, if a point is repeated in the diagonal of the above matrix, it is included with the same multiplicity in SP$\psi_{y}$. **We shall call SP$\psi_{y}$ the spectrum of $\psi$ at the point** $y$. Let us define the **spectrum** of $\psi$, denoted by SP$\psi$, to be the closed subset $$SP\psi:= \overline{\bigcup\limits_{y\in Y}SP\psi_{y}}(=\bigcup\limits_{y\in Y}SP\psi_{y})\subset X.$$ Alternatively, SP$\psi$ is the complement of the specturm of the kernel of $\psi$, considered as a closed ideal of $C(X)$. The map $\psi$ can be factored as $$C(X)\xrightarrow{i^{\ast}}C(SP\psi)\xrightarrow{\psi_{1}}PM_{k_{1}}(C(Y))P$$ with $\psi_{1}$ an injective homomorphism, where $i$ denotes the inclusion $SP\psi\hookrightarrow X.$
Also, if $A=PM_{k_{1}}(C(Y))P$, then we shall call the space $Y$ the spectrum of the algebra $A$, and write $SP A=Y(=SP(id))$.
**2.10.** In 2.9, if we group together all the repeated points in $\{x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\}$, and sum their corresponding projections, we can write $$\psi(f)(y)=\sum\limits^{l}\limits_{i=1}f(\lambda_{i}(y))P_{i},~~~~(l\leqslant k),$$ where $\{\lambda_{1}(y),\lambda_{2}(y),\cdots,\lambda_{l}(y)\}$ is equal to $\{x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\}$ as a set, but $\lambda_{i}(y)\neq \lambda_{j}(y)$ if $i\neq j$; and each $P_{i}$ is the sum of the projections corresponding to $\lambda_{i}(y)$. If $\lambda_{i}(y)$ has multiplicity $m$ (i.e., it appears $m$ times in $\{x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\}$), then rank($P_{i}$)=$m$.
**2.11.** Set $P^{k}(X)=\underbrace{X\times X\times\cdots\times X}\limits_{k}/\thicksim$, where the equivalence relation $\thicksim$ is defined by $(x_{1},x_{2},\cdots,x_{k})\thicksim(x^{\prime}_{1},x^{\prime}_{2},\cdots,x^{\prime}_{k})$ if there is a permutation $\sigma$ of $\{1,2,\cdots,k\}$ such that $x_{i}=x^{\prime}_{\sigma(i)}$, for each $1\leq i\leq k$. A metric $d$ on $X$ can be extended to a metric on $P^{k}(X)$ by $$d([x_{1},x_{2},\cdots,x_{k}],[x^{\prime}_{1},x^{\prime}_{2},\cdots,x^{\prime}_{k}])=\min\limits_{\sigma}\max\limits_{1\leq i\leq k}d(x_{i},x^{\prime}_{\sigma(i)}),$$ where $\sigma$ is taken from the set of all permutations, and $[x_{1},x_{2},\cdots,x_{k}]$ denotes the equivalence class in $P^{k}(X)$ of $(x_{1},x_{2},\cdots,x_{k})$. Two k-tuples of (possible repeating) points $\{x_{1},x_{2},\cdots,x_{k}\}\subset X$ and $\{x^{\prime}_{1},x^{\prime}_{2},\cdots,x^{\prime}_{k}\}\subset X$ are said to be paired within $\eta$ if $$d([x_{1},x_{2},\cdots,x_{k}],[x^{\prime}_{1},x^{\prime}_{2},\cdots,x^{\prime}_{k}])<\eta$$ when one regards $(x_{1},x_{2},\cdots,x_{k})$ and $(x^{\prime}_{1},x^{\prime}_{2},\cdots,x^{\prime}_{k})$ as two points in $P^{k}(X)$.
**2.12.** Let $\psi: C(X)\longrightarrow P M_{k_{1}}(C(Y))P$ be a unital homomorphism as in 2.9. Then $$\psi^{\ast}: y\mapsto SP \psi_{y}$$ defines a map $Y\longrightarrow P^{k}(X)$, if one regards $SP \psi_{y}$ as an element of $P^{k}(X)$. This map is continuous. In terms of this map and the metric $d$, let us define the **spectral variation** of $\psi$:\
$$SPV(\psi):= the\; diamerter\; of\; image\; of\; \psi^{\ast}.$$
**Definition 2.13** We shall call $P_{i}$ in 2.10 the spectral projection of $\phi$ at $y$ with respect to the spectral element $\lambda_{i}(y)$. For a subset $X_{1}\subset X$, we shall call $$\sum\limits_{\lambda_{i}(y)\in X_{1}}P_{i}$$ the spectral projection of $\phi$ at $y$ corresponding to the subset $X_{1}$ (or with respect to the subset $X_{1}$).\
**2.14.** Let $\phi: M_{k}(C(X))\longrightarrow P M_{l}(C(Y))P$ be a unital homomorphism. Set $\phi(e_{11})=p$, where $e_{11}$ is the canonical matrix unit corresponding to the upper left corner. Set $$\phi_{1}=\phi\mid_{e_{11}M_{k}(C(X))e_{11}}: C(X)\longrightarrow p M_{l}(C(Y))p.$$ Then $P M_{l}(C(Y))P$ can be identified with $p M_{l}(C(Y))p\otimes M_{k}$ in such a way that $$\phi=\phi_{1}\otimes id_{k}.$$ Let us define $$SP \phi_{y}:=SP(\phi_{1})_{y},$$ $$SP \phi:=SP\phi_{1},$$ $$SPV(\phi):=SPV(\phi_{1}).$$
Suppose that $X$ and $Y$ are connected. Let $Q$ be a projection in $M_{k}(C(X))$ and $\phi:QM_{k}(C(X))Q\longrightarrow PM_{l}(C(Y))P$ be a unital map. By the Dilation lemma(Lemma 2.13 of \[EG2\]), there are an $n$, a projection $P_{1}\in M_{n}(C(Y))$, and a unital homomorphism $$\widetilde{\phi}:M_{k}(C(X))\longrightarrow P_{1}M_{n}(C(Y))P_{1}$$ such that $$\phi=\widetilde{\phi}\mid_{QM_{k}(C(X))Q}.$$ (Note that this implies that $P$ is a subprojection of $P_{1}.$) We define: $$SP \phi_{y}:=SP\widetilde{\phi}_{y},$$ $$SP \phi:=SP\widetilde{\phi},$$ $$SPV(\phi):=SPV(\widetilde{\phi}).$$ (Note that these definitions do not depend on the choice of the dilation $\widetilde{\phi}$.)
**2.15.** Let $\phi: M_{k}(C(X))\longrightarrow P M_{l}(C(Y))P$ be a (not necessarily unital) homomorphism, where $X$ and $Y$ are connected finite simplicial complexes. Then $$\#(SP \phi_{y})=\frac{rank\phi(\textbf{1}_{k})}{rank(\textbf{1}_{k})},~~~for\; any\; y\in Y,$$ where again $\#(\cdot)$ denotes the number of elements in the set counting multiplicity. It is also true that for any nonzero projection $p\in M_{k}(C(X))$, $\#(SP \phi_{y})=\frac{rank\phi(p)}{rank(p)}$.
**2.16.** Let $X$ be a compact connected space and let $Q$ be a projection of rank $n$ in $M_{N}(C(X))$. The **weak variation of a finite set** $F\subset QM_{N}(C(X))Q$ is defined by $$\omega(F)=\sup\limits_{\Pi_{1},\Pi_{2}}\inf\limits_{u\in U(n)}\max\limits_{a\in F}\parallel u\Pi_{1}(a)u^{\ast}-\Pi_{2}(a)\parallel,$$ where $\Pi_{1},\Pi_{2}$ run through the set of irreducible representations of $QM_{N}(C(X))Q$ into $M_{n}(\mathbb{C})$.
Let $X_{i}$ be compact connected spaces and $Q_{i}\in M_{n_{i}}(C(X_{i}))$ be projections. For a finite set $F\subset\bigoplus_{i}Q_{i}M_{n_i}(C(X_{i}))Q_{i}$, define the **weak variation** $\omega(F)$ to be $\max_{i} \omega(\pi_{i}(F))$, where $\pi_{i}: \bigoplus_{j}Q_{j}M_{n_{j}}(C(X_{j}))Q_{j}\longrightarrow Q_{i}M_{n_{i}}(C(X_{i}))Q_{i}$ is the natural projection map onto the $i$-th block.
The set $F$ is said to be **weakly approximately constant to within** $\varepsilon$ if $\omega(F)<\varepsilon$.
**2.17.** The following notations will be frequently used in this article.\
(a) As in 2.15, we use notation $\#(\cdot)$ to denote the cardinal number of the set counting multiplicity.\
(b) For any metric space $X$, any $x_{0}\in X$ and any $c>0$, let $$B_{c}(x_{0}):= \{x\in X \mid d(x,x_{0})<c\}$$ denote the open ball with radius $c$ and center $x_{0}$.\
(c) Suppose that $A$ is a $C^{\ast}$-algebra, $B\subset A$ is a sub-$C^{*}$-algebra, $F\subset A$ is a (finite) subset and let $\varepsilon>0$. If for each element $f\in F$, there is an element $g\in B$ such that $\parallel f-g\parallel<\varepsilon$, then we shall say that $F$ is approximately contained in $B$ to within $\varepsilon$, and denote this by $F\subset_{\varepsilon} B$.\
(d) Let $X$ be a compact metric space. For any $\delta>0$, a finite set $\{x_{1},x_{2},\cdots,x_{n}\}$ is said to be $\delta$-dense in $X$, if for any $x\in X$, there is $x_{i}$ such that dist$(x,x_{i})<\delta$.\
(e) We shall use $\bullet$ to denote any possible positive integer. To save notation, $a_{1},a_{2},\cdots$ may be used for a finite sequence if we do not care how many terms are in the sequence. Similarly, $A_{1}\cup A_{2}\cup\cdots$ or $A_{1}\cap A_{2}\cap\cdots$ may be used for a finite union or a finite intersection. If there is a danger of confusion with an infinite sequence, union or intersection, we will write them as $a_{1},a_{2},\cdots,a_{\bullet}$, $A_{1}\cup A_{2}\cup\cdots\cup A_{\bullet}$, $A_{1}\cap A_{2}\cap\cdots\cap A_{\bullet}$.\
(f) In this paper, we often use $\textbf{1}$ to denote the units of different unital $C^{\ast}$-algebras. In particular, if $\textbf{1}$ appears in $\phi(\textbf{1})$, where $\phi$ is a homomorphism, then $\textbf{1}$ is the unit of the domain algebra. For example for a homomorphism $\phi: \bigoplus^{r}_{i=1}A^{i}\longrightarrow B$, then $\textbf{1}$ in $\phi(\textbf{1})$ means $\textbf{1}_{\bigoplus^{r}_{i=1}A^{i}}$ and $\textbf{1}$ in $\phi^{i}(\textbf{1})$ means $\textbf{1}_{A^{i}}$.\
(g) For any two projections $p,q\in A$, we use the notation $[p]\leqslant[q]$ to denote that $p$ is unitarily equivalent to a subprojection of $q$. And we use $p\thicksim q$ to denote that $p$ is unitarily equivalent to $q$.
**2.18.** For any $\eta>0$, $\delta>0$, a unital homomorphism $\phi: PM_{k}(C(X))P\longrightarrow Q M_{k^{'}}(C(Y))Q$ is said to have the property $sdp(\eta,\delta)$(spectral distribution property with respect to $\eta$ and $\delta$) if for any $\eta$-ball $B_{\eta}(x)$ and any point $y\in Y$, $$\#(SP \phi_{y}\cap B_{\eta}(x))\geqslant\delta\#(SP \phi_{y})(=\delta\frac{rank(Q)}{rank(P)})$$ counting multiplicity. Any homomorphism $$\phi:\bigoplus_{i}P_{i}M_{k_{i}}(C(X_{i}))P_{i}\longrightarrow \bigoplus_{j}Q_{j}M_{l_{j}}(C(Y_{j}))Q_{j}$$ is said to have the property $sdp(\eta,\delta)$ if each partial map $$\phi^{i,j}:P_{i}M_{k_{i}}(C(X_{i}))P_{i}\longrightarrow \phi^{i,j}(P_{i})M_{l_{j}}(C(Y_{j}))\phi^{i,j}(P_{i})$$ has the property $sdp(\eta,\delta)$ as a unital homomorphism. Note that by definition, a nonunital homomorphism $\phi: M_{k}(C(X))\longrightarrow M_{l}(C(Y))$ has the property $sdp(\eta,\delta)$ if the corresponding unital map $$\phi: M_{k}(C(X))\longrightarrow \phi(\textbf{1}_{k})M_{l}(C(Y))\phi(\textbf{1}_{k})$$ has the property $sdp(\eta,\delta)$.
The following Lemma is Lemma 2.8 of \[GJLP\].
**Lemma 2.19.** Let $A=\lim\limits_{\longrightarrow}(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be an $AH$ algebra with the ideal property. For $A_{n}$ and any $\eta>0$, there exist a $\delta>0$, a positive integer $m>n$, connected finite simplicial complexes $Z^{1}_{i},Z^{2}_{i},\cdots,Z^{\bullet}_{i}\subset X_{n,i}, i=1,2,\cdots,t_{n}$, and a homomorphism $$\phi: B=\bigoplus\limits_{i=1}\limits^{t_{n}}\bigoplus\limits_{s}M_{[n,i]}(C(Z^{s}_{i}))\longrightarrow A_{m}$$ such that\
(1) $\phi_{n,m}$ factors as $A_{n}\xrightarrow{\pi}B\xrightarrow{\phi}A_{m}$, where $\pi$ is defined by $$\pi(f)=(f\mid_{Z^{1}_{i}},f\mid_{Z^{2}_{i}},\cdot\cdot\cdot,f\mid_{Z^{\bullet}_{i}})\in \bigoplus\limits_{s}M_{[n,i]}(C(Z^{s}_{i}))\subset B,$$ for any $f\in M_{[n,i]}(C(X_{n,i})).$\
(2) The homomorphism $\phi$ satisfies the dichotomy condition ($\ast$): for each $Z^{s}_{i}$, the partial map $$\phi^{(i,s),j}: M_{[n,i]}(C(Z^{s}_{i}))\longrightarrow A_{m}^j$$ either has the property $sdp(\frac{\eta}{32},\delta)$ or is the zero map. Furthermore for any $m^{'}>m$, each partial map of $\phi_{m,m^{'}}\circ\phi$ satisfies the dichotomy condition $(\ast)$: either it has the property $sdp(\frac{\eta}{32},\delta)$ or is the zero map.
The following result is quoted from \[Pa2\].
**Lemma 2.20.** Let $A=lim(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be an $AH$ inductive limit with the ideal property and with no dimension growth (as in 2.5, we assume $X_{n,i}$ path connected). For any $A_{n}$, finite set $F_{n}=\bigoplus F^{i}_{n}\subset A_{n}$, $\varepsilon>0$, and positive integer $L$, there is an $A_{m}$, such that for each pair $(i,j)$, one of the following conditions holds\
(i) $\frac{rank(\phi^{i,j}_{n,m}(\textbf{1}_{A^{i}_{n}}))}{rank(\textbf{1}_{A^{i}_{n}})}\geqslant L$, or\
(ii) there is a homomorphism $$\psi: A^{i}_{n}\longrightarrow \phi^{i,j}_{n,m}(\textbf{1}_{A^{i}_{n}})A^{j}_{m}\phi^{i,j}_{n,m}(\textbf{1}_{A^{i}_{n}})$$ with finite dimensional image such that $\psi$ is homotopic to $\phi^{i,j}_{n,m}$ and $$\parallel \psi(f)-\phi^{i,j}_{n,m}(f)\parallel<\varepsilon~~~~~\forall f\in F^{i}_{n}.$$ The following proposition is Theorem 2.12 of \[GJLP\].
**Proposition 2.21.** Let $A=\lim\limits_{\longrightarrow}(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be an $AH$ algebra with the ideal property and with no dimension growth. For any $A_{n}$, finite set $F=\bigoplus_{i=1}^{t_{n}}F^{i}_{n}\subset A_{n}$, positive integer $J$ and $\varepsilon>0$, there exists $m$ and there exist projections $Q_{0},Q_{1},Q_{2}\in A_{m}$ with $Q_{0}+Q_{1}+Q_{2}=\phi_{n,m}(\textbf{1}_{A_{n}})$, a unital map $\psi_{0}\in Map(A_{n},Q_{0}A_{m}Q_{0})_{1}$ and two unital homomorphisms $\psi_{1}\in Hom(A_{n},Q_{1}A_{m}Q_{1})_{1}$, $\psi_{2}\in Hom(A_{n},Q_{2}A_{m}Q_{2})_{1}$ such that the following statement are true\
(1) $\parallel \phi_{n,m}(f)-(\psi_{0}(f)\oplus\psi_{1}(f)\oplus\psi_{2}(f))\parallel<\varepsilon$, for all $f\in F$\
(2) $\omega((\psi_{0}\oplus\psi_{1})(F))<\varepsilon$\
(3) The homomorphism $\psi_{2}$ factors through $C$—a finite direct sum of matrix algebras over $C[0,1]$, or $\mathbb{C}$ as $$\psi_{2}: A_{n}\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}Q_{2}A_{m}Q_{2}$$ where $\xi_{1}, \xi_{2}$ are unital homomorphisms\
(4) Let $\psi^{i,j}_{0}:
A^{i}_{n}\longrightarrow\psi^{i,j}_{0}(\textbf{1}_{A^{i}_{n}})A^{j}_{m}\psi^{i,j}_{0}(\textbf{1}_{A^{i}_{n}})$ and $\psi^{i,j}_{1}: A^{i}_{n}\longrightarrow\psi^{i,j}_{1}(\textbf{1}_{A^{i}_{n}})A^{j}_{m}\psi^{i,j}_{1}(\textbf{1}_{A^{i}_{n}})$ be the corresponding partial maps of $\psi_{0}$ and $\psi_{1}$. For each pair $(i,j)$, one of the following is true.\
(i) Both $\psi^{i,j}_{0}$ and $\psi^{i,j}_{1}$ are zero, or\
(ii) $\psi^{i,j}_{1}$ is a homomorphism with finite dimensional image and for each non zero projections $e\in A^{i}_{n}$ (including any rank 1 projection)$$[\psi^{i,j}_{1}(e)]>J[\psi^{i,j}_{0}(\textbf{1}_{A^{i}_{n}})](\in K_{0}(A_{m}^j)).$$
Furthermore, we can assume that $Q^{j}_{0}, Q^{j}_{1}$ are trivial projections in $A^{j}_{m}$.
The following Corollary is also in \[GJLP\],
**Corollary 2.22.** We use the notation from 2.21. For any $A_{n}$, any projection $P=\bigoplus P^{i}\in\bigoplus A^{i}_{n}$, any finite set $F=\bigoplus F^{i}\in\bigoplus P^{i}A^{i}_{n}P^{i}=PA_{n}P$, any positive integer $J$, and any number $\varepsilon>0$, there are an $A_{m}$, mutually orthogonal projections $Q_{0},Q_{1},Q_{2}\in A_{m}$ with $Q_{0}+Q_{1}+Q_{2}=\phi_{n,m}(\textbf{1}_{A_{n}})$, a unital map $\psi_{0}\in Map(A_{n},Q_{0}A_{m}Q_{0})_{1}$ and two unital homomorphisms $\psi_{1}\in Hom(A_{n},Q_{1}A_{m}Q_{1})_{1}$, $\psi_{2}\in Hom(A_{n},Q_{2}A_{m}Q_{2})_{1}$ such that for each pair $(i,j)$, $\psi^{i,j}_{0}(P^{i})$ and $\psi^{i,j}_{0}(\textbf{1}_{A_{n}^{i}}-P^{i})$ are mutually orthogonal projections and there is an approximate decomposition of $\phi{'}_{n,m}:=\phi_{n,m}\mid_{PA_{n}P}$ as a direct sum of $\psi{'}_{0}:=\psi_{0}\mid_{PA_{n}P},\psi{'}_{1}:=\psi_{1}\mid_{PA_{n}P}$ and $\psi{'}_{2}:=\psi_{2}\mid_{PA_{n}P}$, satisfying the following conditions:\
(1) $\parallel \phi{'}_{n,m}(f)-(\psi{'}_{0}(f)\oplus\psi{'}_{1}(f)\oplus\psi{'}_{2}(f))\parallel<\varepsilon$, for all $f\in F$.\
(2) $\psi{'}_{1}$ has finite dimensional image and $\psi{'}_{2}$ factors through a finite direct sum of matrix algebras over $C[0,1]$ or $\mathbb{C}$.\
(3) If $\psi'^{i,j}_{0}\neq 0$, then for any nonzero projection $e\in P^{i}A^{i}_{n}P^{i}, [\psi'^{i,j}_{1}(e)]>J[\psi'^{i,j}_{0}(P^{i})](\in K_0( A^{j}_{m}))$.\
(4) $\psi{'}_{0}$ is $F-\varepsilon$ multiplicative.
**2.23.** Let $X$ be a connected finite simplicial complex, $A=M_{k}(C(X))$. A unital $\ast$-monomorphism $\phi:A\longrightarrow M_{l}(A)$ is called a $\textbf{(unital) simple embedding} $ if it is homotopic to the homomorphism $id\oplus\lambda$, where $\lambda: A\longrightarrow M_{l-1}(A)$ is defined by $$\lambda(f)=diag(\underbrace{f(x_{0}),f(x_{0}),\cdots,f(x_{0})}\limits_{l-1}),$$ for a fixed base point $x_{0}\in X$.\
Let $A=\bigoplus_{i=1}^{{n}}M_{k_{i}}(C(X_{i}))$, where $X_{i}$ are connected finite simplicial complexes. A unital $\ast$-monomorphism $\phi:A\longrightarrow M_{l}(A)$ is called a (unital) simple embedding, if $\phi$ is of the form $\phi=\oplus\phi^{i}$ defined by $$\phi(f_{1},f_{2},\cdots,f_{n})=(\phi^{1}(f_{1}),\phi^{2}(f_{2}),\cdots,\phi^{n}(f_{n})),$$ where the homomorphisms $\phi^{i}: A^{i}(=M_{k_{i}}(C(X_{i})))\longrightarrow M_{l}(A^{i})$ are unital simple embeddings.
**§3. Additional decomposition theorems and factorization theorems**
In this section, we will prove certain decomposition theorems which say that the map $\phi_{n,m}$ (in an $AH$ inductive limit with the ideal property) can be decomposed into two parts roughly described as below:\
(a) the major part factors through as $$A_{n}\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}A_{m},$$ where $C$ is a direct sum of matrix algebras over interval \[0,1\] or $\{pt\}$ and $\xi_{1},\xi_{2}$ are $\ast$ homomorphisms, and\
(b) the other part factors through as $$A_{n}\xrightarrow{\beta}B\xrightarrow{\alpha}A_{m},$$ where $B$ is a direct sum of matrix algebras over $C(S^{1}), C( T_{\uppercase\expandafter{\romannumeral2}, k}), C( T_{\uppercase\expandafter{\romannumeral3}, k})$ or $C(S^{2})$, $\alpha$ is a $\ast$ homomorphisms, but $\beta$ is a sufficiently multiplicative completely positive contraction. These theorems will play the roles of Theorem $5.32a$ and Theorem $5.32b$ in \[G4\] in our proof. Since our limit algebra $A$ is no longer simple, we can not make each partial map to satisfy the dichotomy condition: each partial map $\phi^{i,j}_{n,m}$ is either injective or has finite dimensional image.
**3.1.** Recall that the total K-theory of $A$ is defined by $$\underline{K}(A)=K_{\ast}(A)\oplus\bigoplus_{n=2}^{\infty}K_{\ast}(A,\mathbb{Z}/n).$$ Any $KK$ element $\alpha\in KK(A,B)$ determine a homomorphism $$\underline{\alpha}: \underline{K}(A)\longrightarrow\underline{K}(B).$$ Now we let $W_{k}=T_{II,k}$ and let ${\mathcal{P}}\subset\mathop{\cup}\limits_{k}M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$ be any finite set of projections. Then each element $p\in{\mathcal{P}}$ defines an element $[p]\in\underline{K}(A)$. We will use ${\mathcal{P}}\underline{K}(A)$ to denote the subgroup of $\underline{K}(A)$ generated by $\{[p]\in\underline{K}(A), p\in{\mathcal{P}}\}$. For each finite set $\mathcal{P}$, there is a finite set $G(\mathcal{P})\subset A$ (large enough) and $\delta(\mathcal{P})>0$ (small enough) such that if $\phi: A\longrightarrow B$ is $G(\mathcal{P})-\delta(\mathcal{P})$ multiplicative completely positive contraction, then $\phi$ induces $$\phi_{\ast}: \mathcal{P}\underline{K}(A)\longrightarrow \underline{K}(B).$$ (See \[GL\].)
Let $A=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i}))$, where $X_{n,i}$ are connected finite simplicial complexes. Then $K_{\ast}(A)$ is finitely generated, and there is a finite set $\mathcal{P}\subset\mathop{\cup}\limits_{k}M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$ such that if two element $\alpha, \beta\in KK(A,B)$ satisfy $$\underline{\alpha}\mid_{\mathcal{P}\underline{K}(A)}=\underline{\beta}\mid_{\mathcal{P}\underline{K}(A)}$$ then $\alpha=\beta\in KK(A,B)$.
The following is 5.24 of \[G4\].
**Definition 3.2.** For any finite set of projections $\mathcal{P}\subset\mathop{\cup}\limits_{k=2}\limits^{\infty}M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$, let $G(\mathcal{P}), \delta(\cal{P})$ be as in 3.1. A $G(\mathcal{P})-\delta(\cal{P})$ multiplicative map $\phi:A\longrightarrow B$ is called quasi-$\mathcal{P}\underline{K}$-homomorphism if there is a homomorphism $\psi:A\longrightarrow B$ with $\phi(\textbf{1}_{A})=\psi(\textbf{1}_{A})$ such that $[\phi]_{\ast}=[\psi]_{\ast}: \mathcal{P}\underline{K}(A)\longrightarrow \underline{K}(B)$.
**3.3.** Let $B=M_{\bullet}(C(Y))$, $Y$ is one of the space $T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$, let $\mathcal{P}\subset\mathop{\cup}\limits_{k=2}\limits^{\infty} M_{\bullet}(B\otimes C(W_{k}\times S^{1}))$ be a finite set as in the end of 3.1 (one can choose $\mathcal{P}$ as in 5.16 of \[G4\]). Let $\phi:B\longrightarrow A$ be a homomorphism and let $\mathcal{P}_{1}\subset\mathop{\cup}\limits_{k=2}\limits^{\infty} M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$, where $A$ and $\mathcal{P}_{1}$ satisfy the condition in 3.1. Furthermore we assume $\mathcal{P}_{1}\supseteq\phi(\mathcal{P})$. Then there is a finite set $G\subset A$ and $\delta>0$, such that if a $G-\delta$ multiplicative map $\psi: A\longrightarrow A_{1}$ is a quasi-$\mathcal{P}_{1}\underline{K}$-homomorphism, then $\psi\circ\phi:B\rightarrow A_{1}$ is a quasi-$\mathcal{P}\underline{K}$-homomorphism.
**Lemma 3.4.** Let $A=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i}))$. For any $\mathcal{P}_{1}\subseteq\mathop{\cup}\limits^{+\infty}_{k=2}M_{\bullet}(A\otimes C(W_{k}\times S^{1})),$ there is a finite set $F\subset A$ and $\varepsilon>0$, such that the following is true.
If $\phi: A\longrightarrow A^{\prime}=PM_{\bullet}(C(X))P$ is a unital homomorphism, $Q_{1},Q_{2}\in PM_{\bullet}(C(X))P$ are two projections with $Q_{1}+Q_{2}=P$, and $\phi_{1}\in Map(A, Q_{1}A^{\prime}Q_{1})_{1}$, $\phi_{2}\in Hom(A, Q_{2}A^{\prime}Q_{2})_{1}$ satisfy\
(i) $\parallel \phi(f)-(\phi_{1}\oplus\phi_{2}(f))\parallel<\varepsilon$ $\forall f\in F$ and\
(ii) for each $i$, $\phi^{i}_{1}\in Map(A^{i}, Q_{1}A^{\prime}Q_{1})$ is either zero, or $$rank(\phi^{i}_{1}(\textbf{1}_{A^{i}}))\geqslant3(dim(X)+1)rank(\textbf{1}_{A^{i}})$$\
then $\phi_{1}$ is a quasi-$\mathcal{P}_{1}\underline{K}$-homomorphism, where $A^{i}=M_{[n,i]}(C(X_{n,i}))$.
First, by Lemma 4.40 of \[G4\], if $F$ is large enough and $\varepsilon>0$ is small enough, then the condition (i) implies that $\phi_{1}$ is $G(\mathcal{P}_{1})-\delta(P_{1})$ multiplicative and induces maps on $\mathcal{P}_{1}\underline{K}(A)$.
We can assume that $A$ has only one block $M_{[n,1]}(C(X_{n,1}))$. Using Lemma 1.6.8 of \[G4\], one can assume that $\phi_1|_{M_{[n,1]}(\mathbb{C})}$ is a homomorphism, where $M_{[n,1]}(\mathbb{C})\subset M_{[n,1]}(C(X_{n,1}))$. Then one can reduce the proof of the lemma to the case that $[n,1]=1$, that is, $A=C(X_{n,1})$. Then both $\phi\in Hom(C(X_{n,1}), PM_{\bullet}C(X)P)$ and $\phi_{2}\in Hom(C(X_{n,1}), Q_{2}M_{\bullet}(C(X))Q_{2})$ induce $[\phi]\in kk(X,X_{n,1})$ and $[\phi_{2}]\in kk(X,X_{n,1})$ (see 2.8 of \[EG2\], also see \[DN\] for notation $kk$). Since $kk(X,X_{n,1})$ is an Abelian group, and $rank (Q_{1})\geqslant3(dim(X)+1)rank(\textbf{1}_{A^{i}})$, by 6.4.4 of \[DN\] (see also Proposition 3.16 of \[EG2\]), there is a unital homomorphism $\psi_{1}\in Hom(A, Q_{1}A^{\prime}Q_{1})_{1}$ such that $$[\psi_{1}]=[\phi]-[\phi_{2}]\in kk(X,X_{n,1}).$$ Hence $[\psi_{1}]\mid_{\mathcal{P}_{1}\underline{K}(A)}=[\phi_{1}]\mid_{\mathcal{P}_{1}\underline{K}(A)}$, which implies $\phi_{1}$ is a quasi-$\mathcal{P}_{1}\underline{K}$-homomorphism.\
**Lemma 3.5.** Let $A=PM_{\bullet}(C(X))P$, where $X$ is connected finite simplicial complex. Let $F\subset A$ be approximately constant to within $\varepsilon$ (i.e. $\omega(F)<\varepsilon$). Then for any two homomorphisms $\phi,\psi: A\longrightarrow B=QM_{l}C(Y)Q$ defined by point evaluations with $K_{0}\phi=K_{0}\psi$ and assuming that for any $p\in A$, $rank(\phi(p))\geqslant rank(p)\cdot dim(Y)$, there exists a unitary $u\in B$ such that $$\parallel\phi(f)-u\psi(f)u^{\ast}\parallel<2\varepsilon,~~~~~\forall f\in F.$$
Let $x_{0}\in X$ be a base point of $X$. There are finitely many points $\{x_{1},x_{2},\cdots,x_{n}\}\subset X$ such that $\phi$ factors through as $$A\xrightarrow{\pi}A\mid_{\{x_{1},x_{2},\cdots,x_{n}\}}=\bigoplus_{i=1}^{n}M_{rank(P)}(\mathbb{C})\xrightarrow{\phi^{''}}B.$$ But for each $i$, there is a unitary $u_{i}\in M_{rank(P)}(\mathbb{C})$ such that $$\parallel f(x_{0})-u_{i}f(x_{i})u^{\ast}_{i}\parallel<\varepsilon~~~~\forall f\in F,$$ since $\omega(F)<\varepsilon$. Let $\pi^{\prime}: A\longrightarrow \bigoplus_{i=1}^{n}M_{rank(P)}(\mathbb{C})$ defined by $$\pi^{\prime}(f)=(u^{\ast}_{1}f(x_{0})u_{1},u^{\ast}_{2}f(x_{0})u_{2},\cdots,u^{\ast}_{n}f(x_{0})u_{n}).$$ Then $\parallel\pi(f)-\pi^{\prime}(f)\parallel<\varepsilon$ for all $f\in F$. Evidently there is a homomorphism $\phi^{\prime}: M_{rank(P)}(\mathbb{C})\longrightarrow B$ such that $\parallel\phi(f)-\phi^{\prime}(f(x_{0}))\parallel<\varepsilon ,~\forall f\in F$. Similarly there is a $\psi^{\prime}: M_{rank(P)}(\mathbb{C})\longrightarrow B$ with $\parallel\psi(f)-\psi^{\prime}(f(x_{0}))\parallel<\varepsilon,~\forall f\in F$. On the other hand\
$K_{0}\phi=K_{0}\psi$, since $e_{\ast}: K_{0}(P M_{\bullet}(C(X))P)\longrightarrow K_{0}(M_{rank}(\mathbb{C}))$ is surjective for\
$e: P M_{\bullet}(C(X))P\longrightarrow P(x_{0})M_{\bullet}(\mathbb{C})P(x_{0})\cong M_{rank(P)}(\mathbb{C})$. Furthermore by the condition that $rank(\phi(P))\geq rank(P)\cdot dim(Y)$ and the fact that for any two projections $Q_{1},Q_{2}$ (of rank at least $dim(Y)$) over $Y$, $[Q_{1}]=[Q_{2}]\in K_{0}(M_{\bullet}(C(Y)))$ implies $Q_{1}$ unitarily equivalent to $Q_{2}$, we know that there is a unitary $u\in B$ such that $\phi^{\prime}=u\psi^{\prime}u^{\ast}$.\
**Lemma 3.6.** Fix a positive integer $M\geq3$, suppose that $B=\bigoplus_{i=1}^{s}M_{l_{i}}(C(Y_{i}))$, where $Y_{i}$ are the spaces: $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$. Let $\varepsilon>0$ and $$\widetilde{G}(=\bigoplus\widetilde{G}^{i})\subset G(=\bigoplus G^{i})\subset \bigoplus B^{i}$$ be two finite sets satisfying that if $Y_{i}$ is one of $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}_{k=1}^{\infty}, \{T_{\uppercase\expandafter{\romannumeral3}, k}\}_{k=1}^{\infty}$ and $S^{2}$, then $\omega(\widetilde{G}^{i})<\varepsilon$, and if $Y_{i}$ is one of $\{pt\}$, \[0,1\], $S^{1}$, then $\widetilde{G}^{i}=G^{i}$. Then there is a subset $G_{1}\subset B$ with $G_{1}\supset G(\mathcal{P})$, ($\mathcal{P}\subset\mathop{\cup}\limits^{\infty}\limits_{k=2}M_{\bullet}(B\otimes C(W_{k}\times S^{1}))$ as in the end of 3.1) and $\delta_{1}>0$ with $\delta_{1}<\delta(\mathcal{P})$, and a positive integer $L>0$ such that the following is true.
If a map $\alpha=\alpha_{0}\oplus\alpha_{1}: B\longrightarrow A=\bigoplus_{j=1}^{t}M_{k_{j}}(C(X_{j}))$, where $X_{j}$ are connected finite simplicial complexes with $dim(X_{j})\leqslant M$, satisfying the following conditions:\
(1) $\alpha_{0}$ is $G_{1}-\delta_{1}$ multiplicative, $\{\alpha_{0}(\textbf{1}_{B^{i}})\}^{s}_{i=1}$ are mutually othogonal projections, and for any block $B^{i}$ with $Y_{i}=T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$ and any block $A^{j}$, the partial map $\alpha^{i,j}_{0}$ is quasi-$\mathcal{P}\underline{K}$-homomorphism; and\
(2) $\alpha_{1}$ is a homomorphism defined by point evaluations and for each block $B^{i}$, with $Y_{i}=T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$ and any block $A^{j}$, $$\alpha^{i,j}_{1}(\textbf{1}_{B_{i}})\geqslant L\alpha^{i,j}_{0}(\textbf{1}_{B_{i}}),$$ then there is a unital homomorphism $\alpha^{\prime}: B\longrightarrow \alpha(\textbf{1}_{B}) A\alpha (\textbf{1}_{B})$ such that $$\parallel\alpha^{\prime}(g)-\alpha(g)\parallel<3\varepsilon~~~\forall g\in\widetilde{G}.$$
Without loss of generality, we can assume $B$ has only one block: $B=M_{l}(C(Y))$. If $Y=\{pt\},[0,1],S^{1}$, the Lemma is true since $B$ is stably generated (see Lemma 1.6.1 of \[G4\]).
Suppose that $Y=T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$. Let $G_{1}$ and $\delta_{1}$, and $\eta$ be as Lemma 5.30 of \[G4\] (note that, for a positive integer $L$ and positive number $\eta$, the notion $PE(L,\eta)$ (used in Lemma 5.30 of \[G4\]) is defined in Definition 4.38 of \[G4\]). Let $\{y_{j}\}^{K}_{j=1}$ be an $\eta$-dense subset of $Y$. Let $L=4KM$. Let us verify the conclusion holds for such a choice. If $\alpha_{0}=0$, then one can choose $\alpha^{\prime}=\alpha_{1}$ as desired. Suppose $\alpha_{0}\neq0$ and therefore $rank$ $\alpha_{0}(\textbf{1}_{B})>0$. Using $ rank$ $\alpha_{1}(\textbf{1}_{B})\geqslant L rank(\alpha_{0}
(\textbf{1}_{B}))$, there is a unital homomorphism $\alpha^{\prime}_{1}: B\longrightarrow \alpha_{1}(\textbf{1}_{B})A\alpha_{1}(\textbf{1}_{B})$ which satisfies the following conditions\
(i) $\alpha^{\prime}_{1}$ is homotopy to $\alpha_{1}$\
(ii) $\alpha^{\prime}_{1}$ is defined by direct sum of point evaluations at different points\
(iii) all the points in the $\eta$-dense set $\{y_{j}\}^{K}_{j=1}$ are among those point evaluations that define $\alpha^{\prime}_{1}$ and each point evaluation at $y_{i}$ satisfies that the rank of the image of $\textbf{1}_{B}$ is at least rank $\alpha_{0}(\textbf{1}_{B})$—that is $\alpha^{\prime}_{1}$ has property $PE(rank$ $\alpha_{0}(\textbf{1}_{B}), \eta)$.
By 5.30 of \[G4\], there is a homomorphism $\alpha{''}: B\longrightarrow \alpha(\textbf{1}_{B})A\alpha(\textbf{1}_{B})$ such that $$\parallel\alpha{''}(f)-(\alpha_{0}\oplus\alpha^{\prime}_{1}(f))\parallel<\varepsilon,~~~\forall f\in \widetilde{G}.$$ On the other hand by Lemma 3.5, there is a unitary $u\in \alpha^{\prime}_{1}(\textbf{1}_{B})
A\alpha^{\prime}_{1}(\textbf{1}_{B})$ (note that $\alpha^{\prime}_{1}(\textbf{1}_{B})=\alpha_{1}(\textbf{1}_{B})$) such that $$\parallel u^{\ast}\alpha^{\prime}_{1}(f)u-\alpha_{1}(f)\parallel<2\varepsilon, ~~~\forall f\in\widetilde{G}$$ Let $\alpha^{\prime}(t)=diag(\alpha_{0}(\textbf{1}_{B}),u^{\ast})\alpha{''}(f)diag(\alpha_{0}(\textbf{1}_{B}),u)$.\
Evidently, we have $$\parallel\alpha{'}(f)-(\alpha_{0}\oplus\alpha_{1})(f)\parallel<3\varepsilon, ~~~\forall f\in \widetilde{G}.$$
**Lemma 3.7.** Let $M$ be a fixed positive integer. Let $B=M_{l}(C(Y))$, $Y=T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$. Let the set of projection $\mathcal{P}\subset M_{\bullet}(B\otimes C(W_{k}\times S^{1}))$ be as in 3.1 (also see 5.16 of \[G4\]).
Let $A=RM_{l}(C(X))R$ with $X$ a connected finite simplicial complex and let $\alpha: B\longrightarrow A$ be a homomorphism. Let $\mathcal{P}^{\prime}\subset\mathop{\cup}\limits^{\infty}\limits_{k=2}M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$ be a set of projections (chosen for $A$) as in the end of 3.1 and $\mathcal{P}^{\prime}\supseteq(\alpha\otimes id)(\mathcal{P})$.
For any finite sets $\widetilde{G}_{1}\subset G_{1}\subset B$ and numbers $\varepsilon_{1}>0, \delta_{1}>0,$ with $\omega(\widetilde{G}_{1})<\varepsilon_{1},$ there are a finite set $G_{2}\subset A$ a number $\delta_{2}>0$, and positive integer $L$, such that the following are true. Let $C=M_{\bullet}(C(Z))$ with $Z$ connected simplicial complex and $dim(Z)\leqslant M$, and let $Q_{0},Q_{1}\in C$ be two orthogonal projections.\
(1) If $\psi_{0}: A\longrightarrow Q_{0}CQ_{0}$ is $G_{2}-\delta_{2}$ multiplicative quasi- $\mathcal{P}^{\prime}\underline{K}$-homomorphism and $\psi_{0}(\alpha(\textbf{1}_{B}))$ is a projection, then $\psi_{0}\circ\alpha$ is a $G_{1}-\delta_{1}$ multiplicative quasi-$\mathcal{P}\underline{K}$-homomorphism.\
(2) If $\psi_{0}$ as in (1) and $\psi_{1}: A\longrightarrow Q_{1}CQ_{1}$ is defined by point evaluations with\
rank $(\psi_{1}(\textbf{1}_{A}))\geq L\mbox{rank}(\psi_{0}(\textbf{1}_{A}))$, then there is a homomorphism\
$\psi: B\longrightarrow(Q_{0}\oplus Q_{1})C(Q_{0}\oplus Q_{1})$ such that $$\parallel \psi(g)-(\psi_{0}\oplus \psi_{1})(\alpha(g))\parallel<3\varepsilon.$$
The proof is the same as the proof of Lemma 5.31 of \[G4\] using Lemma 3.6 above to replace 5.30 of \[G4\].\
**Theorem 3.8.** Let $M\geqslant3$ be a positive integer. Let $\lim\limits_{n\rightarrow\infty}\limits^{}(A_{n}=\bigoplus_{i=1}^{k_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be an $AH$ inductive limit such that the limit algebra has the ideal property, where $X_{n,i}$ are connected simplicial complexes with $dim(X_{n,i})\leq M$, for all $n,i$. Let $B=\bigoplus_{i=1}^{s}M_{l_{i}}(C(Y_{i})),$ where $Y_{i}$ are spaces $\{pt\}$, \[0,1\], $S^{1}$, $T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k},$ and $S^{2}$. Suppose that $\widetilde{G}(=\bigoplus\widetilde{G}^{i})\subset G(=\bigoplus G^{i})\subset B(=\bigoplus B^{i})$ are two finite sets and $\varepsilon_{1}>0$ is a positive number with $\omega(\widetilde{G}^{i})<\varepsilon_{1}$, if $Y_{i}=T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k},$ or $S^{2}$. Suppose that $L$ is any positive integer. Let $\alpha: B\longrightarrow A_{n}$ be any homomorphism. Denote $\alpha(\textbf{1}_{B}):= R(=\bigoplus R^{i})\in A_{n}(=\bigoplus A^{i}_{n})$. Let $F\subset RA_{n}R$ be any finite set and $\varepsilon<\varepsilon_{1}$ be any positive number.
It follows that there are $A_{m}$, and mutually orthogonal projections $Q_{0},Q_{1},Q_{2}\in A_{m}$ with $\phi_{n,m}(R)=Q_{0}+Q_{1}+Q_{2},$ a unital map $\theta_{0}\in Map(RA_{n}R, Q_{0}A_{m}Q_{0})_{1}$, two unital homomorphisms $\theta_{1}\in Hom(RA_{n}R, Q_{1}A_{m}Q_{1})_{1}$, $\xi\in Hom(RA_{n}R, Q_{2}A_{m}Q_{2})_{1}$ such that\
(1) $\parallel \phi_{n,m}(f)-(\theta_{0}(f)\oplus\theta_{1}(f)\oplus \xi(f))\parallel<\varepsilon~~~~\forall f\in F$\
(2) there is a homomorphism $\alpha_{1}: B\longrightarrow(Q_{0}+Q_{1})A_{m}(Q_{0}+Q_{1})$ such that
$\parallel \alpha_{1}(g)-(\theta_{0}\oplus \theta_{1})\circ\alpha(g)\parallel<3\varepsilon_{1}~~~~\forall g\in \widetilde{G}^{i},
~~if\; Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $ S^{2}$
and
$\parallel \alpha_{1}(g)-(\theta_{0}\oplus\theta_{1})\circ\alpha(g)\parallel<\varepsilon~~~~\forall g\in G^{i},~~ if\; Y_{i}=\{pt\}, [0,1]$ or $S^{1}.$\
\(3) $\theta_{0}$ is $F-\varepsilon$ multiplicative and $\theta_{1}$ satisfies that for any nonzero projections $e\in R^{i}A^{i}_{n}R^{i}$ $$\theta^{i,j}_{1}([e])\geqslant L\cdot[\theta^{i,j}_{0}(R^{i})].$$ (4) $\xi$ factors through a $C^{\ast}$-algebra $C$—a direct sum of matrix algebras over C\[0,1\] or $\mathbb{C}$ as $$\xi:RA_{n}R\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}Q_{2}A_{m}Q_{2}.$$
Let $D\subset A_{n}=\oplus A^{i}_{n}$ be defined by $$D=\bigoplus\limits_{j}(\bigoplus\limits_{i}\alpha^{i,j}(\mathbb{C}\cdot\textbf{1}_{B^{i}}))\subset \bigoplus\limits_{j}A^{j}_{n}$$ which is a finite dimensional subalgebra of $A_{n}$ containing the set of mutually orthogonal projections $\{E^{i,j}=\alpha^{i,j}(\textbf{1}_{B^{i}})\}_{i,j}$.
Apply Corollary 2.22 for sufficiently large set $F^{\prime}\subset RA_{n}R$, sufficiently small number $\varepsilon^{\prime}>0$ and sufficiently large integer $J>0$, to obtain $A_{m}$ and the decomposition $\theta_{0}\oplus\theta_{1}\oplus\xi$ of $\phi_{n,m}\mid_{RA_{n}R}$ as $\psi^{\prime}_{0}\oplus\psi^{\prime}_{1}\oplus\psi^{\prime}_{2}$ in the corollary. By Lemma 1.6.8 of \[G4\], we can assume $\theta_{0}\mid_{D}$ is a homomorphism. The condition (1) follows if we choose $F^{\prime}\supset F$, and $\varepsilon^{\prime}<\varepsilon$. The $F-\varepsilon$ multiplicative of $\theta_{0}$ in (3) follows from Lemma 4.40 of \[G4\], if $F^{\prime}$ is large enough and $\varepsilon^{\prime}$ is small enough; and property of $\theta_{1}$ in (3) follows if we choose $J>L$.
To construct $\alpha_{1}$ as desired in the condition (2), we need to construct $$\alpha^{i,j,k}_{1}: B^{i}\longrightarrow\theta^{j,k}(E^{i,j})A^{k}_{m}\theta^{j,k}(E^{i,j})$$ where $\theta=\theta_{0}\oplus \theta_{1}$, to satisfy $$\parallel \alpha^{i,j,k}_{1}(g)-\theta^{j,k}\circ\alpha^{i,j}(g)\parallel<3\varepsilon_{1}~~~~\forall g\in \widetilde{G}^{i},
~~if\; Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$$ and $$\parallel \alpha^{i,j,k}_{1}(g)-\theta^{j,k}\circ\alpha^{i,j}(g)\parallel<\varepsilon~~~~\forall g\in {G}^{i},~~ if\; Y_{i}=\{pt\}, [0,1], S^{1}.$$\
For the case of $Y_{i}=\{pt\}, [0,1], S^{1}$, the existence of $\alpha^{i,j,k}_{1}$ follows from Lemma 1.6.1 and Lemma 4.40 of \[G4\]—that is $\theta^{j,k}_{0}\circ\alpha^{i,j}$ itself can be perturbed to a homomorphism as $B^{i}$ is stably generated.\
For the case $ Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$, the existence of $\alpha^{i,j,k}_{1}$ follows from Lemma 3.7 and $\omega(\widetilde{G}^{i})<\varepsilon_{1}$ provided $J$ is large enough, $F^{\prime}$ large enough, $\varepsilon^{\prime}$ small enough.\
**Theorem 3.9.** Let $M$ be a positive integer. Let $\lim\limits_{n\rightarrow\infty}\limits^{}(A_{n}=\bigoplus_{i=1}^{k_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be an $AH$ inductive limit such that the limit algebra has the ideal property, where $X_{n,i}$ are connected simplicial complexes with $dim(X_{n,i})\leq M$, for all $n,i$. Let $B=\bigoplus_{i=1}^{s}M_{l_{i}}(C(Y_{i})),$ where $Y_{i}$ are spaces $\{pt\}$, \[0,1\], $S^{1}$, $T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$. Suppose that $\widetilde{G}(=\bigoplus\widetilde{G}^{i})\subset G(=\bigoplus G^{i}\subset B(=\bigoplus B^{i})$ are two finite subsets and $\varepsilon_{1}$ is a positive number such that $\omega(\widetilde{G}^{i})<\varepsilon_{1}$, if $Y_{i}$ is one of $T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k},$ or $ S^{2}$. Let $\alpha: B\longrightarrow A_{n}$ be a homomorphism and $F(\supset\alpha(G))$ be a finite subset of $A_{n}$ and $\varepsilon<\varepsilon_{1}$ be any positive number.
It follows that there are $A_{m}$ and mutually orthogonal projections $P,Q\in A_{m}$ with $\phi_{n,m}(\textbf{1}_{A_{n}})=P+Q$, a unital map $\theta\in Map(A_{n}, PA_{m}P)_{1}$, and a unital homomorphism $\xi\in Hom(A_{n}, QA_{m}Q)_{1}$ such that\
(1) $\parallel \phi_{n,m}(f)-(\theta(f)\oplus\xi(f))\parallel<\varepsilon~~~~\forall f\in F$\
(2) there is a homomorphism $\alpha_{1}: B\longrightarrow PA_{m}P$ such that $$\parallel \alpha^{i,j}_{1}(g)-(\theta\circ\alpha)^{i,j}(g)\parallel<3\varepsilon_{1}~~~~\forall g\in \widetilde{G}^{i},
~~if\; Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2},$$ $$\parallel \alpha^{i,j}_{1}(g)-(\theta\circ\alpha)^{i,j}(g)\parallel<\varepsilon~~~~\forall g\in G^{i},~~ if\; Y_{i}=\{pt\}, [0,1], S^{1}.$$\
(3) $\omega(\theta(F))<\varepsilon$ and $\theta$ is $F-\varepsilon$ multiplicative.\
(4) $\xi$ factors through a $C^{\ast}$-algebra $C$—a direct sum of matrix algebras over $C[0,1]$ or $\mathbb{C}$ as $$\xi: A_{n}\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}QA_{m}Q.$$\
The proof is similar to but slightly easier than that of Theorem 3.8, and we omit it.
The following is Corollary 1.6.15 of \[G4\].
**Proposition 3.10.** Let $A=\bigoplus_{k=1}^{l}M_{s(k)}(C(X_{k}))$, where $X_{k}$ are connected finite simplicial complexes and $s(k)$ are positive integers. Let $F\subset A$ be a finite set and $\varepsilon>0$. There are a finite set $G\subset A$ and a number $\delta>0$ with the following property. If $B$ is a unital $C^{\ast}$-algebra, $p\in B$ is a projection, $\phi_{t}: A\longrightarrow pBp, 0\leqslant t\leqslant1$ is a continuous path of $G-\delta$ multiplicative maps, then there are a positive integer $L$, and $\eta>0$ such that for a homomorphism $\lambda: A\longrightarrow B\otimes \mathcal{K}$ (with finite dimensional image), there is a unitary $u\in(p\oplus\lambda(\textbf{1}))(B\otimes \mathcal{K})(p\oplus\lambda(\textbf{1}))$ satisfying $$\parallel \phi_{0}(f)\oplus\lambda(f)-u(\phi_{1}(f)\oplus\lambda(f))u^{\ast}\parallel<\varepsilon,~~~~\forall f\in F,$$ provided that $\lambda$ is of the following form: there are an $\eta$-dense subset $\{x_{1},x_{2},\cdots,x_{\bullet}\}\subset \amalg^{l}_{k=1}X_{k}(=Sp(A))$, and a set of mutually orthogonal projections $$\{p_{1},p_{2},\cdots,p_{\bullet}\}\subset \lambda(\bigoplus_{k}e^{k}_{11})(B\otimes \mathcal {K})\lambda(\bigoplus_{k}e^{k}_{11})$$ with $[p_{i}]\geqslant L\cdot[p]$, such that $$\lambda(f)=\sum\limits^{n}\limits_{i=1}p_{i}\otimes f(x_{i})\oplus\lambda_{1}(f),~~~~\forall f\in A,$$ where $\lambda_{1}$ is also a homomorphism, under the identification $$\lambda(\textbf{1}_{A^{k}})B\lambda(\textbf{1}_{A^{k}})\cong \lambda(e^{k}_{11})B\lambda(e^{k}_{11})\otimes M_{s(k)}(\mathbb{C}).$$
**Corollary 3.11.** Let $A=\bigoplus_{k=1}^{l}M_{s(k)}(C(X_{k}))$, where $X_{k}$ are connected finite simplicial complexes and $s(k)$ are positive integers. Let $\varepsilon>0$ and $F\subset A$ be a finite set with\
$\omega(F)<\varepsilon$. There are a finite set $G\subset A$ and a number $\delta>0$ with the following property. If $B=\bigoplus\limits_{j=1}^{}M_{r(j)}(C(Z_{j}))$, where $Z_{j}$ are connected finite simplicial complexes and $p\in B$ is a projection, $\phi_{t}: A\longrightarrow pBp, 0\leqslant t\leqslant1$, is a continuous path of $G-\delta$ multiplicative maps, then there is a positive integer $L$ such that for a homomorphism $\lambda: A\longrightarrow B\otimes \mathcal{K}$ with finite dimensional image, there is a unitary $u\in(p\oplus\lambda(\textbf{1}))(B\otimes \mathcal{K})(p\oplus\lambda(\textbf{1}))$ satisfying $$\parallel \phi_{0}(f)\oplus\lambda(f)-u(\phi_{1}(f)\oplus\lambda(f))u^{\ast}\parallel<5\varepsilon~~~~\forall f\in F,$$ provided $\lambda^{k}=\lambda\mid_{A^{k}}: M_{s(k)}(C(X_{k}))\longrightarrow B\otimes \mathcal{K}$ has finite dimensioned image (or equivalently, is defined by point evaluation) with $$[\lambda^{k}(\textbf{1}_{A^{k}})]\geqslant L\cdot[p]\in K_{0}(B).$$
Let $L^{\prime}$ and $\eta$ be $L$ and $\eta$ as in Proposition 3.10. Let $\{x_{1},x_{2},\cdots,x_{K}\}\subset\mathop{\amalg}\limits^{l}\limits_{k=1}X_{k}$ be a $\eta$-dense subset. Choose $L=4L^{\prime}\cdot K \max(\mathop{dim}\limits_{j}(Z_{j})+1)$. If $\lambda$ satisfies the condition in our corollary then it is easy to find $\lambda^{\prime}: A\longrightarrow \lambda(\textbf{1})(B\otimes\mathcal{K})\lambda(\textbf{1})$ satisfies the condition in Proposition 3.10 and $K_{0}\lambda=K_{0}\lambda^{\prime}$. Then the corollary follows from Proposition 3.10 and Lemma 3.5.\
**Theorem 3.12.** Let $B_{1}=\bigoplus_{j=1}^{s}M_{k(j)}(C(Y_{j}))$, where $Y_{j}$ are spaces $\{pt\}$, \[0,1\], $S^{1}$, $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$, $\{T_{\uppercase\expandafter{\romannumeral3}, k}\}^{\infty}_{k=2}$ and $S^{2}$. Let $X$ be a connected finite simplicial complex and let $A=M_{N}(C(X))$. Let $\widetilde{G}_{1}(=\bigoplus\widetilde{G}^{i}_{1})\subset G_{1}(=\bigoplus G^{i}_{1})\subset B_1(=\bigoplus B_1^{i})$ be two finite sets with $\omega(\widetilde{G}^{i}_{1})<\varepsilon_{1}$ for certain $\varepsilon_{1}>0$ and any $i$ with $Y_{i}$ being $T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$. Let $\alpha_{1}:B_{1}\rightarrow A$ be a homomorphism, and let $F_{1}\subset A$ be a finite set and any positive number $\varepsilon<\varepsilon_{1}$ and $\delta>0$. Then there exists a diagram $$\xymatrix{
A\ar[rdrd]^{\beta}\ar[rr]^{\phi} & & A^{\prime} \\
& & & & & & \\
B_{1}\ar[uu]_{\alpha_{1}}\ar[rr]^{\psi} & & B_{2}\ar[uu]_{\alpha_{2}} \\
}$$ where $A^{\prime}=M_{L}(A)$, and $B_{2}$ is a direct sum of matrix algebras over space: $\{pt\}$, \[0,1\], $S^{1}$, $T_{\uppercase\expandafter{\romannumeral2}, k}$, $T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$, with the following conditions\
(1) $\psi$ is a homomorphism, $\alpha_{2}$ is a unital injective homomorphism and $\phi$ is a unital simple embedding (see 2.23 for the definition of unital simple embedding);\
(2) $\beta\in Map(A,B_{2})_{1}$ is $F_{1}-\delta$ multiplicative;\
(3) $\parallel\psi(g)-\beta\circ\alpha_{1}(g)\parallel<5\varepsilon_{1}~~~\forall g\in\widetilde{G}^{i}_{1}$, if $Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}, S^{2}$,\
$~~~~~\parallel \psi(g)-\beta\circ\alpha_{1}(g)\parallel<\varepsilon~~~\forall g\in G^{i}_{1}$, if $Y_{i}=\{pt\}, [0,1], S^{1}$,\
$~~~~~~ \mbox{and}$\
$~~~~~\parallel \phi(f)-\alpha_{2}\circ\beta(f)\parallel<\varepsilon$ $\forall f\in F_{1}$; and\
(4) $\beta(F_{1})\cup \psi(G_{1})\subset B_{2}$ satisfies that $~\omega(\beta(F_{1})\cup \psi(G_{1}))<\varepsilon.$
The proof is similar to the proof of Theorem 1.6.26 (see 1.6.25) of \[G4\]. The differences are the following. First, we do not have the condition that $\alpha_{1}$ is injective, so we need to use Corollary 3.11 to deal with blocks $B^{i}_{1}$ with $Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$. For those block $B^{i}_{1}$ with $Y_{i}=\{pt\}, [0,1], S^{1}$, we use the condition that $B^{i}_{1}$ is stably generated (see \[G4\] 1.6.1). Secondly, we need to make the condition (4) hold.
Without loss of generality, we can assume $\alpha_{1}(B^{i}_{1})\neq 0$ for each block $B^{i}_{1}$, otherwise we simply take $\psi$ to be zero on this block.
Apply Corollary 3.11 to $\widetilde{G}^{i}_{1}\subset B^{i}_{1}$ for blocks $Y_{i}= T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$, to obtain $G^{i}\subset B^{i}_{1}$ and $\delta_{1}$ as in the corollary. Apply Lemma 1.6.1 of \[G4\] to $G^{i}_{1}\subset B^{i}_{1}$ for $Y_{i}=\{pt\}, [0,1], S^{1}$, to obtain $G^{i}$ and $\delta^{\prime}_{1}$. We assume $G^{i}\supset G_{1}^{i}$. Let $G=\bigoplus G^{i}$. Let $F^{\prime}_{1}\subset A$ and $\delta_{2}>0$ be such that if $\beta: A\longrightarrow C$ (any $C^{\ast}$-algebra) is $F^{\prime}_{1}-\delta_{2}$ multiplicative, then $\beta\circ\alpha_{1}$ is $G-\min(\delta_{1},\delta^{\prime}_{1})$ multiplicative.
Apply Proposition 3.10 to $F^{\prime}_{1}\cup F_{1}$ (as $F\subset A$) and $\min(\delta_{2},\delta)$ (as $\varepsilon>0$) to obtain $F\subset A$ and $\delta_{3}$ (in place of the set $G$ and the number $\delta$ there). We can assume $\delta_{3}<\min(\delta_{2},\delta,\delta_{1},\delta^{\prime}_{1})$ and $F\supset F^{\prime}_{1}\cup F_{1}$.
Apply 1.6.24 of \[G4\] to obtain the following diagram $$\xymatrix{
A\ar[rdrd]^{\beta}\ar[rr]^{\phi} & & A^{\prime} \\
& & & & & & \\
B_{1}\ar[uu]_{\alpha_{1}}\ar[rr]^{\psi} & & B_{2}\ar[uu]_{\alpha_{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\displaystyle\huge{(I)}}}
}$$ where $A^{\prime}=M_{L}(A)$, and $B_{2}$ is a direct sum of matrix algebras over the spaces $\{pt\}, [0,1], S^{1}$, $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}, \{T_{\uppercase\expandafter{\romannumeral3}, k}\}$ and $S^{2}$ with the following conditions:\
(i) $\psi$ is homomorphism, $\alpha_{2}$ is a unital injective homomorphism and $\phi$ is a unital simple embedding\
(ii) $\beta\in Map(A,B_{2})_{1}$ is $F-\delta_{3}$ multiplicative and therefor $\beta\circ\alpha_{1}$ is $G-min(\delta_{1},\delta^{'}_{1})$ multiplicative\
(iii) there exist two homotopies $\Psi\in Map(B_{1},B_{2}[0,1])$ and $\Phi\in Map(A,A^{'}[0,1])$ such that $\Psi$ is $G-\delta_{3}$ multiplicative and $\Phi$ is $F-\delta_{3}$ multiplicative.\
Note that we can choose a simple embedding $\xi:B_{2}\rightarrow M_{k}(B_{2})$ such that $\omega(\xi(\beta(F)\cup\psi(G)))<\varepsilon$ by adding to the identity the homomorphism defined by point evaluations at all points in a sufficiently dense finite set. Let $\xi^{'}:A^{'}\rightarrow M_{k}(A^{'})$ be any simple embedding. Since $\alpha_{2}$ take trivial projections to trivial projections(see Remark 1.6.20 of \[G4\]), we know that $$\xymatrix{
A^{'}\ar[rr]^{\xi^{'}} & & M_{k}(A^{'}) & & \\
& & & & & & \\
B_{2}\ar[rr]^{\xi}\ar[uu]^{\alpha_{2}} & & M_{k}(B_{2})\ar[uu]_{\alpha_{2}\otimes id_{k}}
& & \\
}$$ commutes up to homotopy. Therefor replacing $\beta$, $\phi$, $\psi$, and $\alpha_2$, by $\xi\circ\beta$, $\xi^{'}\circ\phi$, $\xi\circ\psi$, and $\alpha_{2}\otimes id_{k}$, respectively, we can assume our original diagram (I) also satisfies\
(iv) $\omega(\beta(F)\cup\psi(G))<\varepsilon$.
Without loss of generality, we can assume $B_{1}$ has only one block $Y=T_{II,k}, T_{III,k}$ or $S^{2}$. Regarding the homotopy $\Psi$ as the homotopy path $\phi_{t}$ in Corollary 3.11, we can obtain $L_{1}$ as the number $L$ in Corollary 3.11. Similarly regarding the homotopy $\Phi$ as $\phi_{t}$ in Proposition 3.10 we obtain $L_{2}$ (as $L$) and $\eta$.
Choose $\{x_{1},x_{2},\cdot\cdot\cdot,x_{m}\}\subset X$ to be a $\eta$-dense subset. Define $$\lambda_{1}:A(=M_{N}(C(X)))\rightarrow M_{mN}(B_{2})$$ by $$\lambda_{1}(f)=diag(\textbf{1}_{B_{2}}\otimes f(x_{1}),\textbf{1}_{B_{2}}\otimes f(x_{2}),\cdot\cdot\cdot,\textbf{1}_{B_{2}}\otimes f(x_{m}))$$
Let $L{'}=max\{L_{1},L_{2}\}$ and $n=mNL{'}$. Define $$\lambda:A\longrightarrow M_{n}(B_{2})=M_{L{'}}(M_{mN}(B_{2}))\;\;by$$ $$\lambda=diag(\underbrace{\lambda_{1},\lambda_{1},\cdot\cdot\cdot,\lambda_{1}}\limits_{L^{\prime}})$$\
Then $\lambda\circ\alpha_{1}:B_{1}\longrightarrow M_{n}(B_{2})$ satisfies the condition for $\lambda$ in Corollary 3.11 for the homotopy $\Psi$ and the positive integer $L_1$. Also $(\alpha_{2}\otimes id_{n})\circ \lambda:A\longrightarrow M_{n}(A^{\prime})$ satisfies the condition for $\lambda$ in Proposition 3.10 for the homotopy $\Phi, L_{2}$ and $\eta$. Therefor there are $u_{1}\in M_{n+1}(B_{2})$ and $u_{2}\in M_{n+1}(A^{\prime})$ such that $$\|(\beta\oplus\lambda)\circ\alpha_{1}(g)-u_{1}((\psi\oplus\lambda\circ\alpha_{1})(g))u^{*}_{1}\|\leq5\varepsilon_{1},~~~\forall g\in \widetilde{G_{1}} \eqno (\ast)$$\
and $$\|(\phi\oplus((\alpha_{2}\otimes id_{n})\circ \lambda))(f)-u_{2}((\alpha_{2}\otimes id_{n+1})\circ(\beta\oplus\lambda)(f))u^{*}_{2}\|<\varepsilon,~~~ \forall f\in F_{1}$$ Note that if $Y_{i}=\{pt\}, [0,1]$ or $S^{1}$, then $\beta\circ\alpha_{1}$ itself is close to a homomorphism $\psi{'}:B_{1}\rightarrow B_{2}$. That is, we can replace the above $(*)$ by $$\|\beta\circ\alpha_{1}(g)-\psi{'}(g)\|<\varepsilon,~~~ \forall g\in G_{1}. \eqno (\ast\ast)$$ In the diagram (I) if we replace $B_{2}$ by $M_{n+1}(B_{2}), A^{\prime}$ by $M_{n+1}(A^{\prime}), \psi$ by $Adu_{1}\circ(\psi\oplus\lambda\circ\alpha_{1})$ (or $\psi$ by $\psi{'}\oplus(\lambda\circ\alpha_{1})$ for the case $Y_{i}=\{pt\}, [0,1],S^{1}$ using $(**)$), $\beta$ by $\beta\oplus\lambda, \alpha_{2}$ by $Adu_{2}\circ(\alpha_{2}\otimes id_{n+1})$ and finally $\phi$ by $\phi\oplus((\alpha_{2}\otimes id_{n})\circ\lambda)$, we get the desired diagram.\
**3.13** Recall that in 1.1.7(h) of \[G4\], for $A=\bigoplus^{t}_{i=1}M_{k_{i}}(C(X_{i}))$, where $X_{i}$ are path connected simplicial complexes, we used the notation $r(A)$ to denote $\bigoplus^{t}_{i=1}M_{k_{i}}(\mathbb{C})$, which could be considered to be the subalgebra consisting of t-tuples of constant functions from $X_{i}$ to $M_{k_{i}}(\mathbb{C})(i=1,2,...,t)$. Fixed a base point $x^{0}_{i}\in X_{i}$ for each $X_{i}$, one defines a map $r:A\rightarrow r(A)$ by $$r(f_{1},f_{2},...,f_{t})=(f_{1}(x^{0}_{1}),f_{2}(x^{0}_{2}),\cdot\cdot\cdot,f_{t}(x^{0}_{t}))\in r(A)$$ We have the following corollary.
**Corollary 3.14** Let $B_{1}=\bigoplus^{s}_{j=1}M_{k(j)}(C(Y_{j}))$, where $Y_{j}$ are the spaces $\{pt\}, [0,1], S^{1}$, $\{T_{II,k}\}_{k}$, $\{T_{III,k}\}_{k}$ and $S^{2}$. Let $A=\bigoplus^{t}_{j=1}M_{l(j)}(C(X_{j}))$, where $X_{j}$ are connected simplicial complexes. Let $\alpha_{1}:B_{1}\rightarrow A$ be any homomorphism. Let $\varepsilon_{1}>\varepsilon_{2}>0$ be any two positive numbers. Let $\widetilde{E}(=\bigoplus\widetilde{E}^{i})\subset E(=\bigoplus E^{i})\subset B_{1}(=\bigoplus B_1^{i})$ be two finite subsets with the condition $$\omega(\widetilde{E}^{i})<\varepsilon_{1}\;\; for\; all \;Y_{i}=T_{II,k}, T_{III,k}\; or\; S^{2}.$$ Let $F\subset A$ be any finite subset, $\delta>0$. Then there exists a diagram $$\xymatrix{
A\ar[rr]^{\phi\oplus r}\ar[rdrd]^{\beta\oplus r} & & A^{'}\oplus r(A) & & \\
& & & & & & \\
B_{1}\ar[rr]^{\psi\oplus(r\circ\alpha_{1})}\ar[uu]^{\alpha_{1}} & & B_{2}\oplus r(A)\ar[uu]_{\alpha_{2}\oplus id} & & \\
}$$ where $A^{'}=M_{L}(A), B_{2}$ is a direct sum of matrix algebras over the spaces $\{pt\}, [0,1]$, $S^{1}$, $\{T_{II,k}\}, \{T_{III,k}\}$ and $S^{2}$, with the following properties.\
(1) $\psi$ is a homomorphism, $\alpha_{2}$ is injective homomorphism and $\phi$ is a unital simple embedding (see 2.23).\
(2) $\beta\in Map(A,B_{2})_{1}$ is $F-\delta$ multiplicative.\
(3) For $g\in\widetilde{E}^{i}$ with $Y_{i}=T_{II,k}, T_{III,k}$ or $S^{2}$, $$\|(\beta\oplus r)(\alpha_1(g))-(\psi\oplus(r\circ\alpha_{1}))(g)\|<5\varepsilon_{1};$$ for $g\in E^{i}(\supset\widetilde{E}^{i})$ with $Y_{i}=\{pt\}, [0,1], S^{1}$, $$\|(\beta\oplus r)(\alpha_1(g))-(\psi\oplus r\circ\alpha_{1})(g)\|<\varepsilon_{1};$$ for all $f\in F$, $$\|(\alpha_{2}\oplus id)\circ(\beta\oplus r)(f)-(\phi\oplus r)(f)\|<\varepsilon_{1}.$$ (4) $\omega(\beta(F)\cup\psi(E))<\varepsilon_{2}$.
The following lemma will be used in the proof of our main theorem.
**Lemma 3.15** Let $lim(A_{n}=\bigoplus^{k_{n}}_{i=1}M_{[n,i]}C(X_{n,i}),\phi_{n,m})$ be an $AH$ inductive system for which the limit $C^{*}$-algebra has the ideal property, with $X_{n,i}$ connected simplicial complexes and with $\sup\limits_{n,i}dim(X_{n,i})<+\infty$. Let $P_{0}\in A_{n}$ be a trivial projection. Let $A=P_{0}A_{n}P_{0}(=\bigoplus_{i}P^{i}_{0}A^{i}_{n}P^{i}_{0})$. Let $A^{'}=M_{L}(A), r(A)$ be as in 3.13 and 3.14 with $r:A\longrightarrow r(A)$. Let $\phi:A\longrightarrow A^{'}$ be a unital simple embedding (see 2.23). Then there exist $m>n$, and a unital homomorphism $$\lambda:M_{L}(A)\oplus r(A)\longrightarrow\phi_{n,m}(P_{0})A_{m}\phi_{n,m}(P_{0})$$ such that $\lambda\circ(\phi\oplus r)$ is homotopic to $\phi_{n,m}|_{P_{0}A_{n}P_{0}}:$ $P_{0}A_{n}P_{0}\longrightarrow\phi_{n,m}(P_{0})A_{m}\phi_{n,m}(P_{0})$. That is the following diagram commutes up to homotopy $$\xymatrix{
A(=P_{0}A_{n}P_{0})\ar[rr]^{\phi_{n,m}}\ar[rdrd]^{\phi\oplus r} & & \phi_{n,m}(P_{0})A_{m}\phi_{n,m}(P_{0}) & & \\
& & & & & & \\
& & M_{L}(A)\oplus r(A)\ar[uu]_{\lambda} & & \\
}$$
One can construct the maps inside each block for each $\phi^{i,j}_{n,m}:P^{i}_{0}A^{i}_{n}P^{i}_{0}\longrightarrow\phi^{i,j}_{n,m}(P^{i}_{0})A^{j}_{m}\phi^{i,j}_{n,m}(P^i_0)$. Then the lemma follows from Lemma 1.6.30 of \[G4\] and Lemma 2.20. Note that we use the fact that if $\xi:M_{\bullet}(C(X))\longrightarrow B$ has finite dimensional image, then there is a map $\xi^{'}:r(M_{\bullet}C(X))\longrightarrow B$ such that $\xi^{'}\circ r$ is homotopic to $\xi$.\
**§4. The proof of the main theorem**
The following lemma is one of the versions of the Elliott intertwining argument (see \[Ell1\]) or Proposition 3.1 of \[D\].
**Proposition 4.1** Consider the diagram $$\xymatrix{
A_{1}\ar[rr]^{\phi_{1,2}}\ar[rdrd]^{\beta_{1}} & & A_{2}\ar[rr]^{\phi_{2,3}}\ar[rdrd]^{\beta_{2}} & & A_{3}\ar[rr]
& & \cdot\cdot\cdot\ar[rr] & & A_{n}\ar[rr]^{\phi_{n,n+1}}\ar[rdrd]^{\beta_{n}} & & \cdot\cdot\cdot & & \\
& & & & & & \\
B_{1}\ar[rr]_{\psi_{1,2}}\ar[uu]_{\alpha_{1}} & & B_{2}\ar[rr]_{\psi_{2,3}}
\ar[uu]^{\alpha_{2}} & & B_{3} \ar[rr] & & \cdot\cdot\cdot\ar[rr] & & B_{n}\ar[rr]_{\psi_{n,n+1}}\ar[uu]^{\alpha_{n}} & & \ar[uu]_{\alpha_{n+1}}\cdot\cdot\cdot & & \\
}$$ where $A_{n}$,$B_{n}$ are $C^{*}$-algebras $\phi_{n,n+1}, \psi_{n,n+1}$ are homomorphisms and $\alpha_{n}, \beta_{n}$ are linear $*$-contraction. Suppose that $F_{n}\subset A_{n}, \widetilde{E_{n}}\subset E_{n}\subset B_{n}$ are finite sets satisfying the following condition $$\phi_{n,n+1}(F_{n})\cup\alpha_{n+1}(E_{n+1})\subset F_{n+1}, \psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\subset \widetilde{E_{n+1}}$$ and $\overline{\bigcup^{\infty}_{n=1}\phi_{n,\infty}(F_{n})}$ and $\overline{\bigcup^{\infty}_{n=1}\psi_{n,\infty}(E_{n})}$ are the unit balls of $A=lim(A_{n},\phi_{n,m})$ and $B=lim(B_{n},\psi_{n,m})$, respectively. Suppose that there is a sequence $\varepsilon_{1}, \varepsilon_{2}...$ of positive numbers with $\sum\varepsilon_{n}<+\infty$ such that $\alpha_{n}$ and $\beta_{n}$ are $E_{n}-\varepsilon_{n}$ and $F_{n}-\varepsilon_{n}$ multiplicative, respectively, and $$\|\phi_{n,n+1}(f)-\alpha_{n+1}\circ\beta_{n}(f)\|<\varepsilon_{n},$$ and $$\|\psi_{n,n+1}(g)-\beta_{n}\circ\alpha_{n}(g)\|<\varepsilon_{n},$$ for all $f\in F_{n}$ and $g\in\widetilde{E_{n}}$.\
Then $A$ is isomorphic to $B$.\
The following is the main theorem of this article.
**Theorem 4.2** Suppose that $lim(A_{n}=\bigoplus^{t_{n}}_{i=1}M_{[n,i]}C(X_{n,i}),\phi_{n,m})$ is an $AH$ inductive limit with $dim(X_{n,i})\leq M$ for a fixed positive integer $M$ such that the limit algebra has the ideal property. Then there is another inductive limit system$(B_{n}=\bigoplus^{s_{n}}_{i=1}M_{\{n,i\}}C(Y_{n,i}), \psi_{n,m})$ with the same limit algebra as the above system, where all the $Y_{n,i}$ are spaces of the form $\{pt\}, [0,1], S^{1}, S^{2}$, $T_{\uppercase\expandafter{\romannumeral2},k}$, $T_{\uppercase\expandafter{\romannumeral3},k}$.
Without loss of generality, assume that the spaces $X_{n,i}$ are connected finite simplicial complexes and $\phi_{n,m}$ are injective (see \[EGL2\]).
Let $\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}>\cdot\cdot\cdot>0$ be a sequence of positive numbers satisfying $\sum\varepsilon_{n}<+\infty$. We need to construct the intertwining diagram $$\xymatrix@R=0.5ex{
F_{1} & & F_{2} & & & & F_{n} & & F_{n+1}\\
\bigcap & & \bigcap & & & & \bigcap & & \bigcap\\
A_{s(1)} \ar[rdrd]^{\beta_{1}} \ar[rr]^{\phi_{s(1),s(2)}} & & A_{s(2)} \ar[rdrd]^{\beta_{2}} \ar[rr]^{\phi_{s(2),s(3)}} & &\cdots \ar[rr] & & A_{s(n)} \ar[rdrd]^{\beta_{n}} \ar[rr]^{\phi_{s(n),s(n+1)}} & & A_{s(n+1)} \ar[rr] \ar[rdrd] & & \cdots\\
& & & & & &\\
B_{1} \ar[rr]^{\psi_{1,2}} \ar[uu]^{\alpha_{1}} & & B_{2} \ar[uu]^{\alpha_{2}} \ar[rr]^{\psi_{2,3}} & &\cdots \ar[rr] & & B_{n}\ar[rr]^{\psi_{n,n+1}} \ar[uu]^{\alpha_{n}} & & B_{n+1} \ar[uu]^{\alpha_{n+1}} \ar[rr] & & \cdots\\
\bigcup & & \bigcup & & & & \bigcup & & \bigcup\\
E_{1} & & E_{2} & & & & E_{n} & & E_{n+1}\\
\bigcup & & \bigcup & & & & \bigcup & & \bigcup\\
\widetilde{E_{1}} & & \widetilde{E_{2}} & & & & \widetilde{E_{n}} & & \widetilde{E_{n+1}}
}$$
satisfying the following conditions\
(0.1) $(A_{s(n)},\phi_{s(n),s(m)})$ is a subinductive system of $(A_{n}, \phi_{n,m})$, and $(B_{n}, \psi_{n,m})$ is an inductive system of matrix algebras over the spaces $\{pt\}, [0,1], S^{1}, {T_{\uppercase\expandafter{\romannumeral2},k}} , {T_{\uppercase\expandafter{\romannumeral3},k}},S^{2}.$\
(0.2) Choose $\{a_{ij}\}^{\infty}_{j=1}\subset A_{s(i)}$ and $\{b_{ij}\}^{\infty}_{j=1}\subset B_{i}$ to be countable dense subsets of the unit balls of $A_{s(i)}$ and $B_{i}$, respectively. $F_{n}$ are subsets of the unit balls of $A_{s(n)}$, and $\widetilde{E_{n}}\subset E_{n}$ are both subsets of the unit balls of $B_{n}$ satisfying $$\phi_{s(n),s(n+1)}(F_{n})\cup\alpha_{n+1}(E_{n+1})\cup\bigcup^{k+1}_{i=1}\phi_{s(i),s(n+1)}(\{a_{i1},a_{i2}\cdot\cdot\cdot a_{in+1}\})\subset F_{n+1}$$ $$\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\subset \widetilde{E}_{n+1}\subset E_{n+1}$$ and $$\bigcup^{n+1}_{i=1}\psi_{i,n+1}(\{b_{i1},b_{i2}\cdot\cdot\cdot b_{in+1}\})\subset E_{n+1}$$ (0.3) $\beta_{n}$ are $F_{n}-2\varepsilon_{n}$ multiplicative and $\alpha_{n}$ are homomorphisms\
(0.4) $\|\psi_{n,n+1}(g)-\beta_{n}\circ\alpha_{n}(g)\|<8\varepsilon_{n}$ for all $g\in\widetilde{E}_{n}$ and $\|\phi_{s(n),s(n+1)}(f)-\alpha_{n+1}\circ\beta_{n}(f)\|<14\varepsilon_{n}$ for all $f\in F_{n}$\
(0.5) For any block $B^{i}_{n}$ with spectrum $T_{\uppercase\expandafter{\romannumeral2},k},T_{\uppercase\expandafter{\romannumeral3},k},S^{2},\omega(\widetilde{E}_{n}^{i})<\varepsilon_{n}$, where $\widetilde{E}_{n}^{i}=\pi_{i}(\widetilde{E}_{n})$ for $\pi_{i}:B_{n}\longrightarrow B^{i}_{n}$ the canonical projections.
The diagram will be constructed inductively. First, let $B_{1}=\{0\}, A_{s(1)}=A_{1}, \alpha_{1}=0$. Let $b_{1j}=0\in B_{1}$ for $j=1,2,...$ and let $\{a_{1j}\}^{\infty}_{j=1}$ be a countable dense subset of the unit ball of $A_{s(1)}$. And let $\widetilde{E}_{1}=E_{1}=\{b_{11}\}=B_{1}$ and $F_{1}=\bigoplus^{t_{1}}_{i=1}F^{i}_{1}$, where $F^{i}_{1}=\pi_{i}(\{a_{11}\})\subset A^{i}_{1}$.\
As inductive assumption, assume that we already have the diagram
$$\xymatrix@R=0.5ex{
F_{1} & & F_{2} & & & & F_{n}\\
\bigcap & & \bigcap & & & & \bigcap\\
A_{s(1)} \ar[rdrd]^{\beta_{1}} \ar[rr]^{\phi_{s(1),s(2)}} & & A_{s(2)} \ar[rdrd]^{\beta_{2}} \ar[rr]^{\phi_{s(2),s(3)}} & &\cdots \ar[rr] \ar[rdrd]^{\beta_{n-1}} & & A_{s(n)}\\
& & & & & &\\
B_{1} \ar[rr]^{\psi_{1,2}} \ar[uu]^{\alpha_{1}} & & B_{2} \ar[uu]^{\alpha_{2}} \ar[rr]^{\psi_{2,3}} & &\cdots \ar[rr] & & B_{n}\ar[uu]^{\alpha_{n}}\\
\bigcup & & \bigcup & & & & \bigcup\\
E_{1} & & E_{2} & & & & E_{n}\\
\bigcup & & \bigcup & & & & \bigcup\\
\widetilde{E_{1}} & & \widetilde{E_{2}} & & & & \widetilde{E_{n}}
}$$ and for each $i=1,2,\cdot\cdot\cdot,n$, we have dense subsets $\{a_{ij}\}^{\infty}_{j=1}\subset$ the unit ball of $A_{s(i)}$ and $\{b_{ij}\}^{\infty}_{j=1}\subset$ the unit ball of $B_{i}$ satisfying the conditions (0.1)-(0.5) above. We have to construct the next piece of the diagram $$\xymatrix@!C{
F_{n}\subset A_{s(n)}\ar[rr]^{\phi_{s(n),s(n+1)}}\ar[rdrd]^{\beta_{n}} & & A_{s(n+1)}\supset F_{n+1} & & \\
& & & & & & \\
\widetilde{E}_{n}\subset E_{n}
\subset B_{n}\ar@<-8mm>[uu]^{\alpha_{n}}\ar[rr]_{\psi_{n,n+1}} & &
\;\;B_{n+1}\ar@<10mm>[uu]_{\alpha_{n+1}}\supset
E_{n+1}\supset\widetilde{E}_{n+1} & & \\
}$$ to satisfy the conditions (0.1)-(0.5).
Among the conditions for the induction assumption, we will only use the conditions that $\alpha_{n}$ is a homomorphism and (0.5) above.\
**Step 1**. We enlarge $\widetilde{E}_{n}$ to $\bigoplus_{i}\pi_{i}(\widetilde{E}_{n})$ and $E_{n}$ to $\bigoplus_{i}\pi_{i}(E_{n})$. Then $$\widetilde{E}_{n}(=\bigoplus\widetilde{E}_{n}^{i})\subset E_{n}(=\bigoplus E_{n}^i)$$ and for each $B^{i}_{n}$ with spectrum $T_{\uppercase\expandafter{\romannumeral2},k},T_{\uppercase\expandafter{\romannumeral3},k},S^{2}$, we have $\omega(\widetilde{E}_{n}^{i})<\varepsilon_{n}$ from the induction assumption (0.5). By Theorem 3.9 applied to $\alpha_{n}:B_{n}\rightarrow A_{s(n)},\widetilde{E}_{n}\subset E_{n}\subset B_{n},F_{n}\subset A_{s(n)}$ and $\varepsilon_{n}>0$, there are $A_{m_{1}}(m_{1}>s(n))$, two orthogonal projections $P_{0},P_{1}\in A_{m_{1}}$ with $\phi_{s(n),m_{1}}(\textbf{1}_{A_{s(n)}})=P_{0}+P_{1}$ and $P_{0}$ trivial, a $C^{*}$-algebra $C$—a direct sum of matrix algebras over $C[0,1]$ or $\mathbb{C}$, a unital map $\theta\in Map(A_{s(n)},P_{0}A_{m_{1}}P_{0})_{1}$, a unital homomorphism $\xi_{1}\in Hom(A_{s(n)},C)_{1}$, a unital homomorphism $\xi_{2}\in Hom(C,P_{1}A_{m_{1}}P_{1})_{1}$ and a homomorphism $\alpha\in Hom(B_n, P_0A_{m_1}P_0)$ such that\
(1.1) $\|\phi_{s(n),m_{1}}(f)-\theta(f)\oplus(\xi_{2}\circ\xi_{1})(f)\|<\varepsilon_{n}$ for all $f\in F_{n}$.\
(1.2) $\theta$ is $F_{n}-\varepsilon_{n}$ multiplicative and $F:=\theta(F_{n})$ satisfies $\omega(F)<\varepsilon_{n}$.\
(1.3) $\|\alpha(g)-\theta\circ\alpha_{n}(g)\|<3\varepsilon_{n}$ for all $g\in\widetilde{E}_{n}$.
Let all the blocks of $C$ be parts of the $C^{\ast}$-algebra $B_{n+1}$. That is $$B_{n+1}=C\oplus(some\; other\; blocks).$$ The map $\beta_{n}:A_{s(n)}\rightarrow B_{n+1}$, and the homomorphism $\psi_{n,n+1}:B_{n}\rightarrow B_{n+1}$ are defined by $\beta_{n}=\xi_{1}:A_{s(n)}\rightarrow C(\subset B_{n+1})$ and $\psi_{n,n+1}=\xi_{1}\circ\alpha_{n}:B_{n}\rightarrow C(\subset B_{n+1})$ for the blocks of $C(\subset B_{n+1})$. For this part, $\beta_{n}$ is also a homomorphism.\
**Step 2**. Let $A=P_{0}A_{m_{1}}P_{0},F=\theta(F_{n})$. Since $P_{0}$ is a trivial projection, $$A\cong\oplus M_{l_{i}}(C(X_{m_{1},i})).$$ Let $r(A):=\bigoplus M_{l_{i}}(\mathbb{C})$ and $r:A\rightarrow r(A)$ be as in 3.13. Applying Corollary 3.14 to $\alpha:B_{n}\rightarrow A,\widetilde{E}_{n}\subset E_{n}\subset B_{n}$ and $F\subset A$, we obtain the following diagram $$\xymatrix{
A\ar[rr]^{\phi\oplus r}\ar[rdrd]^{\beta} & &M_{L}(A)\oplus r(A) & & \\
& & & & & & \\
B_{n}\ar[rr]^{\psi}\ar[uu]^{\alpha} & & B\ar[uu]_{\alpha{'}} & & \\
}$$ such that\
(2.1) $B$ is a direct sum of matrix algebras over $\{pt\},[0,1],S^{1},T_{\uppercase\expandafter{\romannumeral2},k},T_{\uppercase\expandafter{\romannumeral3},k}$ or $S^{2}$.\
(2.2) $\alpha^{\prime}$ is an injective homomorphism and $\beta$ is $F-\varepsilon_{n}$ multiplicative.\
(2.3) $\phi:A\rightarrow M_{L}(A)$ is a unital simple embedding and $r:A\rightarrow r(A)$ is as in 3.13.\
(2.4) $\|\beta\circ\alpha(g)-\psi(g)\|<5\varepsilon_{n}$ for all $g\in \widetilde{E}_{n}$ and $\|(\phi\oplus r)(f)-\alpha{'}\circ\beta(f)\|<\varepsilon_{n}$ for all $f\in F(:=\theta(F_{n}))$.\
(2.5) $\omega(\psi(E_{n})\cup\beta(F))<\varepsilon_{n+1}$ (note that $\beta(F)=\beta\circ\theta(F_{n})$).
Let all the blocks B be also part of $B_{n+1}$, that is $$B_{n+1}=C\oplus B\oplus(some\; other\; blocks)$$ The maps $\beta_{n}:A_{s(n)}\longrightarrow B_{n+1},\psi_{n,n+1}:B_{n}\longrightarrow B_{n+1}$ are defined by $$\beta_{n}:=\beta\circ\theta:A_{s(n)}\xrightarrow{\theta}A\xrightarrow{\beta}B(\subset B_{n+1})$$ and $$\psi_{n, n+1}:=\psi:B_{n}\rightarrow B(\subset B_{n+1})$$ for the blocks of $B(\subset B_{n+1})$. This part of $\beta_{n}$ is $F_{n}-2\varepsilon_{n}$ multiplicative, since $\theta$ is $F_{n}-\varepsilon_{n}$ multiplicative, $\beta$ is $F-\varepsilon_{n}$ multiplicative and $F=\theta(F_{n})$.\
**Step 3**. By Lemma 3.15 applied to $\phi\oplus r:A\rightarrow M_{L}(A)\oplus r(A)$, there is an $A_{m_{2}}$ and there is a unital homomorphism $$\lambda:M_{L}(A)\oplus r(A)\rightarrow RA_{m_{2}}R,$$ where $R=\phi_{m_{1},m_{2}}(P_{0})$ (write $R$ as $\bigoplus_{j}R^{j}\in\bigoplus_{j}A^{j}_{m}$) such that the diagram $$\xymatrix{
A(=P_{0}A_{m}P_{0})\ar[rr]^{\phi_{m_{1},m_{2}}}\ar[rdrd]^{\phi\oplus r} & & RA_{m_{2}}R & & \\
& & & & & & \\
& & M_{L}(A)\oplus r(A)\ar[uu]_{\lambda} & & \\
}$$ satisfies the following condition:\
(3.1) $\lambda\circ(\phi\oplus r)$ is homotopy equivalent to $$\phi{'}:=\phi_{m_{1},m_{2}}|_{A}.$$ **Step 4**. Applying Theorem 1.6.9 of \[G4\] to the finite set $F\subset A$ (with $\omega(F)<\varepsilon_{n}$) and to the two homotopic homomorphisms $\phi{'}$ and $\lambda\circ(\phi\oplus r):A\rightarrow RA_{m_{2}}R$ (with $RA_{m_{2}}R$ in place of $C$ in Theorem 1.6.9 of \[G4\]), we obtain a finite set $F{'}\subset RA_{m_{2}}R, \delta>0$ and $L>0$ as in the Theorem.
Let $G:=\psi(E_{n})\cup\beta(F)$. From (2.5), we have $\omega(G)<\varepsilon_{n+1}<\varepsilon_{n}$. By Theorem 3.8 applied to $RA_{m_{2}}R$ and $$\lambda\circ\alpha{'}:B\rightarrow RA_{m_{2}}R$$ finite set $G\subset B$, $F{'}\cup\phi^{\prime}(F)\subset RA_{m_{2}}R$, $min \{\varepsilon_{n},\delta\}>0$ (in place of $\varepsilon$) and $L>0$, there are $A_{s(n+1)}$, mutually orthogonal projections $Q_{0},Q_{1},Q_{2}\in A_{s(n+1)}$ with $\phi_{m_{2},s(n+1)}(R)=Q_{0}\oplus Q_{1}\oplus Q_{2}$, a $C^{*}$-algebra $D$—a direct sum of matrix algebras over C\[0,1\] or $\mathbb{C}$—, a unital map $\theta_{0}\in$ Map$(RA_{m_{2}}R,Q_{0}A_{s(n+1)}Q_{0})_1$, and four unital homomorphisms $$\theta_{1}\in Hom(RA_{m_{2}}R,Q_{1}A_{s(n+1)}Q_{1})_{1},\xi_{3}\in Hom(RA_{m_{2}}R,D)_{1},\xi_{4}\in Hom(D,Q_{2}A_{s(n+1)}Q_{2})_{1}$$ and $\alpha{''}\in Hom(B,(Q_{0}+Q_{1})A_{s(n+1)}(Q_{0}+Q_{1}))_{1}$ such that the following statements are true.\
(4.1) $\|\phi_{m_{2},s(n+1)}(f)-((\theta_{0}+\theta_{1})\oplus\xi_{4}\circ\xi_{3})(f)\|<\varepsilon_{n}$ for all $f\in\phi_{m_{1},m_{2}}|_{A}(F)\subset RA_{m_{2}}R$.\
(4.2) $\|\alpha{''}(g)-(\theta_{0}+\theta_{1})\circ\lambda\circ\alpha{'}(g)\|<3\varepsilon_{n+1}<3\varepsilon_{n}$ for all $g\in G$.\
(4.3) $\theta_{0}$ is $F{'}-min(\varepsilon_{n},\delta)$ multiplicative and $\theta_{1}$ satisfies that $$\theta^{i,j}_{1}([q])>L\cdot[\theta^{i,j}_{0}(R^{i})]$$ for any non zero projection $q\in R^{i}A_{m_{1}}R^{i}$.\
By Theorem 1.6.9 of \[G4\], there is a unitary $u\in(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}\oplus Q_{1})$ such that $$\|(\theta_{0}+\theta_{1})\circ\phi{'}(f)-Adu\circ(\theta_{0}+\theta_{1})\circ\lambda\circ(\phi\oplus r)(f)\|<8\varepsilon_{n}$$ for all $f\in F$.\
Combining with second inequality of (2.4), we have\
(4.4) $\|(\theta_{0}+\theta_{1})\circ\phi{'}(f)-Adu\circ(\theta_{0}+\theta_{1})\circ\lambda\circ\alpha{'}\circ\beta(f)\|<9\varepsilon_{n}$ for all $f\in F$.\
**Step 5**. Finally let all blocks of $D$ be the rest of $B_{n+1}$. Namely, let $$B_{n+1}=C\oplus B\oplus D,$$ where $C$ is from Step 1, $B$ is from Step 2 and $D$ is from Step 4.
We already have the definition of $\beta_{n}:A_{s(n)}\rightarrow B_{n+1}$ and $\psi_{n,n+1}: B_{n}\rightarrow B_{n+1}$ for those blocks of $C\oplus B\subset B_{n+1}$ (from Step 1 and Step 2). The definition of $\beta_{n}$ and $\psi_{n,n+1}$ for blocks of $D$ and the homomorphism $\alpha_{n+1}:C\oplus B\oplus D\rightarrow A_{s(n+1)}$ will be given below.
The part of $\beta_{n}:A_{s(n)}\rightarrow D(\subset B_{n+1})$ is defined by $$\beta_{n}=\xi_{3}\circ\phi^{'}\circ\theta:A_{s(n)}\xlongrightarrow{\theta}A\xlongrightarrow{\phi^{\prime}}RA_{m_{2}}R\xlongrightarrow{\xi_{3}}D$$ (Recall that $A=P_{0}A_{m_{2}}P_{0}$ and $\phi{'}=\phi_{m_{1},m_{2}}|_{A})$. Since $\theta$ is $F_{n}-\varepsilon_{n}$ multiplicative, and $\phi{'}$ and $\xi_{3}$ are homomorphisms, we know this part of $\beta_{n}$ is $F_{n}-\varepsilon_{n}$ multiplicative.
The part of $\psi_{n,n+1}:B_{n}\rightarrow D(\subset B_{n+1})$ is defined by $$\psi_{n,n+1}=\xi_{3}\circ\phi{'}\circ\alpha:B_{n}\xlongrightarrow{\alpha}A\xlongrightarrow{\phi{'}}RA_{m}R\xlongrightarrow{\xi_{3}}D$$ which is a homomorphism.
The homomorphism $\alpha_{n+1}:C\oplus B\oplus D\rightarrow A_{s(n+1)}$ is defined as following.
Let $\phi{''}=\phi_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}}P_{1}}:P_{1}A_{m_{1}}P_{1}\longrightarrow
\phi_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}\phi_{m_{1},s(n+1)}(P_{1})$, where $P_{1}$ is from Step 1. Define $$\alpha_{n+1}|_{C}=\phi{''}\circ\xi_{2}:C\xlongrightarrow{\xi_{2}}P_{1}A_{m_{1}}P_{1}\xlongrightarrow{\phi{''}}\phi_{m_{1},s(n+1)}(P_{1})A_{s(m+1)}\phi_{m_{1},s(n+1)}(P_{1})$$ where $\xi_{2}$ is from Step 1.\
$\alpha_{n+1}|_{B}=Adu\circ\alpha{''}:B\xlongrightarrow{\alpha{''}}(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}+Q_{1})\xlongrightarrow{Adu}(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}+Q_{1})$,
where $\alpha{''}$ is from Step 4, and define $$\alpha_{n+1}|_{D}=\xi_{4}:D\rightarrow Q_{2}A_{s(n+1)}Q_{2}.$$ Finally choose $\{a_{n+1,j}\}^{\infty}_{j=1}\subset A_{s(n+1)}$ and $\{b_{n+1,j}\}^{\infty}_{j=1}\subset B_{n+1}$ to be countable dense subsets of the unit balls of $A_{s(n+1)}$ and $B_{n+1}$, respectively. And choose\
$$F{'}_{n+1}=\phi_{s(n),s(n+1)}(F_{n})\cup\alpha_{n+1}(E_{n+1})\cup\bigcup^{n+1}_{i=1}\phi_{s(i),s(n+1)}(\{a_{i1},a_{i2},\cdot\cdot\cdot, a_{in+1}\})$$\
$$E{'}_{n+1}=\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\cup\bigcup^{n+1}_{i=1}\psi_{i,n+1}(\{b_{i1},b_{i2},\cdot\cdot\cdot, b_{in+1}\})$$\
$$\widetilde{E}'_{n+1}=\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\subset E'_{n+1}.$$ Define $F^{i}_{n+1}=\pi_{i}(F^{\prime}_{n+1})$ and $F_{n+1}=\bigoplus_{i}F^{i}_{n+1}$, $E^{i}_{n+1}=\pi_{i}(E{'}_{n+1})$ and $E_{n+1}=\oplus_{i}E^{i}_{n+1}$. For those blocks $B^{i}_{n+1}$ inside the algebra $B$ define $\widetilde{E}_{n+1}^{i}=\pi_{i}(\widetilde{E}'_{n+1})$. For those blocks inside $C$ and $D$, define $\widetilde{E}_{n+1}^{i}=E^{i}_{n+1}$. And finally let $\widetilde{E}_{n+1}=\bigoplus_{i}\widetilde{E}_{n+1}^{i}$. Note that all the blocks with spectrum $T_{\uppercase\expandafter{\romannumeral2},k},T_{\uppercase\expandafter{\romannumeral3},k}$ and $S^{2}$ are in $B$. And hence (2.5) tells us that for those blocks, $\omega(\widetilde{E}_{n+1}^{i})<\varepsilon_{n+1}$.\
Thus we obtain the following diagram $$\xymatrix@!C{
F_{n}\subset A_{s(n)}\ar[rr]^{\phi_{s(n),s(n+1)}}\ar[drdr]^{\beta_{n}} & & A_{s(n+1)}\supset F_{n+1} & & \\
& & & & & & \\
\widetilde{E}_{n}\subset E_{n}
\subset B_{n}\ar@<-8mm>[uu]^{\alpha_{n}}\ar[rr]_{\psi_{n,n+1}} & &
\;\;B_{n+1}\ar@<12mm>[uu]_{\alpha_{n+1}}\supset
E_{n+1}\supset\widetilde{E}_{n+1}. & & \\
}$$ **Step 6**. Now we need to verify all the conditions (0.1)-(0.5) for the above diagram.
From the end of Step 5, we know (0.5) holds, (0.1)-(0.2) hold from the construction (see the construction of $B, C, D$ in Step 1, 2 and 4, and $\widetilde{E}_{n+1}\subset E_{n+1}, F_{n+1}$ is the end of Step 5). (0.3) follows from the end of Step 1, the end of Step 2 and the part of definition of $\beta_{n}$ for $D$ from Step 5.
So we only need to verify (0.4).
Combining (1.1) with (4.1), we have $$\|\phi_{s(n),s(n+1)}(f)-[(\phi^{''}\circ\xi_{2}\circ\xi_{1})\oplus(\theta_{0}+\theta_{1})\circ\phi^{'}\circ\theta\oplus(\xi_{4}\circ\xi_{3}\circ\phi^{'}\circ\theta)](f)\|
<\varepsilon_{n}+\varepsilon_{n}=2\varepsilon_{n}$$ for all $f\in F_{n}$ (recall that $\phi^{''}=\phi_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}}P_{1}},\phi^{'}:=\phi_{m_{1},m_{2}}|_{P_{0}A_{m_{1}}P_{0}}$).
Combining with (4.2) and (4.4), and the definitions of $\beta_{n}$ and $\alpha_{n+1}$, the above inequality yields $$\|\phi_{s(n),s(n+1)}(f)-(\alpha_{n+1}\circ\beta_{n})(f)\|< 9\varepsilon_{n}+3\varepsilon_{n}+2\varepsilon_{n}=14\varepsilon_{n},~~~\forall f\in F_{n}.$$ Combining (1.3), the first inequality of (2.4) and the definition of $\beta_{n}$ and $\psi_{n,n+1}$, we have $$\|\psi_{n,n+1}(g)-(\beta_{n}\circ\alpha_{n})(g)\|<5\varepsilon_{n}+3\varepsilon_{n}=8\varepsilon_{n},~~~\forall g\in \widetilde{E}_{n}.$$ So we obtain (0.4).\
The theorem follows from Proposition 4.1.\
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Guihua Gong, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei, 050024, China, and\
Department of Mathematics, University of Puerto Rico at Rio Piedras, PR 00936, USA\
email address: guihua.gong@upr.edu\
Chunlan Jiang, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei, 050024, China\
email address: cljiang@hebtu.edu.cn\
Liangqing Li, Department of Mathematics, University of Puerto Rico at Rio Piedras, PR 00936, USA\
email address: liangqing.li@upr.edu\
Cornel Pasnicu, Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA\
email address: Cornel.Pasnicu@utsa.edu
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---
abstract: 'Large, complex, active regions may produce multiple flares within a certain period of one or two days. These flares could occur in the same location with similar morphologies, commonly referred to as “homologous flares”. In 2011 September, active region NOAA 11283 produced a pair of homologous flares on the 6th and 7th, respectively. Both of them were white-light (WL) flares, as captured by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory in visible continuum at 6173 Å which is believed to originate from the deep solar atmosphere.We investigate the WL emission of these X-class flares with HMI’s seeing-free imaging spectroscopy. The durations of impulsive peaks in the continuum are about 4 minutes. We compare the WL with hard X-ray (HXR) observations for the September 6 flare and find a good correlation between the continuum and HXR both spatially and temporally. In absence of RHESSI data during the second flare on September 7, the derivative of the GOES soft X-ray is used and also found to be well correlated temporally with the continuum. We measure the contrast enhancements, characteristic sizes, and HXR fluxes of the twin flares, which are similar for both flares, indicating analogous triggering and heating processes. However, the September 7 flare was associated with conspicuous sunquake signals whereas no seismic wave was detected during the flare on September 6. Therefore, this comparison suggests that the particle bombardment may not play a dominant role in producing the sunquake events studied in this paper.'
author:
- 'Yan Xu, Ju Jing, Shuo Wang, and Haimin Wang'
title: COMPARISON OF EMISSION PROPERTIES OF TWO HOMOLOGOUS FLARES IN AR 11283
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Introduction
============
Observations and modeling have demonstrated that flare energy is released in current sheets where magnetic reconnection occurs (e.g., recent review by @Hudson2011). As a consequence, particles, including electrons and ions, can be accelerated and propagate upward along the open field lines or spiral downward along the closed field lines. The latter group of energetic particles can penetrate down to the chromosphere or even photosphere and generate flare footpoint emissions in HXR and visible continua, which is also known as white-light (WL). It is hard to detect the WL signal during a flare because the flare emission is much weaker than the background of solar radiation. Apparently, most WL emissions are identified in large flares, although they are believed to exist in all flares [@Neidig1989; @Zirin1988]. During the era dominated by ground-based observations, only about 120 WL flares were reported and most of them were above X2 class [@Hudson2006; @Neidig1993b]. @Wang2009 systematically investigated all WLFs with Hinode observations in the G-band at 4305 Å, and found that the cut-off visibility was reached for M1 flares. Using the 1-meter ground-based telescope, @Jess2008 detected the WL emission from a C2 flare. There are limitations on observing WL flares, such as the observing durations, spatial/temporal resolutions, limited field-of-view (FOV), dynamic range of detectors, choice of filters (wavelengths) and seeing conditions for ground-based observations. The newly launched space telescope, Solar Dynamics Observatory (SDO) [@SDO], provides full disk and imaging spectroscopy capabilities and hence increases the chance of catching WL flares.
A substantial amount of work has been undertaken in an attempt to understand the energy transport and release processes. Two fundamental questions need to be answered: 1) Where does the WL emission originate? and 2) What is the energy source of the WL emission? In the literature, two groups of models have been proposed to address these issues concerning WL flares. Considering the fact that WL emission is always associated with HXR emission, the direct heating model, a straight forward model, assumes that the accelerated electrons reach the deeper atmosphere and deposit their energy by collision [@Najita1970; @Hudson1972; @Ding2003b]. According to @Vernazza1981, the photospheric column density exceeds some $10^{23}$ cm$^{−2}$ implying that only electrons with initial energy higher than 600 keV can contribute to the heating of the photosphere [@Xu2012a]. However, there is no sufficient electron flux derived from HXR observations reaching the $\tau_{5000} = 1$ level. Given the energetic difficulties associated with the direct heating model, some models involving secondary effects have been proposed, such as the chromospheric backwarming model [@Hudson1972; @Aboudarham1986; @Metcalf1990] and the H$^{-}$ emission model [@Aboudarham1987; @Machado1989; @Metcalf1990; @Metcalf2003; @Ding1994; @Ding2003b]. Both direct and non-direct heating mechanisms can contribute to a single event. This idea has been demonstrated by the studies of core-halo structures by @Xu2006 [@Xu2012a] and @Isobe2007.
A major flare could release energy exceeding 10$^{32}$ erg [@Hudson2011]. However, this may be a small amount of the total magnetic free energy stored in a huge and complex Sunspot group. As an evidence, such a strong Sunspot group can produce multiple flares during its life cycle. These flares, which originate in the same site are also known as homologous flares. The observations and analyses of homologous flares can be traced back to the time when the data were recorded in films. [@Zirin1967]. presented an observation of a sequence of at least 10 flares occurring in one single active region during a 25-hour window. In the digital era, examples of homologous flares can be found in many studies, @Zhang2002 [@Sui2004; @Takasaki2004; @Luoni2006; @Meshalkina2009; @Kumar2010; @Chandra2011]. Active region NOAA 10486 produced the famous Halloween events in 2003, among which at least two flares occurred at the same location on 2003 October 29 and November 02 [@Xu2006].
To further understand the energy transport and release processes, it is crucial to investigate the size and brightness of WL flare kernels [@Fletcher2007]. Two conjugate ribbons are commonly observed for most flares in H$\alpha$ or EUV wavelengths. In radio and HXR observations, which are the the direct diagnostics of electron beams, footpoint sources are commonly detected except for a few special cases [@LiuC2007b] due to relative coarse resolution and non-focus imaging methods. With WL observations, the flare kernels, if detected, are usually fully resolved as footpoint-like cores and ribbon-like halos [@Xu2006]. The characteristic size of the flare core could be as small as 0.7 in the near Infrared and increases to 2 in G-band. This trend in principle resembles a converging flare loop and favors the direct heating model because other models do not predict such a converging variation [@Xu2012a]. Therefore, multi-wavelength observations in WL will provide a diagnosis of direct heating model.
On the other hand, solar flares are not isolated events. They are always associated with other eruptive phenomena, such as filament eruptions, CMEs, Moreton waves and sometimes sunquakes. In fact, all of these phenomena are different manifestations of a single eruptive event [@Hudson2011]. Similar to WL flares, sunquakes are photospheric phenomena observed as propagating wavefronts. @Wolff1972 predicted the existence of sunquakes generated by energetic particles during the impulsive phase of flares. @Kosovichev1998 obtained the first observational evidence of sunquakes associated with an X-class flare on 1996 July 9. Further observations were reported by @Besliu-Ionescu2005; @Donea1999 [@Donea2005; @Donea2006]; @Kosovichev2007 and @Zharkova2007. Previous observations have shown that the sunquake sources were cospatial with HXR or $\gamma$-ray flare kernels [@Besliu-Ionescu2005; @Moradi2007; @Zharkova2007; @Martinez-Oliveros2008], indicating a close relationship between sunquakes and energetic particles. @Donea2006 found that many sunquakes coincide with WL flares. The authors believed that the back-warming mechanism is not only responsible for the WL flare emission, but also the energy source of sunquakes. Besides the models related to electron beams, some other mechanisms can contribute to generate sunquakes as well. @Zharkova2005 [@Hudson2008] believe magnetic reconfiguration or perturbation of flux ropes [@Zharkov2011] can generate sunquakes.
In this paper, we present the study of a pair of homologous X-class WL flares in 2011 September, observed by the Helioseismic and Magnetic Imager (HMI) [@HMI] onboard SDO. These flares occurred close to the disk center and are therefore good candidates for morphological studies. We perform a comprehensive investigation of the first flare (hereafter Flare I) and a comparison to the second flare (hereafter Flare II) associated with a sunquake [@Zharkov2013a], focusing on the following two topics: 1) The basic flare information, such as the contrast enhancement and correlation between WL and HXR emission; 2) The association with sunquakes.
The HMI/SDO observations and data reduction are discussed in §2. Detailed analysis is presented in §3, followed by the summary and discussion in §4.
Observations
============
The primary data that we use for this study are images from SDO observations. There are two channels for WL observations available onboard SDO. The broad band observations centered at 4500 Å ($\pm$ 250 Å) are provided by the Atmospheric Imaging Assembly (AIA) [@AIA]. However, this channel is operating in a guiding/alignment mode while providing very low cadence (about one frame per hour), that is not useful for flare studies.
The observations in visible light near 6173 Å presented in this study are based on the second channel, HMI/SDO observations. This instrument is an imaging spectrometer that takes images at six different spectral points at $\pm$ 34 mÅ, $\pm$ 103 mÅ and $\pm$ 172 mÅ around the Fe I absorption line at 6173.34 Å (referred as 6-point data hereafter). Two types of calibrated data sets are analyzed, namely visible continuum and near real time (NRT) data. [*The visible continuum images*]{} are actually [*derived*]{} from the 6-point data by fitting the line-profile of Fe I 6173.34 Å. During the reconstruction process, one set of 6-point data are integrated using a specific weighting function. Consequently, artificial features may be introduced when generating difference images for flare studies [@MartinezOliveros2011]. To avoid ambiguity, we select reference images at least four minutes before the flare and emphasize more on the qualitative analysis of the continuum emission. Nevertheless, the HMI’s continuum images are good proxies of WL usually obtained using broad band filters. We will use the words ‘continuum’ and ‘WL’ interchangeably hereafter. The image scale of the WL maps is 0.5 per pixel and the effective cadence is about $45$ seconds. The WL data is used for comparison with HXR emission and study of temporal evolution of the flares. [*The NRT data*]{} contains calibrated 6-point line profiles. Different from WL data, it provides spectral information. HMI has two camera systems, the front one is used to produce the line-of-sight observables by scanning six wavelengths at two polarizations (LCP and RCP). For each scan, twelve images are obtained and are spatially aligned via a linearly interpolation with positive weightings [@MartinezOliveros2011; @MartinezOliveros2014]. The side camera is used for retrieving the full Stokes vectors. The NRT data investigated in this paper was taken by the front camera after spatial alignment provided by the HMI team. It is used for the multilayer analysis during the flare peak times.
As the direct diagnosis of electron beams, RHESSI [@RHESSI] HXR data is used to provide supplementary spatial/temporal information of the footpoints in flare I. RHESSI detects HXR emission using nine rotating modulation collimators (RMC). The rotation period is four seconds, which is basically the shortest time period to obtain an image. The spatial resolution depends on the choice of RMC combinations. By selecting the finest RMC \#1, one can achieve a spatial resolution of 2.2 [@Dennis2009]. RMCs with larger numbers are thicker and able to absorb more HXR photons to get better statistics for imaging. In this study, RMC 1-7 are selected for HXR imaging using CLEAN method. In absence of RHESSI during the second flare, we use the time derivative of GOES soft X-ray (SXR) light curve as the proxy of the HXR temporal variation.
Results and Analysis
====================
Flare I
-------
Active region (AR) NOAA 11283 produced two large flares from an identical location on September 6 and 7, respectively. Flare I on September 6 was classified as an X2.1 event. According to GOES SXR record, it was initiated around 22:12 UT and peaked at 22:20 UT. In the HXR, there were two consecutive peaks about three minutes apart. In the continuum, we see two flare kernels which are typical for major flares. At the flare time, the active region was approximately located at N126$\arcsec$W290$\arcsec$ from the disk center.
Figure \[f1\] shows HMI continuum images with a FOV (75 by 75) covering the sunspot group in AR 11283. The upper-right panel was taken during the peak of the flare at 22:18:37 UT, on which the flare signal is not obvious in the raw data. By subtracting the reference image obtained before the flare (upper-left panel), we see two flare kernels on the difference image (lower-left panel). Unlike the UV or H$\alpha$ observations, the duration of continuum emission is much shorter. For this flare, the WL flare kernels can only be identified from five frames using the subtraction method. The light curve in WL is plotted together with the RHESSI HXR and GOES SXR light curves in Figure \[lc\]. In HXR, there are two major impulsive peaks three minutes apart. In the WL, creating a light curve is more complicated. Usually, the time sequence of difference images is obtained by subtracting a reference image. At this point, the selection of reference images is crucial. For instance, a light curve with a pre-flare reference could differ significantly from a light curve with a post-flare reference due to some non-flare variations. In addition, selection of “average light curve” or “maximum light curve” is arbitrary, the former represents the variation of overall emission and the latter represents the time profile of core emission [@Xu2006]. To avoid any randomness of selecting reference and reduce the uncertainties involved in the alignments and normalization, we adopt a method of generating light curves by using the high order moments [@Veronig2000]. In Figure \[lc\], the third order moment (skewness), is used as a proxy of the intensity variation. We see one peak in the WL clearly but no fine structures due to the relatively low cadence (45 seconds) comparing to HXR’s four-second time resolution. Nevertheless, the temporal correlation between WL and HXR is confirmed. To verify the spatial correlation between continuum and HXR sources, RHESSI CLEAN images are reconstructed using collimators 1 - 7 in an energy range of 50 - 100 keV. The time interval of each CLEAN image is 30 seconds and overlaps with the HMI observing time. Figure \[WLHXR\] shows the continuum images with the HXR contours, from which we see that the source locations are almost identical. The slight off-set is probably due to the projection effect because the formation heights are different for HXR and WL emission[^1]. We conclude that at HMI’s resolution of $0\arcsec.5$ per pixel, the HXR and WL sources occurred co-spatially and simultaneously. This result is expected as most of the previous observations have shown such a correlation [e.g., @Rust1975; @Hudson1992; @Xu2004b; @MartinezOliveros2011].
The WL emission reached its maximum at 19:19 UT. The strongest radiation came from the south kernel with a contrast enhancement of 24%, which is somehow lower than the previous observations by [@Xu2004b], in which the core enhancements were 45% in the green continuum at 5200 Å. Note that @Xu2004b observed an X10 flare which was much more energetic than this X2.1 flare. Consequently, one would expect stronger electron flux penetrating down to the lower atmosphere and generating intensive emission in the continuum during that X10 flare.
During HXR and WL observations, we usually observe only one pair of the flare kernels, though exceptions are found in some special cases [@LiuC2007b]. This pair of conjugate footpoints could be the most intensive site of energy dissipation at a certain time. Figure \[size\] presents the flare kernels at the peak time in six spectral positions. The NRT spectral data is provided by the HMI team after proper alignment and normalization by their exposure time. Both flare kernels are fitted by a two-dimensional Gaussian function. Similar to @Xu2012a, the FWHM of the minor axis is calculated and defined as the ‘size’ of each flare kernel. As we can see from Table \[kernelsize1\], there is an obvious trend by which the source size increases toward the line center for both the north and south kernels. Such a wavelength-dependent size variation is consistent with the electron heating model as discussed by [@Xu2012a].
[lccccccr]{}\
Spectral & +172 mÅ& +103 mÅ& +34 mÅ& -34 mÅ& -103 mÅ& -172 mÅ& Average\
Position & & & & & & &\
Size (N) & 1.14 & 1.54 & 1.55 & 1.27 & 1.01 & 1.05 & 1.26 $\pm$ 0.24\
Size (S) & 1.00 & 1.46 & 1.87 & 1.55 & 1.14 & 1.04 & 1.34 $\pm$ 0.34\
Flare II in comparison with Flare I
-----------------------------------
Figure \[flare2lc\] presents the light curve of Flare II, which is also a WL flare. The red curve with asterisks shows the temporal profile of HMI WL variation. Unfortunately, this flare occurred during RHESSI’s night time and therefore HXR data is not available. We used the derivative of GOES SXR as a proxy for the HXR light curve. Again, similar to the flare I, we see that the WL emission is temporally correlated with electron precipitation. Figure \[flare2dif\] shows the WL images and flare signals using the subtraction method. The centroid of the flaring area at the peak time is around $N138\arcsec$, $W495\arcsec$. In the two right panels of Figure \[f1\] and Figure \[flare2dif\], the 50% contours of flare sources fitted using a two-dimensional Gaussian function, are plotted on the WL images. It is clear that the northern flare kernels of both flares are located directly above the same sunspot and the southern kernels reside at a ‘gap’ area close to the center of the sunspot group. Table \[homologous\] gives a comparison of the two homologous flares. Flare II was an X1.8 flare and relatively weaker than Flare I that has a GOES SXR class of X2.1. As a result, it is not surprising that Flare II has a relatively low contrast in the continuum.
[lcccccc]{}\
Date & WL Intensity & WL & WL & X-ray & Sunquake & Location\
& Enhancement & Peak Time & Duration & & &\
Sep-06 & 24% & 22:19 UT & $\sim$ 4 min & RHESSI & No & N126, W290\
& & & & GOES & &\
Sep-07 & 20% & 22:37 UT & $\sim$ 4 min & GOES & Yes & N138, W495\
& & & & derivative & &\
However, it is puzzling that the ‘weak’ Flare II coincides with a clear sunquake [@Zharkov2013a], which could not be identified during Flare I based on private communication with Dr. J. Zhao and Dr. S. Zharkov. Besides the seismic waves, there is no significant difference in emission between the two flares. In other words, the emission magnitudes and durations are similar in WL and HXR/SXR derivative (Figure \[goes2\]). The known possible causes of sunquakes include direct particle precipitation, backwarming, shock waves and the Lorentz force. The former three are associated with particle beams. Considering the structure of hosting AR and properties of accelerated particles, these two events should have similar seismic responses. Therefore, we suspect that besides the particle precipitation, there are some other combined effects in generating seismic emission, such as ambient atmospheric condition, and three-dimensional topology of surrounding magnetic fields, which are not well understood at present time.
Figure \[sizeII\] presents the difference images of Flare II at six spectral positions as same as in Figure \[size\] for Flare I. Again, we see that the flare kernels are relatively compact in line wing and overspread in line center observations. Quantitative measurements of the characteristic sizes are listed in Table \[kernelsize2\].
[lccccccr]{}\
Spectral & +172 mÅ& +103 mÅ& +34 mÅ& -34 mÅ& -103 mÅ& -172 mÅ& Average\
Position & & & & & & &\
Size (N) & 1.06 & 1.37 & 1.47 & 1.48 & 1.29 & 1.15 & 1.30 $\pm$ 0.17\
Size (S) & 1.06 & 1.22 & 1.38 & 1.29 & 1.21 & 1.20 & 1.23 $\pm$ 0.11\
Summary and Discussion
======================
In this paper, we studied two homologous X-class WL flares, which occurred on 2011 September 6 and 7 using HMI, RHESSI and GOES observations. We performed a detailed study of Flare I concerning several important aspects, and compared the emission properties of Flare I with Flare II. The findings are summarized and discussed as follows:\
1. The continuum emission obtained by HMI was well correlated with RHESSI HXR observations in Flare I. Once again, this result confirms the close relationship between the WL emission and energetic electrons.
2\. The maximum intensity enhancements were 24% and 20% for the twin flares, which are moderate comparing with previous WL observations, [e.g., @Lin1996; @Xu2004b]. Note that the continuum around Fe I 6173 Å was rarely used for flare studies prior to the launch of HMI/SDO, we have not established a comprehensive database for the WL flares and can not perform detailed statistical analysis until more flares are observed.
3\. Using the 6-point data, the characteristic sizes of flare kernels were measured for both flares. At a certain time, for instance the flare maximum, the source size increased from the line wing to the line center. It is well known that the radiation from the line wing is formed lower than that from the line center. Therefore, the wavelength-dependent size variation indicates that the flare kernels are smaller at lower atmosphere than those at higher layers. This result favors the direct heating model as discussed in @Xu2012a.
4\. Flares I and II have similar properties in WL but Flare I was not accompanied by a sunquake whereas Flare II was. In the literature, there are several models used to explain the physics of sunquakes: (1) Earlier theories [e.g., @Wolff1972] and observations [e.g., @Besliu-Ionescu2005; @Donea2006; @Zharkova2007] find close relationship between sunquake and energetic particles, which are also believed to responsible for WL flares. These energetic particles normally refer to electrons, as protons are much less in number and the $\gamma$-ray sources are found not co-spatial with WL and HXR flare sources [@Hurford2006]. The electron beams can penetrate down to photosphere and generate sunquakes directly as predicted by @Wolff1972. (2) On the other hand, these electrons may affect photosphere through a secondary effect, such as back-warming effect or shock waves, and produce sunquakes [@Donea2006; @Hudson2008]. (3) Besides the electron-related models, there are some models that do not require electron beams to play an important role in producing sunquakes. @Zharkov2011 analyzed the sunquake associated with an X2.2 flare on 2011 February 15. The authors found that the sunquake sources are located far away from the flare center. The discovery suggests that the erupting flux ropes may have an effect on photosphere and generates sunquakes. (4) Based on private communications with Dr. Donea, we learned that sometimes pre-flare heating can create a favorable environment, such as appropriate temperature and density, for sunquakes. With this hypothesis, a flare with pre-heating is more likely to be followed by a sunquake. (5) Another model that is not related to energetic particles is proposed by @Hudson2008 assuming that the magnetic reconfiguration may lead to a sunquake.
In summary, we present two flares with similar pre-flare conditions and WL emission but only one flare is associated with the sunquake. We do not intend to distinguish a particular model from all the models discussed above. Instead, we conclude that the particle precipitation may not be the only cause of the sunquake associated with Flare II. There are some other effects that may also work together in generating seismic emission, such as an ambient atmospheric condition, and topology of surrounding magnetic fields, which are not well understood at present time.
We thank the referee for valuable comments. Obtaining the excellent data would not have been possible without the help of the HMI/SDO and RHESSI teams. YX thanks Dr. Sebastien Couvidat for providing the aligned HMI/SDO 6-point data. This work is supported by NSF-AGS-1153424, NSF-AGS-1250374, NSF-AGE-1153226, and NASA grants NNX13AG13G, NNX13AF76G and NNX11AQ55G.
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![WL images and Magnetogram of Flare I on 2011 September 6. Upper-left panel: HMI continuum image taken before the flare which is used as the reference frame. Upper-right panel: HMI continuum image taken during the X2.1 flare. Lower-left panel: Difference image by subtracting the reference frame from the middle panel. Lower-right panel: HMI line-of-sight magnetogram taken before the flare at 22:08 UT. The red contours in two right panels show the positions of flare sources relative to the sunspot group. \[f1\]](f1.eps)
![Light curves of Flare I on 2011 September 6. Green curve: GOES SXR light curve in 1 - 8 Å. Purple curve: RHESSI HXR light curve in energy band of 50 - 100 keV. Red curve with asterisks: HMI WL light curve. The cadence of HXR light curves is four seconds. The cadence of WL light curve is 45 seconds. All of the light curves are normalized to their peak counts. \[lc\]](f2.eps)
![Difference image in WL with HXR contours of Flare I. The WL images were taken at 22:18:45, 22:19:30 and 22:20:15 UT, respectively. On each WL image, the corresponding HXR contours (60% and 80%), in 50 - 100 keV, are plotted. The integration periods of HXR images are \[22:18:30 UT $\sim$ 22:19:00 UT\], \[22:19:15 UT $\sim$ 22:19:45 UT\] and \[22:20:00 UT $\sim$ 22:20:30 UT\]. This figure illustrates the spatial and temporal correlation between WL and HXR flare emission. \[WLHXR\]](f3.eps)
![Difference images of Flare I at six spectral positions, namely +174 mÅ, +103 mÅ, +34 mÅ, -34 mÅ, -103 mÅ, and -174 mÅ from the Fe I line center. The first row shows images in the red wing and the second row shows images in the blue wing. The contours represent the half maximum level from a two-dimensional Gaussian fitting. The size listed in Table \[kernelsize1\] is the FWHM of the minor axis. \[size\]](f4.eps)
![Light curves of Flare II on 2011 September 7. Green curve: GOES SXR light curve in 1 - 8 Å. Purple curve: Derivative of GOES SXR light curve in 1 - 8 Å. Red curve with asterisks: HMI WL light curve. The cadence of WL light curve is 45 second. All of the light curves are normalized to their peak counts.\[flare2lc\]](f5.eps)
![WL images and Magnetogram of Flare II on 2011 September 7. Upper-left panel: HMI continuum image taken before the flare which is used as the reference frame. Upper-right panel: HMI continuum image taken during the flare. Lower-left panel: Difference image by subtracting the reference frame from the middle panel. Lower-right panel: HMI line-of-sight magnetogram taken before the flare at 22:32 UT. The red contours in two right panels show the positions of flare sources relative to the sunspot group.\[flare2dif\]](f6.eps)
![Difference images of Flare II at six spectral positions, namely +174 mÅ, +103 mÅ, +34 mÅ, -34 mÅ, -103 mÅ, and -174 mÅ from the Fe I line center. The first row shows images in the red wing and the second row shows images in the blue wing. The contours represent the half maximum level from a two-dimensional Gaussian fitting. The size listed in Table \[kernelsize2\] is the FWHM of the minor axis. \[sizeII\]](f7.eps)
![Top panel: GOES SXR light curves for Flare I (green) and Flare II (purple). Bottom panel: Time derivatives of GOES SXR light curves for Flare I (green) and Flare II (purple). They have similar peak flux and time duration. \[goes2\]](f8.eps)
[^1]: For instance, an electron with 100 keV can reach a layer with a column density of $2.5 \times 10^{21}$cm$^{-2}$ (estimated using Equation 9 in @Brown2002). However the WL emission originates in the photosphere, where the column density reaches $10^{23}$cm$^{-2}$ [@Vernazza1981]. According to the VAL-F model [@Vernazza1981], the height difference is at least 500 km.
|
---
abstract: 'Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing. The most common approaches are time decomposition — and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control — and scenario decomposition — like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed “state” variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove a reduced dynamic programming equation. Then, we apply the reduction method by time blocks to *two time-scales* stochastic optimization problems and to a novel class of so-called *decision-hazard-decision* problems, arising in many practical situations, like in stock management. The *time blocks decomposition* scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises.'
author:
- 'P. Carpentier'
- 'J.-P. Chancelier'
- 'M. De Lara'
- 'T. Rigaut'
title: Time Blocks Decomposition of Multistage Stochastic Optimization Problems
---
**Keywords:** multistage stochastic optimization, dynamic programming, decomposition, time blocks, two time-scales, decision-hazard-decision.
**MSC:** 90C06,90C39,93E20.
Introduction {#Introduction}
============
Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties. Their large scale nature makes decomposition methods appealing. The most common approaches are time decomposition — and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control — and scenario decomposition — like progressive hedging in stochastic programming.
On the one hand, stochastic programming deals with an underlying random process taking a finite number of values, called scenarios [@Shapiro-Dentcheva-Ruszczynski:2009]. Solutions are indexed by a scenario tree, the size of which explodes with the number of stages, hence generally few in practice. However, to overcome this obstacle, stochastic programming takes advantage of scenario decomposition methods (progressive hedging [@Rockafellar-Wets:1991]). On the other hand, stochastic control deals with a state model driven by a white noise, that is, the noise is made of a sequence of independent random variables. Under such assumptions, stochastic dynamic programming is able to handle many stages, as it offers reduction of the search for a solution among state feedbacks (instead of functions of the past noise) [@Bellman:1957; @Puterman:1994].
In a word, dynamic programming is good at handling multiple stages — but at the price of assuming that noises are stagewise independent — whereas stochastic programming does not require such assumption, but can only handle a few stages. Could we take advantage of both methods? Is there a way to apply stochastic dynamic programming at a slow time scale — a scale at which noise would be statistically independent — crossing over short time scale optimization problems where independence would not hold? This question is one of the motivations of this paper.
We will provide a method to decompose multistage stochastic optimization problems by time blocks. In Sect. \[Stochastic\_Dynamic\_Programming\_and\_State\_Reduction\_by\_Time\_Blocks\], we present a mathematical framework that covers both stochastic programming and stochastic dynamic programming. First, in §\[Background\_on\_Stochastic\_Dynamic\_Programming\], we sketch the literature in stochastic dynamic programming, in order to locate our contribution. Second, in §\[Stochastic\_Dynamic\_Programming\_with\_History\_Feedbacks\], we formulate multistage stochastic optimization problems over a so-called history space, and we obtain a general dynamic programming equation. Then, we lay out the basic brick of time blocks decomposition, by revisiting the notion of “state” in Sect. \[State\_Reduction\_by\_Time\_Blocks\]. We lay out conditions under which we can reduce the history using a compressed “state” variable, but with a reduction done by time blocks, that is, at stages that are not necessarily all the original unit stages. We prove a reduced dynamic programming equation, and apply it to two classes of problems in Sect. \[Applications\_of\_Time\_Blocks\_Dynamic\_Programming\]. In §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\], we detail the case of two time-scales stochastic optimization problems. In §\[Decision\_Hazard\_Decision\_Dynamic\_Programming\], we apply the reduction method by time blocks to a novel class consisting of decision-hazard-decision models. In the appendix, we relegate technical results, as well as the specific case of optimization with noise process.
Stochastic Dynamic Programming with Histories {#Stochastic_Dynamic_Programming_and_State_Reduction_by_Time_Blocks}
=============================================
We recall the standard approaches used to deal with a stochastic optimal control problem formulated in discrete time, and we highlight the differences with the framework used in this paper.
Background on Stochastic Dynamic Programming {#Background_on_Stochastic_Dynamic_Programming}
--------------------------------------------
We first recall the notion of stochastic kernel, used in the modeling of stochastic control problems. Let $(\mathbb{X},\tribu{X})$ and $(\mathbb{Y},\tribu{Y})$ be two measurable spaces. A *stochastic kernel* from $(\mathbb{X},\tribu{X})$ to $(\mathbb{Y},\tribu{Y})$ is a mapping $\rho : \mathbb{X}\times\tribu{Y} \to [0,1]$ such that
- for any $Y \in \tribu{Y}$, $\rho(\cdot,Y)$ is $\tribu{X}$-measurable;
- for any $x\in\mathbb{X}$, $\rho(x,\cdot)$ is a probability measure on $\tribu{Y}$.
By a slight abuse of notation, a stochastic kernel is also denoted as a mapping $ \rho : \mathbb{X} \to \Delta\np{\mathbb{Y}} $ from the measurable space $(\mathbb{X},\tribu{X})$ towards the space $\Delta\np{\mathbb{Y}}$ of probability measures over $\np{\mathbb{Y},\tribu{Y}}$, with the property that the function $ x \in \mathbb{X} \mapsto \int_{Y} \rho\np{x,\mathrm{d}y} $ is measurable for any $ Y \in \tribu{Y} $.
We now sketch the most classical frameworks for stochastic dynamic programming.
The most general stochastic dynamic programming principle is sketched by Witsenhausen in [@Witsenhausen:1975b]. However, we do not detail it as its formalism is too far from the following ones. We present here what Witsenhausen calls an optimal stochastic control problem in *standard form* (see [@Witsenhausen:1973]). The ingredients are the following:
1. time $t=\tinitial, \tinitialplusone, \ldots,\horizon-1, \horizon$ is discrete, with integers $ \tinitial < \horizon $;
2. $(\espacea{\State}_{\tinitial},\tribu{\State}_{\tinitial})$, …, $(\espacea{\State}_{\horizon},\tribu{\State}_{\horizon})$ are measurable spaces (“state” spaces);
3. $(\espacea{\Control}_{\tinitial},\tribu{\Control}_{\tinitial})$,…, $(\espacea{\Control}_{\horizon-1},\tribu{\Control}_{\horizon-1})$ are measurable spaces (decision spaces);
4. $\tribu{\Information}_t$ is a subfield of $\tribu{\State}_t$, for $t={\tinitial},\ldots,\horizon-1$ (information); \[it:standard\_form\_information\]
5. $\dynamics_t :(\espacea{\State}_{t} \times \espacea{\Control}_t, \tribu{\State}_{t} \otimes \tribu{\Control}_t) \rightarrow (\espacea{\State}_{t+1},\tribu{\State}_{t+1})$ is measurable, for $t={\tinitial},\ldots,\horizon-1$ (dynamics);
6. $\pi_{\tinitial}$ is a probability on $(\espacea{\State}_{\tinitial},\tribu{\State}_{\tinitial})$;
7. $\criterion :
(\espacea{\State}_{\horizon},\tribu{\State}_{\horizon}) \rightarrow
\RR$ is a measurable function (criterion).
With these ingredients, Witsenhausen formulates a stochastic optimization problem, whose solutions are to be searched among adapted feedbacks, namely $ \policy_{t} : (\espacea{\State}_{t},\tribu{\State}_{t})
\rightarrow (\espacea{\Control}_{t},\tribu{\Control}_{t}) $ with the property that $ \policy_{t}^{-1}(\tribu{\Control}_{t}) \subset
\tribu{\Information}_{t} $ for all $t={\tinitial},\ldots,{\horizon-1}$. Then, he establishes a dynamic programming equation, where the Bellman functions are function of the (unconditional) distribution of the original state $\state_t \in \espacea{\State}_t$, and where the minimization is done over adapted feedbacks.
The main objective of Witsenhausen is to establish a dynamic programming equation for nonclassical information patterns.
The ingredients of the approach developed in [@Evstigneev:1976] are the following:
1. time $t=\tinitial, \tinitialplusone, \ldots,\horizon-1$ is discrete, with integers $ \tinitial < \horizon $;
2. $(\espacea{\Control}_{\tinitial},\tribu{\Control}_{\tinitial})$,…, $(\espacea{\Control}_{\horizon-1},\tribu{\Control}_{\horizon-1})$ are measurable spaces (decision spaces);
3. $ \np{\Omega,\tribu{F}} $ is a measurable space (Nature);
4. $ \sequence{ \tribu{F}_{t} }{ {\tinitial},\ldots,\horizon-1 } $ is a filtration of $\tribu{F}$ (information);
5. $ \PP $ is a probability on $ \np{\Omega,\tribu{F}} $;
6. $ \criterion : \np{
\produit{\Om}{\prod_{t={\tinitial},\ldots,\horizon-1} \UU_{t}} ,
\oproduit{\tribu{F}}{\bigotimes_{t={\tinitial},\ldots,\horizon-1} \tribu{\Control}_{t}}
} \to \RR $ is a measurable function (criterion).
With these ingredients, Evstigneev formulates a stochastic optimization problem, whose solutions are to be searched among adapted processes, namely random processes with values in $ \prod_{t={\tinitial},\ldots,\horizon-1} \UU_{t} $ and adapted to the filtration $ \sequence{ \tribu{F}_{t} }{ {\tinitial},\ldots,\horizon-1 } $. Then, he establishes a dynamic programming equation, where the Bellman function at time $t$ is an $\tribu{F}_{t}$-integrand depending on decisions up to time $t$ (random variables) and where the minimization is done over $\tribu{F}_{t}$-measurable random variables at time $t$.
The main objective of Evstigneev is to establish an existence theorem for an optimal adapted process (under proper technical assumptions, especially on the function $\criterion$, that we do not detail here).
The ingredients of the approach developed in [@Bertsekas-Shreve:1996] are the following:
1. time $t=\tinitial, \tinitialplusone, \ldots,\horizon-1, \horizon$ is discrete, with integers $ \tinitial < \horizon $;
2. $(\espacea{\State}_{\tinitial},\tribu{\State}_{\tinitial})$, …, $(\espacea{\State}_{\horizon},\tribu{\State}_{\horizon})$ are measurable spaces (state spaces);
3. $(\espacea{\Control}_{\tinitial},\tribu{\Control}_{\tinitial})$,…, $(\espacea{\Control}_{\horizon-1},\tribu{\Control}_{\horizon-1})$ are measurable spaces (decision spaces);
4. $(\espacea{\Uncertain}_{\tinitial},\tribu{\Uncertain}_{\tinitial})$,…, $(\espacea{\Uncertain}_{\horizon},\tribu{\Uncertain}_{\horizon})$ are measurable spaces (Nature);
5. $\dynamics_t :(\espacea{\State}_{t} \times \espacea{\Control}_t \times \espacea{\Uncertain}_t,
\tribu{\State}_{t} \otimes \tribu{\Control}_t \otimes \tribu{\Uncertain}_t)
\rightarrow (\espacea{\State}_{t+1},\tribu{\State}_{t+1})$ is a measurable mapping, for $t={\tinitial},\ldots,\horizon-1$ (dynamics);
6. $ {\rho_{t-1:t}} : \espacea{\State}_{t-1} \times
\espacea{\Control}_{t-1} \to \Delta\np{\UNCERTAIN_{t}} $ is a stochastic kernel, for $t={\tinitial},\ldots,\horizon-1$;
7. $ \coutint_{\tter} : \STATE_{\tter} \times \CONTROL_{\tter} \times \UNCERTAIN_{\tter+1}
\to \RR $, for $t={\tinitial},\ldots,\horizon-1$ and $ \coutfin : \STATE_{\horizon} \to \RR $, measurable functions (instantaneous and final costs).
With these ingredients, Bertsekas and Shreve formulate a stochastic optimization problem with time additive additive cost function over given state spaces, action spaces and uncertainty spaces (note that state and action spaces are assumed to be of fixed sizes when time varies, thus a “state” is a priori given). They introduce the notion of history at time $t$ which consists in the states and the actions prior to $t$ and study optimization problems whose solutions (policies) are to be searched among history feedbacks (or relaxed history feedbacks), namely sequences of mappings $ \STATE_{\tinitial} \times \prod_{\tbis=\tinitial}^{t-1}
\np{ \CONTROL_{\tbis} \times \STATE_{\tbis+1} } \to \CONTROL_t $. They identify cases where no loss of optimality results from reducing the search to (relaxed) Markovian feedbacks $ \STATE_t \to \CONTROL_t $. Then, they establish a dynamic programming equation, where the Bellman functions are function of the state $\state_t \in \espacea{\State}_t$, and where the minimization is done over controls $ \control_t \in \espacea{\Control}_t $. For finite horizon problems, the mathematical challenge is to set up a mathematical framework (the Borel assumptions) for which optimal policies exists.
The main objective of Bertsekas and Shreve is to state conditions under which the dynamic programming equation is mathematically sound, namely with universally measurable Bellman functions and with universally measurable relaxed control strategies in the context of Borel spaces. The interested reader will find all the subtleties about Borel spaces and universally measurable concepts in [@Bertsekas-Shreve:1996 Chapter 7].
The ingredients of the approach developed in [@Puterman:1994] are the following:
1. time $t=\tinitial, \tinitialplusone, \ldots,\horizon-1, \horizon$ is discrete, with integers $ \tinitial < \horizon $;
2. $(\espacea{\State}_{\tinitial},\tribu{\State}_{\tinitial})$, …, $(\espacea{\State}_{\horizon},\tribu{\State}_{\horizon})$ are measurable spaces (state spaces);
3. $(\espacea{\Control}_{\tinitial},\tribu{\Control}_{\tinitial})$,…, $(\espacea{\Control}_{\horizon-1},\tribu{\Control}_{\horizon-1})$ are measurable spaces (decision spaces);
4. $ {\rho_{t-1:t}} : \espacea{\State}_{t-1} \times
\espacea{\Control}_{t-1} \to \Delta\np{\espacea{\State}_{t}} $ is a stochastic kernel, for $t={\tinitial},\ldots,\horizon-1$;
5. $ \coutint_{\tter} : \STATE_{\tter} \times \CONTROL_{\tter}
\to \RR $, for $t={\tinitial},\ldots,\horizon-1$ and $ \coutfin : \STATE_{\horizon} \to \RR $, measurable functions (instantaneous and final costs).
Puterman shares most of his ingredients with Bertsekas and Shreve, but he does not require uncertainty sets and dynamics, as he directly considers state transition stochastic kernels. With these ingredients, Puterman formulates a stochastic optimization problem, whose solutions are to be searched among history feedbacks, namely sequences of mappings $ \STATE_{\tinitial} \times \prod_{\tbis=\tinitial}^{t-1}
\np{ \CONTROL_{\tbis} \times \STATE_{\tbis+1} } \to \CONTROL_t $. Then, he establishes a dynamic programming equation, where the Bellman functions are function of the history $ \history_t \in \STATE_{\tinitial} \times \prod_{\tbis=\tinitial}^{t-1}
\np{ \CONTROL_{\tbis} \times \STATE_{\tbis+1} } $. He identifies cases where no loss of optimality results from reducing the search to Markovian feedbacks $ \STATE_t \to \CONTROL_t $. In such cases, the Bellman functions are function of the state $\state_t \in \espacea{\State}_t$, and the minimization in the dynamic programming equation is done over controls $ \control_t \in \espacea{\Control}_t $.
The main objective of Puterman is to explore infinite horizon criteria, average reward criteria, the continuous time case, and to present many examples.
The ingredients that we will use are the following:
1. time $t=\tinitial, \tinitialplusone, \ldots,\horizon-1, \horizon$ is discrete, with integers $ \tinitial < \horizon $;
2. $(\espacea{\Control}_{\tinitial},\tribu{\Control}_{\tinitial})$,…, $(\espacea{\Control}_{\horizon-1},\tribu{\Control}_{\horizon-1})$ are measurable spaces (decision spaces);
3. $(\espacea{\Uncertain}_{\tinitial},\tribu{\Uncertain}_{\tinitial})$,…, $(\espacea{\Uncertain}_{\horizon},\tribu{\Uncertain}_{\horizon})$ are measurable spaces (Nature);
4. $ {\rho_{t-1:t}} :
\UNCERTAIN_{0} \times \prod_{\tbis=0}^{t-1}
\np{ \CONTROL_{\tbis} \times \UNCERTAIN_{\tbis+1} }
\to \Delta\np{\UNCERTAIN_{t}} $ is a stochastic kernel, for $t={\tinitial},\ldots,\horizon-1$,
5. $ \criterion : \np{
\UNCERTAIN_{0} \times \prod_{\tbis=0}^{\horizon-1}
\np{ \CONTROL_{\tbis} \times \UNCERTAIN_{\tbis+1} },
\tribu{\Uncertain}_{0} \otimes \bigotimes_{\tbis=0}^{\horizon-1}
\np{ \tribu{\Control}_{\tbis} \otimes \tribu{\Uncertain}_{\tbis+1}} }
\to \RR $ is a measurable function (criterion).
The main features of the framework developed in this paper are the following: the history at time $t$ consists of all uncertainties and actions prior to time $t$ (rather than states and actions); the cost is a unique function depending on the whole history, from initial time $\tinitial$ to the horizon $T$; the probability distribution of uncertainty at time $t$ depends on the history up to time $t-1$. We will state a dynamic programming equation, where the Bellman functions are function of the history $ \history_t \in \UNCERTAIN_{0} \times \prod_{\tbis=0}^{t}
\np{ \CONTROL_{\tbis} \times \UNCERTAIN_{\tbis+1} } $ and where the minimization is done over controls $ \control_t \in \espacea{\Control}_t $.
Our main objective is to establish a dynamic programming equation with a state, not at any time $t\in\{0,\ldots,T\}$, but at some specified instants $ 0 = t_{0}<t_{1} < \cdots <t_{N} = \horizon $. The state spaces are not given a priori, but introduced a posteriori as image sets of history reduction mappings. With this, we can mix dynamic programming and stochastic programming.
Our framework is rather distant with the one of Evstigneev in [@Evstigneev:1976]. It falls in the general framework developed by Witsenhausen (see [@Witsenhausen:1973] and [@Carpentier-Chancelier-Cohen-DeLara:2015 § 4.5.4]), *except* for the stochastic kernels (we are more general) and for the information structure (we are less general). Finally, our framework is closest to the one found in Bertsekas and Shreve [@Bertsekas-Shreve:1996] and Puterman [@Puterman:1994], *except* for the state spaces, not given a priori, and for the criterion, function of the whole history.
Stochastic Dynamic Programming with History Feedbacks {#Stochastic_Dynamic_Programming_with_History_Feedbacks}
-----------------------------------------------------
We now present a framework that is adapted to both stochastic programming and stochastic dynamic programming. Time is discrete and runs among the integers $t=0,1,2\ldots, \horizon-1, \horizon$, where $\horizon \in \NN^*$. For $ 0 \leq \tun \leq \tbis \leq \horizon $, we introduce the interval $\interval{\tun}{\tbis}= \defset{\tter \in \NN}{\tun \leq \tter \leq \tbis } $.
### Histories and Feedbacks {#Histories_and_Feedbacks}
We first define the basic and the composite spaces that we need to formulate multistage stochastic optimization problems. Then, we introduce a class of solutions called history feedbacks.
\[Histories\_and\_History\_Spaces\]
For each time $t=0,1,2\ldots, \horizon-1$, the decision $\control_{t}$ takes its values in a measurable set $\CONTROL_{t}$ equipped with a $\sigma$-field $\tribu{\Control}_{t}$. For each time $t=0,1,2\ldots, \horizon$, the uncertainty $\uncertain_{t}$ takes its values in a measurable set $\UNCERTAIN_{t}$ equipped with a $\sigma$-field $\tribu{\Uncertain}_{t}$.
For $t=0,1,2\ldots, \horizon$, we define the *history space* $\HISTORY_{t}$ equipped with the *history field* $\tribu{\History}_{t}$ by $$\HISTORY_{t} =
\UNCERTAIN_{0} \times \prod_{\tbis=0}^{t-1}
\np{ \CONTROL_{\tbis} \times \UNCERTAIN_{\tbis+1} }
\mtext{ and }
\tribu{\History}_{t} =
\tribu{\Uncertain}_{0} \otimes \bigotimes_{\tbis=0}^{t-1}
\np{ \tribu{\Control}_{\tbis} \otimes \tribu{\Uncertain}_{\tbis+1}} \eqsepv
t=0,1,2\ldots, \horizon \eqfinv
\label{eq:history_space}$$ with the particular case $ \HISTORY_{0} = \UNCERTAIN_{0} $, $ \tribu{\History}_{0} = \tribu{\Uncertain}_{0} $. A generic element $\history_{t} \in \HISTORY_{t}$ is called a *history*: $$\history_{t}= \np{ \uncertain_{0},
\np{\control_{\tbis},\uncertain_{\tbis+1}}_{\tbis=0,\ldots,t-1} } =
\np{\uncertain_{0},\control_{0},\uncertain_{1},\control_{1},
\uncertain_{2},\ldots, \control_{t-2}, \uncertain_{t-1},
\control_{t-1},\uncertain_{t}} \in \HISTORY_{t} \eqfinp$$ For $1 \leq \tun \leq \tbis \leq \tter$, we introduce the $\interval{\tun}{\tbis}$-*history subpart* $$\history_{\tun:\tbis} =
\np{ \control_{\tun-1},\uncertain_{\tun}, \ldots,
\control_{\tbis-1},\uncertain_{\tbis} }
\eqfinv$$ so that we have $\history_{\tter} = \np{ \history_{\tun-1}, \history_{\tun:\tter} }$.
When $ 0 \leq \tun \leq \tter \leq \horizon-1 $, we define a $\interval{\tun}{\tter}$-*history feedback* as a sequence $ \sequence{{\gamma}_{\tbis}}{\tbis=\tun,\ldots,\tter} $ of measurable mappings $${\gamma}_{\tbis} :
\np{\HISTORY_{\tbis},\tribu{\History}_{\tbis}} \to
\np{\CONTROL_{\tbis},\tribu{\Control}_{\tbis}} \eqfinp$$ We call ${\Gamma}_{\tun:\tter}$ the set of $\interval{\tun}{\tter}$-history feedbacks.
The history feedbacks reflect the following information structure. At the end of the time interval $[t-1,t[$, an uncertainty variable $\uncertain_{t}$ is produced. Then, at the beginning of the time interval $[t,t+1[$, a decision-maker takes a decision $\control_{t}$, as follows $$\label{eq:interplay_Noise_Control}
\uncertain_{0} \rightsquigarrow \control_{0} \rightsquigarrow \uncertain_{1}
\rightsquigarrow \control_{1} \rightsquigarrow
\quad \dots \quad \rightsquigarrow
\uncertain_{\horizon-1} \rightsquigarrow \control_{\horizon-1}
\rightsquigarrow \uncertain_{\horizon} \eqfinp$$
### Optimization with Stochastic Kernels {#sect:History_Feedback_Case}
We introduce a family of optimization problems with stochastic kernels. Then, we show how such problems can be solved by stochastic dynamic programming.
In what follows, we say that a function is *numerical* if it takes its values in $ [-\infty,+\infty] $ (also called *extended* or *extended real-valued* function). \[Family\_of\_Optimization\_Problems\_with\_Stochastic\_Kernels\]
To build a family of optimization problems over the time span $\{0,\ldots,\horizon-1\}$, we require two ingredients:
- a family $ \sequence{ {\rho_{\tbis-1:\tbis}}}{1 \leq \tbis \leq \horizon}$ of stochastic kernels $${\rho_{\tbis-1:\tbis}} :
\np{\HISTORY_{\tbis-1},\tribu{\History}_{\tbis-1}}
\to \Delta\np{\UNCERTAIN_{\tbis}}
\eqsepv \tbis = 1, \ldots, \horizon
\eqfinv
\label{eq:family_of_stochastic_kernels}$$ that represents the distribution of the next uncertainty $ \uncertain_{\tbis} $ parameterized by past history $ \history_{\tbis-1} $ (see the chronology in ),
- a numerical function, playing the role of a cost to be minimized, $$\criterion : \np{\HISTORY_{\horizon},\tribu{\History}_{\horizon}}
\to {[0,+\infty]}\eqfinv
\label{eq:criterion}$$ assumed to be nonnegative[^1] and measurable with respect to the field $\tribu{\History}_{\horizon}$.
We define, for any $ \sequence{{\gamma}_{\tbis}}{ \tbis=\tter,\ldots,\horizon\!-\!1}
\in {\Gamma}_{\tter:\horizon\!-\!1} $, a new family of stochastic kernels $${\rho_{\tter:\horizon}^{{\gamma}}} :
\np{\HISTORY_{\tter},\tribu{\History}_{\tter}} \to \Delta\np{ \HISTORY_{\horizon} } \eqfinv$$ that capture the transitions between histories when the dynamics $\history_{\tbis+1} = \bp{\history_{\tbis},\control_{\tbis},\uncertain_{\tbis+1}}$ is driven by $ \control_{\tbis} = {\gamma}_{\tbis}\np{\history_{\tbis}} $ for $\tbis = \tter,\ldots,\horizon-1$ (see Definition \[de:stochastic\_kernels\_rho\] in §\[Stochastic\_Kernels\] for the detailed construction of ${\rho_{\tun:\tter}^{{\gamma}}}$; note that ${\rho_{\tter:\horizon}^{{\gamma}}}$ generates a probability distribution on the space $\HISTORY_{\horizon}$ of histories over the whole horizon $\{0,\dots,\horizon\}$).
We consider the family of optimization problems, indexed by $t=0,\ldots,\horizon-1$ and parameterized by the history $ \history_{t} \in \HISTORY_{t} $: $$\inf_{{\gamma}_{t:\horizon-1} \in {\Gamma}_{t:\horizon-1}}
\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}}
\eqsepv \forall \history_{\tter} \in \HISTORY_{\tter}
\eqfinv
\label{eq:Optimization_Problem_Over_History_Feedbacks_Value}$$ the integral in the right-hand side of the above equation corresponding to the cost induced by the feedback ${\gamma}_{t:\horizon-1}$ when starting at time $t$ with a given history $\history_{t}$. For all $t=0,\ldots,\horizon-1$, we define the minimum value of Problem by
$$\begin{aligned}
\Value_{t}(\history_{t})
& = \inf_{{\gamma}_{t:\horizon-1} \in {\Gamma}_{t:\horizon-1}}
\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}}
\eqsepv \forall \history_{\tter} \in \HISTORY_{\tter}
\eqfinv
\intertext{ and we also define}
\Value_{\horizon}(\history_{\horizon}) &= \criterion(\history_{\horizon})
\eqsepv \forall \history_{\horizon} \in \HISTORY_{\horizon}
\eqfinp\end{aligned}$$
\[eq:value\_functions\] The numerical function $ \Value_{t} : \HISTORY_{t} \to {[0,+\infty]}$ is called the *value function* at time $t$.
\[Resolution\_by\_Dynamic\_Programming\]
We show that the value functions in are *Bellman functions*, in that they are solution of the Bellman or dynamic programming equation.
For $t\inic{0}{\horizon}$, let $ \espace{L}^{0}_{+}(\HISTORY_{t},\tribu{\History}_{t}) $ be the space of universally measurable nonnegative numerical functions over $\HISTORY_{t}$ (see [@Bertsekas-Shreve:1996] for further details). For $t\inic{0}{\horizon-1}$, we define the *Bellman operator* by, for all $\varphi\in
\espace{L}^{0}_{+}(\HISTORY_{t+1},\tribu{\History}_{t+1})$ and for all $\history_{t} \in \HISTORY_{t}$, $$\bp{ {\mathcal{B}_{t+1:t}}\varphi }\np{\history_{t}} =
\inf_{\control_{t}\in\CONTROL_{t}} \int_{\UNCERTAIN_{t+1}}
\varphi\np{\history_{t},\control_{t},\uncertain_{t+1}}
{\rho_{t:t+1}} (\history_{t},d\uncertain_{t+1})
\eqfinp
\label{eq:Bellman_operators_rho}$$ Since $\varphi\in \espace{L}^{0}_{+}(\HISTORY_{t+1},\tribu{\History}_{t+1})$, we have that ${\mathcal{B}_{t+1:t}}\varphi$ is a well defined nonnegative numerical function.
The proof of the following theorem is inspired by [@Bertsekas-Shreve:1996], and given in §\[proof:DP\_withoutstate\_third\].
Assume that all the spaces introduced in §\[Histories\_and\_Feedbacks\] are Borel spaces, the stochastic kernels in are Borel-measurable, and that the criterion $\criterion$ in is a nonnegative lower semianalytic function.
Then, the Bellman operators in map $ \espace{L}^{0}_{+}(\HISTORY_{t+1},\tribu{\History}_{t+1}) $ into $ \espace{L}^{0}_{+}(\HISTORY_{t},\tribu{\History}_{t}) $ $${\mathcal{B}_{t+1:t}} : \espace{L}^{0}_{+}(\HISTORY_{t+1},\tribu{\History}_{t+1})
\to \espace{L}^{0}_{+}(\HISTORY_{t},\tribu{\History}_{t})
\eqfinv$$ and the value functions $\Value_{t}$ defined in are universally measurable and satisfy the *Bellman equation*, or *(stochastic) dynamic programming equation*,
$$\begin{aligned}
\Value_{\horizon} &=\criterion \eqfinv \\
\Value_{t} &={\mathcal{B}_{t+1:t}}\Value_{t+1} \eqsepv
\mtext{ for } t=\horizon\!-\!1,\ldots, 1, 0 \eqfinp
\end{aligned}$$
\[eq:Bellman\_equation\]
\[pr:DP\_withoutstate\_third\]
This theorem is mainly inspired by [@Bertsekas-Shreve:1996], with the feature that the state $\state_t$ is in our case the history $\history_t$, with the dynamics: $$\history_{t+1} = \bp{\history_{t},\control_{t},\uncertain_{t+1}} \eqfinp
\label{eq:canonical_dynamics}$$ This very general dynamic programming result will be the basis of all future developments in this paper. In the sequel, we assume that all the assumptions of Theorem \[pr:DP\_withoutstate\_third\] are fulfilled, that is,
- all the spaces (like the ones introduced in §\[Histories\_and\_Feedbacks\]) will be supposed to be Borel spaces,
- all the stochastic kernels (like the ones introduced in ) will be supposed to be Borel-measurable,
- all the criteria (like the one introduced in ) will be supposed to be nonnegative lower semianalytic functions.
State Reduction by Time Blocks and Dynamic Programming {#State_Reduction_by_Time_Blocks}
======================================================
In this section, we consider the question of reducing the history using a compressed “state” variable. Differing with traditional practice, such a variable may be not available at any time $t\in\{0,\ldots,T\}$, but at some specified instants $ 0 = t_{0}<t_{1} < \cdots <t_{N} = \horizon $. We have see in the previous section that the history $\history_{t}$ is itself a canonical state variable in our framework with associated dynamics . However the size of this canonical state increases with $t$, which is a nasty feature for dynamic programming.
State Reduction on a Single Time Block
--------------------------------------
We first present the case where the reduction only occurs at two instants denoted by $\tun$ and $\tter$: $$0 \leq \tun < \tter \leq \horizon \eqfinp$$
Let $(\STATE_{\tun},\tribu{\State}_{\tun})$ and $(\STATE_{\tter},\tribu{\State}_{\tter})$ be two measurable *state spaces*, $\theta_{\tun}$ and $\theta_{\tter}$ be two measurable *reduction mappings*
$$\theta_{\tun} : \HISTORY_{\tun} \to \STATE_{\tun} \eqsepv
\theta_{\tter} : \HISTORY_{\tter} \to \STATE_{\tter} \eqfinv
\label{eq:reduction_mappings}$$
and $\Dynamics{\tun}{\tter}$ be a measurable *dynamics* $$\Dynamics{\tun}{\tter} : \STATE_{\tun} \times \HISTORY_{\tun+1:\tter}
\to \STATE_{\tter} \eqfinp$$ The triplet $\np{\theta_{\tun},\theta_{\tter},\Dynamics{\tun}{\tter}}$ is called a *state reduction across $\interval{\tun}{\tter}$* if we have $$\theta_{\tter}\bp{\np{\history_{\tun}, \history_{\tun+1:\tter} }} =
\Dynamics{\tun}{\tter} \bp{ \theta_{\tun}\np{\history_{\tun}},\history_{\tun+1:\tter} }
\eqsepv \forall \history_{\tter} \in \HISTORY_{\tter} \eqfinp
\label{eq:reduction-dynamics}$$
The state reduction $\np{\theta_{\tun},\theta_{\tter},\Dynamics{\tun}{\tter}}$ is said to be *compatible* with the family $\na{{\rho_{\tbis-1:\tbis}}}_{\tun+1 \leq \tbis \leq \tter}$ of stochastic kernels if
- there exists a *reduced stochastic kernel* $${{\tilde\rho}_{\tun:\tun+1}} : \STATE_{\tun} \to \Delta\np{\UNCERTAIN_{\tun+1}} \eqfinv$$ such that the stochastic kernel ${\rho_{\tun:\tun+1}}$ in can be factored as $${\rho_{\tun:\tun+1}}\np{\history_{\tun},\mathrm{d}\uncertain_{\tun+1}} =
{{\tilde\rho}_{\tun:\tun+1}}\bp{\theta_{\tun}\np{\history_{\tun}},
\mathrm{d}\uncertain_{\tun+1}} \eqsepv
\forall \history_{\tun} \in \HISTORY_{\tun} \eqfinv$$
- for all $ \tbis={\tun+2}, \ldots, {\tter} $, there exists a *reduced stochastic kernel* $${{\tilde\rho}_{\tbis-1:\tbis}} :
\STATE_{\tun} \times \HISTORY_{\tun+1:\tbis-1}
\to \Delta\np{\UNCERTAIN_{\tbis}} \eqfinv$$ such that the stochastic kernel ${\rho_{\tbis-1:\tbis}}$ can be factored as $${\rho_{\tbis-1:\tbis}}\bp{\np{\history_{\tun},\history_{\tun+1:\tbis-1}},
\mathrm{d}\uncertain_{\tbis}} =
{{\tilde\rho}_{\tbis-1:\tbis}}
\Bp{\bp{\theta_{\tun}\np{\history_{\tun}},\history_{\tun+1:\tbis-1}},
\mathrm{d}\uncertain_{\tbis}} \eqsepv
\forall \history_{\tbis-1} \in \HISTORY_{\tbis-1} \eqfinp$$
\[eq:reduction-dynamics\_compatible\]
\[de:reduction-dynamics\]
According to this definition, the triplet $\np{\theta_{\tun},\theta_{\tter},\Dynamics{\tun}{\tter}}$ is a state reduction across $\interval{\tun}{\tter}$ if and only if the diagram in Figure \[fig:state\_reduction\] is commutative; it is compatible if and only if the diagram in Figure \[fig:state\_reduction\_compatible\] is commutative.
$$\begin{tikzcd}[column sep=large, labels={font=\everymath\expandafter{\the\everymath\textstyle}}]
\HISTORY_{\tun} \times \HISTORY_{\tun+1:\tter} \arrow[d, xshift=-0.8cm ,"\theta_{\tun}" ] \arrow[d,xshift=+0.5cm, "I_{d}" ]
\arrow[r, "I_{d}" ]
&\HISTORY_{\tter} \arrow[d, "\theta_{\tter}"] \\[1.0cm]
\STATE_{\tun} \times \HISTORY_{\tun+1:\tter} \arrow[r, "\Dynamics{\tun}{\tter} " ]
&\STATE_{\tter}
\end{tikzcd}$$
$$\begin{tikzcd}[column sep=large, labels={font=\everymath\expandafter{\the\everymath\textstyle}}]
\HISTORY_{\tun} \times \HISTORY_{\tun+1:\tbis-1} \arrow[d, xshift=-0.8cm ,"\theta_{\tun}" ] \arrow[d,xshift=+0.5cm, "I_{d}" ]
\arrow[r, " {\rho_{\tbis-1:\tbis}} " ]
& \Delta\np{\UNCERTAIN_{\tbis}} \\[1.0cm]
\STATE_{\tun} \times \HISTORY_{\tun+1:\tbis-1} \arrow[ur, "{{\tilde\rho}_{\tbis-1:\tbis}}"', xshift=0.4cm] &
\end{tikzcd}$$
We define the *Bellman operator across $\interval{\tter}{\tun}$* ${\mathcal{B}_{\tter:\tun}} : \espace{L}^{0}_{+}(\HISTORY_{\tter},\tribu{\History}_{\tter})
\to \espace{L}^{0}_{+}(\HISTORY_{\tun},\tribu{\History}_{\tun})$ by $${\mathcal{B}_{\tter:\tun}} = {\mathcal{B}_{\tter:\tter-1}} \circ \cdots \circ {\mathcal{B}_{\tun+1:\tun}} \eqsepv
\label{eq:BelOp_t:r}$$ where the one time step operators $ {\mathcal{B}_{\tbis:\tbis-1}} $, for $ \tun+1 \leq \tbis \leq \tter $ are defined in .
The following proposition, whose proof is given in §\[proof:DPB\], is the key ingredient to formulate dynamic programming equations with a reduced state.
Suppose that there exists a state reduction $\np{\theta_{\tun},\theta_{\tter},\Dynamics{\tun}{\tter}}$ that is compatible with the family $\na{{\rho_{\tbis-1:\tbis}}}_{\tun+1 \leq \tbis \leq \tter}$ of stochastic kernels (see Definition \[de:reduction-dynamics\]). Then, there exists a *reduced Bellman operator across $\interval{\tter}{\tun}$* $${{\tilde{\mathcal{B}}}_{\tter:\tun}} : \espace{L}^{0}_{+}(\STATE_{\tter},\tribu{\State}_{\tter})
\to \espace{L}^{0}_{+}(\STATE_{\tun},\tribu{\State}_{\tun}) \eqfinv
\label{eq:reduced_Bellman_operator_across}$$ such that, for all $\tilde\varphi_{\tter} \in \espace{L}^{0}_{+}(\STATE_{\tter},\tribu{\State}_{\tter})$, we have that $$\bp{{{\tilde{\mathcal{B}}}_{\tter:\tun}} \tilde\varphi_{\tter}} \circ \theta_{\tun} =
{\mathcal{B}_{\tter:\tun}} \np{\tilde\varphi_{\tter} \circ \theta_{\tter}}
\eqfinp
\label{eq:DPB}$$ For all measurable nonnegative numerical function $\tilde\varphi_{\tter}
: \STATE_{\tter} \to {[0,+\infty]}$ and for all $\state_{\tun}\in\STATE_{\tun}$, we have that $$\begin{gathered}
\bp{{{\tilde{\mathcal{B}}}_{\tter:\tun}}\tilde\varphi_{\tter}}(\state_{\tun}) =
\inf_{\control_{\tun}\in\CONTROL_{\tun}} \int_{\UNCERTAIN_{\tun+1}}
{{\tilde\rho}_{\tun:\tun+1}}
\np{\state_{\tun},\mathrm{d}\uncertain_{\tun+1}} \\
\inf_{\control_{\tun+1}\in\CONTROL_{\tun+1}} \int_{\UNCERTAIN_{\tun+2}}
{{\tilde\rho}_{\tun+1:\tun+2}}
\np{\state_{\tun},\control_{\tun},\uncertain_{\tun+1},\mathrm{d}\uncertain_{\tun+2}}
\ldots \\
\qquad\qquad\qquad\qquad
\inf_{\control_{\tter-1}\in\CONTROL_{\tter-1}} \int_{\UNCERTAIN_{\tter}}
\tilde\varphi_{\tter}\bp{\Dynamics{\tun}{\tter}
\np{\state_{\tun}, \control_{\tun},\uncertain_{\tun+1},\ldots,
\control_{\tter-1},\uncertain_{\tter}}} \\
\qquad \qquad \qquad
{{\tilde\rho}_{\tter-1:\tter}}
\np{\state_{\tun},
\control_{\tun},\uncertain_{\tun+1},\ldots,\control_{\tter-2},\uncertain_{\tter-1},
\mathrm{d}\uncertain_{\tter}}
\label{eq:tildeBelOp-expression} \eqfinp
\end{gathered}$$ \[thm:DPB\]
Proposition \[thm:DPB\] can be interpreted as follows. Denoting by $\theta_{\tter}^{\star} : \espace{L}^{0}_{+}(\STATE_{\tter},\tribu{\State}_{\tter})
\to \espace{L}^{0}_{+}(\HISTORY_{\tter},\tribu{\History}_{\tter})$ the operator defined by $$\theta_{\tter}^{\star}(\tilde\varphi_{\tter}) =
\tilde\varphi_{\tter} \circ \theta_{\tter}
\eqsepv \forall \tilde\varphi_{\tter} \in \espace{L}^{0}_{+}(\STATE_{\tter},\tribu{\State}_{\tter})
\eqfinv$$ the relation rewrites $$\theta_{\tun}^{\star} \circ {{\tilde{\mathcal{B}}}_{\tter:\tun}} =
{\mathcal{B}_{\tter:\tun}} \circ \theta_{\tter}^{\star} \eqfinv$$ that is, Proposition \[thm:DPB\] states that the diagram in Figure \[fig:state\_reduction\_compatible\_Bellman\_operators\] is commutative.
$$\begin{tikzcd}[column sep=large, labels={font=\everymath\expandafter{\the\everymath\textstyle}}]
\espace{L}^{0}_{+}(\HISTORY_{\tter},\tribu{\History}_{\tter})
\arrow[r, "{\mathcal{B}_{\tter:\tun}}" ]
& \espace{L}^{0}_{+}(\HISTORY_{\tun},\tribu{\History}_{\tun}) \\[1cm]
\espace{L}^{0}_{+}(\STATE_{\tter},\tribu{\State}_{\tter})
\arrow[u ,"\theta_{\tter}^{\star}" ]
\arrow[r, "{{\tilde{\mathcal{B}}}_{\tter:\tun}}" ]
& \espace{L}^{0}_{+}(\STATE_{\tun},\tribu{\State}_{\tun})
\arrow[u, "\theta_{\tun}^{\star}"]
\end{tikzcd}$$
State Reduction on Multiple Consecutive Time Blocks and Dynamic Programming Equations
-------------------------------------------------------------------------------------
Proposition \[thm:DPB\] can easily be extended to the case of multiple consecutive time blocks $ [t_{i},t_{i+1}] $, $i=0,\ldots,N-1$, where $$0 = t_{0}<t_{1} < \cdots <t_{N} = \horizon \eqfinp
\label{eq:multiple_consecutive_time_blocks}$$
Let $\sequence{\np{\STATE_{t_{i}},\tribu{\State}_{t_{i}}}}{i=0,\ldots,N}$ be a family of measurable state spaces, $\sequence{\theta_{t_{i}}}{i=0,\ldots,N}$ be a family of measurable reduction mappings $ \theta_{t_{i}} : \HISTORY_{t_{i}} \to \STATE_{t_{i}} $, and $\sequence{ \Dynamics{t_{i}}{t_{i+1}} }{i=0,\ldots,N-1}$ be a family of measurable dynamics $ \Dynamics{t_{i}}{t_{i+1}} : \STATE_{t_{i}} \times \HISTORY_{t_{i}+1:t_{i+1}}
\to \STATE_{t_{i+1}} $.
The triplet $ \np{ \sequence{\STATE_{t_{i}}}{i=0,\ldots,N} ,
\sequence{\theta_{t_{i}}}{i=0,\ldots,N} ,
\sequence{ \Dynamics{t_{i}}{t_{i+1}} }{i=0,\ldots,N-1} } $ is called a *state reduction across the consecutive time blocks $ [t_{i},t_{i+1}] $, $i=0,\ldots,N-1$* if every triplet $ \np{ \theta_{t_{i}} , \theta_{t_{i+1}} ,
\Dynamics{t_{i}}{t_{i+1}} } $ is a state reduction, for $i=0,\ldots,N-1$.
The state reduction across the consecutive time blocks $ [t_{i},t_{i+1}] $ is said to be *compatible* with the family $\na{{\rho_{\tbis-1:\tbis}}}_{1 \leq \tbis \leq \horizon}$ of stochastic kernels given in if every triplet $ \np{ \theta_{t_{i}} , \theta_{t_{i+1}} ,
\Dynamics{t_{i}}{t_{i+1}} } $ is compatible with the family $\na{{\rho_{\tbis-1:\tbis}}}_{t_{i}+1 \leq \tbis \leq t_{i+1}}$, for $i=0,\ldots,N-1$. \[de:reduction-dynamics\_family\]
Assuming the existence of a state reduction across the consecutive time blocks $[t_{i},t_{i+1}]$ compatible with the family of stochastic kernels , we obtain the existence of a *family of reduced Bellman operators* across the consecutive $\interval{t_{i+1}}{t_{i}}$ as an immediate consequence of multiple applications of Proposition \[thm:DPB\], that is, $${{\tilde{\mathcal{B}}}_{t_{i+1}:t_{i}}} :
\espace{L}^{0}_{+}(\STATE_{t_{i+1}},\tribu{\State}_{t_{i+1}})
\to \espace{L}^{0}_{+}(\STATE_{t_{i}},\tribu{\State}_{t_{i}})
\eqsepv i=0,\ldots,N-1 \eqfinv$$ such that, for any function $\tilde\varphi_{t_{i+1}} \in
\espace{L}^{0}_{+}(\STATE_{t_{i+1}},\tribu{\State}_{t_{i+1}}) $, we have that $$\bp{ {{\tilde{\mathcal{B}}}_{t_{i+1}:t_{i}}} \tilde\varphi_{t_{i+1}} } \circ \theta_{t_{i}} =
{\mathcal{B}_{t_{i+1}:t_{i}}} \np{\tilde\varphi_{t_{i+1}} \circ \theta_{t_{i+1}}} \eqfinp$$
We now consider the family of optimization problems and the associated value functions . Thanks to the state reductions, we are able to state the following theorem which establishes dynamic programming equations *across* consecutive time blocks. Its proof is an immediate consequence of multiple applications of Theorem \[pr:DP\_withoutstate\_third\] and Proposition \[thm:DPB\].
Suppose that a state reduction $ \np{ \sequence{\STATE_{t_{i}}}{i=0,\ldots,N} ,
\sequence{\theta_{t_{i}}}{i=0,\ldots,N} ,
\sequence{ \Dynamics{t_{i}}{t_{i+1}} }{i=0,\ldots,N-1} } $ exists across the consecutive time blocks $ [t_{i},t_{i+1}] $, $i=0,\ldots,N-1$ as in , that is compatible with the family $\na{{\rho_{\tbis-1:\tbis}}}_{1 \leq \tbis \leq \horizon}$ of stochastic kernels given in .
Assume that there exists a *reduced criterion*
$$\tilde\criterion : \STATE_{\horizon} \to {[0,+\infty]}\eqfinv$$
such that the cost function $\criterion$ in can be factored as $$\criterion = \tilde\criterion \circ \theta_{t_{N}} \eqfinp$$
We define the family of *reduced value functions* $\{\tilde\Value_{t_{i}}\}_{i=0,\ldots,N}$ by
$$\begin{aligned}
\tilde\Value_{t_{N}}
& = \tilde\criterion \eqfinv \\
\tilde\Value_{t_{i}}
& = {{\tilde{\mathcal{B}}}_{t_{i+1}:t_{i}}} \tilde\Value_{t_{i+1}}
\eqsepv \text{ for } i = N-1,\ldots,0
\eqfinp\end{aligned}$$
Then, the family $\{\Value_{t_{i}}\}_{i=0,\ldots,N}$ in satisfies $$\Value_{t_{i}} = \tilde\Value_{t_{i}} \circ \theta_{t_{i}}
\eqsepv i = 0,\ldots,N \eqfinp
\label{eq:value_functions_factorization_blocks}$$
\[thm:DPB\_family\]
To obtain such a dynamic programming equation across time blocks, we needed the detour of Sect. \[Stochastic\_Dynamic\_Programming\_and\_State\_Reduction\_by\_Time\_Blocks\], with a dynamic programming equation over the history space. Thus equipped, it is now possible to propose a decomposition scheme for optimization problems with multiple time scales, using both stochastic programming and stochastic dynamic programming. We detail applications of this scheme in Sect. \[Applications\_of\_Time\_Blocks\_Dynamic\_Programming\].
Applications of Time Blocks Dynamic Programming {#Applications_of_Time_Blocks_Dynamic_Programming}
===============================================
We present in this section two applications of the state reduction result stated in Theorem \[thm:DPB\_family\].
The first one corresponds to a *two time-scales* optimization problem. A typical instance of such a problem is to optimize long-term investment decisions (slow time-scale) — for example the renewal of batteries in an energy system — but the optimal long-term decisions highly depend on short-term operating decisions (fast time-scale) — for example the way the battery is operated in real-time.
The second application corresponds to a class of stochastic multistage optimization problems arising often in practice, especially when managing stocks (dams for instance). The decision-maker takes two decisions at each time step $t$: at the beginning of the time interval $[t,t+1[$, the first decision (quantity of water to be turbinated to produce electricity for instance) is taken without knowing the uncertainty that will occur during the time step (decision-hazard framework); at the end of the time interval $[t,t+1[$, an uncertainty variable $\uncertain_{t+1}$ is produced and the second decision (quantity of water to be released to avoid dam overflow for instance) is taken once the uncertainty at time step $t$ is revealed (hazard-decision framework). This new class of problems is called *decison-hazard-decision* optimization problems.
Two Time-Scales Multistage Optimization Problems {#Stochastic_Dynamic_Programming_by_Time_Blocks}
------------------------------------------------
In this class of problems, each time index $t$ is represented by a couple $({d},{m})$ of indices, with ${d}\in \{ 0, \ldots , {\MakeUppercase{{d}}}+1 \}$ and ${m}\in \{ 0, \ldots, {\MakeUppercase{{m}}}\}$: we can think of the index ${d}$ as an index of days (slow time-scale), and ${m}$ as an index of minutes (fast time-scale). The corresponding set of time indices is thus $$\TT = \{ 0, \ldots , {\MakeUppercase{{d}}}\} \times
\{ 0, \ldots, {\MakeUppercase{{m}}}\} \cup \{ \np{{\MakeUppercase{{d}}}+1,0} \}
\eqfinp
\label{eq:couples_of_indices}$$ At the end of every minute ${m}-1$ of every day ${d}$, that is, at the end of the time interval $\big[({d},{m}-1),({d},{m})\big)$, $0 \leq {d}\leq {\MakeUppercase{{d}}}$ and $1 \leq {m}\leq {\MakeUppercase{{m}}}$, an uncertainty variable $\uncertain_{{d},{m}}$ becomes available. Then, at the beginning of the minute ${m}$, a decision-maker takes a decision $\control_{{d}, {m}}$. Moreover, at the beginning of every day ${d}$, an uncertainty variable $\uncertain_{{d},0}$ is produced, followed by a decision $\control_{{d}, 0}$. The interplay between uncertainties and decision is thus as follows (compare the chronology with the one in ): $$\begin{gathered}
\uncertain_{0,0} \rightsquigarrow \control_{0,0}
\rightsquigarrow \uncertain_{0,1}
\rightsquigarrow \control_{0,1} \rightsquigarrow \cdots \\
\cdots \rightsquigarrow \uncertain_{0,{\MakeUppercase{{m}}}-1}
\rightsquigarrow \control_{0,{\MakeUppercase{{m}}}-1}
\rightsquigarrow \uncertain_{0,{\MakeUppercase{{m}}}}
\rightsquigarrow \control_{0,{\MakeUppercase{{m}}}}
\rightsquigarrow \uncertain_{1,0}
\rightsquigarrow \control_{1,0}
\rightsquigarrow \uncertain_{1,1} \cdots \\
\cdots \rightsquigarrow
\uncertain_{{\MakeUppercase{{d}}},{\MakeUppercase{{m}}}} \rightsquigarrow \control_{{\MakeUppercase{{d}}},{\MakeUppercase{{m}}}}
\rightsquigarrow \uncertain_{{\MakeUppercase{{d}}}+1,0} \eqfinp\end{gathered}$$ We assume that a state reduction (as in Definition \[de:reduction-dynamics\_family\]) is available at the beginning of each day $d$, so that it becomes possible to write dynamic programming equations by time blocks as stated by Theorem \[thm:DPB\_family\]. Such state reductions will be for example available when the noises of the different days are stochastically independent.
We present the mathematical formalism to handle such type of problems. In this application, the difficulty to apply Theorem \[thm:DPB\_family\] is mainly notational.
We consider the set $\TT$ equipped with the *lexicographical order*
$$(0,0)<(0,1)<\cdots < ({d},{\MakeUppercase{{m}}})<({d}+1,0)< \cdots
< ({\MakeUppercase{{d}}},{\MakeUppercase{{m}}}-1)< ({\MakeUppercase{{d}}},{\MakeUppercase{{m}}})
< ({\MakeUppercase{{d}}}+1,0)
\eqfinp
\label{eq:lexicographically_ordered_time_span}$$
The set $\TT$ of couples in is in one to one correspondence with the (linear) *time span* $ \{0, \ldots, \horizon \} $, where $$\horizon = ({\MakeUppercase{{d}}}+1) \times ({\MakeUppercase{{m}}}+1)+1 \eqfinv$$ by the *lexicographic mapping* $\tau$ $$\begin{aligned}
\tau : \{0, \ldots, \horizon \} & \to \TT \\
t &\mapsto \tau(t) = \np{{d},{m}} \eqfinp
\end{aligned}$$ In the sequel, we will denote by $ \np{{d},{m}} \in \TT $ the element of $\{0, \ldots, \horizon \}$ given by $ \tau^{-1}\np{{d},{m}} =
{d}\times ({\MakeUppercase{{m}}}+1) + {m}$: $$\TT \ni \np{{d},{m}} \leftrightarrow \tau^{-1}\np{{d},{m}}=
{d}\times ({\MakeUppercase{{m}}}+1) + {m}\in \{0, \ldots, \horizon \}
\eqfinp
\label{eq:lexicographically_ordered_time_span_d_0_identification}$$ \[eq:lexicographically\_ordered\_time\_span\_d\_0\]
For $ ({d}, {m}) \leq ({d}', {m}') $, as ordered by the lexicographical order , we introduce the time interval $ \interval{({d}, {m})}{({d}', {m}')} =
\defset{({d}'', {m}'') \in \TT}{ ({d}, {m}) \leq
({d}'', {m}'') \leq ({d}', {m}') } $.
For all $ \np{{d},{m}} \in
\{ 0, \ldots , {\MakeUppercase{{d}}}\} \times \{ 0, \ldots, {\MakeUppercase{{m}}}\} $, the decision $\control_{{d},{m}}$ takes its values in a measurable set $\CONTROL_{{d},{m}}$ equipped with a $\sigma$-field $\tribu{\Control}_{{d},{m}}$. For all $ \np{{d},{m}} \in
\{ 0, \ldots , {\MakeUppercase{{d}}}\} \times \{ 0, \ldots, {\MakeUppercase{{m}}}\}
\cup \{ \np{{\MakeUppercase{{d}}}+1,0} \} $, the uncertainty $\uncertain_{{d},{m}}$ takes its values in a measurable set $\UNCERTAIN_{{d},{m}}$ equipped with a $\sigma$-field $\tribu{\Uncertain}_{{d},{m}}$.
With the identification , for all $\np{{d},{m}} \in \TT$, we define the *history space* $\HISTORY_{({d},{m})}$ $$\HISTORY_{({d},{m})} =
\UNCERTAIN_{0,0} \times \CONTROL_{0,0} \times \UNCERTAIN_{0,1}
\times \cdots
\times \CONTROL_{{d},{m}-1} \times \UNCERTAIN_{{d},{m}}
\eqfinv$$ equipped with the *history field* $\tribu{\History}_{({d},{m})}$ as in . For all ${d}\in \{ 0, \ldots , {\MakeUppercase{{d}}}+1 \} $, we define the *slow scale history* $\history_{{d}}$ element of the *slow scale history space* $\HISTORY_{{d}}$ $$\begin{aligned}
\history_{{d}}=\history_{({d},0)} \in \HISTORY_{{d}}
= \HISTORY_{({d},0)}\eqfinv
\label{eq:slow_scale_history_space}
\end{aligned}$$ equipped with the *slow scale history field* $\tribu{\History}_{{d}} = \tribu{\History}_{({d},0)}$. For all ${d}\in \{ 1, \ldots , {\MakeUppercase{{d}}}\} $, we define the *slow scale partial history space* $\HISTORY_{{d}:{d}+1}$ $$\begin{aligned}
&\HISTORY_{{d}:{d}+1} = \HISTORY_{({d},1):({d}+1,0)}
= \CONTROL_{{d},0} \times \UNCERTAIN_{{d},1}
\times \cdots
\times \CONTROL_{{d},{\MakeUppercase{{m}}}-1} \times \UNCERTAIN_{{d},{\MakeUppercase{{m}}}}
\times \CONTROL_{{d},{\MakeUppercase{{m}}}} \times \UNCERTAIN_{{d}+1,0}
\eqfinv
\end{aligned}$$ equipped with the associated *slow scale partial history field* $\tribu{\History}_{{d}:{d}+1}$, the case ${d}=0$ being $$\HISTORY_{0:1} = \HISTORY_{(1,0)} =
\UNCERTAIN_{0,0} \times \CONTROL_{0,0} \times \UNCERTAIN_{0,1} \times \cdots
\times \CONTROL_{0,{\MakeUppercase{{m}}}-1} \times \UNCERTAIN_{0,{\MakeUppercase{{m}}}}
\times \CONTROL_{0,{\MakeUppercase{{m}}}} \times \UNCERTAIN_{1,0} \eqfinp$$
Because of the jump from one day to the next, we introduce two families of stochastic kernels[^2]:
- a family $ \sequence{ {\rho_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}}}{ 0 \leq {d}\leq {\MakeUppercase{{d}}}} $ of stochastic kernels *across* consecutive slow scale steps $${\rho_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}} :
\HISTORY_{({d},{\MakeUppercase{{m}}})} \to \Delta\np{\UNCERTAIN_{{d}+1,0}}
\eqsepv {d}= 0, \ldots,{\MakeUppercase{{d}}}\eqfinv
\label{eq:family_of_stochastic_kernels_2ts_across}$$
- a family $ \sequence{ {\rho_{({d},{m}-1):({d},{m})}}}{ 0 \leq {d}\leq {\MakeUppercase{{d}}}, 1 \leq {m}\leq {\MakeUppercase{{m}}}} $ of stochastic kernels *within* consecutive slow scale steps $${\rho_{({d},{m}-1):({d},{m})}} :
\HISTORY_{({d},{m}-1)} \to \Delta\np{\UNCERTAIN_{{d},{m}}}
\eqsepv {d}= 0, \ldots,{\MakeUppercase{{d}}}\eqsepv {m}= 1,\ldots,{\MakeUppercase{{m}}}\eqfinp
\label{eq:family_of_stochastic_kernels_2ts_within}$$
A history feedback at index $\np{{d},{m}} \in \TT$ is a measurable mapping $${\gamma}_{({d},{m})}: \HISTORY_{({d},{m})}
\to \CONTROL_{({d},{m})} \eqfinp$$ For $ ({d}, {m}) \leq ({d}', {m}') $, as ordered by the lexicographical order , we denote by ${\Gamma}_{({d}, {m}):({d}', {m}')}$ the set of $\interval{({d}, {m})}{({d}', {m}')}$-history feedbacks.
We suppose given a nonnegative numerical function $$\criterion : \HISTORY_{{\MakeUppercase{{d}}}+1} \to {[0,+\infty]}\eqfinv
\label{eq:criterion_two_scales}$$ assumed to be measurable with respect to the field $\tribu{\History}_{{\MakeUppercase{{d}}}+1}$ associated to $\HISTORY_{{\MakeUppercase{{d}}}+1}$.
For $ {d}=0,\ldots,{\MakeUppercase{{d}}}$, we build the new stochastic kernels ${\rho_{({d},0):({\MakeUppercase{{d}}}+1,0)}^{{\gamma}}}
: \HISTORY_{{d}} \to \Delta\np{ \HISTORY_{{\MakeUppercase{{d}}}+1} }$ (see Definition \[de:stochastic\_kernels\_rho\] in §\[Stochastic\_Kernels\] for their construction), and we define the *slow scale value functions* $$\begin{aligned}
\Value_{{d}}(\history_{{d}}) &=
\inf_{{\gamma}\in {\Gamma}_{({d},0):({\MakeUppercase{{d}}},{\MakeUppercase{{m}}})}}
\int_{\HISTORY_{{\MakeUppercase{{d}}}+1}}
\criterion\np{\history'_{{\MakeUppercase{{d}}}+1}}
{\rho_{({d},0):({\MakeUppercase{{d}}}+1,0)}^{{\gamma}}}
\np{\history_{{d}},\mathrm{d}\history'_{{\MakeUppercase{{d}}}+1}}
\eqsepv \forall \history_{{d}} \in \HISTORY_{{d}}
\eqfinv \\
\Value_{{\MakeUppercase{{d}}}+1} &= \criterion \eqfinp\end{aligned}$$ \[eq:value\_function\_2ts\]
For $ {d}=0,\ldots,{\MakeUppercase{{d}}}$, we define a *family of slow scale Bellman operators across $\interval{{d}+1}{{d}}$* $${\mathcal{B}_{{d}+1:{d}}} :
\espace{L}^{0}_{+}(\HISTORY_{{d}+1},\tribu{\History}_{{d}+1})
\to \espace{L}^{0}_{+}(\HISTORY_{{d}},\tribu{\History}_{{d}})
\eqsepv {d}=0,\ldots,{\MakeUppercase{{d}}}\eqfinv$$ by $${\mathcal{B}_{{d}+1:{d}}}
= {\mathcal{B}_{({d}+1,0):({d},0)}} \\
= {\mathcal{B}_{({d}+1,0):({d},{\MakeUppercase{{m}}})}} \circ
{\mathcal{B}_{({d},{\MakeUppercase{{m}}}):({d},{\MakeUppercase{{m}}}-1)}}
\circ \ldots \circ {\mathcal{B}_{({d},1):({d},0)}} \eqfinp$$ \[eq:Bellman\_operators\_2ts\]
Then, applying repeatedly Theorem \[pr:DP\_withoutstate\_third\] leads to the fact that the family $\{\Value_{{d}}\}_{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ of slow scale value functions satisfies $$\begin{aligned}
\Value_{{\MakeUppercase{{d}}}+1}
& = \criterion \eqfinv \\
\Value_{{d}}
& = {\mathcal{B}_{{d}+1:{d}}} \Value_{{d}+1} \eqsepv
\text{ for } {d}= {\MakeUppercase{{d}}},{\MakeUppercase{{d}}}-1,\ldots,0 \eqfinp\end{aligned}$$ \[thm:DPB\_2ts\]
We now rewrite Definition \[de:reduction-dynamics\_family\] in the context of the two time-scales problem.
Let $\sequence{\np{\STATE_{{d}},\tribu{\State}_{{d}}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ be a family of measurable state spaces, $\sequence{\theta_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ be family of measurable reduction mappings such that
$$\theta_{{d}} : \HISTORY_{{d}} \to \STATE_{{d}} \eqfinv$$
and $\sequence{ \Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}}$ be a family of measurable dynamics such that $$\Dynamics{{d}}{{d}+1} : \STATE_{{d}} \times \HISTORY_{{d}:{d}+1}
\to \STATE_{{d}+1} \eqfinp$$ The triplet $\bp{\sequence{\STATE_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}, \sequence{\theta_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},\sequence{ \Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}}}$ is said to be a *slow scale state reduction* if for all ${d}= 0, \ldots, {\MakeUppercase{{d}}}$ $$\theta_{{d}+1}\bp{\np{\history_{{d}}, \history_{{d}:{d}+1} }}
= \Dynamics{{d}}{{d}+1}
\bp{ \theta_{{d}}\np{\history_{{d}}},\history_{{d}:{d}+1} }
\eqsepv \forall \np{\history_{{d}}, \history_{{d}:{d}+1} } \in \HISTORY_{{d}+1}
\eqfinp
$$
The slow scale state reduction $\bp{\sequence{\STATE_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{\theta_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{ \Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}}}$ is said to be *compatible with the two families $\sequence{{\rho_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}}}
{0\leq{d}\leq{\MakeUppercase{{d}}}}$ and $\sequence{{\rho_{({d},{m}-1):({d},{m})}}}
{0\leq{d}\leq{\MakeUppercase{{d}}}, 1\leq{m}\leq{\MakeUppercase{{m}}}}$ of stochastic kernels* defined in – if for any ${d}= 0,\ldots, {\MakeUppercase{{d}}}$, we have that
- there exists a *reduced stochastic kernel* $${{\tilde\rho}_{({d}, {\MakeUppercase{{m}}}):({d}+1,0)}} :
\STATE_{{d}} \times \HISTORY_{({d},0):({d},{\MakeUppercase{{m}}})}
\to \Delta\np{\UNCERTAIN_{{d}+1,0}} \eqfinv$$ such that the stochastic kernel ${\rho_{({d}, {\MakeUppercase{{m}}}):({d}+1,0)}}$ in can be factored as $$\begin{gathered}
{\rho_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}}
\np{\history_{{d}, {\MakeUppercase{{m}}}},\mathrm{d}\uncertain_{{d}+1,0}} =
{{\tilde\rho}_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}}
\bp{\theta_{{d}}\np{\history_{{d}}},
\history_{({d},0):({d}, {\MakeUppercase{{m}}})},
\mathrm{d}\uncertain_{{d}+1,0}} \eqsepv
\forall \history_{{d}, {\MakeUppercase{{m}}}}
\in \HISTORY_{({d},{\MakeUppercase{{m}}})} \eqfinv\end{gathered}$$
- for each ${m}= 1,\ldots, {\MakeUppercase{{m}}}$ , there exists a *reduced stochastic kernel* $${{\tilde\rho}_{({d},{m}-1):({d},{m})}} :
\STATE_{{d}} \times \HISTORY_{({d},0):({d},{m}-1)}
\to \Delta\np{\UNCERTAIN_{{d}, {m}}} \eqfinv$$ such that the stochastic kernel ${\rho_{({d},{m}-1):({d},{m})}}$ in can be factored as $$\begin{gathered}
{\rho_{({d},{m}-1):({d},{m})}}
\np{\history_{{d},{m}-1},
\mathrm{d}\uncertain_{{d}, {m}}} =
{{\tilde\rho}_{({d},{m}-1):({d},{m})}}
\bp{\theta_{{d}}\np{\history_{{d}}},\history_{({d}, 0):({d},{m}-1)},
\mathrm{d}\uncertain_{{d}, {m}}} \eqsepv
\forall \history_{{d}, {m}-1} \in \HISTORY_{({d}, {m}-1)}
\eqfinp
\end{gathered}$$
\[de:Compatible\_slow\_scale\_reduction\]
Using the reduced stochastic kernels of Definition \[de:Compatible\_slow\_scale\_reduction\], we apply Proposition \[thm:DPB\] and obtain a *family of slow scale reduced Bellman operators across $\interval{{d}+1}{{d}}$* $${{\tilde{\mathcal{B}}}_{{d}+1:{d}}} :
\espace{L}^{0}_{+}(\STATE_{{d}+1},\tribu{\State}_{{d}+1})
\to \espace{L}^{0}_{+}(\STATE_{{d}},\tribu{\State}_{{d}})
\eqsepv d=0,\ldots,{\MakeUppercase{{d}}}\eqfinp
\label{eq:Bellman_operators_2ts_reduced}$$ We are now able to state the main result of this section.
Assume that there exists a compatible slow scale state reduction\
$\bp{\sequence{\STATE_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{\theta_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{ \Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}}}$ and that there exists a *reduced criterion*
$$\tilde\criterion : \STATE_{{\MakeUppercase{{d}}}+1} \to {[0,+\infty]}\eqfinv$$
such that the cost function $\criterion$ in can be factored as $$\criterion = \tilde\criterion \circ \theta_{{\MakeUppercase{{d}}}+1} \eqfinp$$
We define the family of *reduced value functions* $\{\tilde\Value_{{d}}\}_{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ by
$$\begin{aligned}
\tilde\Value_{{\MakeUppercase{{d}}}+1}
& = \tilde\criterion \eqfinv \\
\tilde\Value_{{d}}
& = {{\tilde{\mathcal{B}}}_{{d}+1:{d}}} \tilde\Value_{{d}+1}
\eqsepv \text{ for } {d}= {\MakeUppercase{{d}}},\ldots,0
\eqfinp\end{aligned}$$
Then, the family $\{\Value_{{d}}\}_{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ of slow scale value functions satisfies $$\Value_{{d}} = \tilde\Value_{{d}} \circ \theta_{{d}}
\eqsepv {d}= 0,\ldots,{\MakeUppercase{{d}}}\eqfinp
\label{eq:value_functions_factorization_2ts}$$
\[thm:DPB\_2ts\_reduced\]
Since the triplet $ \np{ \sequence{\STATE_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{\theta_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1},
\sequence{ \Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}} }$ is a state reduction across the time blocks $[({d},0),({d}\!+\!1,0)]$, which is compatible with the family $ \sequence{ {\rho_{({d},0):({d}+1,0)}} }{ 0 \leq {d}\leq {\MakeUppercase{{d}}}} $ of stochastic kernels, the proof is an immediate consequence of Theorem \[thm:DPB\_family\].
Thanks to Theorem \[thm:DPB\_2ts\_reduced\], we are able to replace the optimization problem formulated on the whole time set $\TT$ by a sequence of ${\MakeUppercase{{d}}}$ optimization subproblems formulated each on a single time block $[({d},0),({d}\!+\!1,0)]$. Moreover, the numerical burden of the method remains reasonable provided that the dimensions of the spaces $\STATE_{{d}}$ remain small, thus avoiding the curse of dimensionality. This is the benefit induced by *dynamic programming* which makes possible a time decomposition of the problem. However, to make the method operational, we need to compute the functions $\tilde\Value_{{d}}$, whose expression is available thanks to Proposition \[thm:DPB\]: $$\begin{aligned}
&
\tilde\Value_{{d}} \np{\state_{{d}}} =
\inf_{\control_{{d},0}\in\CONTROL_{{d},0}}
\int_{\UNCERTAIN_{{d},1}}
{{\tilde\rho}_{({d},0):({d},1)}}\np{{\state_{{d}}},
\mathrm{d}\uncertain_{{d}, 1}} \ldots
\nonumber \\
& \qquad\qquad
\inf_{\control_{{d},{\MakeUppercase{{m}}}-1}\in\CONTROL_{{d},{\MakeUppercase{{m}}}-1}}
\int_{\UNCERTAIN_{{d},{\MakeUppercase{{m}}}}}
{{\tilde\rho}_{({d},{\MakeUppercase{{m}}}-1):({d},{\MakeUppercase{{m}}})}}
\np{\state_{{d}},\control_{{d},0},\uncertain_{{d}, 1}, \cdots,
\uncertain_{{d}, {\MakeUppercase{{m}}}-1},
\mathrm{d}\uncertain_{{d}, {\MakeUppercase{{m}}}}}
\nonumber \\
& \qquad\qquad\qquad\quad
\inf_{\control_{{d},{\MakeUppercase{{m}}}}\in\CONTROL_{{d},{\MakeUppercase{{m}}}}}
\int_{\UNCERTAIN_{{d}+1,0}}
\tilde\Value_{{d}+1}\bp{\tildeDynamics{{d}}{{d}+1}
\np{\state_{{d}},
\control_{{d},0},\uncertain_{{d}, 1}, \cdots, \control_{{d},{\MakeUppercase{{m}}}-1},\uncertain_{{d}, {\MakeUppercase{{m}}}}, \control_{{d},{\MakeUppercase{{m}}}}, \uncertain_{{d}+1, 0}}} \nonumber \\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
{{\tilde\rho}_{({d},{\MakeUppercase{{m}}}):({d}+1,0)}}
\np{\state_{{d}},
\control_{{d},0},\uncertain_{{d}, 1}, \cdots,
\uncertain_{{d}, {\MakeUppercase{{m}}}},
\mathrm{d}\uncertain_{{d}+1,0}}
\eqfinp
\label{eq:fast-scale_optimization_problem}\end{aligned}$$ In many practical situations, this computation is tractable by using *stochastic programming*. For example, if the stochastic kernels ${{\tilde\rho}_{({d},{m}):({d},{m}+1)}}$ do not depend on the past controls $\np{\control_{{d},0},\cdots,\control_{{d},{m}-1}}$, then it is possible to approximate the optimization problem by using scenario tree techniques. Note that these last techniques do not require stagewise independence of the noises. We are thus able to take advantage of both the dynamic programming world and the stochastic programming world:
- use dynamic programming at slow time scale across consecutive slow time steps, when the slow time scale noises are supposed to be stochastically independent; produce slow time scale Bellman functions;
- use stochastic programming at short time scale, within two consecutive slow time steps; the final short time scale cost is given by the slow time scale Bellman functions; no stagewise independence assumption is required for the short time scale noises.
Decision-Hazard-Decision Optimization Problems {#Decision_Hazard_Decision_Dynamic_Programming}
----------------------------------------------
We apply the reduction by time blocks to the so-called *decision-hazard-decision* dynamic programming.
### Motivation for the Decision-Hazard-Decision Framework {#Decision_Hazard_Decision_Motivation}
We illustrate our motivation with a single dam management problem. We can model the dynamics of the water volume in a dam by $$\volume_{t+1} = \min \{ \volume\upper, \volume_{t} - \turbined_{t} +
\inflow_{t+1} \}
\eqfinv
\label{eq:dam_model_decision-hazard}$$ where $ t=\tinitial,\tinitialplusone,\ldots,\horizon-1 $ and
- $\volume\upper$ is the maximal dam volume,
- $\volume_{t}$ is the volume (stock) of water at the beginning of period $[t,t+1[$,
- $\inflow_{t+1}$ is the inflow water volume (rain, etc.) during $[t,t+1[$,
- $\turbined_{t}$ is the turbined outflow volume during $[t,t+1[$ (control variable),
- decided at the *beginning* of period $[t,t+1[$,
- chosen such that $ 0\leq \turbined_{t} \leq \volume_{t} $,
- supposed to depend on the stock $\volume_{t}$ but not on the inflow water $\inflow_{t+1}$.
The $\min$ operation in Equation ensures that the dam volume always remains below its maximal capacity, but induces a non linearity in the dynamics.
Alternatively, we can model the dynamics of the water volume in a dam by $$\volume_{t+1}= \volume_{t}-\turbined_{t}-\inflow_{t+1}-\spilled_{t+1}
\eqfinv
\label{eq:dam_model_decision-hazard-decision}$$ where $t=\tinitial,\tinitialplusone,\ldots,\horizon-1$ and
- $ \spilled_{t+1} $ is the spilled volume
- decided at the *end* of period $[t,t+1[$,
- supposed to depend on the stock $\volume_{t}$ and on the inflow water $\inflow_{t+1}$,
- and chosen such that $0\leq \volume_{t} - \turbined_{t} + \inflow_{t+1}
- \spilled_{t+1} \leq \volume\upper$.
Thus, with the formulation , we pay the price to add one control $ \spilled_{t+1} $, but we obtain a linear model instead of the nonlinear model . This is especially interesting when using the stochastic dual dynamic programming (SDDP), for which the linearity of the dynamics is used to obtain the convexity properties required by the algorithm.
### Decision-Hazard-Decision Framework
We consider stochastic optimization problems where, during the time interval between two time steps, the decision-maker takes two decisions. At the end of the time interval $[{s}-1,{s}[$, an uncertainty variable $\uncertain^{\flat}_{{s}}$ is produced, and then, at the beginning of the time interval $[{s},{s}+1[$, the decision-maker takes a *head decision* $\control_{{s}}^{\sharp}$. What is new is that, at the end of the time interval $[{s},{s}+1[$, when an uncertainty variable $\uncertain^{\flat}_{{s}+1}$ is produced, the decision-maker has the possibility to make a *tail decision* $\control_{{s}+1}^{\flat}$. This latter decision $\control_{{s}+1}^{\flat}$ can be thought as a *recourse* variable for a two stage stochastic optimization problem that would take place inside the time interval $[{s},{s}+1[$. We call $\uncertain^{\sharp}_0$ the uncertainty happening right before the first decision. The interplay between uncertainties and decisions is thus as follows (compare the chronology with the one in ): $$\begin{gathered}
\uncertain^{\sharp}_{0} \rightsquigarrow \control^{\sharp}_{0}
\rightsquigarrow \uncertain^{\flat}_{1} \rightsquigarrow \control^{\flat}_{1}
\rightsquigarrow \control^{\sharp}_{1} \rightsquigarrow \uncertain^{\flat}_{2}
\rightsquigarrow \;\; \dots \;\; \rightsquigarrow
\uncertain^{\flat}_{{S}-1} \rightsquigarrow \control^{\flat}_{{S}-1} \rightsquigarrow \control^{\sharp}_{{S}-1}
\rightsquigarrow \uncertain^{\flat}_{{S}} \rightsquigarrow \control^{\flat}_{{S}} \eqfinp\end{gathered}$$
Let $ {S}\in \NN^* $. For each time ${s}=0,1,2\ldots, {S}-1$, the *head decision* $ \control_{{s}}^{\sharp}$ takes values in a measurable set $\CONTROL^{\sharp}_{{s}}$, equipped with a $\sigma$-field $\tribu{\Control}^{\sharp}_{{s}}$. For each time ${s}=1,2\ldots, {S}$, the *tail decision* $\control_{{s}}^{\flat}$ takes values in measurable set $\CONTROL^{\flat}_{{s}}$, equipped with a $\sigma$-field $\tribu{\Control}^{\flat}_{{s}}$. For each time ${s}=1,2\ldots, {S}$, the uncertainty $\uncertain^{\flat}_{{s}}$ takes its values in a measurable set $\UNCERTAIN^{\flat}_{{s}}$, equipped with a $\sigma$-field $\tribu{\Uncertain}^{\flat}_{{s}}$. For time ${s}=0$, the uncertainty $\uncertain^{\sharp}_{0}$ takes its values in a measurable set $\UNCERTAIN^{\sharp}_{0}$, equipped with a $\sigma$-field $\tribu{\Uncertain}^{\sharp}_{0}$.
Again, in this application, the difficulty to apply Theorem \[thm:DPB\_family\] is mainly notational.
For ${s}=0,1,2\ldots, {S}$, we define the *head history space*
$$\begin{aligned}
\HISTORY^{\sharp}_{{s}}
&=
\UNCERTAIN_{0}^{\sharp}\times \prod_{{s'}=0}^{{s}-1}
\bp{ \CONTROL^{\sharp}_{{s'}} \times \UNCERTAIN_{{s'}+1}^{\flat}\times \CONTROL^{\flat}_{{s'}+1} } \eqfinv
\label{eq:head_history_space}
\intertext{ and its associated \emph{head history field}
$\tribu{\History}^{\sharp}_{{s}}$. We also define,
for ${s}=1,2\ldots, {S}$,
the \emph{tail history space} }
\HISTORY^{\flat}_{{s}}
&=
\HISTORY^{\sharp}_{{s}-1}
\times \CONTROL^{\sharp}_{{s}-1}
\times \UNCERTAIN_{{s}}^{\flat}\eqfinv
\label{eq:tail_history_space}
\end{aligned}$$
and its associated *tail history field* $\tribu{\History}^{\flat}_{{s}}$.
We introduce a family of stochastic kernels $\na{{\rho_{{s}-1:{s}}}}_{1 \leq {s}\leq {S}}$, with $${\rho_{{s}-1:{s}}} :
\HISTORY^{{\sharp}}_{{s}-1}
\to \Delta\np{\UNCERTAIN_{{s}}^{\flat}}
\eqfinp
\label{eq:rhodhd}$$
For ${s}= 0,\ldots,{S}-1$, a *head history feedback* at time ${s}$ is a measurable mapping $${\gamma}^{\sharp}_{s}:
\HISTORY^{\sharp}_{{s}} \to \CONTROL^{\sharp}_{s}\eqfinp$$ We call ${\Gamma}^{\sharp}_{{s}}$ the *set of head history feedbacks at time ${s}$*, and we define ${\Gamma}^{\sharp}_{{s}:{S}} =
{\Gamma}^{\sharp}_{{s}} \times \cdots
\times {\Gamma}^{\sharp}_{{S}}$. We also define, for all ${s}= 1,2\ldots,{S}$, a *tail history feedback* at time ${s}$ as a measurable mapping $${\gamma}^{\flat}_{s}:
\HISTORY^{\flat}_{{s}} \to \CONTROL^{\flat}_{s}\eqfinp$$ We call ${\Gamma}^{\flat}_{{s}}$ the *set of tail history feedbacks at time ${s}$*, and we define ${\Gamma}^{\flat}_{{s}:{S}} =
{\Gamma}^{\flat}_{{s}} \times \cdots
\times {\Gamma}^{\flat}_{{S}}$.
We consider a nonnegative numerical function $$\criterion : \HISTORY^{{\sharp}}_{{S}} \to {[0,+\infty]}\eqfinv
\label{eq:criterion_dhd}$$ assumed to be measurable with respect to the head history field $\tribu{\History}^{{\sharp}}_{S}$.
For $s = 0,\ldots, {S}$ , we define *value functions* by $$\Value_{{s}}(\history^{\sharp}_{{s}}) =
\inf_{{\gamma}^{\sharp}\in {\Gamma}^{\sharp}_{{s}:{S}-1},
{\gamma}^{\flat}\in {\Gamma}^{\sharp}_{{s}+1:{S}}}
\int_{\HISTORY^{\sharp}_{{S}}}
\criterion\np{\history'_{{S}}}
{{\rho_{{s}:{S}}^{{\gamma}^{\sharp},
{\gamma}^{\flat}}}}
\np{\history^{\sharp}_{s},\mathrm{d}\history'_{{S}}}
\eqsepv \forall \history^{\sharp}_{{s}} \in \HISTORY^{\sharp}_{{s}}
\eqfinv
\label{eq:value_function_dhd}$$ where $ {{\rho_{{s}:{S}}^{{\gamma}^{\sharp},
{\gamma}^{\flat}}}} $ has to be understood as $ {\rho_{{s}:{S}}^{{\gamma}}} $ (see Definition \[de:stochastic\_kernels\_rho\]), with
$$\begin{aligned}
{\gamma}_{{s}}(\history_{{s}}^{\sharp})
&=
{\gamma}^{\sharp}_{{s}}(\history_{{s}}^{\sharp})
\eqsepv
\forall \history_{{s}}^{\sharp}\in \HISTORY_{{s}}^{\sharp}\eqfinv
\\
{\gamma}_{{s'}}(\history_{{s'}}^{\flat})
&=
\Bp{{\gamma}^{\flat}_{{s'}}\np{\history_{{s'}}^{\flat}},
{\gamma}^{\sharp}_{{s'}}\bp{\history_{{s'}}^{\flat},
{\gamma}^{\flat}_{{s'}}\np{\history_{{s'}}^{\flat}} } }
\eqsepv
\forall {s'}= {s}+1,\ldots, {S}-1
\eqsepv
\forall \history_{{s'}}^{\flat}\in \HISTORY_{{s'}}^{\flat}\eqfinv
\\
{\gamma}_{{S}}(\history_{{S}}^{\flat})
&=
{\gamma}^{\flat}_{{S}}(\history_{{S}}^{\flat})
\eqsepv
\forall \history_{{S}}^{\flat}\in \HISTORY_{{S}}^{\flat}\eqfinp\end{aligned}$$
\[eq:feedback\_dhd\]
The following proposition, whose proof has been relegated in \[proof:DPB\_dhd\], characterizes the dynamic programming equations in the decision-hazard-decision framework.
\[thm:DPB\_dhd\] For ${s}\inic{0}{{S}}-1$, we define the *Bellman operator*
$${\mathcal{B}_{{s}+1:{s}}} : \espace{L}^{0}_{+}(\HISTORY^{\sharp}_{{s}+1},\tribu{\History}^{\sharp}_{{s}+1})
\to \espace{L}^{0}_{+}(\HISTORY^{\sharp}_{{s}},\tribu{\History}^{\sharp}_{{s}})$$
such that, for all $\varphi\in \espace{L}^{0}_{+}(\HISTORY^{\sharp}_{{s}+1},\tribu{\History}^{\sharp}_{{s}+1})$ and for all $\history^{\sharp}_{{s}} \in \HISTORY^{\sharp}_{{s}}$, $$\bp{ {\mathcal{B}_{{s}+1:{s}}}\varphi }\np{\history^{\sharp}_{{s}}} =
\inf_{\control^{\sharp}_{{s}}\in\CONTROL^{\sharp}_{{s}}} \int_{\UNCERTAIN_{{s}+1}^{\flat}}
\Bp{\inf_{\control_{{s}+1}^{\flat}\in\CONTROL_{{s}+1}^{\flat}} \varphi\np{\history^{\sharp}_{{s}},\control_{{s}}^{\sharp},\uncertain_{{s}+1}^{\flat}, \control_{{s}+1}^{\flat}}}
{\rho_{{s}:{s}+1}} (\history_{{s}}^{\sharp},d\uncertain_{{s}+1}^{\flat})
\eqfinp
\label{eq:Bellman_operators_rho_dhd}$$
Then the value functions satisfy $$\begin{aligned}
\Value_{S}&=
\criterion \eqfinv
\\
\Value_{s}&= {\mathcal{B}_{{s}+1:{s}}}\Value_{{s}+1}
\eqsepv \forall {s}= 0, \ldots, {S}-1
\eqfinp
\end{aligned}$$
We now rewrite Definition \[de:reduction-dynamics\_family\] in the context of a decision-hazard-decision problem.
Let $\sequence{\STATE_{{s}}}{{s}=0,\ldots,{S}}$ be a family of state spaces, $\sequence{\theta_{{s}}}{{s}=0,\ldots,{S}}$ be a family of measurable reduction mappings such that
$$\theta_{{s}} : \HISTORY^{\sharp}_{{s}} \to \STATE_{{s}} \eqfinv$$
and $\sequence{ \Dynamics{{s}}{{s}+1} }{{s}=0,\ldots,{S}-1}$ be a family of measurable dynamics such that $$\Dynamics{{s}}{{s}+1} :
\STATE_{{s}} \times \CONTROL_{{s}}^{\sharp}\times \UNCERTAIN_{{s}+1}
\times \CONTROL_{{s}+1}^{\flat}\to \STATE_{{s}+1} \eqfinp$$ The triplet $\bp{\sequence{\STATE_{{s}}}{{s}=0,\ldots,{S}},
\sequence{\theta_{{s}}}{{s}=0,\ldots,{S}},
\sequence{ \Dynamics{{s}}{{s}+1} }{{s}=0,\ldots,{S}-1}}$ is said to be a *decision-hazard-decision state reduction* if, for all ${s}= 0, \ldots, {S}-1$, we have that $$\begin{gathered}
\theta_{{s}+1}\bp{\np{\history_{{s}},
\control_{{s}}^{\sharp}, \uncertain_{{s}+1}, \control_{{s}+1}^{\flat}}}
= \Dynamics{{s}}{{s}+1}
\bp{ \theta_{{s}}\np{\history_{{s}}},
\control_{{s}}^{\sharp}, \uncertain_{{s}+1}, \control_{{s}+1}^{\flat}}
\eqsepv \\
\forall \np{\history_{{s}},
\control_{{s}}^{\sharp}, \uncertain_{{s}+1}, \control_{{s}+1}^{\flat}}
\in \HISTORY_{{s}}^{\sharp}\times
\CONTROL_{{s}}^{\sharp}\times \UNCERTAIN_{{s}+1} \times \CONTROL_{{s}+1}^{\flat}\eqfinp
$$
The decision-hazard-decision state reduction is said to be *compatible with the family $\na{{\rho_{{s}:{s}+1}}}_{0 \leq {s}\leq {S}-1}$ of stochastic kernels* in if there exists a family $\na{{{\tilde\rho}_{{s}:{s}+1}}}_{0 \leq {s}\leq {S}-1} $ of *reduced stochastic kernels*
$${{\tilde\rho}_{{s}:{s}+1}} :
\STATE_{{s}} \to \Delta\np{\UNCERTAIN_{{s}+1}} \eqfinv$$
such that, for each ${s}= 0,\ldots, {S}-1$, the stochastic kernel ${\rho_{{s}:{s}+1}}$ in can be factored as $${\rho_{{s}:{s}+1}}
\np{\history_{{s}}^{\sharp},\mathrm{d}\uncertain_{{s}+1}} =
{{\tilde\rho}_{{s}:{s}+1}}
\bp{\theta_{{s}}\np{\history_{{s}}^{\sharp}},
\mathrm{d}\uncertain_{{s}+1}} \eqsepv
\forall \history_{{s}}^{\sharp}\in \HISTORY_{{s}}^{\sharp}\eqfinp$$
We state the main result of this section.
Assume that there exists a decision-hazard-decision state reduction\
$\bp{\sequence{\STATE_{{s}}}{{s}=0,\ldots,{S}},
\sequence{\theta_{{s}}}{{s}=0,\ldots,{S}},\sequence{
\Dynamics{{s}}{{s}+1} }{{s}=0,\ldots,{S}-1}}$ and that there exists a *reduced criterion*
$$\tilde\criterion : \STATE_{{S}} \to {[0,+\infty]}\eqfinv$$
such that the cost function $\criterion$ in can be factored as $$\criterion = \tilde\criterion \circ \theta_{{S}} \eqfinp$$
We define a *family of reduced Bellman operators across $\interval{{s}+1}{{s}}$*
$${{\tilde{\mathcal{B}}}_{{s}+1:{s}}} :
\espace{L}^{0}_{+}(\STATE_{{s}+1},\tribu{\State}_{{s}+1})
\to \espace{L}^{0}_{+}(\STATE_{{s}},\tribu{\State}_{{s}})
\eqsepv {s}=1,\ldots,{S}-1 \eqfinv$$
by, for any measurable function $\tilde\varphi : \STATE_{{s}+1}\to{[0,+\infty]}$, $$\np{{{\tilde{\mathcal{B}}}_{{s}+1:{s}}} \tilde\varphi}(\state_{s}) =
\inf_{\control^{\sharp}_{{s}}\in\CONTROL^{\sharp}_{{s}}}
\int_{\UNCERTAIN_{{s}+1}}
\Bp{\inf_{\control_{{s}+1}^{\flat}\in\CONTROL_{{s}+1}^{\flat}}
\tilde\varphi\bp{\Dynamics{{s}}{{s}+1}
(\state_{{s}},\control_{{s}}^{\sharp},\uncertain_{{s}+1}, \control_{{s}+1}^{\flat})}}
{{\tilde\rho}_{{s}:{s}+1}} (\state_{{s}},d\uncertain_{{s}+1})
\eqfinp
\label{eq:Bellman_operators_dhd_reduced}$$
\*We define the family of *reduced value functions* $\{\tilde\Value_{{s}}\}_{{s}=0,\ldots,{S}}$ by
$$\begin{aligned}
\tilde\Value_{{S}}
& = \tilde\criterion \\
\tilde\Value_{{s}}
& = {{\tilde{\mathcal{B}}}_{{s}+1:{s}}} \tilde\Value_{{s}+1}
\quad \text{for } {s}= {S}-1,\ldots,0 \eqfinp\end{aligned}$$
Then, the value functions $\Value_{{s}}$ defined by satisfy $$\Value_{{s}} = \tilde\Value_{{s}} \circ \theta_{{s}}
\eqsepv {s}= 0,\ldots,{S}\eqfinp
\label{eq:value_functions_factorization_dhd}$$ \[thm:DPB\_dhd\_reduced\]
It has been shown in the proof of Proposition \[thm:DPB\_dhd\] that the setting of a decision-hazard-decision problem was a particular kind of two time-scales problem. The proof of the theorem is then a direct application of Theorem \[thm:DPB\_2ts\_reduced\].
Theorem \[thm:DPB\_dhd\_reduced\] allows to develop dynamic programming equations in the decision-hazard-decision framework. Such equations can be solved using the stochastic dual dynamic programming (SDDP) algorithm provided that convexity of the value functions is preserved. This requires linearity in the dynamics, a feature that may be recovered by modeling the problem in the decision-hazard-decision framework as illustrated in §\[Decision\_Hazard\_Decision\_Motivation\].
Conclusion and Perspectives
===========================
As said in the introduction, decomposition methods are appealing to tackle multistage stochastic optimization problems, as they are naturally large scale. The most common approaches are time decomposition (and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control), and scenario decomposition (like progressive hedging in stochastic programming). One also finds space decomposition methods [@Barty_RAIRO_2010].
This paper is part of a general research program that consists in *mixing* different decomposition bricks. Here, we tackled the issue of mixing time decomposition (stochastic dynamic programming) with scenario decomposition. For this purpose, we have revisited the notion of state, and have provided a way to perform time decomposition but only accross specified time blocks. Inside a time block, one can then use stochastic programming methods, like scenario decomposition. Our time blocks decomposition scheme is especially adapted to multi time-scales stochastic optimization problems. In this vein, we have shown its application to two time-scales and to the novel class of decision-hazard-decision problems.
We are currently working on how to mix time decomposition (stochastic dynamic programming) with space/units decomposition.
**Acknowledgements.** We thank Roger Wets for the fruitful discussions about the possibility of mixing stochastic dynamic programming with progressive hedging. We thank an anonymous reviewer for challenging our first version of the paper: the current version has been deeply restructured.
Technical Details and Proofs
============================
In this section, we provide technical details, constructions and proofs of results in the paper.
Histories, Feedbacks and Flows {#Histories_Feedbacks_Flows}
------------------------------
We introduce the notations
$$\begin{aligned}
\UNCERTAIN_{\tun:\tter}
& = \prod_{\tbis=\tun}^{\tter} \UNCERTAIN_{\tbis} \eqsepv
0 \leq \tun \leq \tter \leq \horizon \\
\CONTROL_{\tun:\tter}
& = \prod_{\tbis=\tun}^{\tter} \CONTROL_{\tbis} \eqsepv
0 \leq \tun \leq \tter \leq \horizon-1 \\
\HISTORY_{\tun:\tter} & =
\prod_{\tbis=\tun-1}^{\tter-1} \np{ \CONTROL_{\tbis} \times \UNCERTAIN_{\tbis+1}}
= \CONTROL_{\tun-1} \times \UNCERTAIN_{\tun} \times \cdots \times
\CONTROL_{\tter-1} \times \UNCERTAIN_{\tter}
\eqsepv 1 \leq \tun \leq \tter \leq \horizon \eqfinp\end{aligned}$$
\[eq:partial\_history\_space\]
Let $ 0 \leq \tun \leq \tbis \leq \tter \leq \horizon $. From a history $\history_{\tter} \in \HISTORY_{\tter}$, we can extract the $\interval{\tun}{\tbis}$-*history uncertainty part* $$\nc{\history_{\tter}}_{\tun:\tbis}^{\UNCERTAIN}=
\np{\uncertain_{\tun},\ldots, \uncertain_{\tbis} } = \uncertain_{\tun:\tbis}
\in \UNCERTAIN_{\tun:\tbis}
\eqsepv 0 \leq \tun \leq \tbis \leq \tter
\eqfinv
\label{eq:uncertain_t:r}$$ the $\interval{\tun}{\tbis}$-*history control part* (notice that the indices are special) $$\nc{\history_{\tter}}_{\tun:\tbis}^{\CONTROL}=
\np{\control_{\tun-1},\ldots, \control_{\tbis-1} } = \control_{\tun-1:\tbis-1}
\in \CONTROL_{\tun-1:\tbis-1}
\eqsepv 1 \leq \tun \leq \tbis \leq \tter
\eqfinv
\label{eq:control_t:r}$$ and the $\interval{\tun}{\tbis}$-*history subpart* $$\nc{\history_{\tter}}_{\tun:\tbis} =
\np{ \control_{\tun-1},\uncertain_{\tun}, \ldots,
\control_{\tbis-1},\uncertain_{\tbis} }
= \history_{\tun:\tbis} \in \HISTORY_{\tun:\tbis}
\eqsepv 1 \leq \tun \leq \tbis \leq \tter
\eqfinv
\label{eq:history_t:r}$$ so that we obtain, for $ 0 \leq \tun+1 \leq \tbis \leq \tter $, $$\history_{\tter} =
\np{\underbrace{ \uncertain_{0},\control_{0},\uncertain_{1},\ldots,
\control_{\tun-1},\uncertain_{\tun} }_{\history_{\tun}},
\underbrace{ \control_{\tun},\uncertain_{\tun+1} ,\ldots,
\control_{\tter-2}, \uncertain_{\tter-1},
\control_{\tter-1},\uncertain_{\tter} }_{\history_{\tun+1:\tter}} }
= \np{ \history_{\tun}, \history_{\tun+1:\tter} }
\eqfinp$$
Let $\tun$ and $\tter$ be given such that $ 0 \leq \tun < \tter \leq \horizon $. For a $\interval{\tun}{\tter-1}$-history feedback ${\gamma}=\sequence{{\gamma}_{\tbis}}{\tbis=\tun,\ldots,\tter-1}
\in{\Gamma}_{\tun:\tter-1} $, we define the *flow* ${\Phi_{\tun:\tter}^{{\gamma}}}$ by
$$\begin{aligned}
{\Phi_{\tun:\tter}^{{\gamma}}}
: \HISTORY_{\tun} \times \UNCERTAIN_{\tun+1:\tter}
& \to \HISTORY_{\tter} \\
\np{ \history_{\tun}, \uncertain_{\tun+1:\tter} }
& \mapsto
\np{\history_{\tun},
{\gamma}_{\tun}(\history_{\tun}),\uncertain_{\tun+1},
{\gamma}_{\tun+1}\np{\history_{\tun},
{\gamma}_{\tun}(\history_{\tun}),
\uncertain_{\tun+1}}, \uncertain_{\tun+2}, \cdots,
{\gamma}_{\tter-1}(\history_{\tter-1}),
\uncertain_{\tter}} \label{eq:flow-def}
\eqfinv
\end{aligned}$$
that is, $$\begin{aligned}
{\Phi_{\tun:\tter}^{{\gamma}}}
\np{ \history_{\tun}, \uncertain_{\tun+1:\tter} } &=
\np{\history_{\tun}, \control_{\tun},\uncertain_{\tun+1},
\control_{\tun+1}, \uncertain_{\tun+2},
\ldots, \control_{\tter-1},\uncertain_{\tter}} \eqfinv
\\
\mtext{with } \history_{\tbis}
&=\np{\history_{\tun}, \control_{\tun},\uncertain_{\tun+1},
\ldots, \control_{\tbis-1}, \uncertain_{\tbis} }
\eqsepv \tun < \tbis \leq \tter \eqfinv
\\
\mtext{and } \control_{\tbis} &= {\gamma}_{\tbis}
\np{\history_{\tbis}}
\eqsepv \tun < \tbis \leq \tter-1
\eqfinp\end{aligned}$$ When $ 0 \leq \tun = \tter \leq \horizon $, we put $$\begin{aligned}
{\Phi_{\tun:\tun}^{{\gamma}}}
: \HISTORY_{\tun}
& \to \HISTORY_{\tun} \eqsepv
\history_{\tun}
\mapsto
\history_{\tun}\eqfinp
\label{eq:added_convention}
\end{aligned}$$ \[eq:flow\]
With this convention, the expression ${\Phi_{\tun:\tter}^{{\gamma}}}$ makes sense when $ 0 \leq \tun \leq \tter \leq \horizon $: when $\tun=\tter$, no $\interval{\tun}{\tun-1}$-history feedback exists, but none is needed. The mapping ${\Phi_{\tun:\tter}^{{\gamma}}}$ gives the history at time $\tter$ as a function of the initial history $\history_{\tun}$ at time $\tun$ and of the history feedbacks $ \sequence{{\gamma}_{\tbis}}{\tbis=\tun,\ldots,\tter-1}\in{\Gamma}_{\tun:\tter-1} $.
An immediate consequence of this definition are the *flow properties*: $$\begin{aligned}
{\Phi_{\tun:\tter+1}^{{\gamma}}}
\np{ \history_{\tun}, \uncertain_{\tun+1:\tter+1}} &=
\Bp{ {\Phi_{\tun:\tter}^{{\gamma}}}
\np{ \history_{\tun}, \uncertain_{\tun+1:\tter}},
{\gamma}_{\tter}\bp{{\Phi_{\tun:\tter}^{{\gamma}}}
\np{ \history_{\tun}, \uncertain_{\tun+1:\tter}}},\uncertain_{\tter+1}}
\eqsepv 0 \leq \tun \leq \tter \leq \horizon-1
\eqfinv
\label{eq:flowconsequence1}
\\
{\Phi_{\tun:\tter}^{{\gamma}}}\np{\history_{\tun},\uncertain_{\tun+1:\tter}} &=
{\Phi_{\tun+1:\tter}^{{\gamma}}}
\bp{\np{\history_{\tun},{\gamma}_{\tun}(\history_{\tun}),
\uncertain_{\tun+1}},\uncertain_{\tun+2:\tter}}
\eqsepv 0 \leq \tun < \tter \leq \horizon
\eqfinp
\label{eq:flowconsequence2}
\end{aligned}$$ \[eq:flowconsequences\]
Building Stochastic Kernels from History Feedbacks {#Stochastic_Kernels}
--------------------------------------------------
Let $\tun$ and $\tter$ be given such that $ 0 \leq \tun \leq \tter \leq \horizon $.
When $ 0 \leq \tun < \tter \leq \horizon $, for
1. a $\interval{\tun}{\tter-1}$-history feedback ${\gamma}=\sequence{{\gamma}_{\tbis}}{\tbis=\tun,\ldots,\tter-1}
\in{\Gamma}_{\tun:\tter-1} $,
2. a family $ \sequence{ {\rho_{\tbis-1:\tbis}}} {\tun+1 \leq \tbis \leq \tter} $ of stochastic kernels $${\rho_{\tbis-1:\tbis}} : \HISTORY_{\tbis-1} \to \Delta\np{\UNCERTAIN_{\tbis}}
\eqsepv \tbis = \tun+1, \ldots, \tter
\eqfinv$$
we define a stochastic kernel
$${\rho_{\tun:\tter}^{{\gamma}}} :
\HISTORY_{\tun} \to \Delta\np{ \HISTORY_{\tter} }
\label{eq:gamma_stoch_kernel}$$
by, for any $ \varphi : \HISTORY_{\tter} \to {[0,+\infty]}$, measurable nonnegative numerical function, that is, $ \varphi \in \espace{L}^{0}_{+}(\HISTORY_{t},\tribu{\History}_{t}) $, [^3] $$\begin{aligned}
\int_{\HISTORY_{\tter}} \varphi\np{\history'_{\tun},\history'_{\tun+1:\tter}}
& {\rho_{\tun:\tter}^{{\gamma}}}\np{\history_{\tun}, \mathrm{d}\history'_{\tter}}
= \nonumber \\
&\int_{\UNCERTAIN_{\tun+1:\tter}}
\varphi \bp{ {\Phi_{\tun:\tter}^{{\gamma}}} \np{\history_{\tun}, \uncertain_{\tun+1:\tter} }}
\prod_{\tbis=\tun+1}^{\tter}
{\rho_{\tbis-1:\tbis}}\bp{ {\Phi_{\tun:\tbis-1}^{{\gamma}}}\np{\history_{\tun},
\uncertain_{\tun+1:\tbis-1} },
\mathrm{d}\uncertain_{\tbis} }
\eqfinp
\label{eq:stochastic_kernels_rho_b}
\end{aligned}$$
When $ 0 \leq \tun = \tter \leq \horizon $, we define $${\rho_{\tun:\tun}^{{\gamma}}} :
\HISTORY_{\tun} \to \Delta\np{ \HISTORY_{\tun} } \eqsepv
{\rho_{\tun:\tun}^{{\gamma}}} \np{\history_{\tun} , \mathrm{d}\history'_{\tun} } =
\delta_{\history_{\tun}}\np{ \mathrm{d}\history'_{\tun} } \eqfinp
\label{eq:stochastic_kernels_rho_c}$$ \[eq:stochastic\_kernels\_rho\]
\[de:stochastic\_kernels\_rho\]
The stochastic kernels $ {\rho_{\tun:\tter}^{{\gamma}}} $ on $\HISTORY_{\tter} $, given by , are of the form $${\rho_{\tun:\tter}^{{\gamma}}}
\np{\history_{\tun},\mathrm{d}\history'_{\tter}} =
{\rho_{\tun:\tter}^{{\gamma}}}
\np{\history_{\tun},\mathrm{d}\history'_{\tun}\mathrm{d}\history'_{\tun+1:\tter}} =
\delta_{\history_{\tun}}\np{ \mathrm{d}\history'_{\tun} }
\otimes {\varrho_{\tun:\tter}^{{\gamma}}}
\np{\history_{\tun},\mathrm{d}\history'_{\tun+1:\tter}} \eqfinv$$ where, for each $ \history_{\tun} \in \HISTORY_{\tun} $, the probability distribution $ {\varrho_{\tun:\tter}^{{\gamma}}}
\np{\history_{\tun}, \mathrm{d}\history'_{\tun+1:\tter}} $ only charges the histories visited by the flow from $\tun+1$ to $\tter$. The construction of the stochastic kernels $ {\rho_{\tun:\tter}^{{\gamma}}}$ is developed in [@Bertsekas-Shreve:1996 p. 190] for relaxed history feedbacks and obtained by using [@Bertsekas-Shreve:1996 Proposition 7.45].
Following Definition \[de:stochastic\_kernels\_rho\], we can define a family $ \sequence{ {\rho_{\tbis:\tter}^{{\gamma}}}}{\tun \leq \tbis \leq \tter } $ of stochastic kernels. This family has the flow property, that is, for $ \tbis < \tter $, $${\rho_{\tbis:\tter}^{{\gamma}}} \np{\history_{\tbis},
\mathrm{d}\history'_{\tter}} =
\int_{\UNCERTAIN_{\tbis+1}}
{\rho_{\tbis:\tbis+1}}\bp{\history_{\tbis},\mathrm{d}\uncertain_{\tbis+1} }
{\rho_{\tbis+1:\tter}}^{{\gamma}} \Bp{
\bp{\history_{\tbis},{\gamma}_{\tbis}\np{\history_{\tbis}},\uncertain_{\tbis+1}},
\mathrm{d}\history'_{\tter}}
\eqfinp
\label{eq:stochastic_kernels_rho_property}$$ \[pr:stochastic\_kernels\_rho\]
Let $ \tbis < \tter $. For any $\varphi : \HISTORY_{\tter} \to {[0,+\infty]}$, we have that
$$\begin{aligned}
\int_{\HISTORY_{\tter}}
\varphi
& \np{\history'_{\tbis},\history'_{\tbis+1:\tter}}
{\rho_{\tbis:\tter}^{{\gamma}}}\np{\history_{\tbis}, \mathrm{d}\history'_{\tter}}
\label{eq:StochasticKernel_Flow_Ini} \\
=& \int_{\UNCERTAIN_{\tbis+1:\tter}}
\varphi \bp{ {\Phi_{\tbis:\tter}^{{\gamma}}}
\np{\history_{\tbis}, \uncertain_{\tbis+1:\tter} } }
\prod_{{\tbis'}=\tbis+1}^{\tter}
{\rho_{{\tbis'}-1:{\tbis'}}}\bp{ {\Phi_{\tbis:{\tbis'}-1}^{{\gamma}}}\np{\history_{\tbis},
\uncertain_{\tbis+1:{\tbis'}-1} }, \mathrm{d}\uncertain_{{\tbis'}} }
\nonumber \\
\intertext{by the definition~\eqref{eq:stochastic_kernels_rho_b} of the stochastic
kernel~${\rho_{\tbis:\tter}^{{\gamma}}}$,}
=&
\int_{\UNCERTAIN_{\tbis+1:\tter}}
\varphi \bp{ {\Phi_{\tbis:\tter}^{{\gamma}}}
\np{\history_{\tbis}, \uncertain_{\tbis+1:\tter} } }
{\rho_{\tbis:\tbis+1}}\bp{ \history_{\tbis}, \mathrm{d}\uncertain_{\tbis+1} }
\prod_{{\tbis'}=\tbis+2}^{\tter}
{\rho_{{\tbis'}-1:{\tbis'}}}\bp{ {\Phi_{\tbis:{\tbis'}-1}^{{\gamma}}}\np{\history_{\tbis},
\uncertain_{\tbis+1:{\tbis'}-1} }, \mathrm{d}\uncertain_{{\tbis'}} }
\nonumber \\
\intertext{by the property~\eqref{eq:added_convention} of the flow
\( {\Phi_{\tbis:\tbis}^{{\gamma}}}\),}
=&
\int_{\UNCERTAIN_{\tbis+1:\tter}}
\varphi \bp{ {\Phi_{\tbis+1:\tter}^{{\gamma}}}
\bp{\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},\uncertain_{\tbis+2:\tter}} }
\nonumber \\
& \qquad
{\rho_{\tbis:\tbis+1}}\bp{ \history_{\tbis}, \mathrm{d}\uncertain_{\tbis+1} }
\prod_{{\tbis'}=\tbis+2}^{\tter}
{\rho_{{\tbis'}-1:{\tbis'}}}\bp{ {\Phi_{\tbis+1:{\tbis'}-1}^{{\gamma}}}
\bp{\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},\uncertain_{\tbis+2:{\tbis'}-1}}
, \mathrm{d}\uncertain_{{\tbis'}} }
\nonumber \\
\intertext{by the flow property~\eqref{eq:flowconsequence2},}
=&
\int_{\UNCERTAIN_{\tbis+1}}
{\rho_{\tbis:\tbis+1}}\bp{ \history_{\tbis} , \mathrm{d}\uncertain_{\tbis+1} }
\int_{\UNCERTAIN_{\tbis+2:\tter}}
\varphi \bp{ {\Phi_{\tbis+1:\tter}^{{\gamma}}}
\bp{\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},\uncertain_{\tbis+2:\tter}} }
\nonumber \\
& \qquad \prod_{{\tbis'}=\tbis+2}^{\tter}
{\rho_{{\tbis'}-1:{\tbis'}}}\bp{ {\Phi_{\tbis+1:{\tbis'}-1}^{{\gamma}}}
\bp{\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},\uncertain_{\tbis+2:{\tbis'}-1}}
, \mathrm{d}\uncertain_{{\tbis'}} }
\nonumber \\
\intertext{by Fubini Theorem \cite[p.137]{Loeve:1977},}
=& \int_{\UNCERTAIN_{\tbis+1}} {\rho_{\tbis:\tbis+1}}\bp{\history_{\tbis},\mathrm{d}\uncertain_{\tbis+1} }
\int_{\HISTORY_{\tter}} \varphi\bp{
\np{\history'_{\tbis},{\gamma}_{\tbis}(\history'_{\tbis}),\uncertain'_{\tbis+1}},
\history'_{\tbis+2:\tter}}
{\rho_{\tbis+1:\tter}}^{{\gamma}} \bp{
\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},
\mathrm{d}\history'_{\tter}}
\nonumber \\
\intertext{by definition~\eqref{eq:stochastic_kernels_rho_b} of
\( {\rho_{\tbis+1:\tter}}^{{\gamma}} \),}
=&
\int_{\HISTORY_{\tter}} \varphi\bp{
\np{\history'_{\tbis},{\gamma}_{\tbis}(\history'_{\tbis}),\uncertain'_{\tbis+1}},
\history'_{\tbis+2:\tter}}
\int_{\UNCERTAIN_{\tbis+1}}
{\rho_{\tbis:\tbis+1}}\bp{\history_{\tbis},\mathrm{d}\uncertain_{\tbis+1}}
{\rho_{\tbis+1:\tter}}^{{\gamma}}
\bp{\np{\history_{\tbis},{\gamma}_{\tbis}(\history_{\tbis}),\uncertain_{\tbis+1}},
\mathrm{d}\history'_{\tter}} \label{eq:StochasticKernel_Flow_Last}
\end{aligned}$$
by Fubini Theorem and by definition of $ {\rho_{\tbis:\tter}^{{\gamma}}} $.
As the two expressions and are equal for any $\varphi : \HISTORY_{\tter} \to {[0,+\infty]}$, we deduce the flow property . This ends the proof.
Proofs
------
### Proof of Theorem \[pr:DP\_withoutstate\_third\] {#proof:DP_withoutstate_third}
We only give a sketch of the proof, as it is a variation on different results of [@Bertsekas-Shreve:1996], the framework of which we follow.
We take the history space $\HISTORY_{t}$ for state space, and the state dynamics $$f\bp{\history_{t},\control_{t},\uncertain_{t+1}}
=\bp{\history_{t},\control_{t},\uncertain_{t+1}}=
\history_{t+1}
\in \HISTORY_{t+1} = \HISTORY_{t} \times \CONTROL_{t} \times \UNCERTAIN_{t+1}
\eqfinp
\label{eq:history=state_dynamics}$$ Then, the family $ \sequence{ {\rho_{\tbis-1:\tbis}}}{1 \leq \tbis \leq \horizon}$ of stochastic kernels gives a family of disturbance kernels that do not depend on the current control. The criterion to be minimized is a function of the history at time $\horizon$, thus of the state at time $\horizon$. Problem is thus a finite horizon model with a final cost and we are minimizing over the so-called state-feedbacks. Then, the proof of Theorem \[pr:DP\_withoutstate\_third\] follows from the results developed in Chap. 7, 8 and 10 of [@Bertsekas-Shreve:1996] in a Borel setting. Since we are considering a finite horizon model with a final cost, we detail the steps needed to use the results of [@Bertsekas-Shreve:1996 Chap. 8].
The final cost at time $\horizon$ can be turned into an instantaneous cost at time $\horizon-1$ by inserting the state dynamics in the final cost. Getting rid of the disturbance in the expected cost by using the disturbance kernel is standard practice. Then, we can turn this non-homogeneous finite horizon model into a finite horizon model with homogeneous dynamics and costs by following the steps of [@Bertsekas-Shreve:1996 Chap. 10]. Using [@Bertsekas-Shreve:1996 Proposition 8.2], we obtain that the family of optimization problems , when minimizing over the relaxed state feedbacks, satisfies the Bellman equation ; we conclude with [@Bertsekas-Shreve:1996 Proposition 8.4] which covers the minimization over state feedbacks.
To summarize, Theorem \[pr:DP\_withoutstate\_third\] is valid under the general Borel assumptions of [@Bertsekas-Shreve:1996 Chap. 8] and with the specific $(F^{-})$ assumption needed for [@Bertsekas-Shreve:1996 Proposition 8.4]; this last assumption is fulfilled here since we have assumed that the criterion is nonnegative.
### Proof of Proposition \[thm:DPB\] {#proof:DPB}
Let $\tilde\varphi_{\tter}: \STATE_{\tter} \to {[0,+\infty]}$ be a given measurable nonnegative numerical function, and let $\varphi_{\tter}: \HISTORY_{\tter} \to {[0,+\infty]}$ be $$\varphi_{\tter} = \tilde\varphi_{\tter} \circ \theta_{\tter} \eqfinp
\label{eq:varphi}$$ Let $\varphi_{\tun} : \HISTORY_{\tun} \to {[0,+\infty]}$ be the measurable nonnegative numerical function obtained by applying the Bellman operator ${\mathcal{B}_{\tter:\tun}}$ across $\interval{\tter}{\tun}$ (see ) to the measurable nonnegative numerical function $\varphi_{\tter}$: $$\varphi_{\tun} = {\mathcal{B}_{\tter:\tun}} \varphi_{\tter} =
{\mathcal{B}_{\tun+1:\tun}} \circ \cdots \circ
{\mathcal{B}_{\tter:\tter-1}} \varphi_{\tter} \eqfinp
\label{eq:varphi_tun}$$ We will show that there exists a measurable nonnegative numerical function $$\tilde\varphi_{\tun} : \STATE_{\tun} \to {[0,+\infty]}$$ such that $$\varphi_{\tun} = \tilde\varphi_{\tun} \circ \theta_{\tun} \eqfinp
\label{eq:proof_DPB}$$
First, we show by backward induction that, for all $\tbis \in \{\tun,\dots,\tter\}$, there exists a measurable nonnegative numerical function $\overline\varphi_{\tbis}$ such that $\varphi_{\tbis}(h_{\tbis})= \overline\varphi_{\tbis}
( \theta_{\tun} \np{\history_{\tun}},\history_{\tun+1:\tbis})$. Second, we prove that the function $\tilde\varphi_{\tun}=\overline\varphi_{\tun}$ satisfies .
- For $\tbis=\tter$, we have, by and by , that $$\varphi_{\tter}(\history_{\tter}) =
\tilde\varphi_{\tter}\bp{\theta_{\tter} \np{\history_{\tter}}} =
\tilde\varphi_{\tter}\bp{\Dynamics{\tun}{\tter}\np{\theta_{\tun}\np{\history_{\tun}},
\history_{\tun+1:\tter}}} \eqfinv$$ so that the measurable nonnegative numerical function $\overline\varphi_{\tter}$ is given by $\tilde\varphi_{\tter} \circ \Dynamics{\tun}{\tter}$.
- Assume that, at $\tbis+1$, the result holds true, that is, $$\varphi_{\tbis+1}(h_{\tbis+1})=
\overline\varphi_{\tbis+1}(\theta_{\tun}\np{\history_{\tun}},
\history_{\tun+1:\tbis+1})
\eqfinp
\label{eq:proof_DPB_induction}$$ Then, by ,
$$\begin{aligned}
\varphi_{\tbis}(\history_{\tbis})
& =
\bp{{\mathcal{B}_{\tbis+1:\tbis}} \varphi_{\tbis+1}}(\history_{\tbis}) \\
& =
\inf_{\control_{\tbis}\in\CONTROL_{\tbis}} \int_{\UNCERTAIN_{\tbis+1}}
\varphi_{\tbis+1}\bp{\np{\history_{\tbis},\control_{\tbis},\uncertain_{\tbis+1}}}
{\rho_{\tbis:\tbis+1}}\np{\history_{\tbis},\mathrm{d}\uncertain_{\tbis+1} }
\intertext{by definition~\eqref{eq:Bellman_operators_rho} of the Bellman
operator }
& =
\inf_{\control_{\tbis}\in\CONTROL_{\tbis}} \int_{\UNCERTAIN_{\tbis+1}}
\overline\varphi_{\tbis+1}\bp{\np{\theta_{\tun} \np{\history_\tun},
\np{\history_{\tun+1:\tbis},\control_{\tbis},\uncertain_{\tbis+1}}}}
{\rho_{\tbis:\tbis+1}}\np{\history_{\tbis},\mathrm{d}\uncertain_{\tbis+1}}
\intertext{ by induction assumption~\eqref{eq:proof_DPB_induction} }
& =
\inf_{\control_{\tbis}\in\CONTROL_{\tbis}} \int_{\UNCERTAIN_{\tbis+1}}
\overline\varphi_{\tbis+1}\bp{\np{\theta_{\tun}\np{\history_\tun},
\np{\history_{\tun+1:\tbis},\control_{\tbis},\uncertain_{\tbis+1}}}}
{{\tilde\rho}_{\tbis:\tbis+1}}
\bp{\np{\theta_{\tun}\np{\history_{\tun}},\history_{\tun+1:\tbis}},
\mathrm{d}\uncertain_{\tbis+1}}
\intertext{by compatibility~\eqref{eq:reduction-dynamics_compatible}
of the stochastic kernel}
& =
\overline\varphi_{\tbis}\bp{\theta_{\tun}\np{\history_{\tun}},\history_{\tun+1:\tbis}}
\eqfinv
\end{aligned}$$
where $$\overline\varphi_{\tbis}\bp{\state_{\tun},\history_{\tun+1:\tbis}}
= \inf_{\control_{\tbis}\in\CONTROL_{\tbis}} \int_{\UNCERTAIN_{\tbis+1}}
\overline\varphi_{\tbis+1}\bp{\np{\state_{\tun},
\np{\history_{\tun+1:\tbis},\control_{\tbis},\uncertain_{\tbis+1}}}}
{{\tilde\rho}_{\tbis:\tbis+1}}\bp{\np{\state_{\tun},\history_{\tun+1:\tbis}},
\mathrm{d}\uncertain_{\tbis+1}} \eqfinp$$ The result thus holds true at time $\tbis$.
The induction implies that, at time $\tun$, the expression of $\varphi_{\tun}(\history_{\tun})$ is $$\varphi_{\tun}(\history_{\tun}) =
\overline\varphi_{\tun}\bp{\theta_{\tun}\np{\history_{\tun}}} \eqfinv$$ since the term $\history_{\tun+1:\tun}$ vanishes. Choosing $\tilde\varphi_{\tun}=\overline\varphi_{\tun}$ gives the expected result.
### Proof of Proposition \[thm:DPB\_dhd\] {#proof:DPB_dhd}
We now show that the setting in §\[Decision\_Hazard\_Decision\_Dynamic\_Programming\] is a particular kind of two time scales problem as seen in §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\]. For this purpose, we introduce a *spurious uncertainty variable* $\uncertain^{\sharp}_{s}$ taking values in a singleton set $\UNCERTAIN^{\sharp}_{{s}} =
\{\bar\uncertain^{\sharp}_{{s}}\}$, equipped with the trivial $\sigma$-field $\{\emptyset, \UNCERTAIN^{\sharp}_{{s}} \}$, for each time ${s}=1,2\ldots, {S}$. Now, we obtain the following sequence of events: $$\begin{gathered}
\uncertain^{\sharp}_{0} \rightsquigarrow \control^{\sharp}_{0}
\rightsquigarrow \uncertain^{\flat}_{1} \rightsquigarrow \control^{\flat}_{1}
\rightsquigarrow \uncertain^{\sharp}_{1} \rightsquigarrow \control^{\sharp}_{1}
\rightsquigarrow \uncertain^{\flat}_{2} \rightsquigarrow \control^{\flat}_{2}
\rightsquigarrow \uncertain^{\sharp}_{2} \rightsquigarrow \control^{\sharp}_{2}
\rightsquigarrow \quad \dots \\ \quad
\rightsquigarrow \uncertain^{\flat}_{{S}-1}
\rightsquigarrow \control^{\flat}_{{S}-1}
\rightsquigarrow \uncertain^{\sharp}_{{S}-1}
\rightsquigarrow \control^{\sharp}_{{S}-1}
\rightsquigarrow \uncertain^{\flat}_{{S}}
\rightsquigarrow \control^{\flat}_{{S}}
\rightsquigarrow \uncertain^{\sharp}_{{S}} \eqfinv\end{gathered}$$
which coincides with a two time scales problem:
$$\begin{gathered}
\underbrace{\uncertain_{0,0}=\uncertain^{\sharp}_{0} \rightsquigarrow
\control_{0,0}=\control^{\sharp}_{0} \rightsquigarrow
\uncertain_{0,1}=\uncertain^{\flat}_{1} \rightsquigarrow
\control_{0,1}=\control^{\flat}_{1}}_{\textrm{slow time cycle} } \rightsquigarrow \\
\underbrace{\uncertain_{1,0}=\uncertain^{\sharp}_{1} \rightsquigarrow
\control_{1,0}=\control^{\sharp}_{1} \rightsquigarrow
\uncertain_{1,1}=\uncertain^{\flat}_{2} \rightsquigarrow
\control_{1,1}=\control^{\flat}_{2}}_{\textrm{slow time cycle} } \rightsquigarrow \\
\cdots \rightsquigarrow
\underbrace{\uncertain_{{S}-1,0}=\uncertain^{\sharp}_{{S}-1} \rightsquigarrow
\control_{{S}-1,0}=\control^{\sharp}_{{S}-1} \rightsquigarrow
\uncertain_{{S}-1,1}=\uncertain^{\flat}_{{S}} \rightsquigarrow
\control_{{S}-1,1}=\control^{\flat}_{{S}}}_{\textrm{slow time cycle} } \rightsquigarrow
\uncertain_{{S},0}=\uncertain^{\sharp}_{{S}} \eqfinp\end{gathered}$$
We introduce the sets $$\begin{aligned}
\UNCERTAIN_{{d},0} &= \UNCERTAIN_{d}^{\sharp},
\text{ for }~{d}\in \{0,\ldots,{S}\},
\\
\UNCERTAIN_{{d},1} &= \UNCERTAIN_{{d}+1}^{\flat},
\text{ for }~{d}\in \{0,\ldots,{S}-1\},
\\
\CONTROL_{{d},0} &= \CONTROL_{d}^{\sharp},
\text{ for }~{d}\in \{0,\ldots,{S}-1\},
\\
\CONTROL_{{d},1} &= \CONTROL_{{d}+1}^{\flat},
\text{ for }~{d}\in \{0,\ldots,{S}-1\}.\end{aligned}$$
As a consequence, we observe that the two time scales history spaces in §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\] are in one to one correspondence with the decision-hazard-decision history spaces and fields in – as follows:
$$\begin{aligned}
\intertext{ for ${d}=0,1,2\ldots, {S}$, }
\HISTORY_{{d},0}
&=
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL_{{d'},0} \times \UNCERTAIN_{{d'},1}
\times \CONTROL_{{d'},1} \times \UNCERTAIN_{{d'}+1,0} }
\\
&=
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL^{\sharp}_{{d'}} \times \UNCERTAIN_{{d'}+1}^{\flat}\times \CONTROL^{\flat}_{{d'}+1} \times \UNCERTAIN_{{d'}+1}^{\sharp}} \\
&
\equiv
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL^{\sharp}_{{d'}} \times \UNCERTAIN_{{d'}+1}^{\flat}\times \CONTROL^{\flat}_{{d'}+1} } = \HISTORY^{\sharp}_{{d}}
\eqfinv
\intertext{ for ${d}=0,1,2\ldots, {S}$,}
\tribu{\History}_{{d},0}
&=
\tribu{\Uncertain}_{0}^{\sharp}\otimes \bigotimes_{{d'}=0}^{{d}-1}
\bp{ \tribu{\Control}^{\sharp}_{{d'}} \otimes
\tribu{\Uncertain}_{{d'}+1}^{\flat}\otimes
\tribu{\Control}^{\flat}_{{d'}+1} \otimes \tribu{\Uncertain}_{{d'}+1}^{\sharp}}
\eqfinv
\intertext{ for ${d}=0,1,2\ldots, {S}-1$,}
\HISTORY_{{d},1}
&=
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL_{{d'},0} \times \UNCERTAIN_{{d'},1}
\times \CONTROL_{{d'},1} \times \UNCERTAIN_{{d'}+1,0}}
\times \CONTROL_{{d},0} \times \UNCERTAIN_{{d},1}
\\
&=
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL^{\sharp}_{{d'}} \times \UNCERTAIN_{{d'}+1}^{\flat}\times \CONTROL^{\flat}_{{d'}+1} \times \UNCERTAIN_{{d'}+1}^{\sharp}}
\times \CONTROL^{\sharp}_{{d}} \times \UNCERTAIN_{{d}+1}^{\flat}\\
&\equiv
\UNCERTAIN_{0}^{\sharp}\times \prod_{{d'}=0}^{{d}-1}
\bp{ \CONTROL^{\sharp}_{{d'}} \times \UNCERTAIN_{{d'}+1}^{\flat}\times \CONTROL^{\flat}_{{d'}+1} }
\times \CONTROL^{\sharp}_{{d}} \times \UNCERTAIN_{{d}+1}^{\flat}= \HISTORY_{{d}+1}^{\flat}\eqfinv
\intertext{ for ${d}=0,1,2\ldots, {S}-1$,}
\tribu{\History}_{{d},1}
&=
\tribu{\Uncertain}_{0}^{\sharp}\otimes \bigotimes_{{d'}=0}^{{d}-1}
\bp{ \tribu{\Control}^{\sharp}_{{d'}} \otimes
\tribu{\Uncertain}_{{d'}+1}^{\flat}\otimes
\tribu{\Control}^{\flat}_{{d'}+1} \otimes \tribu{\Uncertain}_{{d'}+1}^{\sharp}}
\otimes \tribu{\Control}^{\sharp}_{{d}} \otimes
\tribu{\Uncertain}_{{d}+1}^{\flat}\eqfinp
\end{aligned}$$
For any element $\history$ of $\HISTORY_{{d}, 0}$ or $\HISTORY_{{d},1}$ we call $\bc{\history}^{\sharp}$ the element of $\HISTORY_{{d}}^{\sharp}$ or $\HISTORY_{{d}}^{\flat}$ corresponding to $\history$ with all the spurious uncertainties removed. By a slight abuse of notation, the criterion $\criterion$ in (decision-hazard-decision setting) corresponds to $ \criterion \circ \bc{\cdot}^{\sharp}$ in the two time scales setting in §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\]. The feedbacks in the two time scales setting in §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\] are in one to one correspondence with the same elements in the decision-hazard-decision setting, namely $${\gamma}_{{d},0} =
{\gamma}^{\sharp}_{d}\circ \bc{\cdot}^{\sharp}\eqsepv
{\gamma}_{{d},1} =
{\gamma}^{\flat}_{{d}+1} \circ \bc{\cdot}^{\sharp}\eqfinp$$
Now we define two families of stochastic kernels
- a family $ \sequence{ {\rho_{({d},0):({d},1)}} }{0 \leq {d}\leq {\MakeUppercase{{d}}}} $ of stochastic kernels within two consecutive slow scale indexes
$$\begin{aligned}
{\rho_{({d},0):({d},1)}}:
\HISTORY_{{d},0} &\to \Delta\np{\UNCERTAIN_{{d},1}} \eqfinv\\
\history_{{d},0} &\mapsto {\rho_{{d}:{d}+1}}
\circ \bc{\cdot}^{\sharp}\eqfinp
\end{aligned}$$
- a family $ \sequence{ {\rho_{({d},1):({d}+1,0)}}}{0 \leq {d}\leq {\MakeUppercase{{d}}}-1} $ of stochastic kernels across two consecutive slow scale indexes
$$\begin{aligned}
{\rho_{({d},1):({d}+1,0)}}:
\HISTORY_{{d},1} &\to \Delta\np{\UNCERTAIN_{{d}+1,0}} \eqfinv\\
\history_{{d},1} &\mapsto \delta_{\bar\uncertain_{{d}+1}^{\sharp}}(\cdot)
\eqfinv
\end{aligned}$$
where we recall that $ \UNCERTAIN_{{d}+1,0} =\UNCERTAIN_{{d}+1}^{\sharp}=\{ \bar\uncertain_{{d}+1}^{\sharp}\} $.
With these notations, we obtain Equation , where only one integral appears because of the Dirac in the stochastic kernels ${\rho_{({d},1):({d}+1,0)}}$. Indeed, for any measurable function $\varphi : \HISTORY_{{d}+1, 0}\to{[0,+\infty]}$, we have that $$\begin{gathered}
\bp{{\mathcal{B}_{{d}+1:{d}}}\varphi} \np{\history_{{d},0}}
= \inf_{\control_{{d},0}\in\CONTROL_{{d},0}}
\int_{\UNCERTAIN_{{d},1}} {\rho_{({d},0):({d},1)}}
\Bp{{\history_{{d},0}},
\mathrm{d}\uncertain_{{d}, 1}}
\\
\inf_{\control_{{d},1}\in\CONTROL_{{d},1}}
\int_{\UNCERTAIN_{{d}+1,0}}
\varphi\bp{\history_{{d},0},\control_{{d},0},
\uncertain_{{d},1}, \control_{{d},1},\uncertain_{{d}+1,0}}
{\rho_{({d},1):({d}+1,0)}}
\Bp{{\history_{{d},0},\history_{{d}:{d}+1}},
\mathrm{d}\uncertain_{{d}+1, 0}}
\eqfinp\end{gathered}$$ Now, if there exists $ \tilde \varphi : \HISTORY_{{d}+1}^{\sharp}\to{[0,+\infty]}$ such that $\varphi = \tilde \varphi \circ \bc{\cdot}^{\sharp}$, we obtain that $$\begin{aligned}
\bp{{\mathcal{B}_{{d}+1:{d}}}\varphi} \np{\history_{{d},0}}
&=
\inf_{\control_{{d},0}\in\CONTROL_{{d},0}}
\int_{\UNCERTAIN_{{d},1}} {\rho_{({d},0):({d},1)}}
\Bp{{\history_{{d},0}},
\mathrm{d}\uncertain_{{d}, 1}}
\inf_{\control_{{d},1}\in\CONTROL_{{d},1}}
\tilde\varphi(\bc{\history_{{d},0}}^{\sharp}, \control_{{d},0},
\uncertain_{{d},1}, \control_{{d},1}) \\
& \qquad \qquad \qquad
\int_{\UNCERTAIN_{{d}+1,0}}{\rho_{({d},1):({d}+1,0)}}
\Bp{{\history_{{d},0},\history_{{d}:{d}+1}},
\mathrm{d}\uncertain_{{d}+1, 0}}\\
&= \inf_{\control_{{d},0}\in\CONTROL_{{d},0}}
\int_{\UNCERTAIN_{{d},1}} {\rho_{({d},0):({d},1)}}
\Bp{{\history_{{d},0}},
\mathrm{d}\uncertain_{{d}, 1}}
\inf_{\control_{{d},1}\in\CONTROL_{{d},1}}
\tilde\varphi(\bc{\history_{{d},0}}^{\sharp}, \control_{{d},0}, \uncertain_{{d},1}, \control_{{d},1})
\intertext{ by the Dirac probability of the stochastic kernels
${\rho_{({d},1):({d}+1,0)}}$, }
&= \inf_{\control_{{d}}^{\sharp}\in\CONTROL_{{d}}^{\sharp}}
\int_{\UNCERTAIN_{{d}+1}^{\flat}} {\rho_{({d},0):({d},1)}}
\Bp{{\history_{{d}}^{\sharp}},
\mathrm{d}\uncertain_{{d}+1}^{\flat}}
\inf_{\control_{{d}+1}^{\flat}\in\CONTROL_{{d}+1}^{\flat}}
\tilde\varphi(\history_{{d}}^{\sharp}, \control_{{d}}^{\sharp}, \uncertain_{{d}+1}^{\flat}, \control_{{d}+1}^{\flat})\end{aligned}$$ This ends the proof.
Dynamic Programming with Unit Time Blocks {#Dynamic_Programming_with_Unit_Time_Blocks}
=========================================
Here, we recover the classical dynamic programming equations when a state reduction exists at each time $t\inic{0}{\horizon-1}$, with associated dynamics. Following the setting in §\[sect:History\_Feedback\_Case\], we consider a family $\na{{\rho_{\tter-1:\tter}}}_{1 \leq \tter \leq \horizon}$ of stochastic kernels as in and a measurable nonnegative numerical cost function $\criterion$ as in .
The General Case of Unit Time Blocks\[sec:UTB\_general\]
--------------------------------------------------------
First, we treat the general criterion case. We assume the existence of a family of measurable state spaces $\sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon}$ and the existence of a family of measurable mappings $\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon}$ with $\theta_{\tter} : \HISTORY_{\tter} \to \STATE_{\tter}$. We suppose that there exists a family of measurable dynamics $\sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1}$ with $\Dynamics{\tter}{\tter+1} : \STATE_{\tter} \times \CONTROL_{\tter}
\times \UNCERTAIN_{\tter+1} \to \STATE_{\tter+1}$, such that $$\theta_{\tter+1}\bp{\np{\history_{\tter}, \control_{\tter}, \uncertain_{\tter+1} } }
=
\Dynamics{\tter}{\tter+1}\bp{ \theta_{\tter}\np{\history_{\tter}},
\control_{\tter}, \uncertain_{\tter+1} } \eqsepv
\forall \np{\history_{\tter},\control_{\tter},\uncertain_{\tter+1}}
\in \HISTORY_{\tter} \times \CONTROL_{\tter} \times \UNCERTAIN_{\tter+1}
\eqfinp
\label{eq:reduction-dynamics_UNIT}$$ The following proposition is a immediate application of Theorem \[thm:DPB\_family\] and Proposition \[thm:DPB\].
Suppose that the triplet $ \np{ \sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} } $, which is a state reduction across the consecutive time blocks $[\tter,\tter+1]_{\tter = 0,\ldots,\horizon-1}$ of the time span, is compatible with the family $\na{{\rho_{\tter-1:\tter}}}_{\tter=1,\ldots,\horizon}$ of stochastic kernels in (see Definition \[de:reduction-dynamics\_family\]).
Suppose that there exists a measurable nonnegative numerical function
$$\tilde\criterion : \STATE_{\horizon} \to {[0,+\infty]}\eqfinv$$
such that the cost function $\criterion$ in can be factored as $$\criterion = \tilde\criterion \circ \theta_{\horizon} \eqfinp$$
Define the family $\sequence{\tilde\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of functions by the backward induction $$\begin{aligned}
\tilde\Value_{\horizon}\np{\state_{\horizon}}
& =
\tilde\criterion\np{\state_{\horizon}}
\eqsepv \forall \state_{\horizon} \in \STATE_{\horizon} \eqfinv
\\
\tilde\Value_{\tter}\np{\state_{\tter}}
& =
\inf_{\control_{\tter}\in\CONTROL_{\tter}} \int_{\UNCERTAIN_{\tter+1}}
\tilde\Value_{\tter+1}\bp{\Dynamics{\tter}{\tter+1}
\np{\state_{\tter},\control_{\tter},\uncertain_{\tter+1}}}
{{\tilde\rho}_{\tter:\tter+1}} (\state_{\tter},\mathrm{d}\uncertain_{\tter+1})
\eqsepv \forall \state_{\tter} \in \STATE_{\tter} \eqfinv
\label{eq:DPB-unit_Bellman_equation_b}\end{aligned}$$ \[eq:DPB-unit\_Bellman\_equation\]
for $ \tter = \horizon-1,\ldots,0 $.
Then, the family $\sequence{\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of value functions defined by the family of optimization problems satisfies $$\Value_{\tter} = \tilde\Value_{\tter} \circ \theta_{\tter}
\eqsepv \tter = 0,\ldots,\horizon \eqfinp
\label{eq:value_functions_factorization}$$ \[thm:DPB-unit\]
The Case of Time Additive Cost Functions {#The_Case_of_Time_Additive_Cost_Functions}
----------------------------------------
A time additive stochastic optimal control problem is a particular form of the stochastic optimization problem presented previously. As in §\[sec:UTB\_general\], we assume the existence of a family of measurable state spaces $\sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon}$, the existence of a family of measurable mappings $\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon}$, and the existence of a family of measurable dynamics such that Equation is fulfilled.
We then assume that, for $t\inic{0}{\horizon-1}$, there exist measurable nonnegative numerical functions (*instantaneous cost*)
$$\coutint_{\tter} : \STATE_{\tter} \times \CONTROL_{\tter} \times \UNCERTAIN_{\tter+1}
\to {[0,+\infty]}\eqfinv$$
and that there exists a measurable nonnegative numerical function (*final cost*) $$\coutfin : \STATE_{\horizon} \to {[0,+\infty]}\eqfinv$$ such that the cost function $\criterion$ in writes $$\criterion(\history_{\horizon}) =
\sum_{\tter=0}^{\horizon-1}
\coutint_{\tter}\bp{\theta_{\tter}\np{\history_{\tter}},\control_{\tter},\uncertain_{\tter+1}} +
\coutfin\bp{\theta_{\horizon}\np{\history_{\horizon}}} \eqfinp$$
The following proposition is an immediate consequence of the specific form of the cost function $\criterion$ when applying Proposition \[thm:DPB-unit\].
\[cor:SOC\] Suppose that the triplet $ \np{ \sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} } $, which is a state reduction across the consecutive time blocks $[\tter,\tter+1]_{\tter = 0,\ldots,\horizon-1}$ of the time span, is compatible with the family $\na{{\rho_{\tter-1:\tter}}}_{\tter=1,\ldots,\horizon}$ of stochastic kernels in (see Definition \[de:reduction-dynamics\_family\]).
We inductively define the family of functions $\{\widehat\Value_{\tter}\}_{\tter=0,\ldots,\horizon}$, with $\widehat\Value_{\tter} : \STATE_{\tter} \to {[0,+\infty]}$, by the relations
$$\begin{aligned}
\widehat\Value_{\horizon}(\state_{\horizon})
& =
\coutfin(\state_{\horizon})
\eqsepv \forall \state_{\horizon} \in \STATE_{\horizon}
\intertext{and, for~$t=\horizon-1,\ldots,0$ and
for all \( \state_{\tter} \in \STATE_{\tter} \), }
\widehat\Value_{\tter}(\state_{\tter})
& =
\inf_{\control_{\tter}\in\CONTROL_{\tter}} \int_{\UNCERTAIN_{\tter+1}}
\Bp{\coutint_{\tter}\np{\state_{\tter},\control_{\tter},\uncertain_{\tter+1}}
+ \widehat\Value_{\tter+1}
\bp{\Dynamics{\tter}{\tter+1}
\np{\state_{\tter},\control_{\tter},\uncertain_{\tter+1}}}}
{{\tilde\rho}_{\tter:\tter+1}}
\np{\state_{\tter},\mathrm{d}\uncertain_{\tter+1}}
\eqfinp\end{aligned}$$
\[eq:DPB-unit\_Bellman\_equation\_additive\]
Then, the family $\sequence{\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of value functions defined by the family of optimization problems satisfies $$\begin{aligned}
\Value_{\tter}(\history_{\tter})
& =
\sum_{\tbis=0}^{\tter-1} \coutint_{\tbis}
\bp{\theta_{\tbis}\np{\history_{\tbis}},\control_{\tbis},\uncertain_{\tbis+1}} +
\widehat\Value_{\tter}\bp{\theta_{\tter}\np{\history_{\tter}}}
\eqsepv \tter = 1,\ldots,\horizon \eqfinv \\
\Value_{0}(\history_{0})
& =
\widehat\Value_{0}\bp{\theta_{0}\np{\history_{0}}} \eqfinp\end{aligned}$$ \[eq:value\_functions\_factorization\_additive\]
The Case of Optimization with Noise Process {#The_Case_of_Optimization_with_Noise_Process}
===========================================
In this section, the *noise* at time $t$ is modeled as a random variable $\va{\Uncertain}_{t}$. We suppose given a stochastic process $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ called *noise process*. Then, optimization with noise process becomes a special case of the setting in §\[Stochastic\_Dynamic\_Programming\_with\_History\_Feedbacks\]. Therefore, we can apply the results obtained in Sect. \[State\_Reduction\_by\_Time\_Blocks\].
We moreover assume that, for any $ \tbis=0,\ldots,\horizon\!-\!1 $, the set $\CONTROL_{\tbis}$ in §\[Histories\_and\_History\_Spaces\] is a separable complete metric space.
Optimization with Noise Process {#sect:Noise_Feedback_Case}
-------------------------------
Let $ (\Omega,\tribuomega) $ be a measurable space. For $t=0,\ldots, \horizon$, the *noise* at time $t$ is modeled as a random variable $\va{\Uncertain}_{t}$ defined on $\Omega$ and taking values in $\UNCERTAIN_{t}$. Therefore, we suppose given a stochastic process $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ called *noise process*. The following assumption is made in the sequel.
For any $ 1 \leq \tbis \leq \horizon $, there exists a regular conditional distribution of the random variable $\va{\Uncertain}_{\tbis}$ knowing the random process $\va{\Uncertain}_{0:\tbis-1}$, denoted by $ \PP_{\va{\Uncertain}_{\tbis}}^{ \va{\Uncertain}_{0:\tbis-1} }
(\uncertain_{0:\tbis-1},\mathrm{d}\uncertain_{\tbis}) $. \[ass:regular\_conditional\_distribution\]
Under Assumption \[ass:regular\_conditional\_distribution\], we can introduce the family $ \sequence{ {\rho_{\tbis-1:\tbis}}} {1 \leq \tbis \leq \horizon} $ of stochastic kernels
$${\rho_{\tbis-1:\tbis}} :
\HISTORY_{\tbis-1} \to \Delta\np{\UNCERTAIN_{\tbis}}
\eqsepv \tbis = 1, \ldots, \horizon \eqfinv$$
defined by $${\rho_{\tbis-1:\tbis}}\np{\history_{\tbis-1},\mathrm{d}\uncertain_{\tbis}}
= \PP_{\va{\Uncertain}_{\tbis}}^{ \va{\Uncertain}_{0:\tbis-1}}
\bp{\nc{\history_{\tbis-1}}_{0:\tbis-1}^{\UNCERTAIN},\mathrm{d}\uncertain_{\tbis}}
\eqsepv \tbis = 1, \ldots, \horizon
\eqfinv
\label{eq:optimhistexo}$$ where $\nc{\history_{\tbis-1}}_{0:\tbis-1}^{\UNCERTAIN}=
\np{\uncertain_{0},\uncertain_{1},\ldots, \uncertain_{\tbis-1} }$ is the uncertainty part of the history $\history_{\tbis-1}$ (see Equation ). \[eq:regular\_conditional\_distribution\]
Then, using Definition \[de:stochastic\_kernels\_rho\], the stochastic kernels ${\rho_{\tun:\tter}^{{\gamma}}} :
\HISTORY_{\tun} \to \Delta\np{\HISTORY_{\tter}}$ are defined, for any measurable nonnegative numerical function $ \varphi : \HISTORY_{\tter} \to {[0,+\infty]}$, by $$\begin{aligned}
\int_{\HISTORY_{\tter}}
\varphi\np{\history'_{\tter}}
{\rho_{\tun:\tter}^{{\gamma}}}\np{\history_{\tun},\mathrm{d} \history^{'}_{\tter} }
&=
\int_{\UNCERTAIN_{\tun+1:\tter}}
\varphi \Bp{ {\Phi_{\tun:\tter}^{{\gamma}}} \np{\history_{\tun},\uncertain_{\tun+1:\tter}}}
\PP_{\va{\Uncertain}_{\tun+1:\tter}}^{ \va{\Uncertain}_{0:\tun}}
\bp{ \nc{\history_\tun}_{0:\tun}^{\UNCERTAIN},\mathrm{d}\uncertain_{\tun+1:\tter}} \eqfinp\nonumber \\
&=
\Bespc{\varphi
\bp{ {\Phi_{\tun:\tter}^{{\gamma}}}
\np{\history_{\tun},\va{\Uncertain}_{\tun+1:\tter}}}}
{\va{\Uncertain}_{0:\tun} = \nc{\history_{\tun}}^{\UNCERTAIN}_{0:\tun}} \eqfinv
\label{kernel-versus-esp-cond}\end{aligned}$$ where ${\Phi_{\tun:\tter}^{{\gamma}}} \np{\history_{\tun},\uncertain_{\tun+1:\tter}}=
\np{\history_{\tun},{\gamma}_{\tun}(\history_{\tun}),\uncertain_{\tun+1},
{\gamma}_{\tun+1}\np{\history_{\tun},
{\gamma}_{\tun}(\history_{\tun}),
\uncertain_{\tun+1}}, \uncertain_{\tun+2}, \cdots,
{\gamma}_{\tter-1}(\history_{\tter-1}),
\uncertain_{\tter}}$ is the flow induced by the feedback ${\gamma}$ (see §\[Histories\_Feedbacks\_Flows\]).
Let $\tter$ be given such that $ 0 \leq \tter \leq \horizon-1 $. We introduce $$\tribu{A}_{\tter:\tter} = \{ \emptyset, \Omega \} \eqsepv
\tribu{A}_{\tter:\tter+1} = \sigma\np{\va{\Uncertain}_{\tter+1}} \eqsepv
\ldots, \eqsepv \tribu{A}_{\tter:\horizon-1} = \sigma\np{
\va{\Uncertain}_{\tter+1}, \ldots, \va{\Uncertain}_{\horizon-1} }
\eqfinp$$ Let $ \espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}} $ be the space of $\tribu{A}$-*adapted control processes* $\np{\va\Control_{\tter},\ldots, \va\Control_{\horizon-1}}$ with values in $\CONTROL_{\tter:\horizon-1}$, that is, such that $$\sigma(\va\Control_{\tbis}) \subset \tribu{A}_{\tter:\tbis} \eqsepv
\tbis=\tter, \ldots, \horizon-1 \eqfinp$$
We suppose here that the measurable space $ (\Omega,\tribuomega) $ is equipped with a probability $\PP$, so that $ (\Omega,\tribuomega,\PP) $ is a probability space. Following the setting given in §\[sect:History\_Feedback\_Case\], we consider a measurable nonnegative numerical cost function $\criterion$ as in Equation .
We consider the following family of optimization problems, indexed by $\tter=0,\ldots,\horizon-1$ and by $ \history_{\tter} \in \HISTORY_{\tter} $, $$\widecheck{\Value}_{\tter}(\history_{\tter}) =
\inf_{\np{\va\Control_{\tter:\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}}}
\Bespc{\criterion\np{\history_{\tter},\va{\Control}_{\tter},\va{\Uncertain}_{\tter+1},
\ldots,\va{\Control}_{\horizon-1},\va{\Uncertain}_{\horizon}}}
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
\eqfinp
\label{eq:Optimization_Problem_Over_Adapted_Control_Processes}$$
Let $\tter \in \{0,\ldots,\horizon-1\} $ and $ \history_{\tter} \in \HISTORY_{\tter} $ be given. Problem and Problem coincide, that is, $$\widecheck{\Value}_{\tter}(\history_{\tter}) =
\Value_{\tter}(\history_{\tter}) \eqfinv
\label{eq:U_egal_Gamma}$$ where the family of value functions $\na{\Value_{\tter}}_{\tter=0,\ldots,\horizon}$ is defined by . \[th:optimal\_adapted\_control\_process\]
Let $\tter \in \{0,\ldots,\horizon-1\} $ and $ \history_{\tter} \in \HISTORY_{\tter} $ be given. We show that Problem and Problem are in one-to-one correspondence.
- First, for any history feedback $ {\gamma}_{\tter:\horizon-1}
=\sequence{{\gamma}_{\tbis}}{\tbis=\tter,\ldots,\horizon-1}
\in {\Gamma}_{\tter:\horizon-1} $, we define\
$ \np{\va\Control_{\tter:\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}} $ by $$\np{\va\Control_{\tter},\ldots, \va\Control_{\horizon-1}}
= \bc{ {\Phi_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\va{\Uncertain}_{\tter+1},\ldots,\va{\Uncertain}_{\horizon}}
}_{\tter+1:\horizon}^{\CONTROL}
\eqfinv$$ where the flow $ {\Phi_{\tter:\horizon}^{{\gamma}}} $ has been defined in and the history control part $ \nc{ \cdot }_{\tter+1:\horizon}^{\CONTROL} $ in . By the expression of $ {\rho_{\tbis:\tbis+1}} (\history'_{\tbis},\mathrm{d}\uncertain_{\tbis+1}) $ and by Definition \[de:stochastic\_kernels\_rho\] of the stochastic kernel $ {\rho_{\tter:\horizon}^{{\gamma}}} $, we obtain that $$\begin{aligned}
\Bespc{\criterion\np{\history_{\tter},\va{\Control}_{\tter},
\va{\Uncertain}_{\tter+1},\ldots,
\va{\Control}_{\horizon-1},\va{\Uncertain}_{\horizon}}}
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
& =
\Bespc{\criterion\np{
{\Phi_{\tter:\horizon}^{{\gamma}}}\np{\history_{\tter},
\va{\Uncertain}_{\tter+1}, \ldots, \va{\Uncertain}_{\horizon} } } }
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
\nonumber \\
& =
\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}}
\eqfinp
\label{annexe-noise}
\end{aligned}$$ As a consequence $$\begin{gathered}
\inf_{\np{\va\Control_{\tter:\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}}}
\Bespc{\criterion\np{\history_{\tter},\va{\Control}_{\tter},\va{\Uncertain}_{\tter+1},
\ldots,\va{\Control}_{\horizon-1},\va{\Uncertain}_{\horizon}}}
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
\\
\leq
\inf_{{\gamma}_{t:\horizon-1} \in {\Gamma}_{t:\horizon-1}}
\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}} \eqfinp
\label{eq:U_inferieur_Gamma}\end{gathered}$$
- Second, we define a $\interval{\tter}{\horizon-1}$-*noise feedback* as a sequence $ {\lambda}=
\sequence{{\lambda}_{\tbis}}{\tbis=\tter,\ldots,\horizon-1} $ of measurable mappings (the mapping ${\lambda}_{\tter}$ is constant) $${\lambda}_{\tter} \in \CONTROL_{\tter} \eqsepv
{\lambda}_{\tbis} : \UNCERTAIN_{\tter+1:\tbis} \to \CONTROL_{\tbis}
\eqsepv \tter+1 \leq \tbis \leq \horizon-1
\eqfinp$$ We denote by ${\Lambda}_{\tter:\horizon-1}$ the set of $\interval{\tter}{\horizon-1}$-noise feedbacks. Let $ \np{\va\Control_{\tter},\ldots, \va\Control_{\horizon-1}} \in
\espace{L}^0
\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}} $. As each set $\CONTROL_{\tbis}$ is a separable complete metric space, for $ \tbis=\tter,\ldots,\horizon-1 $, we can invoke Doob Theorem (see [@Dellach Chap. 1, p. 18]). Therefore, there exists a $\interval{\tter}{\horizon-1}$-noise feedback ${\lambda}=\sequence{{\lambda}_{\tbis}}{\tbis=\tter,\ldots,\horizon-1}
\in {\Lambda}_{\tter:\horizon-1}$ such that $$\va{\Control}_{\tter} = {\lambda}_{\tter} \eqsepv
\va{\Control}_{\tbis} = {\lambda}_{\tbis}
\np{\va{\Uncertain}_{\tter+1:\tbis}} \eqsepv
\tter+1 \leq \tbis \leq \horizon-1 \eqfinp$$ Then, we define the history feedback $ {\gamma}_{\tter:\horizon-1}
=\sequence{{\gamma}_{\tbis}}{\tbis=\tter,\ldots,\horizon-1}
\in {\Gamma}_{\tter:\horizon-1} $ by, for any history $\history'_{\tun} \in \HISTORY_{\tun} $, $ \tun=\tter, \ldots, \horizon-1 $: $$\begin{aligned}
{\gamma}_{\tter}\np{\history'_{\tter}}
&= {\lambda}_{\tter} \eqsepv \\
{\gamma}_{\tter+1}\np{\history'_{\tter+1}}
&= {\lambda}_{\tter+1}
\Bp{\bc{ \history'_{\tter+1} }_{\tter+1:\tter+1}^{\UNCERTAIN} }
= {\lambda}_{\tter+1} \np{\uncertain'_{\tter+1}}
\eqsepv \\
& \; \vdots \\
{\gamma}_{\horizon-1}\np{\history'_{\horizon-1}}
&= {\lambda}_{\horizon-1}
\Bp{\bc{ \history'_{\horizon-1} }_{\tter+1:\horizon-1}^{\UNCERTAIN} }
= {\lambda}_{\horizon-1}
\np{\uncertain'_{\tter+1},\cdots,\uncertain'_{\horizon-1}}
\eqfinp\end{aligned}$$ By the expression of $ {\rho_{\tbis:\tbis+1}}
(\history'_{\tbis},\mathrm{d}\uncertain_{\tbis+1}) $ and by Definition \[de:stochastic\_kernels\_rho\] of the stochastic kernel $ {\rho_{\tter:\horizon}^{{\gamma}}} $, we obtain that $$\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}}
=
\Bespc{\criterion\np{\history_{\tter},\va{\Control}_{\tter},\va{\Uncertain}_{\tter+1},
\ldots,\va{\Control}_{\horizon-1},\va{\Uncertain}_{\horizon}}}
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
\eqfinp$$ As a consequence $$\begin{gathered}
\inf_{{\gamma}_{t:\horizon-1} \in {\Gamma}_{t:\horizon-1}}
\int_{\HISTORY_{\horizon}} \criterion\np{\history'_{\horizon}}
{\rho_{\tter:\horizon}^{{\gamma}}}
\np{\history_{\tter},\mathrm{d}\history'_{\horizon}} \\
\leq
\inf_{\np{\va\Control_{\tter},\ldots, \va\Control_{\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}}}
\Bespc{\criterion\np{\history_{\tter},\va{\Control}_{\tter},
\va{\Uncertain}_{\tter+1},
\ldots,\va{\Control}_{\horizon-1},\va{\Uncertain}_{\horizon}}}
{\va{\Uncertain}_{0:\tter} = \nc{\history_{\tter}}_{0:\tter}^{\UNCERTAIN}}
\eqfinp
\label{eq:Gamma_inferieur_U}\end{gathered}$$
Gathering inequalities and leads to . This ends the proof.
The following proposition is an immediate consequence of Theorem \[pr:DP\_withoutstate\_third\] and Proposition \[th:optimal\_adapted\_control\_process\].
The family $\sequence{\widecheck{\Value}_{\tter}}{\tter=0,\ldots,\horizon}$ of functions in satisfies the backward induction
$$\begin{aligned}
\widecheck{\Value}_{\horizon}\np{\history_{\horizon}}
& =
\criterion\np{\history_{\horizon}}
\eqsepv \forall \history_{\horizon} \in \HISTORY_{\horizon} \eqfinv
\\
\intertext{and, for $\tter = \horizon-1,\ldots,0$,}
\widecheck{\Value}_{t}(\history_{t}) &=
\inf_{ \control_{t} } \int_{\UNCERTAIN_{t+1}}
\widecheck{\Value}_{t+1}\bp{\history_{t},\control_{t},\uncertain_{t+1}}
\PP_{\va{\Uncertain}_{t+1}}^{ \va{\Uncertain}_{0:t}}
\bp{\nc{\history_{t}}_{0:t}^{\UNCERTAIN},\mathrm{d}\uncertain_{t+1} } \\
&=
\inf_{ \control_{t} } \: \bespc{
\widecheck{\Value}_{t+1}\bp{\history_{t},\control_{t},\va{\Uncertain}_{t+1}} }{ \va{\Uncertain}_{0:t} = \nc{\history_{t}}_{0:t}^{\UNCERTAIN} }
\eqsepv \forall \history_{\tter} \in \HISTORY_{\tter} \eqfinp
\end{aligned}$$
\[pr:DP\_withoutstate\]
Two Time-Scales Dynamic Programming
-----------------------------------
We adopt the notation of §\[Stochastic\_Dynamic\_Programming\_by\_Time\_Blocks\]. We suppose given a two time-scales noise process $$\va{\Uncertain}_{(0,0):({\MakeUppercase{{d}}}+1,0)} =
\bp{\va{\Uncertain}_{0,0},\va{\Uncertain}_{0,1},\ldots,
\va{\Uncertain}_{0,{\MakeUppercase{{m}}}},\va{\Uncertain}_{1,0},\ldots,
\va{\Uncertain}_{{\MakeUppercase{{d}}},{\MakeUppercase{{m}}}},\va{\Uncertain}_{{\MakeUppercase{{d}}}+1,0}}
\eqfinp$$ For any ${d}\in \{ 0,1, \ldots, {\MakeUppercase{{d}}}\}$, we introduce the $\sigma$-fields $$\tribu{A}_{{d},0} = \{ \emptyset, \Omega \} \eqsepv
\tribu{A}_{{d},{m}} =
\sigma\np{\va{\Uncertain}_{({d},1):({d},{m})}} \eqsepv
{m}=1, \ldots, {\MakeUppercase{{m}}}\eqfinp$$
The proof of the following proposition is left to the reader.
Suppose that there exists a family $\sequence{\STATE_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ of measurable state spaces, with $ \STATE_{0}=\UNCERTAIN_{0,0} $, and a family $ \sequence{\Dynamics{{d}}{{d}+1} }{{d}=0,\ldots,{\MakeUppercase{{d}}}} $ of measurable dynamics $$\Dynamics{{d}}{{d}+1} : \STATE_{{d}} \times \HISTORY_{{d}:{d}+1}
\to \STATE_{{d}+1} \eqfinp$$ Suppose that the slow scale subprocesses $ \va{\Uncertain}_{({d},1):({d}+1,0)}=
\bp{\va{\Uncertain}_{{d},1} , \cdots , \va{\Uncertain}_{{d}+1,0} } $, ${d}=0,\ldots,{\MakeUppercase{{d}}}$, are independent (under the probability law $\PP$).
For a measurable nonnegative numerical cost function $$\tilde\criterion : \STATE_{{\MakeUppercase{{d}}}+1} \to [0,+\infty] \eqfinv$$ we define the family $\sequence{\tilde\Value_{{d}}}{{d}=0,\ldots,{\MakeUppercase{{d}}}+1}$ of functions by the backward induction
$$\begin{aligned}
\tilde\Value_{{\MakeUppercase{{d}}}+1}\np{\state_{{\MakeUppercase{{d}}}+1}}
& = \tilde\criterion\np{\state_{{\MakeUppercase{{d}}}+1}} \eqfinv \\
\tilde\Value_{{d}}\np{\state_{{d}}}
& = \inf_{\va\Control_{({d},0):({d},{\MakeUppercase{{m}}})}
\in \espace{L}^0\np{\Omega,
\tribu{A}_{({d},0):({d},{\MakeUppercase{{m}}})},\CONTROL_{({d},0):({d},{\MakeUppercase{{m}}})}}}
\Besp{ \tilde\Value_{{d}+1}\bp{\Dynamics{{d}}{{d}+1}
\np{\state_{{d}},
\va{\Control}_{{d},0}, \va{\Uncertain}_{{d},1} , \cdots ,
\va{\Control}_{{d},{\MakeUppercase{{m}}}} , \va{\Uncertain}_{{d}+1,0}}}} \eqfinp\end{aligned}$$
Then, the value functions $\tilde\Value_{{d}}$ are the solution of the following family of optimization problems, indexed by ${d}=0,\ldots,{\MakeUppercase{{d}}}$ and by $ \state_{{d}} \in \STATE_{{d}} $,
$$\tilde\Value_{{d}}(\state_{{d}}) =
\inf_{\Control_{({d},0):({\MakeUppercase{{d}}},{\MakeUppercase{{m}}})}
\in \espace{L}^0\np{\Omega,
\tribu{A}_{({d},0):({\MakeUppercase{{d}}},{\MakeUppercase{{m}}})},
\CONTROL_{({d},0):({\MakeUppercase{{d}}},{\MakeUppercase{{m}}})}}}
\besp{ \tilde\criterion\np{\va{\State}_{{\MakeUppercase{{d}}}+1}} } \eqfinv$$
where, for all ${d'}={d}, \ldots, {\MakeUppercase{{d}}}$, $$\va{\State}_{{d}}=\state_{{d}} \eqsepv
\va{\State}_{{d'}+1} =
\Dynamics{{d'}}{{d'}+1}
\bp{ \va{\State}_{{d'}},
\va{\Control}_{{d'},0},\va{\Uncertain}_{{d'},1},\cdots,
\va{\Control}_{{d'},{\MakeUppercase{{m}}}},\va{\Uncertain}_{{d'}+1,0} }
\eqfinp$$ \[eq:Optimization\_Problem\_Over\_Adapted\_Control\_Processes\_2ts\_Mayer\]
\[cor:2ts\_noises\]
Decision-Hazard-Decision Dynamic Programming
--------------------------------------------
We adopt the notation of §\[Decision\_Hazard\_Decision\_Dynamic\_Programming\]. We suppose given a noise process $$\va{\Uncertain}_{0:{S}} =
\bp{\va{\Uncertain}_{0}^{\sharp},\va{\Uncertain}_{1}^{\flat},\ldots,
\va{\Uncertain}_{{S}}^{\flat}}
\eqfinp$$ For any ${s}\in \{ 0,1, \ldots, {S}-1 \}$, we introduce the $\sigma$-fields $$\tribu{A}_{{s}} = \{ \emptyset, \Omega \} \eqsepv
\tribu{A}_{{s'}} =
\sigma\np{\va{\Uncertain}_{{s}+1:{s'}}^{\flat}} \eqsepv
{s'}= {s}+1,\ldots,{S}\eqfinp$$
The proof of the following proposition is left to the reader.
Suppose that there exists a family $\sequence{\STATE_{{s}}}{{s}=0,\ldots,{S}}$ of measurable state spaces, with $ \STATE_{0}=\UNCERTAIN_{0}^{\sharp}$, and a family $ \sequence{\Dynamics{{s}}{{s}+1} }{{s}=0,\ldots,{S}-1} $ of measurable dynamics $$\Dynamics{{s}}{{s}+1} : \STATE_{{s}} \times \CONTROL^{\sharp}_{{s}} \times
\UNCERTAIN_{{s}+1}^{\flat}\times \CONTROL^{\flat}_{{s}+1} \to \STATE_{{s}+1} \eqfinp$$ Suppose that the noise process $ \sequence{\va{\Uncertain}_{{s}}^{\flat}}{{s}=0,\ldots, {S}} $ is made of independent random variables (under the probability law $\PP$).
For a measurable nonnegative numerical cost function $$\tilde\criterion : \STATE_{{S}} \to [0,+\infty] \eqfinv$$ we define the family of functions $\sequence{\tilde\Value_{{s}}}{{s}=0,\ldots,{S}}$ by the backward induction
$$\begin{aligned}
\tilde\Value_{{S}}\np{\state_{{S}}}
& =
\tilde\criterion\np{\state_{{S}}}
\eqfinv
\\
\tilde\Value_{{s}}\np{\state_{{s}}}
& =
\inf_{\control_{{s}}^{\sharp}\in\CONTROL_{{s}}^{\sharp}}
\Besp{
\inf_{
\control_{{s}+1}^{\flat}\in \CONTROL_{{s}+1}^{\flat}}
\tilde\Value_{{s}+1}\Bp{\Dynamics{{s'}}{{s'}+1}
\bp{\state_{{s}},\control_{{s}}^{\sharp},\va{\Uncertain}_{{s}+1}^{\flat}, \control_{{s}+1}^{\flat}} }}
\eqfinp\end{aligned}$$
\[eq:DPB-unit\_Bellman\_equation\_Noise\_dhd\]
Then, the value functions $\tilde\Value_{{s}}$ in are the solution of the following family of optimization problems, indexed by ${s}=0,\ldots,{S}-1$ and by $ \state_{{s}} \in \STATE_{{s}} $,
\[eq:Optimization\_Problem\_Over\_Adapted\_Control\_Processes\_Unit\_Time\_Blocks\_Mayer\_dhd\] $$\tilde\Value_{{s}}(\state_{{s}}) =
\inf_{\va\Control_{{s}:{S}-1}^{\sharp}\in
\espace{L}^0\np{\Omega,\tribu{A}_{{s}:{S}-1},\CONTROL_{{s}:{S}-1}^{\sharp}}}
\inf_{\va\Control_{{s}+1:{S}}^{\flat}\in
\espace{L}^0\np{\Omega,\tribu{A}_{{s}+1:{S}},\CONTROL_{{s}+1:{S}}^{\flat}}}
\besp{ \tilde\criterion\np{\va{\State}_{{S}}} }
\eqfinv$$ where $$\va{\State}_{{s'}}=\state_{{s}} \eqsepv
\va{\State}_{{s'}+1} =
\Dynamics{{s'}}{{s'}+1}
\bp{ \va{\State}_{{s'}},
\va\Control_{{s'}}^{\sharp},
\va{\Uncertain}_{{s'}+1}^{\flat}, \va{\Control}_{{s'}+1}^{\flat}}
\eqsepv \forall {s'}={s}, \ldots, {S}-1 \eqfinp$$
Dynamic Programming with Unit Time Blocks {#dynamic-programming-with-unit-time-blocks}
-----------------------------------------
In the setting of optimization with noise process, we now consider the case where a state reduction exists at each time $t\inic{0}{\horizon-1}$. We will use a standard assumption in Dynamic Programming, that is, $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ is a white noise process.
### The Case of Final Cost Function
We first treat the case of a general criterion, as in §\[sec:UTB\_general\].
Suppose that there exists a family $\sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon}$ of measurable state spaces, with $ \STATE_{0}=\UNCERTAIN_{0} $, and a family $ \sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} $ of measurable dynamics $$\Dynamics{\tter}{\tter+1} : \STATE_{\tter} \times \CONTROL_{\tter} \times
\UNCERTAIN_{\tter+1} \to \STATE_{\tter+1} \eqfinp$$ Suppose that the noise process $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ is made of independent random variables (under the probability law $\PP$).
For a measurable nonnegative numerical cost function $$\tilde\criterion : \STATE_{\horizon} \to [0,+\infty] \eqfinv$$ we define the family $\sequence{\tilde\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of functions by the backward induction
$$\begin{aligned}
\tilde\Value_{\horizon}\np{\state_{\horizon}}
& =
\tilde\criterion\np{\state_{\horizon}}
\eqsepv \forall \state_{\horizon} \in \STATE_{\horizon} \eqfinv
\\
\tilde\Value_{\tter}\np{\state_{\tter}}
& =
\inf_{\control_{\tter}\in\CONTROL_{\tter}} \besp{
\tilde\Value_{t+1}\bp{\state_{t},\control_{t},\va{\Uncertain}_{t+1}} }
\eqsepv \forall \state_{\tter} \in \STATE_{\tter} \eqfinv\end{aligned}$$
\[eq:DPB-unit\_Bellman\_equation\_Noise\] for $ \tter = \horizon-1,\ldots,0 $.
Then, the value functions $\tilde\Value_{\tter}$ are the solution of the following family of optimization problems, indexed by $\tter=0,\ldots,\horizon-1$ and by $ \state_{\tter} \in \STATE_{\tter} $,
$$\tilde\Value_{\tter}(\state_{\tter}) =
\inf_{{\va\Control_{\tter:\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}}}
\besp{ \tilde\criterion\np{\va{\State}_{\horizon}} }
\eqfinv$$
where $$\va{\State}_{\tbis}=\state_{\tter} \eqsepv
\va{\State}_{\tbis+1} =
\Dynamics{\tbis}{\tbis+1}
\bp{ \va{\State}_{\tbis}, \va{\Control}_{\tbis}, \va{\Uncertain}_{\tbis+1} }
\eqsepv \forall \tbis=\tter, \ldots, \horizon-1 \eqfinp$$ \[eq:Optimization\_Problem\_Over\_Adapted\_Control\_Processes\_Unit\_Time\_Blocks\_Mayer\]
We define a family $ \sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon} $ of reduction mappings $ \theta_{\tter} : \HISTORY_{\tter} \to \STATE_{\tter} $ as in by induction. First, as $ \STATE_{0}=\UNCERTAIN_{0}=\HISTORY_{0} $ by assumption, we put $ \theta_{0} = \textrm{I}_{\textrm{d}} : \HISTORY_{0} \to \STATE_{0} $. Then, we use to define the mappings $ \theta_{1}, \ldots, \theta_{\horizon} $. As a consequence, the triplet $ \np{ \sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} } $ is a state reduction across the consecutive time blocks $[\tter,\tter+1]_{\tter = 0,\ldots,\horizon-1}$ of the time span.
Since the noise process $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ is made of independent random variables (under $\PP$), the family $ \sequence{ {\rho_{\tbis-1:\tbis}}} {1 \leq \tbis \leq \horizon} $ of stochastic kernels defined in is given by
$$\begin{aligned}
{\rho_{\tbis-1:\tbis}}:
\HISTORY_{\tbis-1}
& \to \Delta\np{\UNCERTAIN_{\tbis}}
\eqsepv \tbis = 1, \ldots, \horizon \eqfinv
\\
\history_{\tbis-1}
&\mapsto
\PP_{\va{\Uncertain}_{\tbis}}\np{\mathrm{d}\uncertain_{\tbis}}
\eqfinp
\end{aligned}$$
\[eq:regular\_conditional\_distribution\_independent\]
As a consequence, we have by that the triplet $ \np{ \sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\theta_{\tter}}{\tter=0,\ldots,\horizon} ,
\sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} } $ is compatible (see Definition \[de:reduction-dynamics\_family\]) with the family $\na{{\rho_{\tter-1:\tter}}}_{\tter=1,\ldots,\horizon}$ of stochastic kernels in . In addition, the reduced stochastic kernels in coincide with the original stochastic kernels in .
Define the cost function $\criterion$ as $$\criterion = \tilde\criterion \circ \theta_{\horizon} \eqfinp$$ Proposition \[thm:DPB-unit\] applies, so that the family $\sequence{\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of value functions defined for the family of optimization problems satisfies $$\Value_{\tter} = \tilde\Value_{\tter} \circ \theta_{\tter}
\eqsepv t = 0,\ldots,\horizon \eqfinp$$ By means of Proposition \[th:optimal\_adapted\_control\_process\], we deduce that $$\widecheck{\Value}_{\tter}(\history_{\tter}) =
\tilde\Value_{\tter} \circ \theta_{\tter} (\history_{\tter}) \eqfinv$$ for all $t=0,\ldots,\horizon$ and for any $\history_{\tter}\in\HISTORY_{\tter}$. From the definition of the family of functions $\widecheck{\Value}_{\tter}$, we obtain the expression of functions $\tilde\Value_{\tter}$.
### The Case of Time Additive Cost Functions {#the-case-of-time-additive-cost-functions}
We make the same assumptions than in §\[The\_Case\_of\_Time\_Additive\_Cost\_Functions\]. The proof is left to the reader.
Suppose that there exists a family $\sequence{\STATE_{\tter}}{\tter=0,\ldots,\horizon}$ of measurable state spaces, with $ \STATE_{0}=\UNCERTAIN_{0} $, and a family $ \sequence{\Dynamics{\tter}{\tter+1} }{\tter=0,\ldots,\horizon-1} $ of measurable dynamics $$\Dynamics{\tter}{\tter+1} : \STATE_{\tter} \times \CONTROL_{\tter} \times
\UNCERTAIN_{\tter+1} \to \STATE_{\tter+1} \eqfinp$$ Suppose that the noise process $ \sequence{\va{\Uncertain}_{t}}{t=0,\ldots, \horizon} $ is made of independent random variables (under the probability law $\PP$).
We define the family $\sequence{\tilde\Value_{\tter}}{\tter=0,\ldots,\horizon}$ of functions by the backward induction
$$\begin{aligned}
\widehat\Value_{\horizon}\np{\state_{\horizon}}
& =
\coutfin\np{\state_{\horizon}}
\eqsepv \forall \state_{\horizon} \in \STATE_{\horizon} \eqfinv
\intertext{and, for~$t=\horizon-1,\ldots,0$ and
for all \( \state_{\tter} \in \STATE_{\tter} \) }
\widehat\Value_{\tter}\np{\state_{\tter}}
& =
\inf_{\control_{\tter}\in\CONTROL_{\tter}}
\besp{
\coutint_{\tter}\np{\state_{\tter},\control_{\tter},\va{\Uncertain}_{\tter+1}}
+ \widehat\Value_{\tter+1}
\bp{\Dynamics{\tter}{\tter+1}
\np{\state_{\tter},\control_{\tter},\va{\Uncertain}_{\tter+1}} } }
\eqfinp\end{aligned}$$
\[eq:DPB-unit\_Bellman\_equation\_Noise\_Additive\]
Then, the value functions $\widehat\Value_{\tter}$ are the solution of the following family of optimization problems, indexed by $\tter=0,\ldots,\horizon-1$ and by $ \state_{\tter} \in \STATE_{\tter} $,
$$\widehat\Value_{\tter}(\state_{\tter}) =
\inf_{\np{\va\Control_{\tter},\ldots, \va\Control_{\horizon-1}} \in
\espace{L}^0\np{\Omega,\tribu{A}_{\tter:\horizon-1},\CONTROL_{\tter:\horizon-1}}}
\bgesp{ \sum_{\tbis=\tter}^{\horizon-1} \coutint_{\tbis}
\bp{\va{\State}_{\tbis},\va{\Control}_{\tbis},\va{\Uncertain}_{\tbis+1}} +
\coutfin\bp{\va{\State}_{\horizon}} }
\eqfinv$$
where $$\va{\State}_{\tbis}=\state_{\tter} \eqsepv
\va{\State}_{\tbis+1} =
\Dynamics{\tbis}{\tbis+1}
\bp{ \va{\State}_{\tbis}, \va{\Control}_{\tbis}, \va{\Uncertain}_{\tbis+1} }
\eqsepv \forall \tbis=\tter, \ldots, \horizon-1 \eqfinp$$ \[eq:Optimization\_Problem\_Over\_Adapted\_Control\_Processes\_Unit\_Time\_Blocks\_Additive\]
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[^1]: We could also consider any $ \criterion : \HISTORY_{\tter} \to \RR $, measurable bounded function, or measurable and uniformly bounded below function. However, for the sake of simplicity, we will deal in the sequel with measurable nonnegative numerical functions. When $ \criterion\np{\history_{\horizon}}=+\infty $, this materializes joint constraints between uncertainties and controls. \[ft:nonnegative\]
[^2]: These families are defined over the time span $ \{0, \ldots, \horizon \} \equiv \TT $ by the identification in such a way that the notation is consistent with the notation .
[^3]: See Footnote \[ft:nonnegative\].
|
---
abstract: 'We consider the problem of representing multidimensional data where the domain of each dimension is organized hierarchically, and the queries require summary information at a different node in the hierarchy of each dimension. This is the typical case of OLAP databases. A basic approach is to represent each hierarchy as a one-dimensional line and recast the queries as multidimensional range queries. This approach can be implemented compactly by generalizing to more dimensions the $k^2$-treap, a compact representation of two-dimensional points that allows for efficient summarization queries along generic ranges. Instead, we propose a more flexible generalization, which instead of a generic quadtree-like partition of the space, follows the domain hierarchies across each dimension to organize the partitioning. The resulting structure is much more efficient than a generic multidimensional structure, since queries are resolved by aggregating much fewer nodes of the tree.'
author:
- 'Nieves R. Brisaboa'
- 'Ana Cerdeira-Pena'
- 'Narciso López-López'
- |
\
Gonzalo Navarro
- 'Miguel R. Penabad'
- 'Fernando Silva-Coira'
bibliography:
- 'paper.bib'
title: '[Efficient Representation of Multidimensional Data over Hierarchical Domains]{} [^1] '
---
Introduction
============
In many application domains the data is organized into multidimensional matrices. In some cases, like GIS and 3D modelling, the data are actually points that lie in a two- or three-dimensional discretized space. There are, however, other domains such as OLAP systems [@OLAP1; @OLAP2] where the data are sets of tuples that are regarded as entries in a multidimensional cube, with one dimension per attribute. The domains of those attributes are not necessarily numeric, but may have richer semantics. A typical case in OLAP [@MULTIDIM1], in particular in snowflake schemes [@SNOWFLAKE], is that each tuple contains a numeric summary (e.g., amount of sales), which is regarded as the value of a cell in the data cube. The domain of each dimension is hierarchical, so that each value in the dimension corresponds to a leaf in a hierarchy (e.g., countries, cities, and branches in one dimension, and years, months, and days in another). Queries ask for summaries (sums, maxima, etc.) of all the cells that are below some node of the hierarchy across each dimension (e.g., total sales in New York during the previous month).
A way to handle OLAP data cubes is to linearize the hierarchy of the domain of each dimension, so that each internal node corresponds to a range. Summarization queries are then transformed into multidimensional range queries, which are solved with multidimensional indexes [@Samet06]. Such a structure is, however, more powerful than necessary, because it is able to handle [*any*]{} multidimensional range, whereas the OLAP application will only be interested in queries corresponding to combinations of nodes of the hierarchies. There are well-known cases, in one dimension, of problems that are more difficult for general ranges than if the possible queries form a hierarchy. For example, categorical range counting queries (i.e., count the number of different values in a range) requires in general $\Omega(\log n / \log\log n)$ time if using $O(n\,\textrm{polylog}\, n)$ space [@LW13], where $n$ is the array size, but if queries form a hierarchy it is easily solved in constant time and $O(n)$ bits [@Sad07]. A second example is the range mode problem (i.e., find the most frequent value in a range), which is believed to require time $\Omega(n^{1.188})$ if using $O(n^{1.188})$ space [@CDLMW12], but if queries form a hierarchy it is easily solved in constant time and linear space [@HSTV14].
In this paper we aim at a compact data structure to represent data cubes where the domains in each dimension are hierarchical. Following the general idea of the tailored solutions to the problems we mentioned [@Sad07; @HSTV14], our structure partitions the space according to the hierarchies, instead of performing a regular partition like generic multidimensional structures. Therefore, the queries of interest for OLAP applications, which combine nodes of the different hierarchies, will require aggregating the information of just a few nodes in our partitions, much fewer than if we used a generic space partitioning method.
Since we aim at compact representations, our baseline will be an extension to multiple dimensions of a two-dimensional compact summarization structure known as $k^2$-treap [@k2treap:infosis2016], a $k^2$-tree [@k2tree:infosis2014] enriched with summary information on the internal nodes. This $n$-dimensional treap, called $k^n$-treap, will then be extended so that it can follow an arbitrary hierarchy, not only a regular one. The topology of each hierarchy will be represented using a compact tree representation, precisely LOUDS [@louds:Jacobson:1989]. This new structure is called CMHD (Compact representation of Multidimensional data on Hierarchical Domains). Although we focus on sum queries in this paper, it is easy to extend our results to other kinds of aggregations.
The rest of this paper is organized as follows. Sections \[sec:kntreaps\] and \[sec:cmhd\] describe our compact baseline and then how it is extended to obtain our new data structure. An experimental evaluation is given in Section \[sec:experiments\]. Finally, we offer some conclusions and guidelines for future work.
Our Baseline: $k^n$-treaps {#sec:kntreaps}
==========================
The $k^n$-treap is a straightforward extension of the $k^2$-treap to manage multiple dimensions. It uses a $k^n$-tree (in turn a straightforward extension of the $k^2$-tree) to store its topology, and stores separately the list of aggregate values obtained from the sum of all values in the corresponding submatrix. Figure \[fig:kntreap\] shows a matrix and the corresponding $k^n$-treap. The example uses two dimensions, but the same algorithms are used for more dimensions.
Consider a hypercube of $n$ dimensions, where the length of each dimension is $len = k^i$ for some $i$. If the length of the dimensions are different, we can artificially extend the hypercube with empty cells, with a minimum impact in the $k^n$-treap size. The $k^n$-trees, which will be used to represent the $k^n$-treap topology, are very efficient to represent wide empty areas. The algorithm to build the $k^n$-treap starts storing on its root level the sum of all values on the matrix[^2]. It also splits each dimension into $k$ equal-sized parts, thus giving a total of $k^n$ submatrices. We define an ordering to traverse all the submatrices (in the example, rows left-to-right, columns top-to-bottom). Following this ordering, we add a child node to the root for each submatrix. The algorithm works recursively for each child node that represents a nonempty submatrix, storing the sum of the cells in this submatrix, splitting it and adding child nodes. For empty sumatrices, the node stores a sum of $0$.
As we can see in Figure \[fig:kntreap\], the root node stores 51, the sum of all values in the matrix, and it is decomposed into 4 matrices of size $4\times 4$, thus adding 4 children to the root node. Notice that the second submatrix (top-right) is full of zeroes, so this node just stores a sum of $0$ and is not further decomposed. The algorithm proceeds recursively for the remaining 3 children of the root node.
The final data structures used to represent the $k^n$-treap are the following:
- *Values (V)*: Contains the aggregated values (sums) for each (sub)matrix, as they would be obtained by a levelwise traversal of the $k^n$-treap. It is encoded using DACs [@dacs:ipm:2013], which compress small values while allowing direct access.
- *Tree structure (T)*: It is a $k^n$-tree that stores a bitmap $T$ for the whole tree except its leaves. In this case, the usual bitmap $L$ for the leaves in a standard $k^n$-tree is not used, because the information about which cells have or not a value is already represented in $V$. Therefore $L$ is not needed.
[![$k^n$-treap with a highlighted range query[]{data-label="fig:kntreap"}](kntreap "fig:"){width="85.00000%"}]{}
The navigation through the $k^n$-treap is basically a depth first traversal. Finding the child of a node can be done very efficiently by using $rank$ and $select$ operations [@louds:Jacobson:1989] as in the standard $k^2$-tree. The typical queries in this context are: finding the value of an individual cell and finding the sum of the values in a given range of cells, specified by the initial and final coordinates that define the submatrix of interest.
#### **[*Finding the value of a specific cell by its coordinates.*]{}**
To find the value of the cell, for example the cell at coordinates $(4,3)$ in the figure, the search starts at the root node and in each step goes down trough the children of the matrix overlapping the searched cell. In this example, the search goes through the first child node (with value 21 in the figure), then through its third child (with value 6) and finally through the second child, reaching the leaf node with value 4, which is the value returned by the query.
#### **[*Finding the sum of the cells in a submatrix.*]{}**
The second type of query looks for the aggregated value of a range of cells, like the shaded area in Figure \[fig:kntreap\]. This is implemented as a depth-first multi-branch traversal of the tree. If the algorithm finds that the range specified in the query fully contains a submatrix of the $k^n$-treap that has a precomputed sum, it will use this sum and will not descend to its child nodes. The figure highlights the branches of the $k^n$-treap that are used. Notice that this query completely includes the sumatrices of values $\{5,4,0,2\}$ and $\{0,2,1,5\}$, that have their sums (11 and 8) explicitly stored on the third level of the tree. Therefore, the algorithm does not need to reach the leaf levels of the tree for these matrices. Notice also that there is an empty submatrix that intersects with the region of the query (the first child of the third child of the root), so the algorithm also stops before reaching the leaf levels in this submatrix. Only for cells $(3,2)$ (with a value of 4) and $(4,2)$ (with a value of 0) needs the algorithm to reach the leaf levels.
Our proposal: CMHD {#sec:cmhd}
==================
As previously stated, CMHD divides the matrix following the natural hierarchy of the elements in each dimension. In this way we allow the efficient answer of queries that consider the semantic of the dimensions.
Conceptual description
----------------------
Consider an $n$-dimensional matrix where each cell contains a weight (e.g., product sales, credit card movements, ad views, etc.). The CMHD recursively divides the matrix into several submatrices, taking into account the limits imposed by the hierarchy levels of each dimension.
Figure \[fig:cmhd\] depicts an example of a CMHD representation for two dimensions. The matrix records the number of product sales in different locations. For each dimension, a hierarchy of three levels is considered. In particular, cities are aggregated into countries and continents, while products are grouped into sections and good categories. The tree at the right side of the image shows the resulting conceptual CMHD for that matrix. Observe that each hierarchy level leads to an irregular partition of the grid into submatrices (each of them defined by the limits of its elements), having as associated value the sum of product sales of the individual cells inside it. Thus, the root of the tree stores the total amount of sales in the complete matrix. Then the matrix is subdivided by considering the partition corresponding to the first level of the dimension hierarchies (see the bold lines). Each of the submatrices will become a child node of the root, keeping the sum of values of the cells in the corresponding submatrix. The decomposition procedure is repeated for each child, considering subsequent levels of the hierarchies (see the dotted lines), as explained, until reaching the last one. Also notice that, as happens in the $k^n$-treap, the decomposition concludes in all branches when empty submatrices are reached (that is, in this scenario, when a submatrix with no sales is found). See, for example, the second child of the root.
Note that CMHD assumes the same height in all the hierarchies that correspond to the different dimensions. Observe that, for each crossing of elements of the same level from different dimensions, an aggregate value is stored. Notice also that artificial levels can be easily added to a hierarchy of one dimension by subdividing all the elements of a level in just one element (itself), thus creating a new level identical to the previous one. This feature allows us to arbitrarily match the levels of the different hierarchies, and thus to flexibly adapt the representation of aggregated data to particular query needs. That is, by introducing artificial intermediate levels where required, more interesting aggregated values will be precomputed and stored. For example, assume we have two dimensions: ($d_{1}$) with levels for *department*, *section* and *product*; and ($d_{2}$) with levels for *year*, *season*, *month* and *day*. If we were interested in obtaining the number of sales per *section* for *season*s, but also for *months*, we could devise a new level arrangement for $d_{1}$, that will have now the levels *department*, *section*, *section’*, *product*; where each particular *section* of the second hierarchy level results into just one *section’* child, which is actually itself. In this way aggregated values will be computed and stored considering sales for *section* in each *season*, but also sales for *section’* in each *month*.
[![Example of CMHD construction for a two-dimensional matrix.[]{data-label="fig:cmhd"}](cmhd "fig:"){width="100.00000%"}]{}
Data structures
---------------
The conceptual tree that defines the CMHD is represented compactly with different data structures, for the domain hierarchies and for the matrix data itself.
#### **[*Domain hierarchy representation.*]{}**
The hierarchy of a dimension domain is essentially a tree of $C$ nodes. We represent this tree using LOUDS [@louds:Jacobson:1989], a tree representation that uses $2C$ bits, and can efficiently navigate it. Using LOUDS, a tree representing the hierarchy of the elements of a dimension is encoded by appending the degree $r$ of each node in (left-to-right) level-order, in unary: $1^{r}0$. Figure \[fig:cmhd\] illustrates the hierarchy encoding of the dimensions used in that example (see $d_{1}$ and $d_{2}$). For instance, the degree of the first node for the products dimension ($d_{1}$) is 3, so its unary encoding is 1110. Note that each node (i.e., element of a dimension placed at any level of its hierarchy) is associated with one 1 in the encoded representation of the degree of its parent. LOUDS is navigated using $rank$ and $select$ queries: $rank_b(i)$ is the number of bits $b$ up to position $i$, and $select_b(j)$ is the position of the $j$th occurrence of bit $b$. Both queries are computed in constant time using $o(C)$ additional bits [@Cla96]. For example, given a node whose unary representation starts at position $i$, its parent is $p=select_0(t-j)+1$, where $t=select_1(j)$ and $j=rank_0(i)$; and $i$ is the $(t-p+1)$th child of $p$. On the other hand, the $k$th child of $i$ is $select_0(rank_1(i)+k-1)+1$. We also use a hash table to associate the domain nodes (i.e., labels such as “USA” in Figure \[fig:cmhd\]) with the corresponding LOUDS node position.
#### **[*Data representation.*]{}**
To represent the $n$-dimensional matrix, we use the following data structures:
- *Tree structures ($T_{a}$ and $T_{c}$)*: to navigate the CMHD, we need to use two different data structures in conjunction. First, $T_{a}$, a bit array that, similarly to the $k^{n}$-treap, provides a compact representation of the conceptual tree independently of the node values, for all the tree levels, except the last one[^3]. That is, internal nodes whose associated value is greater than 0, will be represented with a 1. In other case, they will be labeled with a 0. Observe that, for the $k^{n}$-treap, the use of this data structure is enough to navigate the tree, taking advantage of the regular partition of the matrix into equal-sized submatrices. Instead, CMHD follows different hierarchy partitions, which results into irregular submatrices. Therefore, a second data structure, $T_{c}$, is also required to traverse the CMHD. This is a bit array aligned to $T_{a}$, which marks the limits of each tree node in $T_{a}$ (this time, it also considers the last tree level). If the next tree node in $T_a$ has $z$ children, we append $1^{z-1}0$ to $T_c$. Notice that each node of $T_a$ is associated with a 0 in $T_c$, which allows navigating the trees using $rank$ and $select$ on $T_a$ and $T_c$: say we are at a node in $T_a$ that starts at position $i$; then it has a $k$th child iff $T_a[i+k-1]=1$, and if so this child starts at position $select_0(T_c,rank_1(T_a,i+k-1))+1$.
- *Values (V)*: the CMHD is traversed levelwise storing the values associated with each node (either corresponding to original matrix cells, or to data aggregations) in a single sequence, which is then represented with DACs [@dacs:ipm:2013].
Queries
-------
Queries in this context give the names of elements of the different dimensions and ask for the sum of the cells defined for those values. Depending on the query, we can answer it by just reporting a single aggregated value already kept in *V*, or by retrieving several stored values, and then adding them up. The first scenario arises when the elements (labels) of the different dimensions specified in the query are all at the same level in their respective hierarchies. The second situation arises from queries using labels of different levels. In both contexts, top-down traversals of the conceptual CMHD are required to fetch the values. The algorithm always starts searching the hash tables for the labels provided by the query for the different dimensions, to locate the corresponding LOUDS nodes. From the LOUDS nodes, we traverse each hierarchy upwards to find out its depth and the child that must be followed at each level to reach it.
This information is then used to find the desired nodes in $T_a$. For example, with two dimensions, we start at the root of $T_a$ and descend to the child number $k_1 + a_1 \cdot k_2$, where $k_i$ is the child that must be followed in the $i$th dimension to reach the queried node, and $a_i$ is the number of children of the root in the $i$th dimension ($a_i$ is easily computed with the LOUDS tree of its dimension). We continue similarly to the node at level 2, and so on, until we reach one of the query nodes in a dimension, say in the first. Now, to reach the other (deeper) node in the second dimension, we must descend by every child in the first dimension, at every level, until reaching the second queried node. Finally, when we have reached all the nodes, we collect and sum up the corresponding values from $V$. Note that, if all the queried nodes are in the same level, we perform a single traversal in $T_a$. Note also that, if we find any zero in a node of $T_a$ along this traversal, we immediately prune that branch, as the submatrix contains no data.
#### Example.
Assume we want to retrieve the total amount of *speaker* sales in *Montreal*, in Figure \[fig:cmhd\]. Since both labels belong to the same level in both dimension hierarchies (the last one), we will have to retrieve a single stored value in that level. The path to reach it has been highlighted in the conceptual tree of the image. To perform the navigation we must start at the root of the tree (position 0 in $T_{a}$). In the first level, we need to fetch the sixth child (offset 5), as it corresponds to the submatrix including the element to search, in that level. Hence we access position 5 in $T_{a}$. Since $T_{a}[5] = 1$, we must continue descending to the next level. Recall that we have a 1 in $T_{a}$ for each node with children, and that each node is associated with just one 0 in $T_{c}$. So the child starts at position $select_{0}(T_{c}, rank_{1}(T_{a},5))+1 =
select_{0}(T_{c}, 4)+1 = 22$ in $T_{a}$. In this level we must access the third child (offset 2), so we check $T_{a}[24] = 1$. Again, as we are in an internal node, we know that its children are located at position $select_{0}(T_{c}, rank_{1}(T_{a}, 24)) + 1 =
select_{0}(T_{c}, 9) + 1 = 59$. Finally, we reach the third and last level of the tree, where we know that the corresponding child is the fourth one (at $T_a[59+3]=T_a[62]$). Recall, however, that this last level is not represented in $T_{a}$. To perform this final step, we directly look into the array $V$: $V[62+1] = V[63] = 7$ is the answer.
In case of queries combining labels of different levels, the same procedure would apply, but having to get the values corresponding to all the possible combinations with the element of the lowest hierarchy level (e.g., if we want to obtain the number of *meal* sales in *America*, we must first recover the values associated with *meal*-*Canada*, *meal*-*USA*, and *meal*-*Chile*, and then sum them up).
Experimental Evaluation {#sec:experiments}
=======================
This section presents the empirical evaluation of the two previously described data structures. Both representations have been implemented in C/C++, and the compiler used was GCC 4.6.1. (option -O$9$). We ran our experiments in a dedicated Intel(R) Core(TM) i7-3820 CPU @ 3.60GHz (4 cores) with 10MB of cache, and 64GB of RAM. The machine runs Ubuntu 12.04.5 LTS with kernel 3.2.0-99 (64 bits).
We generate different datasets, all of them synthetic, to evaluate the performance of the two data structures, varying the number of dimensions and the number of items on each dimension. These datasets have been labeled as `<dim#>D_<item#>`, thus referring to their size specifications in the own name. For example, dataset `5D_16` has 5 dimensions, and the number of items on each dimension is 16. The total size of this dataset is $16^5=1048576$ elements.
In order to show the CMHD advantage of considering the domain semantics, and computing the aggregate values according to the natural limits imposed by the hierarchy of elements in each dimension, the dimensions hierarchies have been generated in two different ways for each dataset. First, the *binary* organization, that corresponds to a regular partition. That is, the hierarchies of each dimension are exactly the same as those produced by a $k^n$-treap matrix partition into equal-sized submatrices. In this way both data structures store exactly the same aggregated values. We named it *binary* because we use a value of $k=2$. Second, the *irregular* organization, which arbitrary groups data, on each dimension, into different and irregular hierarchies (different number of divisions, and also different size at each level). The last scenario simulates what would be a matrix partition following the semantic needs of a given domain. In this case the aggregated values stored by the CMHD will be different from those stored by the $k^n$-treap, and therefore more appropriated to answer queries using the same “semantic”. That means, in our context, queries considering regions that exactly match the natural divisions of each dimension at some level of the hierarchies.
To test the structures behavior, we have also considered three different datasets, with a different number of empty cells, for each size specification: with no empty cells, and with 25% and 50% of empty cells, respectively.
First we analyze the space requirements of both data structures for all the datasets (see Table \[table:sizes\]). Of course, the size decreases as the number of empty cells increases, in both cases. Moreover, we can also observe that the $k^n$-treap size is slightly lower than the CMHD. This is expected, because CMHD has to store the LOUDS representation of each dimension hierarchy, while dimensions are implicit for the $k^n$-treap. Additionally, CMHD uses a second bitmap ($T_c$) to navigate the conceptual tree, which is not necessary when using the $k^n$-treap.
We must also clarify a small issue about the sizes of the $k^n$-treaps: the size of a standard $k^n$-treap for a specific dataset is always the same, regardless of the organization of its dimensions (binary or irregular). However, Table \[table:sizes\] shows some difference in the sizes. For example, for `4D_16`, the size for the binary organization is $44.84$, but it is $44.42$ for the irregular one. The reason for this variation is that all queries are performed by taking dimension labels as input, so we need a vocabulary to translate each label into a range of cells. We have included that vocabulary (dimension labels and cell ranges) into the size of the $k^n$-treaps, and the vocabulary for the irregular organization is usually smaller, as it has less levels and less dimension labels (because each node in the conceptual tree can have more than 2 children in the irregular organization, while the binary organization always has 2).
Regarding query times, we have run several sets of queries for all the datasets. As previously mentioned, queries are posed in this context by giving one element name (label) for each different dimension, as it is the natural way to query a multidimensional matrix defined by hierarchical dimensions. Since the $k^n$-treap does not directly work with labels, each query has been translated into the equivalent ranged query, establishing the initial and final coordinates for each dimension. The following types of queries have been considered:
- *Finding one precomputed value*. This value can be a specific cell of the matrix (so forcing the algorithms to reach the last level of the tree), or a precomputed value that corresponds to an internal node of the conceptual tree.
Following the example of Figure \[fig:cmhd\], a query asking for the amount of *speakers* sales in *Montreal* or the total number of *beberages* sales in *Italy* would be queries of this type, the former accessing an individual cell and the later obtaining a precomputed value in the penultimate level of the tree.
- *Finding the sum of several precomputed values*. This kind of query must obtain a sum that is not precomputed and stored in the data structure itself. In turn, it must access several of these aggregated values and then add them up. Given that we are specifying the queries by dimension labels, this type of query is defined by using labels that belong to different levels of the hierarchies across the dimensions. The lowest level, which corresponds to individual cells, is not used for this scenario.
An example of this query type would be to find the total number of sales of *electronic* products in Chile. Note that *electronic* is located at the first level of its dimension hierarchy, but *Chile* is at the second level of the second dimension (see Figure \[fig:cmhd\]). Hence, the values corresponding to *computers-Chile*, *cameras-Chile*, and *audio-Chile* must be first retrieved to finally sum them up.
Each created set contains $10,000$ queries, randomly generated, of the two previous types, for each dataset. The following tables show the average query times (in microseconds per query) for both data structures, taking into account the two different matrix partitions of the datasets (*binary* or *irregular*) and also the percentage of empty cells.
We first show the results obtained for queries that just need to retrieve one precomputed value, at different levels. On the one hand, Table \[table:queries1\] displays query times for specific matrix cells, that is, located at the last level of the conceptual tree. In this case, the $k^n$-treap performs better than the CMHD in almost all cases. This is an expected outcome as both data structures must reach the leaf level to get the answer, and the depth first navigation of the tree is simpler in the $k^n$-treap (just products and $rank$ operations). In any case, CMHD also performs quite well, using just a few microseconds to answer any of the queries.
On the other hand, Table \[table:queries2\] shows the average query times for queries of the same type, but now considering precomputed values stored in nodes of an intermediate level of the tree (in particular, the penultimate level). Note that this fact holds for both data structures when working with a regular partition of the matrix (that is, the *binary* scenario). Thus, in this case, the $k^n$-treap gets better results than CMHD, but with slight time differences. Yet, observe that this is not the actual scenario when dealing with meaningful application domains, where rich semantics arise. This situation is that corresponding to what we called *irregular* datasets. In this case, CMHD excels, as expected, given that this data structure has been particularly designed to manage hierarchical domains. Results show that CMHD is able to perform up to 12 times faster than $k^n$-treap (for the best case).
To check whether the observed differences are significative (in the cases where times were closer) we performed a statistical significance test. We checked the `4D_16` and `5D_16` datasets, for the irregular organization, with all the different configurations of empty cells.
We show here, as a proof, the details for `4D_16` with 50% of empty cells, which took $5 \mu\mbox{s}$ to the $k^n$-treap, and $2
\mu\mbox{s}$ to CMHD. We ran 20 sets of $10,000$ queries, and measured both the average time and the standard deviation for the $k^n$-treap ($5.100$ and $0.447$, respectively) and for the CMHD ($1.750$ and $0.550$, respectively). With these results, we obtain a critical value of $4.725$, which is greater than $2.580$, so the difference is significative with a 99% of confidence level. The remaining tests also proved the same significance results.
Finally, Table \[table:queries4\] presents the average query times for the second type of queries (that is, those having to recover several precomputed values and then adding them up to provide the final answer). As results show, the $k^n$-treap displays a better performance than CMHD for the *binary* scenario. However, again this is not the most interesting situation in real domains. If we observe the results obtained for the *irregular* datasets, we will appreciate that CMHD clearly outperforms the $k^n$-treap in this scenario, thus demonstrating the good capabilities of our proposal to cope with the aim of this work.
Conclusions and Future Work {#sec:conclusions}
===========================
We have presented a multidimensional compact data structure that is tailored to perform aggregate queries on data cubes over hierarchical domains, rather than general range queries. The structure represents each hierarchy with a succinct tree representation, and then partitions the data cube according to the product of the hierarchies. This partition is represented with an extension of the $k^2$-treap to higher dimensions and to non-regular partitions. The resulting structure, dubbed CMHD, is much faster than a regular multidimensional $k^2$-treap when the queries follow the hierarchical domains. This makes it particularly attractive to represent OLAP data cubes compactly and efficiently answer meaningful aggregate queries.
As future work, we plan to experiment on much larger collections. This would make the vocabulary of hierarchy nodes much less significant compared to the data itself (especially for the CMHD). We also plan to test real datasets (for example, coming from data warehouses) and real query workloads. We also expect to compare our results with established OLAP database management systems, and to enrich our prototype with other kinds of queries and data.
[^1]:
[^2]: The implemented algorithm is recursive and each sum is actually computed only once, when returning from the recursive calls.
[^3]: We do not actually need to represent the nodes of the last level in $T_{a}$. This data structure will be used to first identify a node whose children will be later located in another bit array ($T_{c}$). But these already constitute matrix cells, with no children.
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abstract: 'M87, the active galaxy at the center of the Virgo cluster, is ideal for studying the interaction of a supermassive black hole (SMBH) with a hot, gas-rich environment. A deep Chandra observation of M87 exhibits an approximately circular shock front (13 kpc radius, in projection) driven by the expansion of the central cavity (filled by the SMBH with relativistic radio-emitting plasma) with projected radius $\sim$1.9 kpc. We combine constraints from X-ray and radio observations of M87 with a shock model to derive the properties of the outburst that created the 13 kpc shock. Principal constraints for the model are 1) the measured Mach number ($M$$\sim$1.2), 2) the radius of the 13 kpc shock, and 3) the observed size of the central cavity/bubble (the radio-bright cocoon) that serves as the piston to drive the shock. We find an outburst of $\sim$5$\times$$10^{57}$ ergs that began about 12 Myr ago and lasted $\sim$2 Myr matches all the constraints. In this model, $\sim$22% of the energy is carried by the shock as it expands. The remaining $\sim$80% of the outburst energy is available to heat the core gas. More than half the total outburst energy initially goes into the enthalpy of the central bubble, the radio cocoon. As the buoyant bubble rises, much of its energy is transferred to the ambient thermal gas. For an outburst repetition rate of about 12 Myrs (the age of the outburst), 80% of the outburst energy is sufficient to balance the radiative cooling.'
author:
- 'W. Forman, E. Churazov, C. Jones S. Heinz, R. Kraft, A. Vikhlinin'
title: 'PARTITIONING THE OUTBURST ENERGY OF A LOW EDDINGTON ACCRETION RATE AGN AT THE CENTER OF AN ELLIPTICAL GALAXY: THE RECENT 12 MYR HISTORY OF THE SUPERMASSIVE BLACK HOLE IN M87'
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The Outburst Chronicle of M87’s Supermassive Black Hole
=======================================================
The cavities and shocks observed in cluster, group, and galaxy images of hot gas-rich systems chronicle the mechanical energy release, as distinct from the radiated emission, from supermassive black holes (SMBH) accreting at levels well below the Eddington mass accretion rate ($\dot{M}_{\rm Edd}=4\pi G m_p M_{SMBH}/ \eta c \sigma_T$, $M_{\rm SMBH}$ is the SMBH mass, $G$ is the gravitational constant, $m_p$ is the proton mass, $c$ is the speed of light, $\sigma_T$ is the Thomson electron scattering cross section, and $\eta\approx$10%). For present epoch SMBHs in hot, gas-rich systems, the mechanical power dominates the radiated power (e.g., Churazov et al. 2000, 2005, Fabian et al. 2003, McNamara et al. 2005, Allen et al. 2006, Fabian 2012). The best, and often the only, way to derive the dominant energy release from the SMBH is through the effects of the SMBH on the surrounding hot atmosphere. The Eddington luminosity is given as $L_{\rm Edd} = 1.3\times10^{47} (M_{SMBH}/10^9)$ [ ergs s$^{-1}$]{}. With an SMBH mass of
$3-6\times10^9\>$[[M$_{\odot}$]{}]{} (Harms et al. 1994, Ford et al. 1994, Macchetto et al. 1997, Gebhardt et al. 2011, Walsh et al. 2013), the Eddington luminosity of M87’s SMBH is $L_{\rm Edd} \sim 4-8 \times
10^{47}$[ ergs s$^{-1}$]{}. The currently observed bolometric radiative luminosity $L_{rad}$ of the central AGN is $L_{rad} \approx3\times10^{42}$ [ ergs s$^{-1}$]{}(e.g., Prieto et al. 2016). This $L_{rad}$ is about five orders of magnitude lower than the Eddington limit for M87’s mass, firmly placing the object into the category of low power AGNs. At the same time, the typical estimates of the jet mechanical power $L_{jet}$ of the source are consistently higher, $\sim10^{44}$ [ ergs s$^{-1}$]{}(e.g., Bicknell & Begelman 1996, Owen, Eilek and Kassim, 2000, Stawarz et al. 2006), implying that $L_{rad}/L_{jet}\sim0.03$ or lower. All these properties suggest that we are dealing with a variant of a hot, radiatively inefficient flow (e.g., Ichimaru 1977, Rees et al. 1982, Narayan & Yi 1994, Blandford & Begelman 1999, Yuan & Narayan 2014). M87’s spectral energy distribution also supports this conclusion (Reynolds et al. 1996; Di Matteo et al. 2003; Yuan et al. 2009; Moscibrodzka et al. 2016).
\[tb\]
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\[fig:soft-radio\]
{width="0.99\linewidth"}
\[fig:shock-image\]
X-ray and radio observations of M87 chronicle AGN outbursts over the past 150 Myr. The VLA radio observations from Owen, Eilek & Kassim (2000; see also de Gasperin et al. 2012 observations with LOFAR) show evidence for the oldest outbursts (see Fig. \[fig:xray-radio\]b). The two filamented lobes lying NE and SW of the M87 nucleus have ages of $\sim100-150$ Myr. An eastern “mushroom cloud” with stem and torus and a filamentary southwestern arm (Fig. \[fig:xray-radio\]b) have estimated ages of 40-70 Myr. X-ray filaments of cool gas ($\sim1$ keV) are seen coincident with these radio structures (compare Fig. \[fig:xray-radio\]a, b). In addition, there are several less prominent features including 1) a bubble that is separating from the central cocoon (the “bud”) seen in both X-ray and radio images (see Fig. \[fig:xray-radio\]b and Fig. \[fig:shock-image\]b), 2) a possible weak shock at about 5 kpc (about $10^6$ years old; see Fig. \[fig:surbri\]), 3) a series of filamentary structures extending to the east that are likely the remnants of small bubbles (see Fig. \[fig:xray-radio\]a); 4) a large cavity/bubble to the east (beyond the radio torus labeled as “low-frequency bubble” in Fig. \[fig:xray-radio\]a[^1]; and 5) gas sloshing cold fronts at large radii (33 kpc and 90 kpc; see Simionescu et al. 2010 for a detailed discussion).
Recent major outbursts, in the past 20 Myr, are seen in a combination of X-ray and radio imaging and are the focus of the present paper. The key features of these outbursts include:
- a classical shock at 13 kpc ($2.8'$) from the center of M87, seen in X-rays as a nearly complete azimuthal ring (Fig. \[fig:hardimage\] and Fig. \[fig:surbri\]). This was the first classical shock found in the hot gaseous atmosphere around a central cluster galaxy where both the gas density and gas temperature jumps at the shock could be accurately measured. As Forman et al. (2007; see also Forman et al. 2005, Million et al. 2010) showed, the density and temperature jumps are separately consistent with the Rankine-Hugoniot shock jump conditions (Rankine 1870, Hugoniot 1887) for a shock with a Mach number $M\sim1.2$. The age of the outburst giving rise to the shock is about $12\times10^6$ years.
- the plasma-filled, radio-bright cocoon seen as an elongated bubble in the hard X-ray image (diameter $\sim40''$) that served as the piston to drive the 13 kpc shock and is, most likely, now being re-energized by the present, ongoing outburst (see \[section:two\] and Figs. \[fig:xray-radio\]b, \[fig:hardimage\]b, \[fig:cavity-size-cylinder\]; also Hines et al. 1989).
- the prominent jet, observed over a very broad wavelength range, flaring knots, and variable gamma-ray emission (Hines et al. 1989; Owen et al. 2000; Marshall et al. 2002; Harris et al. 2003, 2006; Shi et al. 2007; Forman et al. 2007; Abdo et al. 2009; Acciari et al. 2010).
\[thb\]
![Surface brightness profiles in four energy bands: broad (0.5-3.5 keV), medium (1.0-3.5 keV), soft (0.5-1.0 keV), and hard (3.5-7.5 keV) from top-most to bottom-most. The surface brightness profiles are extracted from a 90[$^{\circ}$]{} azimuth centered on north with point sources excluded and corrected for vignetting and exposure. The three dashed vertical lines indicate the locations of features seen in the pressure maps (Fig. \[fig:hardimage\]). The inner most and outer most lines mark the strongest features and correspond to the current outburst that is re-inflating the central cavity and the 13 kpc shock. The 13 kpc shock is seen in all energy bands, while the central cavity is best seen in the hard band (lowest) surface brightness profile. A third weaker feature (possible shock) is seen at about $1'$ ($\sim 5$ kpc; see also Forman et al. 2007, Million et al. 2010).[]{data-label="fig:surbri"}](f3){width="0.950\linewidth"}
The prominent 13 kpc shock and its associated “piston” provide a unique opportunity to investigate the energy balance between shock heating and heating from buoyant bubbles inflated by AGN outbursts. Fig. \[fig:surbri\] shows the signature of outbursts in the observed surface brightness profiles. Fig. \[fig:deproj\] shows the same signatures in the deprojected density and temperature profiles. Fig. \[fig:deproj\]a is derived from the 360[$^{\circ}$]{}azimuthal average and provides the cleanestestimate of the mean gas density properties, while Fig. \[fig:deproj\]b, a sector centered on North, where the surface brightness profile is least affected by the projection of cool filaments, provides the best estimates for the shock parameters (see Forman et al. 2007 for the derivation of the density and temperature jumps associated with the shock). This “clean” region in Fig. \[fig:deproj\]b shows the pronounced enhancements in both temperature and density at the 13 kpc shock ($2.8'$) and at the outer edge of the piston at $\sim0.65'$ ($\sim3$ kpc).
\[thb\]
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\[fig:deproj\]
We investigate M87’s recent outburst history by using a 1-D numerical shock model to characterize the observed properties including the gas temperature and density profiles. Because the outburst has occurred in the cool atmosphere of M87, compared to hotter atmospheres in more luminous clusters, we are able to derive the observable quantities of the outburst in considerable detail (see Forman et al. 2007, Churazov et al. 2008). By combining a simple model with the high quality observations of M87, we can determine the parameters of the outburst and the energy partition between the shock and the cavity enthalpy and thus help understand the different heating mechanisms required to suppress strong cooling flows in hot atmospheres in galaxies, groups, and clusters.
Cavity Size {#section:cavity-size}
-----------
One of the key constraints on the outburst model comes from the size/volume of the central cavity produced as the relativistic plasma from the jet displaces the hot X-ray emitting gas in the core of M87. The appropriate size to be used is complicated by the fact that the jet is double-sided and inclined to the plane of the sky. As a result, the jet is probably producing two cavities that together make an elongated structure rather than a single spherical cavity.
{width="0.97\linewidth"}
\[fig:cavity-size-cylinder\]
For a proper comparison with the predictions of the 1D model, it is important to estimate the bubble volume in 3D, since the $P V$ work required to displace the X-ray emitting gas is the most direct proxy for the total energetics of the outburst in the model with “gradual” energy release (see Section 2.5 below). To this end, we have approximated the cavity as an inclined cylinder, co-aligned with the jet axis (Fig. \[fig:cavity-size-cylinder\]). Projected on the sky, the cylinder consists of a circular cross section with radius $0.3'$ and height $1.1'$. Inclination angles for the M87 jet range from 10[$^{\circ}$]{}– 20[$^{\circ}$]{}(e.g., Biretta, Sparks & Macchetto 1999, Wang & Zhou 2009). Taking the volume as the geometric mean from the two extreme inclinations and converting this to a sphere gives a spherical volume with a radius of $\sim3$ kpc (equivalent to $0.65'$). For our 1D model, we use this value in our calculations.
The X-ray cavity size matches that of the radio cocoon/bubble (Fig. \[fig:cavity-image\]) and we typically refer to the “cavity” in the discussion of the model.
Simulations of the M87 13 kpc Shock
===================================
Our simulations are carried out in the context of a simple outburst model that captures the key physics. The radio plasma, ejected from the supermassive black hole by the jet, inflates a central cavity, seen as lobes or a cocoon in M87 radio maps (Fig. \[fig:cavity-size-cylinder\], right panel). The inner radio lobes act as a piston that displaces the X-ray emitting plasma. Our results are uncertain due to projection effects arising from the unknown geometry and since we do not know the precise initial conditions of the M87 atmosphere, prior to these SMBH outbursts. Also, we neglect possible effects of diffusive processes on the weak shock (cf. Fabian et al. 2006). However, as we show, the qualitative features of the density and temperature profiles provide a robust characterization of the outbursts.
Numerical Modeling Details
--------------------------
We have performed a sequence of 1D Lagrangian numerical simulations of a shock propagating into the M87 atmosphere where we vary the energy deposited by the outburst and the timescale over which the energy is injected by the central AGN. The M87 atmosphere is assumed to lie in a static gravitational potential, $\phi (r)$, such that the observed gas density and gas temperature distributions (see section \[section:initial\]) are in hydrostatic equilibrium. We assume, for the initial conditions, that the present M87 gas density and temperature are close to those prior to the outburst, i.e., M87’s atmosphere is in a “steady state” with repeated outbursts that are not unusually violent.
\[thb\]
![The gas density and temperature distributions of the fiducial model as a function of time. The shock is initially strong with both the gas density and gas temperature jumps decaying with time. The eight models shown are snapshots taken at 0.023, 0.061, 0.16, 0.41, 1.07, 2.77, 7.18, $18.6\times10^6$ years after the initial outburst. The particular model shown, with an outburst energy of $5.5\times10^{57}$ ergs and a duration of $2.2\times10^6$ yrs, matches 1) the best fit Mach number ($M=1.2$) at the 13 kpc radius of the observed shock and 2) the estimated central cavity (piston) radius of $\sim3$ kpc. Since this model captures the key parameters of the outburst, it is referred to as the fiducial model. The initial conditions are shown as a solid red line (given in equations 1 and 2). The temperature interior to the piston reflects that for the mixture of very hot relativistic plasma that mixes with the small quantity of thermal gas present in the inner pixels of the model when the outburst begins.[]{data-label="fig:fiducial-evolution"}](f6-75.jpg){width="0.950\linewidth"}
We assume that an outburst from a SMBH deposits an energy $E_0$ uniformly over a time interval $\Delta t$. In the inner cells interior to the boundary of the piston (initially 0.2 kpc), the energy is deposited as a power law in radius to mimic the deposition of energy as a jet fills the central cavity (see Xiang et al. 2009 for additional details). For all gas components, we assumed in the actual calculations that $\gamma=5/3$. For a cavity of radius $R$, pressure $P$ (in pressure equilibrium with the ambient gas) and volume $V$, the minimum total energy required to inflate the cavity is $E_{tot}=\gamma/(\gamma-1) P V$. Since the component interior to the piston is at least partially a relativistic plasma, the appropriate $\gamma$ may be smaller and the input energy larger. For $\gamma = 4/3$ and subsonic expansion, $E_{tot}$ would be 60% larger than for $\gamma = 5/3$. We discuss the implications of different values for $\gamma$ in section \[sec:gamma\].
![Initial radial profiles of density and temperature (the initial conditions) as modified by the outburst. The dashed red lines correspond to the initial conditions. The black solid lines show the density and temperature profiles that characterize the “fiducial 1D model” with total energy release $5.5\times 10^{57}$ ergs and outburst duration $2x10^6$ yr, when the shock front is $\sim13$ kpc from the center of the cluster. For the fiducial model (and for “long” outburst models in general), downstream from the shock, the gas temperature is lower than the initial temperature of the gas at the same radius (for reasons described in the text). []{data-label="fig:fiducial-model"}](f7){width="0.95\linewidth"}
Initial Conditions {#section:initial}
------------------
The initial conditions of the hot gas surrounding M87 are a fundamental input to the model. Despite the high quality Chandra X-ray observations, the conditions of the atmosphere surrounding M87, as they appeared more than 10 Myrs ago, prior to the outburst are uncertain, since the gas surrounding M87 has experienced a variety of outbursts (and possibly even small mergers and the associated “gas sloshing”). However, as a dynamically old system with an old stellar population, we assume that the atmosphere around M87 is in quasi-equilibrium and has not undergone any dramatic changes in recent epochs. If, as seems likely, the SMBH in M87 is able to maintain a quasi-equilibrium between heating and radiative cooling, then the present is a “fair” match to the conditions that were present at the time of the outburst. Therefore, for the region interior to $6'$ ($\sim 30$ kpc), we use the observed gas density and temperature distributions to derive the “unperturbed” gas density and gas temperature profiles that are fit to the deprojected data with the simple analytic functions: $$\begin{aligned}
n_e(r) &= 0.22 (1+(r/r_c)^2)^{-3\beta/2} \\
\label{eq:fiducial1}
kT (r) &= 1.55 (1 +(r/r_T)^2)^{0.18}
\label{eq:fiducial2}\end{aligned}$$ where $r_c= 0.2'$ (0.93 kpc), $\beta=0.33$, and $r_T = 2.2'$ (10.2 kpc). These profiles, derived from the full 360[$^{\circ}$]{} azimuthal average profile are shown in Fig. \[fig:deproj\]a and provide the initial baseline for the simulations. Fig. \[fig:deproj\]b shows the initial conditions compared to the observations of the northern sector where the shock is most clearly seen.
{width="0.49\linewidth"} {width="0.49\linewidth"}
{height="0.49\linewidth"} {height="0.49\linewidth"}
\[fig:energy\]
A Shock in the Atmosphere of M87
--------------------------------
Applying our shock model to the initial conditions described above, we can examine a typical outburst. Our “fiducial” model with total outburst energy and outburst duration $E_{tot}=5.5\times10^{57}$ ergs and $\Delta t=2$ Myr has a temporal evolution shown in Fig. \[fig:fiducial-evolution\]. This temporal evolution is characteristic of all the models. The initial shock weakens with time, because of energy dissipation at the front at early phases, when the shock is still strong, and undergoes pure spherical expansion at later phases. In the last snapshots, the shock is expanding at Mach $M=1.2$ with amplitudes, in both density and temperature, that match the observations. The expansion of the central cavity (the piston) “stalls” at the present observed piston radius of about 3 kpc. In fact, the inertia of the accelerated gas ahead of the piston carries it beyond the pressure equilibrium radius and the piston radius subsequently decreases slightly in the last time steps. This effect also is seen in the 3D simulation that we used to confirm the validity of our 1D models (described in section \[sec:3d\]), but the effect is less pronounced. The final configuration, as we show below, matches the observations and for this reason, the outburst with $E_0 = 5.5\times10^{57}$ ergs and $\Delta t = 2 \times 10^6$ yrs is referred to as the fiducial model.
Fig. \[fig:fiducial-model\] shows the same fiducial model at the moment when the shock front reaches 13 kpc, corresponding to the observed shock radius. For the fiducial model (and for “long” outburst models in general), downstream from the shock, the gas temperature is lower than the initial temperature of the gas at the same radius. This is due to a combination of two effects. First, the rarefaction region behind a shock is a generic feature of weak spherical shocks (as described by Zeldovich & Razier 2002 and Landau & Lifshitz 1959). Second, in these models, the adiabatic expansion of the gas that is displaced from its initial location to lower pressure regions (larger radii) contributes to the temperature decrease. These features can be identified in many of the figures in this paper.
The lack of perfect spherical symmetry, the presence of cool structures (arms), and the uncertainty in the initial conditions complicate any detailed, quantitative comparison of the model and data. However, a qualitative (“factor of 2”) comparison is possible. Since the wedge to the North is less contaminated by cool structures, except for the inner $45''$, we used the deprojected emissivities in the 0.5-3.5 and 3.5-7.5 keV bands for comparison with the model predictions (see Fig. \[fig:model-data\]). The emissivity in these two energy bands was calculated using the predicted density and temperature profiles assuming fixed solar metalicity. For the models shown in Fig. \[fig:model-data\], the fiducial model captures the key parameters measured for the M87 outburst and matches the size of the central cavity, the observed radius and strength of the shock (in both density and temperature), and the emissivity outside the central cavity.
As noted above, none of the 1D models provides a “perfect fit” to the data over the entire radial range. This is especially true for the innermost part, where the 1D model predicts the complete evacuation of the gas as it is pushed away by a spherical piston. In a real cluster, the gas is expected to be evacuated only from regions occupied by the cavities (the radio plasma), while the thermal gas can still be present along other directions. This is why we will compare the size of the cavity predicted by the 1D simulations to that derived in section \[section:cavity-size\], rather than directly comparing the predicted and observed profiles. For the shock front region, which is farther away from the center, the effects of asymmetry should be less severe and the direct comparison of the radial profiles is better justified.
\[thb\]
{width="0.95\linewidth"}
Effects of Outburst Energy and Outburst Duration
------------------------------------------------
To explore the range of allowed outburst parameters, we separately investigate the effects of varying the outburst energy and outburst duration. These two parameters govern the final outburst configuration.
For a given outburst duration, the outburst energy strongly affects the amplitude of the shock. Fig. \[fig:energy\]a shows the gas density and temperature when the shock reaches 13 kpc, for outburst energies of $1.4, 5.5, 22 \times 10^{57}$ ergs. The choice of outburst energy brackets the energy described above as the fiducial value. As Fig. \[fig:energy\]a shows, the amplitude of the shock alone provides a direct diagnostic of the outburst energy. Also, note that the different values of the outburst energy yield different sizes for the central piston – larger energy outbursts drive stronger shocks that reach 13 kpc in a shorter time and have larger central cavities of relativistic plasma.
We also have investigated the effects of varying the outburst duration. Fig. \[fig:duration\]b shows the gas temperature and gas density profiles for models with the outburst energy held fixed at $E_0 = 5.5 \times 10^{57}$ ergs and with outburst durations ranging from $0.1$ to $6.2 \times 10^{6}$ yrs. While the amplitude of the shock at 13 kpc varies only slightly, the size of the piston varies dramatically. The models show the characteristic behavior of “short” and “long” duration outbursts. As we show below, by matching the observations to the models in more detail, we can estimate a quantitative value for the outburst duration. Also, as Fig. \[fig:duration\]b shows, a “short” duration outburst produces a central region with $\sim2-3$ kpc radius), starting just beyond the outer boundary of the piston, that consists of hot, low density gas. In contrast, the longer duration, initially weaker shocks, with the same total outburst energy, are bounded by cool shells and have no extended hot, shocked region (see also Brighenti & Mathews 2002).
Thus, the combination of Fig. \[fig:energy\]a and \[fig:duration\]b shows that the outburst energy is determined (primarily) by the magnitude of the jumps.
\[t\]
{width="0.49\linewidth"} {width="0.49\linewidth"}
\[fig:pjump\]
Short and Long Duration Outbursts
---------------------------------
To further illustrate the principles that drive the models described here and how the duration of the outburst affects the appearance of the hot corona, we select two examples that illustrate the effects of the outburst duration on the properties of M87 – a short duration outburst and a longer duration outburst. The short duration outburst has a duration $\Delta t = 0.1\times10^6$ yrs while the longer duration outburst has $\Delta t = 2.2\times10^6$ yrs (the blue and black curves in Fig. \[fig:duration\]b). Fig. \[fig:models-schematic\] shows graphically the dramatic difference that may arise from the two different duration outbursts.
Quantitatively, the different characters of the short and long outbursts are shown in Fig. \[fig:slow-fast\]a where we label the different regions that characterize the different types of outbursts. We show the gas density and gas temperature profiles of the $0.1\times10^6$ yr duration outburst (blue) and the fiducial $2.2\times10^6$ yr duration outburst (black). We have labeled the key regions – the piston, the hot, low density shocked envelopes (blue text) for the short duration outburst and the piston and the cooler, denser envelope for the fiducial duration outburst (black text). Although the physics of the outbursts are identical, the duration imprints a qualitatively different signature on the surrounding atmosphere with quite different over-pressures and Mach numbers as a function of time (see Fig. \[fig:pjump\]b). For a given shock strength, the longer outburst produces a larger cavity, by a factor of three in volume, that can be used as a proxy for the outburst duration.
The models are shown at the time when the modeled shock reaches 13 kpc. For the two example outbursts (0.1 and 2.2 Myr durations) considered in Fig. \[fig:slow-fast\], the outburst ages (time for the shock to reach 13 kpc) change by only about 10% (11 vs. 12 Myr for the 0.1 and 2.2 Myr durations). Despite the large difference in initial Mach number (Fig.\[fig:pjump\]b) for the outburst energy ($5.5\times 10^{57}$) that yields density and temperature jumps consistent with the observations, the age is dominated by the late phases as the shock approaches 13 kpc.
Also, longer outbursts could be characterized by the absence of a hot, low density envelope around the central cavity that is filled with relativistic plasma. Such an envelope, characteristic of short outburst models, is formed by the gas that has passed through the strong shock. The lack of such an envelope in the data is consistent with the “long outburst scenario”. Whether it can be used as a strong argument against the short outburst model depends on the efficiency of thermal conduction in the gas, which is an open issue.
The Fiducial Model - a single outburst model for the 13 kpc shock {#sec:gamma}
-----------------------------------------------------------------
To quantitatively bound the family of outburst parameters, we examine an ensemble of shock models where we have varied the outburst energy and duration. As described above, we first simulate the primary outburst that produced the 13 kpc shock and assume, for this initial comparison of observations to models, that this is the only outburst that affects the inner 13 kpc of M87. Our outburst model is characterized by two key outburst parameters - the duration (we assume constant power during the outburst event) and the total energy deposited.
The parameters we must match are (a) the shock jump conditions which, as noted above, primarily determine the total outburst energy ($E_{tot}$), and (b) the radius ($r_p$) of the radio cocoon, the piston driving the shock, and (c) the observed radius of the 13 kpc shock. We could use either the temperature jump or the density jump to constrain the model. The density jump is statistically more accurate but has a systematic uncertainty associated with the steep density gradient arising from the “cool core” atmosphere surrounding M87. The temperature jump is less accurate statistically but may provide a more realistic measure of the uncertainties inherent in the complex atmosphere of M87. Mach numbers derived for the density and temperature jumps are fully consistent (see Forman et al. 2007). For the purpose of constraining the model parameters, we choose the less constraining temperature jump to better allow for the systematic uncertainties.
Fig. \[fig:energy-duration\] is a grid of models for two parameters – the outburst energy, $E_{tot}$ and the outburst duration, $\Delta t$. Loci of equal shock temperature jump (blue) and equal cavity size (red) are drawn. The value of the gas temperature jump is $kT_{shock}/kT_{initial} = 1.18\pm0.03$ (Forman et al. 2007). The second constraint arises from the size of the central cavity, the piston. We identify the piston with the central radio cocoon which is labeled in the X-ray image shown in Fig. \[fig:models-schematic\]b (central panel) as well as in Fig. \[fig:xray-radio\]b and \[fig:shock-image\]b.
As noted above and shown in Fig. \[fig:energy-duration\], for outburst durations less than about 3 Myr, the outburst energy is [*independent*]{} of outburst duration (i.e., the loci of equal density jumps are nearly vertical). For durations longer than 3 Myr, the acceptable range of energies does depend on the outburst duration. The second constraint, the radius of the central cocoon, is derived from the X-ray and radio images (see Fig. \[fig:cavity-size-cylinder\] as discussed in section \[section:cavity-size\]). The intersection of the radius and density constraints indicates the most probable locus of points of (energy,duration) for the outburst. The center of this region is $E_{tot}\sim5.5\times10^{57}$ ergs and $\Delta t\sim2$ Myr.
![ A grid of models as a function of outburst energy and outburst duration, for a one dimensional outburst model. The model parameters are taken at the time when the modeled shock reaches 13 kpc, the radius of the observed shock. Within this grid of models, we draw lines of constant temperature jump ($kT_{shock}/kT_{initial}$; blue solid lines) and constant piston size (radio cocoon; red solid lines). The values of the temperature jump and piston size are labeled along the top axis of the figure in the corresponding color. The green region indicates the intersection between regions defined by $kT_{shock}/kT_{initial} = 1.18\pm0.03$ and cavity size of $3\pm0.5$ kpc. []{data-label="fig:energy-duration"}](f12.pdf){height="0.95\linewidth"}
\[fig:grid\]
---------------------------------- ---------------
Outburst Age (Myr) 12
Outburst Duration (Myr) $\sim2$
Outburst Energy ($10^{57}$ ergs) $\sim5$
Energy carried by shock $\lesssim$22%
Thermal energy in cavity $\sim$27%
Change in gravitational energy $\sim$40%
Energy in shock heated gas $\sim$11%
Energy available for heating $\sim$80%
---------------------------------- ---------------
: Fiducial Outburst Model in M87
With the known properties of the surrounding atmosphere and the derived outburst details, we can compute the present epoch energy partition arising from the outburst (Table 1; see also Tang & Churazov (2017) who ran a set of models with varying durations and energetics in a homogeneous medium to determine the energy partition and then mapped the results to more realistic density/temperature profiles.) For the fiducial outburst of $5.5\times10^{57}$ ergs, approximately 11% of the energy resides in the kinetic energy of the shock (and a comparable amount in the thermal energy of the shock, since the shock is weak) that can be carried away from the central region to larger radii since the shock is now relatively weak. At least 50% (and as much as 64%) of the energy is contained in the enthalpy of the central cavity/piston, and about 11% of the energy has been transformed into heating the gas as the shock moved outward to its present position. In summary, in the fiducial model, about 30% of the outburst energy is deposited in the shock. In the model, about 10% of this energy has already been dissipated into heat as the shock traversed the region interior to its present 13 kpc location.
Our 1D simulations assume the adiabatic index $\gamma_g=5/3$ for the gas inside and outside the “piston”. If, in fact, the energy density inside the piston is dominated by relativistic plasma with $\gamma_r=4/3$, the thermal energy inside the cavity $\sim \frac{1}{(\gamma-1)}$ has to be increased by the factor $\frac{\gamma_g-1}{\gamma_r-1}=2$ (see Table 1), while keeping all characteristics of the gas outside the piston unchanged. This would correspond to a moderate increase of the total energy, required to inflate the bubble, and also a reduction in the fraction of energy that goes into the initial shock.
[**Enthalpy of the Central Bubble –**]{} The central bubble, the radio-emitting cocoon, contains a large fraction of the total outburst energy. Much of the enthalpy in a central bubble is available for heating of the central region where radiative cooling is important (e.g., see Churazov et al. 2001, 2002; see also Nulsen et al. 2007). The fractional energy, $f$, retained by the buoyantly rising bubble with adiabatic index $\gamma$, is given as $f=(p_1/p_0)^{(\gamma-1)/\gamma}$ as the pressure changes from $p_0$ to $p_1$. For a relativistic plasma bubble, $\gamma=4/3$ and for a non-relativistic plasma, $\gamma=5/3$. Fig. \[fig:enthalpy\] shows the energy retained by a rising bubble in M87’s atmosphere using the fitted density and temperature profiles given in equations 1) and 2). The enthalpy of the buoyant cocoon is dissipated into a variety of forms including internal waves, sound waves, turbulent motion in the wake of the bubble, potential energy of uplifted (cool) gas, and large scale bulk flows. While sound waves can carry energy away from the central region, most other channels would eventually result in heating the central region (see Churazov et al. 2001 for a more detailed discussion on the containment of SMBH outburst energy in the core region). As Fig. \[fig:enthalpy\] shows, a buoyant bubble rising to about 20 kpc in M87’s atmosphere would lose about 50% of its enthalpy. This energy will eventually be dissipated into heat on a time scale that depends on the plasma microphysics.
![For a buoyantly rising bubble, the fractional enthalpy loss for a plasma with adiabatic indices of 5/3 and 4/3 (upper and lower curves, respectively). Buoyant plasma bubbles rising from the galaxy center to about 20 kpc, would lose approximately 50% of their initial enthalpy which would ultimately be converted into thermal energy of the X-ray emitting plasma on a timescale that depends on the plasma microphysics.[]{data-label="fig:enthalpy"}](f13){height="0.95\linewidth"}
Multiple Outbursts {#section:two}
------------------
\[sec:two\]
The outburst that generated the 13 kpc shock is likely not the most recent one from M87’s SMBH. As the hard band images Fig. \[fig:xray-radio\]b and \[fig:shock-image\]b show, there is a surface brightness enhancement surrounding the radio cocoon (the central bubble) indicating that the cocoon is an overpressurized region which is being driven by the current outburst we see in M87 – that also drives the existing jet.[^2] To understand the effects of the more recent outburst on our derived shock parameters, we add a second ongoing outburst at the present epoch to provide the observed overpressure within the central cocoon.
![Comparison of models with one and two outbursts. We compare two single outburst models with energies of $E_{tot}=5.5\times10^{57}$ ergs but with long (2.2 Myr) and short (0.4 Myr) durations to models with a second outburst of cumulative energy $2\times10^{57}$ ergs and duration 1 Myr that is still ongoing. The models, with differing ages, are shown at the times when the modeled shock reaches 13 kpc, the radius of the observed shock. The single outburst models are shown with dashed lines. The outburst models with a short primary outburst are in magenta and those with long outbursts are in black. The addition of a second outburst injects energy into the existing cavity and makes only a small change to the predicted profiles. To match the observations, the age of the main outburst must be reduced to 11 Myr, while its energy is unchanged. []{data-label="fig:multiple"}](f14-small){width="0.95\linewidth"}
The current (ongoing) outburst has an energy (up to the present) of $2\times10^{57}$ ergs (determined by the weak density jump at $\sim3$ kpc) and a duration of about 1 Myrs. If we include this recent outburst, the age of the main outburst that produced the 13 kpc shock is reduced, since the cavity size is slightly increased by the current outburst. The presence of a second outburst reduces the [outburst age in]{} the fiducial model by about 10% to 11 Myrs.
With the above set of parameters, we find the gas density and gas temperature profile shown in Fig. \[fig:multiple\]. The figure shows the central hot, low density cocoon which acts as the piston. Just beyond the piston is the over-pressurized shell extending to about 4 kpc. In our simple one dimensional model, the presence of an existing cavity at the onset of the second outburst reduces the effects of an initial short period of strong shock heating that might otherwise be present at the beginning of the second outburst. In 3D, if given sufficient time between outbursts, the second outburst will encounter a denser environment as the low density plasma rises buoyantly and is displaced by denser plasma. For short intervals between outbursts, subsequent outbursts will have the effects of their initial expansion mitigated by residual, low density plasma.
The Central Piston
------------------
The only large cavity that is seen interior to the 13 kpc shock is the central cavity, the radio cocoon (Fig. \[fig:xray-radio\]b and \[fig:shock-image\]b). Hence, this $\sim3$ kpc bubble (equivalent 1D size of the 3D bubble) of relativistic plasma must be the piston that drove the 13 kpc shock. However, since the relativistic plasma is buoyant, it will tend to bifurcate into a dumbbell shape and each half will buoyantly rise. Is the presently observed cavity surrounding the M87 nucleus and the jet consistent with having been created about $\sim11$ Myr ago when the shock, presently seen at 13 kpc, was first created? Churazov et al. (2001) simulated the rise of buoyant bubbles in the M87/Virgo system. They found a buoyant velocity over a wide range of radii of about half the sound speed, $v_b\sim c_s/2$. In the M87 core, the gas temperature is about 1 keV, giving a terminal buoyant velocity of about 250 [ km s$^{-1}$]{}. Over a time $t_b\sim11-12$ Myr, the age of the 13 kpc shock, the initial bubble will be pinched, form an elongated (possibly dumbbell-like) shape and rise buoyantly to a distance $d_b \sim v_b
t_b$. The bubble system, at present, would therefore have dimensions $\sim3 \times 8$ kpc, consistent with the highly inclined jet angle with respect to the line of sight (Biretta et al. 1999, Wang & Zhou 2009) and consistent with the 3D simulations presented in the next section (see Figs. \[fig:3d\_map\] and \[fig:3d\_profiles\]).
3D Model of the AGN Outburst {#sec:3d_simulation}
----------------------------
\[sec:3d\]
To test the sensitivity of our results to the simplifying assumption of spherical symmetry, we performed 3D jet simulations to replicate approximately the setup used in our 1D calculations.
Our simulations include a jet driven at a power of $L_{\rm jet} = 1.2\times 10^{44}\,{\rm ergs\,s^{-1}}$ for $\Delta t_{\rm jet} = 2\times 10^{6}\,{\rm yrs}$ into a $\beta$- model atmosphere. Simulations were performed using the FLASH2.4 hydro code, using the PPM solver (Fryxell et al. 2000), and following the same setup described in Heinz et al. (2006) and Morsony et al. (2010). Simulations were run with a central resolution of 50 pc and used AMR to focus computational resources on the volume around the jet axis.
The simulations inject two oppositely directed jets, with the jet axis random-walking within a cone of half-opening angle of ten degrees, following the so-called dentist drill model (Scheuer 1982).
Consistent with the general model employed in this paper, the expanding lobes excavate two cavities that drive an elliptical shock, the semi-major axis of which is aligned with the mean jet direction. The radio cocoon structure has a reasonable shape compared to typical central cluster radio sources, with an aspect ratio of approximately 3:1.
Simulations were run until the semi-minor axis of the shock reached the measured shock size in M87 of 13 kpc. A density slice through the jet axis is shown in Fig. \[fig:3d\_map\], showing the under-dense radio lobes and the shock. The aspect ratio of the shock is approximately 1.3:1.
\[h\] ![Density slice through 3D simulation of the jet-driven shock in M87 (see §\[sec:3d\_simulation\] for details of the setup). []{data-label="fig:3d_map"}](f15.png "fig:"){width="0.95\linewidth"}
The jet viewing angle in M87 is likely close to the line of sight (e.g., Biretta et al. 1999, Wang & Zhou 2009). Thus the elongation of the shock is likely hidden by projection. It is therefore appropriate to use measurements of the shock properties along the semi-minor axis in the simulation for comparisons with observations and the 1D models. Because an elongated shock requires a larger energy (roughly by the aspect ratio of 1.3) compared to a spherical shock, we used a larger total injected energy of $E_{\rm jet} = L_{\rm jet}\Delta
t_{\rm jet} = 7.4\times 10^{57}\,{\rm ergs} = 1.3\times E_{1D}$.
The radial density and temperature profiles along the semi-minor axis of the shock are plotted in Fig. \[fig:3d\_profiles\]. Outward of the 1D piston location, they agree well with the profiles plotted in Fig. 5 for our fiducial 1D model. In particular, the density and temperature jumps at the shock agree well with the 1D model, supporting the use of these measurements as observational diagnostics. The simulations also show a low-temperature post-shock region between 3.5 kpc and the shock, as predicted by the 1D fiducial model. Furthermore, the 3D model reproduces the distinguishing characteristic of the piston-driven expansion: the absence of a large increase in temperature outside the piston (and interior to the shock) that would be produced by impulsive (instantaneous) energy injection in a Sedov-like mode. We note again that this relies on the assumption that thermal conduction is negligible.
![Density and temperature profiles along the semi-minor axis of the shock in the 3D simulations shown in Fig. \[fig:3d\_map\]. The “edge” feature seen at $\sim4$ kpc is produced by emission from gas that has refilled the volume (visible as light region in Fig. \[fig:3d\_map\]) behind the expanding piston (visible as a dark region in Fig. \[fig:3d\_map\]). The depression in density that extends to $\sim8$ kpc arises from the lower density plasma in the expanding cocoon along the line of sight. The decrease in temperature behind the shock is the characteristic of weak shocks discussed in Fig. \[fig:fiducial-model\].[]{data-label="fig:3d_profiles"}](f16-adobe-eps){width="0.95\linewidth"}
The main difference between our 1D and 3D models is the presence of [*two*]{} off-center pistons in the 3D case, which leads to the elongation of the shock. The central region in the simulation is re-filled by gas that falls back and generates a new (slightly hotter) core after the jet is turned off.
It is straightforward to understand why the 1D model is so successful in reproducing the properties of the 3D simulations in the direction perpendicular to the jet: The lateral expansion of a cocoon-driven shock is energy (i.e., pressure) driven, while the initial longitudinal expansion of the shock is driven by the momentum of the jet, as argued by Begelman & Cioffi (1989). The only correction required between 1D and 3D is therefore the total volume of the shock, which increases the energy (by about 30%) required to drive a shock to a given semi-minor axis by the shock aspect ratio, as confirmed by our simulations.
Conclusion
==========
Hot gaseous atmospheres are ideal tracers of major events in the evolution of brightest cluster (or group) galaxies (BCGs), their central supermassive black holes (SMBHs), and their dark matter halos. In addition to evidence of outbursts, X-ray images and temperature maps provide constraints on gas mixing from mergers through shocks, cold fronts, and “gas sloshing” (e.g., Markevitch & Vikhlinin 2007; Markevitch, Vikhlinin & Mazzotta 2001, Markevitch et al. 2002; Vikhlinin, Markevitch & Murray 2001; Johnson et al. 2010). Abundance distributions also show evidence for gas motions and merging events (e.g., Rebusco et al. 2005, 2006; Xiang et al. 2009; Simionescu et al. 2010). Another ICM tracer of the dynamic history is encoded in the X-ray surface brightness fluctuations (Churazov et al. 2012, Zhuravleva et al. 2014). For M87, the obvious outburst history extends over about 100 Myrs. Our discussion above has concentrated on the outburst that produced the nearly circular shock at 13 kpc and the central “bubble” whose inflation drove the shock into the surrounding atmosphere. The relatively simple geometry of the system provided the opportunity to explore the details of the outburst and yielded quantitative estimates of the outburst properties including its age, $\tau\sim11-12$ Myrs, its energy, $E_{tot}\sim5-6\times10^{57}$ ergs, and duration, $\Delta_t\sim1-3$ Myrs. In addition, we are able to estimate the present epoch energy partition with about 80% of the energy available for heating the gas and about 20% carried away, beyond 13 kpc, by the shock as it weakens to a sound wave (see Table 1). Thus, during the outburst, in the fiducial model, about 30% of the outburst energy is deposited in the shock. In this model, $\sim$ 10% of this energy has already been dissipated into heat as the shock traversed the region interior to its present 13 kpc location, while the remaining $\sim 20$% is carried to larger radii. For M87, a large fraction of the outburst energy resides in the central bubble enthalpy. As Churazov et al. (2001, 2002) argued, the bulk of this energy is converted to thermal energy of the X-ray emitting gas in the central region surrounding M87.
In the context of our simple model, we also are able to estimate the properties of the current, ongoing outburst that has only slightly altered the signature left by the preceding outburst. The signature of the current outburst is consistent with having begun about 1 Myr ago and having injected $2\times10^{57}$ ergs into the preexisting cavity. As noted above, the $\sim11-12$ Myr old outburst inflated a cavity that is now elongated, at least partially by buoyancy. While the exact values describing the M87 outbursts are uncertain, with the outburst energy somewhat larger than the 1D model predicts (see section \[sec:3d\_simulation\] and the discussion of the 3D model), the qualitative description of a “slow” (few Myr) outburst remains valid and is confirmed by the more realistic 3D model.
M87 provides a view of a “typical” outburst from a low-Eddington rate accretor with the bulk of the energy liberated as mechanical, rather than radiative, energy. Considerable attention has been given to the very energetic outbursts in luminous clusters (e.g., Nulsen et al. 2005) and to some of the spectacular outbursts in groups (e.g., NGC5813; Randall et al. 2011). However, luminous early type galaxies also have hot coronae (Forman, Jones & Tucker 1985) and, like their more luminous cousins, also harbor mini-“cooling cores”. In the absence of any heating, these systems would have mass deposition rates up to a few solar masses per year (Thomas et al. 1986) and yet they host very little star formation and remain “red and dead” (e.g., Hogg et al. 2002). Outbursts very similar to those discussed for M87 are also present in these systems. NGC4636 (Jones et al. 2002, Baldi et al. 2009), M84 (Finoguenov et al. 2008), and NGC4552 (Machacek et al. 2006) are representative examples of this class.
There are a variety of energy sources suitable for replenishing the radiated energy from the hot gas in galaxy cluster cores. Two of the most prominent are mergers and AGN outbursts which drive gas motions. In M87, we see effects of both processes, e.g., a) ongoing mergers such as M86 (Forman et al. 1979, Randall et al. 2008) and gas sloshing of the entire Virgo core (Simionescu et al. 2010) and b) AGN outbursts from M87 as we have discussed in detail above that inflate buoyant bubbles. In the context of gas sloshing, ZuHone et al. (2010) have discussed the mixing of hotter gas from larger radii with cooler gas from the central regions of the cluster (or galaxy). The mechanism for converting the bulk motions to heat has only recently been probed. Zhuravleva et al. (2014) argued that gas motions, that arise from both merging and SMBH feedback (primarily, motions driven by the rise of buoyant plasma bubbles as discussed for M87 above), are very likely converted to thermal energy via dissipation of turbulence. The turbulent heating inferred for M87 (and Perseus) is sufficient to balance the radiative cooling. Hence, we can now begin to quantitatively understand the feedback process and conversion of gas motions to thermal energy of the gaseous atmosphere.
The outbursts from M87 are characteristic of radiatively inefficient accretion (e.g., Ichimaru 1977, Rees et al. 1982, Narayan & Yi 1994, Abramowicz et al. 1995, Blandford & Begelman 1999, Yuan & Narayan 2014). Early-type galaxy evolution models that include both radiative and mechanical feedback have been explored extensively. Pellegrini, Ciotti & Ostriker (2012, and references therein) have modeled the evolution of isolated early type galaxies over cosmological times. They find episodic outbursts with high quasar-like radiative luminosities ($\sim10^{46}$ [ ergs s$^{-1}$]{}) at early epochs. M87, and most present epoch early-type galaxies, lie in richer environments (cluster or group centers or cluster cores). Although the gas environment is much richer, present epoch early-type galaxies appear to have more moderate outbursts than those at earlier epochs. Future X-ray missions will be able to study the detailed properties of outbursts and probe the conversion of bulk motions to thermal energy (e.g., Croston et al. 2013, Vikhlinin 2012). The ability to probe to high redshift with arc second angular resolution (Vikhlinin 2012 see section 3.1 and Fig. 3) could fully test models of galaxy evolution and the impact of the SMBHs that lie in their nuclei, by tracing the evolution of both the AGN and the surrounding hot gaseous atmosphere and deriving properties (luminosity, temperature, density profile, and abundance) from redshifts of $z\sim6$ to the present.
This work was supported by contracts NAS8-38248, NAS8-01130, NAS8-03060, the Chandra Science Center, the Smithsonian Institution, the Institute for Space Research (Moscow) and Max Planck Institute für Astrophysik (Munich). S.H. acknowledges support through NSF grant AST 1109347. We thank the anonymous referee whose comments significantly improved the paper.
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[^1]: This large cavity/bubble is very clearly detected in the LOFAR images just to the north of the torus, see Figs. 7 and 8 in de Gasperin et al. 2012)
[^2]: We note that there is an indication of a third weak intermediate age outburst with a surface brightness enhancement at $\sim1'$ (see Fig. \[fig:surbri\]) but we have not modeled this weak feature.
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Bose-Einstein condensates (BEC) in dilute systems of trapped neutral atoms [@BEC; @RMP] have offered a new fascinating testing ground for some basic concepts in elementary quantum mechanics and quantum many-body theory as well as for searching new macroscopic quantum coherent phenomena. In particular, much interest has been focused on the possibility of coherent atomic tunneling between two trapped BEC [@Java; @Smerzi; @Leggett; @laser-assist; @Chen-MPLB; @Leggett-RMP], which is analogous to the superconducting Josephson effect.
Recent years have witnessed the rapid development of quantum information theory. One of the central results there is the possibility of quantum computation [@quan-inf; @Nature]. In quantum computers, an elementary building block is a set of two-level systems (“qubits”) storing quantum information. Due to the superposition of quantum information states in qubits, a quantum computer can provide fundamentally more powerful computational ability than its classical counterpart. So far, many systems have been proposed to perform quantum logic, including atoms in traps [@CZ], cavity quantum-electrodynamical systems [@CQED], the nuclear magnetic resonance system [@NMR], and solid state systems (e.g., quantum dots [@dot], nuclear spins of atoms in doped silicon devices [@Kane], and ultrasmall Josephson junctions [@Shnirman; @Averin; @Makhlin; @Ioffe; @eprint]). Recently it has been demonstrated that a single Cooper pair box is a macroscopic two-level system that can be coherently controlled [@box-PRL; @box], thus realizing a [*Josephson charge qubit*]{} [@Shnirman; @Averin; @Makhlin; @eprint]. Making use of superconducting quantum interference loops containing ultrasmall Josephson junctions, the [*Josephson flux qubit*]{} has also been proposed [@Ioffe; @eprint; @Mooij]. Two recent experiments indeed realized the quantum superposition of macroscopic persistent-current states, and demonstrated the small Josephson junction loops as macroscopic quantum two-level systems [@JJcat].
The implementation of qubits in a physical systems is the first step for quantum computing. In this paper we demonstrate that two coupled BEC at zero temperature can be used to realize a qubit. This BEC qubit is the counterpart of the Josephson charge qubits. The two BEC are weakly linked and confined in an asymmetric double-well trap. Typically, the system may display the phenomenon known as quantum coherent tunneling of atoms, or Josephson effect [@Java; @Smerzi; @Leggett; @Chen-MPLB]. However, under the conditions that the “charging energy” of the system is much larger than the Josephson energy and the system is biased near a degeneracy point, the two BEC may represent a qubit with two states differing only by one atom.
The effective many-body Hamiltonian describing atomic BEC in a double-well trapping potential, $V_{trap}({\bf r})$, can be written in the second-quantization form as $$\begin{aligned}
\hat H_{BEC} &=&\int d^3r\hat \psi ^{\dagger }\left[ -\frac{\hbar ^2}{2m}%
\nabla ^2+V_{trap}\right] \hat \psi \nonumber \\
&&\ +\frac{U_0}2\int d^3r\hat \psi ^{\dagger }\hat \psi ^{\dagger }\hat \psi
\hat \psi , \label{ham}\end{aligned}$$ where $m$ is the atomic mass, $U_0=4\pi a\hbar ^2/m$ (with $a$ denoting the $%
s$-wave scattering length) measures the strength of the two-body interaction, and the Heisenberg atomic field operators $\hat \psi ^{\dagger
}({\bf r},t)$ and $\hat \psi ({\bf r},t)$ satisfy the standard bosonic commutation relation $[\hat \psi ({\bf r},t),\hat \psi ^{\dagger }({\bf r}%
^{\prime },t)]=\delta ({\bf r}-{\bf r}^{\prime })$. One can properly choose the trapping potential so that the two lowest states are closely spaced and well separated from the higher energy levels, and that many-body interactions only produce negligibly small modifications of the ground-state properties of the individual wells [@two-mode]. The potential has two minima at ${\bf r}_1$ and ${\bf r}_2$, and can be expanded around each minimum as $$\begin{aligned}
V_{trap}({\bf r}) &=&\sum_{i=1,2}\left[ V_i^{(0)}({\bf r}_i)+V_i^{(2)}({\bf r%
}-{\bf r}_i)\right] +\cdots \nonumber \label{trap} \\
&\equiv &V_1+V_2+\cdots , \label{trap}\end{aligned}$$ where the two constant potentials $V_i^{(0)}({\bf r}_i)$ can be set to zero without loss of generality.
With the above assumptions, one can use a two-mode approximation to the many-body description of the system [@two-mode]. Thus instead of the standard mode expansion over single-particle states, one can approximately expand the field operators $\hat \psi ({\bf r},t)$ in terms of two local modes [@Smerzi; @two-mode] $$\hat \psi ({\bf r},t)\approx \sum_{i=1,2}\hat a_i(t)\Psi _i({\bf r}-{\bf r}%
_i), \label{expan}$$ where $[\hat a_i(t),\hat a_j^{\dagger }(t)]=\delta _{ij}$, and $\Psi _i({\bf %
r}-{\bf r}_i)\equiv \Psi _i({\bf r})$ are two local mode functions (with energies $E_i^0$) of the individual wells and satisfy $$\int d^3r\Psi _i^{*}({\bf r})\Psi _j({\bf r})\approx \delta _{ij}.
\label{ortho}$$ Substituting Eq. (\[expan\]) into the effective Hamiltonian (\[ham\]) yields the two-mode approximation of $\hat H_{BEC}$: $$\begin{aligned}
\hat H &=&\sum_{i=1,2}(E_i^0\hat a_i^{\dagger }\hat a_i+\lambda _i\hat a%
_i^{\dagger }\hat a_i^{\dagger }\hat a_i\hat a_i) \nonumber \\
&&\ \ -(J\hat a_1^{\dagger }\hat a_2+J^{*}\hat a_1\hat a_2^{\dagger }),
\label{h2}\end{aligned}$$ where the parameters are estimated by [@Smerzi; @two-mode] $$\begin{aligned}
E_i^0 &=&\int d^3r\Psi _i^{*}\left[ -\frac{\hbar ^2}{2m}\nabla
^2+V_{trap}\right] \Psi _i, \label{e7} \\
\lambda _i &=&\frac{U_0}2\int d^3r\left| \Psi _i\right| ^4, \label{e8} \\
J &=&-\int d^3r\left[ \frac{\hbar ^2}{2m}{\bf \nabla }\Psi _1^{*}\cdot {\bf %
\nabla }\Psi _2+V_{trap}\Psi _1^{*}\Psi _2\right] . \label{e9}\end{aligned}$$ Without loss of generality, $J$ can be regarded a real number. Notice that interactions between atoms in different wells are neglected in Eq. (\[h2\]) [@two-mode].
Defining two local number operators $\hat n_i=\hat a_i^{\dagger }\hat a_i$, then it is easy to verify that the total number operator $\hat a_1^{\dagger }%
\hat a_1+\hat a_2^{\dagger }\hat a_2=\hat N$ of atoms represents a conservative quantity and thus $\hat N=N_t\equiv 2N=const$. After neglecting the constant term, the two-mode Hamiltonian $\hat H$ can be rewritten as $$\begin{aligned}
\hat H^{\prime } &=&E_c(\hat n-n_g)^2-J(\hat a_1^{\dagger }\hat a_2+\hat a_1%
\hat a_2^{\dagger }), \nonumber \\
\hat n &\equiv &(\hat n_1-\hat n_2)/2,\;\;E_c=\lambda _1+\lambda _2,
\nonumber \\
n_g &=&\frac 1{2E_c}[(E_2^0-E_1^0)+(N_t-1)(\lambda _2-\lambda _1)],
\label{h22}\end{aligned}$$ where $\hat n$ is the number difference operator, $E_c$ is the “charging energy” and $n_g$ acts as a control parameter and is known as the “gate charge” in other context [@eprint]. Due to the conservation of total atom number, the problem can be restricted to the subspace of definite $N$, i.e., $$\begin{aligned}
\left| n_1,n_2\right\rangle &=&\left| N+n,N-n\right\rangle \equiv \left|
n\right\rangle , \nonumber \\
\hat N\left| n\right\rangle &=&2N\left| n\right\rangle ,\;\;\hat n\left|
n\right\rangle =n\left| n\right\rangle , \label{basis}\end{aligned}$$ where $\left| n_1,n_2\right\rangle =\left| n_1\right\rangle _1\left|
n_2\right\rangle _2$, with $\left| n_i\right\rangle _i$ denoting the usual number states of the two local modes ($\hat n_i\left| n_i\right\rangle
_i=n_i\left| n_i\right\rangle _i$). Using the basis $\left| n\right\rangle $ ($n=N$, $N-1$, $\ldots $, $-N$), one easily obtains $$\hat a_1^{\dagger }\hat a_2\left| n\right\rangle =\sqrt{(N+n+1)(N-n)}\left|
n+1\right\rangle . \label{aa}$$ Consequently, $\hat a_1^{\dagger }\hat a_2$ in this subspace permits the following representation in the basis: $$\hat a_1^{\dagger }\hat a_2=\sum_n\sqrt{(N+n+1)(N-n)}\left| n+1\right\rangle
\left\langle n\right| . \label{aa1}$$ Thus the Hamiltonian in Eq. (\[h22\]) takes the form $$\begin{aligned}
\hat H^{\prime } &=&E_c(\hat n-n_g)^2-J\sum_n\sqrt{(N+n+1)(N-n)} \nonumber
\\
&&\times \left[ \left| n+1\right\rangle \left\langle n\right| +\left|
n\right\rangle \left\langle n+1\right| \right] . \label{hnn}\end{aligned}$$
To realize the Josephson charge qubits in a small Josephson junction system (a “single-Cooper-pair box”) [@Shnirman; @box-PRL], two conditions should be met: ($1$) The charging energy of the single-Cooper-pair box is much larger than the Josephson energy; and ($2$) The system is biased near a degeneracy point so that the system represents a qubit with two states differing only by one Cooper-pair charge. Now suppose that two similar conditions are also met in the present context. Then the Hamiltonian in Eq. (\[hnn\]) takes the form $$\begin{aligned}
\hat H^{\prime } &=&E_c(\hat n-n_g)^2 \nonumber \\
&&\ -\frac{E_J}2\sum_n\left[ \left| n+1\right\rangle \left\langle n\right|
+\left| n\right\rangle \left\langle n+1\right| \right] , \label{hn}\end{aligned}$$ in which $E_J\equiv 2J\sqrt{N(N+1)}\approx N_tJ$ is the Josephson energy. In deriving Eq. (\[hn\]), we have simplified Eq. (\[aa1\]) further by $$\hat a_1^{\dagger }\hat a_2\approx \sqrt{N(N+1)}\sum_n\left|
n+1\right\rangle \left\langle n\right| \label{nn1}$$ if $N\gg \left| n\right| $ for most of the relevant number states of the system. Physically, the operators $\left| n+1\right\rangle \left\langle
n\right| $ and $\left| n\right\rangle \left\langle n+1\right| $ transfer a single atom between the two wells.
The form of $\hat H^{\prime }$ in Eq. (\[hn\]) resembles the Hamiltonian that is used to realize the Josephson charge qubits [@Shnirman; @box-PRL]. Similarly, one can introduce the phase-difference operator $\hat \varphi $ that is conjugate with $\hat n$: $\frac 12\left[ \left| n+1\right\rangle
\left\langle n\right| +\left| n\right\rangle \left\langle n+1\right| \right]
\equiv \cos \hat \varphi $. In this case, we recover the Hamiltonian widely used in the literature [@eprint]: $$\hat H^{\prime }=E_c(\hat n-n_g)^2-E_J\cos \hat \varphi .$$ However, it should be noted that there are some subtleties in doing so, mainly in the context of elementary quantum mechanics, as is well known [@phase; @Chen-MPLB]. Yet, identifying $\hat n$ and $\hat \varphi $ as a canonically conjugate pair is very popular and seems to be well justified in the context of condensed matter physics, typically dealing with a large number of particles [@Anderson; @Leggett-RMP].
The first condition stated above implies $E_c\gg E_J$. In this case, the energies of the system’s states are dominated by the charging part of the Hamiltonian (\[hn\]). The second condition means that when $n_g$ is approximately half-integer (say $n_{\deg }+\frac 12$), the charging energies of two adjacent states ($\left| n_{\deg }\right\rangle $ and $\left| n_{\deg
}+1\right\rangle $) are nearly degenerate, and as such the Josephson term mixes the two adjacent states strongly. As a result, the eigenstates of the Hamiltonian (\[hn\]) are now superpositions of $\left| n_{\deg
}\right\rangle $ and $\left| n_{\deg }+1\right\rangle $ with a minimum energy gap $E_J$ between them. In this case, the two coupled BEC form a BEC qubit that is analogous to the Josephson charge qubits of ultrasmall Josephson junctions. Thus the system under study acts as a “single-atom box”, a BEC counterpart of the single-Cooper-pair box [@box]. Similarly to the case of ultrasmall superconducting Josephson junctions [@Shnirman; @eprint], $\left| n_{\deg }\right\rangle \equiv \left| \uparrow
\right\rangle $ and $\left| n_{\deg }+1\right\rangle \equiv \left|
\downarrow \right\rangle $. The effective Hamiltonian in the spin-$\frac 12$ notion is correspondingly $${\cal H}=-\frac{E_c^{\prime }}2\sigma _z-\frac{E_J}2\sigma _x. \label{spin}$$ Here $$\begin{aligned}
\sigma _z &=&\left| \uparrow \right\rangle \left\langle \uparrow \right|
-\left| \downarrow \right\rangle \left\langle \downarrow \right| ,\;\;\sigma
_x=\left| \uparrow \right\rangle \left\langle \downarrow \right| +\left|
\downarrow \right\rangle \left\langle \uparrow \right| , \nonumber \\
E_c^{\prime } &=&E_c[1-2(n_g-n_{\deg })],\;\;(n_g-n_{\deg })\sim 1/2,
\label{segma}\end{aligned}$$ from which the dependence of ${\cal H}$ (or $E_c^{\prime }$) on the gate charge $n_g$ and the degenerate point is evident. The expression of $%
E_c^{\prime }$ shows that though $E_c\gg E_J$, the $E_J$-term may dominate the $E_c^{\prime }$-term near the degenerate point. With the Hamiltonian (\[spin\]) at hand, the single-qubit operations can be achieved for the BEC qubit.
To this end, it is important to notice that in realistic BEC experiments, the double-well potential can be created by using a far-off-resonance optical dipole force to perturb a magnetic-rf trap [@Andrews]. As noted in Ref. [@Smerzi], the population of atoms in the individual well can be monitored by phase-contrast microscopy; the shape of the double-well potential can be tailored by the position and the intensity of the laser sheet partitioning the magnetic trap. Thus it is experimentally feasible to control the charging energy $E_c$, the Josephson energy $E_J$ and the degenerate point. With the above remarks in mind, one can see that the BEC qubit may be manipulated as the superconducting Josephson charge qubits [@Shnirman]. For practically realizing the BEC qubits, one needs to see whether the condition $E_c\gg E_J$ can be met in realistic BEC experiments. The condition is usually known as the Fock regime, which is attainable by adjusting $E_c$ and $E_J$ for the system under study [@Leggett; @Leggett-RMP].
On the other hand, we would like to mention the difficulty of measuring the BEC qubits. However, due to the beautiful technique of loading and detecting individual atoms developed recently [@single-atom], it is possible to overcome this difficulty in a near future such that one can perform experiments on the BEC qubits in which the single atoms can be detected with sufficient efficiency.
Finally, it is worth pointing out that the same (or similar) Hamiltonian as in (\[h2\]) also arises in other context, e.g., in a two-mode nonlinear optical directional coupler [@opt-2m] and in two-species BEC [@BEC-cat]. The latter has been proposed to create macroscopic quantum superpositions (the “Schrödinger cat states”). Thus the present work may be viewed in a wider context.
In conclusion, we have suggested that two coupled BEC at zero temperature can be used to realize the BEC counterpart of the superconducting Josephson charge qubits under the conditions that the charging energy of the system is much larger than the Josephson energy and the system is biased near a degeneracy point. If the conditions can be met, the two BEC may represent a single-atom box, with two states differing only by one atom. It remains to be seen how non-zero temperatures affect the predicted BEC qubits. Due to extremely good coherence of BEC observed so far, we expect decoherence would not render the BEC qubits unobservable. Another interesting issue is how to entangle two BEC qubits.
Note added–Yu Shi recently brought our attention to his related work [@Yu-Shi].
We are grateful to the Referee for valuable suggestion. This work was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences, and by the Laboratory of Quantum Communication and Quantum Computation of USTC.
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PACS numbers: 74.25.Dw, 74.20.Mn
In a recent Letter, Kuroki and Aoki [@Kuroki] presented quantum Monte-Carlo (QMC) results for pairing correlations in the three-band Hubbard model, which describes the $Cu$-$d_{x^2 - y^2}$ and $O$-$p_{x,y} $ orbitals present in the $CuO_2$ planes of high-$T_c$ materials. In this comment, we concentrate on the parameter set: $U_d = 3.2 t_{pd}, \Delta = 2.7 t_{pd}, t_{pp} = -0.4 t_{pd}$. For this parameter choice, Kuroki and Aoki see a maximal increase in the $d_{x^2 - y ^2}$ pairing correlations which they associate with a signature of off-diagonal long-range order (ODLRO). We argue that:
i\) The above parameter set is not appropriate for the description of high- $T_c$ materials since it does [*not satisfy the minimal requirement of a charge-transfer gap at half-filling*]{}. To illustrate this point, we have calculated with QMC methods the average hole number as a function of the chemical potential: $\langle n \rangle (\mu)$. Our results, which are plotted in Fig.1., show a vanishingly small charge-transfer gap (i.e. $\Delta_{ct} < 0.07
t_{pd}$). In contrast, for a physical parameter set [@Dopf], one obtains a sizeable charge-transfer gap which is detectable from the plateau in the $\langle n \rangle (\mu)$ curve (see inset in Fig. 1). For the latter parameter set, a number of normal state properties were shown to successfully reproduce experimental data [@Dopf1]. However, despite intensive numerical efforts, no ODLRO was unambiguously detected [@Assaad].
ii\) The [*observed increase in the $d_{x^2 - y^2}$ channel* ]{} (Fig. 2 in ref [@Kuroki]) [*is dominantly produced by the pair-field correlations without the vertex part*]{} [@Scalapino]. To prove this point, we have calculated the pair-field correlations in the d-wave channel summed over distances $\bf{r}$ with $ |r_x|, |r_y| < R $ ($ S_d(R)$) (See Fig. 2a). As in ref [@Kuroki], an increase as a function of $R$ can be seen. However, the vertex contribution to the pair-field correlations, which is the relevant quantity, is an order of magnitude smaller and shows - within our numerical accuray - no significant increase as a function of $R$ (see Fig. 2b). Hence, the claim of evidence of ODLRO is not justified.
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### Figure captions {#figure-captions .unnumbered}
[Fig. ]{}[ ]{}
$\langle n \rangle (\mu)$ on a $4\times 4$ lattice at $\beta t_{pd} = 10,15$ for the parameter set: $ U_d = 3.2 t_{pd}, \Delta = 2.7 t_{pd}, t_{pp} = -0.4 t_{pd}$.\
Inset: $ \langle n \rangle (\mu)$ on a $4\times 4$ lattice at $\beta t_{pd} = 10$ for $ U_d = 6 t_{pd}, \Delta = 4 t_{pd}, t_{pp} = 0.0$. [@Dopf]
[Fig. ]{}[ ]{}
a\) $S_d(R) $ on an $ 8 \times 8$ lattice at zero temperature and $ U_d = 3.2 t_{pd}, \Delta = 2.7 t_{pd}, t_{pp} = -0.4 t_{pd}$. b) Vertex contribution to $S_d (R)$ shown in a)
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---
abstract: '[We show that for a general four derivative theory of gravity, only the holographic entanglement entropy functionals obey the second law at linearized order in perturbations. We also derive bounds on the higher curvature couplings in several examples, demanding the validity of the second law for higher order perturbations. For the five dimensional Gauss-Bonnet theory in the context of AdS/CFT, the bound arising from black branes coincides with there being no sound channel instability close to the horizon. Repeating the analysis for topological black holes, the bound coincides with the tensor channel causality constraint (which is responsible for the viscosity bound). Furthermore, we show how to recover the holographic c-theorems in higher curvature theories from similar considerations based on the Raychaudhuri equation. ]{}'
author:
- |
Srijit Bhattacharjee$^*$, Arpan Bhattacharyya$^\dagger$,\
Sudipta Sarkar$^*$ and Aninda Sinha$^\dagger$\
*$^*$Indian Institute of Technology, Gandhinagar, Gujarat 382424, India.\
*$^\dagger$Centre for High Energy Physics, *Indian Institute of Science,\
*C.V. Raman Avenue, Bangalore 560012, India.\
****
title: |
**Entropy functionals and c-theorems\
from the second law**
---
Introduction
============
In the context of AdS/CFT, entanglement entropy of the boundary quantum field theory can be calculated using the Ryu-Takayanagi [@Ryu:2006bv; @chm] prescription and its generalizations to higher curvature gravity theories. The derivation of this holographic entanglement entropy for two derivative Einstein gravity was proposed by Lewkowycz and Maldacena in [@LM]. This derivation allows one to obtain a surface equation (for static situations) for the entangling surface which minimizes an entropy functional. There were attempts to extend this calculation to find the entangling surface equation in higher curvature theories in [@bss] and the corresponding entropy functionals were argued to be different from the Wald entropy [@Dong:2013qoa; @Camps:2013zua; @Miao]. In particular, for Lovelock theories the entropy functional is the so-called Jacobson-Myers (JM) [@Jacobson:1993xs; @Hung] one while for general four derivative theories, it coincides with the Fursaev-Patrushev-Solodukhin (FPS) entropy functional [@Fursaev:2013fta]. However, there still exist problems in finding the entangling surface using the Lewkowycz-Maldacena method and there are potential ambiguities related to higher order extrinsic curvature terms in the entropy functionals. Moreover, it leads one to wonder if such entropy functionals could arise independent of AdS/CFT from different and perhaps more fundamental considerations.
Wald and collaborators [@Wald:1993nt; @Iyer:1994ys] had established the first law for black hole mechanics for any diffeomorphism invariant theory of gravity and proposed that the entropy of a black hole is a Noether charge associated with the Killing isometry generating the horizon. The Wald entropy suffers from various ambiguities [@Jacobson:1995uq] and therefore does not provide a unique answer for the horizon entropy. Although, none of these ambiguities contribute to black hole entropy of a stationary Killing horizon with regular bifurcation surface, the study of the second law of black hole mechanics beyond general relativity (GR) shows that these ambiguities need to be carefully included in the expression of black hole entropy to obtain an increase theorem similar to Hawking area theorem in GR. In fact, for black holes in Lovelock gravity, it is the JM functional which leads to an increase theorem for linearized perturbations [@Sarkar:2013swa].
Recently, it has been pointed out that the 2nd law for spherically symmetric black holes is satisfied at linear order in perturbations in general four derivative theories of gravity by the holographic entanglement entropy (HEE) functionals [@Bhattacharjee:2015yaa]. This has been also generalized beyond spherical symmetry and to generic perturbations up to linear order [@Wall:2015raa] and a detailed construction of the entropy functionals has been proposed for any metric theory of gravity such that the linearized second law holds true. For theories with higher curvature terms, the construction produces the HEE functionals. This is a remarkable result since it allows for a way independent of AdS/CFT of deriving entropy functionals–these same entropy functionals therefore find applications in diverse settings. However, what still needs to be addressed is if only the HEE functionals alone do this job. In this paper, we first show that at linear order in perturbations only the HEE functionals can satisfy the second law. We start with an expression for horizon entropy with all possible ambiguous terms relevant for linear order in perturbation. We fix these terms by demanding the validity of the linearized second law and the final entropy coincides with HEE functionals. To be clear, since our analysis is going to be perturbative, we will not be able to rule out higher order extrinsic curvature terms in the functionals (e.g. $O(K^4)$ in the FPS functional)–see section 5.
We will tackle the question keeping in mind two different motivations. First, we are interested in what happens in the case of asymptotically flat black holes. We will consider two different types of perturbations–the first kind will be due to radially symmetric, slowly falling matter and the second kind being a shear perturbation. We will find that for the Gauss-Bonnet (GB) theory, if we go to second order in perturbations, the second law is automatically satisfied for regular horizons if the GB coupling is positive. If the GB coupling is negative then a certain lower bound on the horizon radius ensures that the second law is satisfied; since this suggests that such black holes cannot be formed from collapsing matter, one can take this as an indication that the negative coupling is disfavoured. If we consider a Ricci-square theory then we find that similar extra conditions on the mass of the black hole may be needed for the second law to hold. Second, we are interested in what happens in the context of AdS/CFT. We will find that for the GB theory, the second order analysis leads to a bound on the Gauss-Bonnet coupling. For black branes, this bound coincides with the absence of sound channel instabilities at the black hole horizon. A similar analysis for the Ricci- square theory bounds the corresponding coupling constant. For GB topological black holes [@rgcai], for the zero mass case we find that the second law bound coincides with the tensor channel causality constraint [@Brigante; @holoGB]. This bound is what leads to the lower bound on the ratio of shear viscosity to entropy density in the dual plasma and agrees with the lower bound on the $a/c$ ratio in 4d CFTs [@hm] with $\mathcal N=1$ supersymmetry. We also investigate quadratic theories with $R_{ab}R^{ab}$ and $R^2$ terms. In these theories, the causality constraints following from [@hm] are trivial whereas the second law bound in certain examples we study is nontrivial.
Using the HEE entropy functionals and the Raychaudhuri equation, we then turn to the question of holographic c-theorems. We will argue that the holographic c-functions found in [@ctheorems] arise naturally from such a consideration provided the matter sector satisfies the null energy condition. In [@sahakian], a derivation for the c-functions in Einstein gravity, found in [@cthor], was given using the Raychaudhuri equation. In [@cremades] an attempt was made to extend this to higher curvature theories using the Iyer-Wald prescription. Unfortunately the resulting functions do not give the correct central charge (namely the A-type anomaly coefficient in even dimensions). In [@ctheorems] it was observed that in all sensible holographic models, which can be constructed by demanding the absence of higher derivative terms in the radial direction (to ameliorate the problem of ghosts) allow for a simple c-function which at the fixed points coincide with the A-type anomaly coefficient in even dimensions. This c-function was monotonic under RG flow provided the matter sector satisfied the null energy condition. We will find that using the HEE entropy functionals and considering the Raychaudhuri equation naturally leads to the same c-functions as in [@ctheorems].
The organization of the paper as follows: In the next section we briefly describe the ambiguities in Wald’s Noether charge construction. In the section \[second-set\] set up for proving second law has been described. Next, we determine the coefficients of the ambiguous terms in Wald’s construction using linearized second law. In section \[beyond\], we go beyond linear order in perturbation to study the GB theory and determine bound on the coupling parameter using AdS black hole solution with different horizon topologies as background. We also consider various other examples like critical gravity theories and put bounds on the couplings in those theories. In section 6, we derive holographic c-functions using the HEE functionals and the Raychaudhuri equation. We end this paper by discussing several follow-up questions. We use $\{-,+,+,,\cdots\}$ signature and set $G=1$ throughout the paper.
Ambiguities in Noether charge method for non-stationary horizons {#ambi-wald}
================================================================
Let us start by describing the geometry of the horizon of a stationary black hole in $D$ dimensions. The event horizon is a null hyper-surface ${\cal H}$ parameterized by a non-affine parameter $t$. The vector field $k^a = (\partial_t)^a$ is tangent to the horizon and obeys non-affine geodesic equation $k^a\nabla_ak^b=\kappa~ k^b$ with $\kappa$ is the surface gravity of the horizon ($a,b,...$ are bulk indices). All $t=$ constant slices are co-dimension $2$ space-like surfaces and foliate the horizon. We construct another auxiliary null normal to the $t=$ constant slices $l^a$, and the inner product between $k^a$ and $l^a$ satisfies $k^a l_a = -1$. The induced metric on any $t =$ constant slice of the horizon is now constructed as $h_{ab} = g_{ab} + 2 k_{(a} l_{b)}$. The horizon binormals are then given by $\epsilon_{ab} = \left( k_a l_b - k_b l_a\right)$.
The change in the induced metric along these two null directions can be expressed as a sum of trace part and a trace-free symmetric part (assuming null congruences are hypersurface orthogonal therefore have zero twist). The trace part measures the rate of change of the area of the horizon cross section along the null generators which is known to be the expansion and the other part measures the shear of the null geodesic congruences. We denote $\theta_{k}$ and $\theta_{l}$ to be the two expansion parameters in these two null directions and similarly we have two shears $\sigma_{k}^{ab}$ and $\sigma_{l}^{ab}$. Note that $\theta_{k}$ and $\sigma_{k}^{ab}$ typically vanish on a stationary horizon but $\theta_l$ and $\sigma_l^{ab}$ in general do not vanish.
As mentioned in the introduction, the entropy of a stationary black hole in any general covariant theory is given by the Noether charge associated with the boost symmetry generating Killing vectors at the horizon [@Wald:1993nt; @Iyer:1994ys]. The Wald entropy functional for any general covariant Lagrangian $\cal{L}$ in $D$ dimension takes the form: S\_W = - 2 \_\_[ab]{} \_[cd]{} dA,\[Wald\_Exp\] where $\C$ denotes any horizon slice and $\sqrt{h}$ is the area element. $R_{abcd}$ is the Riemann tensor of bulk geometry and $dA$ is the area of the $D-2$ dimensional cross-sections of horizon. This formula gives a unique expression for entropy so long one is confined to any stationary slice or to the bifurcation surface of the horizon. However, as pointed out in [@Jacobson:1995uq; @Iyer:1994ys], the entropy expression constructed via Noether charge approach has several ambiguities if one tries to apply it for nonstationary slices of the horizon. In fact it turns out, the Wald entropy formula is just one of several possible candidates for the entropy. In [@Jacobson:1995uq], Jacobson, Kang and Myers (JKM) identified three different types of ambiguities which may alter the Wald construction. Among these, the only relevant one for our purpose will be the one which gives us the freedom to add to the Wald entropy a term of the form with arbitrary coefficients: S\_A\^[(JKM)]{}=-2\_X.Y dA \[ambi\] where the integrand is invariant under a Lorentz boost in a plane orthogonal to an arbitrary horizon slice $\C$ but the terms $X$ and $Y$ are not separately boost invariant [@Wall:2015raa; @Sarkar:2010xp]. These ambiguities vanish for Killing horizons but do not necessarily vanish on a nonstationary slice of the horizon. As a result the entropy functional for a nonstationary horizon may be expressed as[^1] §= - 2\_ dA . \[genn1\]0.5cm Here we have used the notation $\sigma_{kab}\sigma^{ab}_l=\sigma_k\sigma_l$. As can be seen from (\[genn1\]) that the ambiguous terms in the entropy formula involve equal number of $k$ and $l$ subscripts which follows from the fact that these are the only boost invariant combinations that can appear in the entropy functional. Also on a stationary slice the ambiguity terms vanish and then the expression coincides with the Wald formula (\[Wald\_Exp\]). The First Law of black hole mechanics [@Iyer:1994ys; @Jacobson:1995uq] doesn’t fix the coefficients $p$ and $q$ uniquely. So, to fix the coefficients of these terms one is thus forced to examine whether $\S$ obeys a local increase law.
Recently in [@Wall:2015raa], the second law has been shown to hold for any arbitrary higher curvature theories if one allows only linear perturbations to a stationary black hole. This analysis also shows, the entropy functional proposed in the context of Holographic Entanglement Entropy (HEE) [@Dong:2013qoa; @Camps:2013zua; @Fursaev:2013fta] remarkably matches with the $\S$. However, it is not apparent from this analysis[^2] how just one condition (entropy increases along $t$) fixes two coefficients in the entropy functional! In this paper, we will show explicitly how this fixing happens for curvature squared gravity theories. Note that the unknown coefficients are sitting in front of products of shear and expansion terms. Since shear and expansion belong to different irreducible parts of a tensor, they will remain separated even when perturbation is turned on at the linearized level. Consequently we will obtain two independent equations when we demand that entropy increase law holds for every slice of the horizon. This fixes all the coefficients of the entropy functional uniquely for generic quadratic curvature gravity.
It is important to mention that we have only considered ambiguities which are quadratic order in expansion and shear. We may also have higher powers of such products added to the entropy functional. However, we cannot fix such terms using linearized second law but in curvature squared theories these terms do not enter at linear order [@Sarkar:2010xp]. Our analysis matches with the result obtained in [@Bhattacharjee:2015yaa] where the coefficients were fixed for Ricci$^2$ theory using a Vaidya like solution in the linearized increase law.
Second law set up {#second-set}
=================
Let us turn to the apparatus needed to verify the second law of black hole mechanics. The equation of motion for a generic higher curvature metric theory of gravity will be of the form, G\_[ab]{}+H\_[ab]{}=8T\_[ab]{}\[eqm\] , where $G_{ab}$ is the Einstein tensor coming from the Einstein-Hilbert part of the action and $H_{ab}$ is the part coming from higher curvature terms–in theories with a cosmological constant there will also be an additional term proportional to the metric which we can absorb into $G_{ab}$. $T_{ab}$ is the energy momentum tensor which we will assume to obey the Null Energy condition (NEC): $T_{ab} k^a k^b>0$ for some null vector $k^a$.
We will use the Raychaudhuri equation for null geodesic congruence which describes the evolution of the expansion along the horizon generating parameter $t$. In nonaffine parametrization this looks like[^3] =\_k --\_k\^2-R\_[kk]{}.\[Ray1\] Our notation is $A_{ab} k^a k^b = A_{kk}$ and $\sigma_{kab}\sigma^{ab}_k=\sigma^2$. Now, we define an entropy density for any generic higher curvature theory as: §= \_(1 + ) d\^[D-2]{}x, \[entropyG\] where $\rho (t)$ contains contribution from the higher curvature terms including ambiguities. For general relativity, $\rho = 0$. In the stationary limit $\rho$ will coincide with the Wald expression (\[Wald\_Exp\]). Now from this expression the change in entropy per unit area gives us the following expression of generalized expansion $\Theta$, = + \_[k]{} (1 + ). \[Theta\] The evolution of $\Theta$ is governed by the following equation, - =&-&8T\_[kk]{}-(1+)-\_[k]{}\^2 (1+)+\_k\
&+&H\_[kk]{}+-R\_[kk]{}-\[Ray2\], where we have used the equation of motion (\[eqm\]) and inserted eq. (\[Ray1\]). We can now write eq, (\[Ray2\]) in a convenient form \[Ray3\] -=-8T\_[kk]{}+E\_[kk]{}, where \[eqn2\] E\_[kk]{}= H\_[kk]{}+\_k -R\_[kk]{}+k\^[a]{}k\^[b]{}\_[a]{}\_[b]{} -( +\_[k]{}\^2)(1+). We have used $$\frac{d^2\rho}{dt^2}=k^{a}k^{b}\nabla_{a}\nabla_{b} \rho +\kappa \frac{d\rho}{dt}$$ and $$\frac{d}{dt}= k^{a}\nabla_{a}.$$ Next, consider a situation when a stationary black hole is perturbed by some matter flux obeying NEC. The perturbation can be parametrized by some dimensionless parameter $\e$. Note that $T_{kk}$ is linear ($\cal{O}(\e)$) in perturbation and so as $\theta_k$, $\sigma_k$ and $d \rho/dt$. Now, to establish a linearized second law we ignore higher order terms in eq. (\[Ray3\]). Then, it is easy to see then eq. (\[eqn2\]) reduces to, E\_[kk]{}\_k \_k - R\_[kk]{} + H\_[kk]{}\[ekkl\]. We have already mentioned that $T_{kk}$ is of order $\e$, so if the rest of the terms in (\[Ray3\]) are also collectively of higher order, i.e.,$ E_{kk} \sim \mathcal{O}(\epsilon^2)$, then we obtain
- = - 8 T\_[kk]{}. The above equation implies $d \Theta / d t - \kappa \Theta < 0$, on every slice of the horizon. We assume that in the asymptotic future, the horizon again settles down to a stationary state, we must have $\Theta \to 0$ in the future. This will imply that $\Theta$ must be positive on every slice prior to the future and as a result the entropy given by (\[entropyG\]) obeys a local increase law.
Therefore, to establish the linearized second law, we only need to show that the linear order terms in $E_{kk}$ exactly cancel each other. Interestingly, this alone will be enough to obtain the values of both the coefficients $p$ and $q$ introduced in the entropy functional (\[gen1\]). In the next section, we will demonstrate this explicitly for the curvature squared gravity theories.
Linearized second law and fixing JKM ambiguities {#linear}
================================================
We start with the most general second order higher curvature theory of gravity in five dimensions. The action of such a theory can be expressed as, S = d\^5x ( R- 2+ R\^2 + R\_[ab]{}R\^[ab]{} + [L\_[GB]{}]{} ) \[Flag\] where ${\cal L_{GB}} = R^2\,-\,4 R_{ab}^2\,+\,R_{abcd}^2$ is the Gauss- Bonnet (GB) combination and $\Lambda$ is the cosmological constant. It is reasonable to assume that such a theory admits a stationary black hole solution as in the case of GR. We also expect to have a non stationary black hole solution with in-falling matter by perturbing this solution. Such a spacetime will be the counterpart of the Vaidya solution in general relativity for spherically symmetric case and can be expressed as,
\[vd1\] ds\^2=-f(r,v) dv\^2+2 dv dr + r\^2d\_[3]{}\^2 $\Sigma_{3}$ can be any three dimensional space with positive, negative or zero curvature. We want to use this solution to investigate the issue of second law of black hole mechanics. Note that, the location of the event horizon $r = r(v)$ for this solution can be obtained by by solving the following equation, \[hor\] r= =. with appropriate boundary condition. Note that, $(\,\dot{}\,)$ and $(\,'\,)$ denote respectively derivative with respect to $v$ and $r$. The null generator of the event horizon is given by $k^{a}=\{1,f(r,v)/2,0,0,0\}$ and the corresponding auxiliary null vector $l_{a}=\{-1,0,0,0,0\}$. The event horizon has nonzero expansion due to the perturbation caused by in falling matter. Next, we will write the entropy associated with the horizon as, §= - 2\_ dA . \[gen1\]We consider several choices of the coefficients $\alpha$, $\beta$ and $\gamma$ etc and study the evolution of this horizon entropy. The aim is to fix the unknown coefficients $\tilde p$ and $\tilde q$ for different gravity theories by demanding the validity of the linearized second law.
Gauss-Bonnet gravity
--------------------
We will first study the local increase law for Gauss-Bonnet (GB) case. This corresponds to the choice $\a=\b=0$ in (\[Flag\]). The action is: S=d\^[5]{}x . where we have also introduced a matter sector obeying the NEC. $\g$ is a coupling constant of dimension $Mass^{-2}$. The equation of motion for this theory is given by,
\[eom\] G\_[ab]{}+g\_[ab]{}+ H\_[ab]{}=8G T\_[ab]{}. With, $$\begin{aligned}
{H}_{ab}\equiv2\g\Bigl[RR_{ab}-2R_{ac}R^c_{~b}
-2R^{cd}R_{acbd} +R_{a}^{~cde}R_{bcde}\Bigr]
-\frac{1}{2}g_{ab} {\cal L}_{GB}.\end{aligned}$$
Next we start with the expression of the entropy functional (\[entropyG\]) and evaluate the entropy density for the EGB theory. This is given by sum of Wald entropy density plus the ambiguity terms,
§&=& \_(1 + ) d\^3x,\
&=& \_(1 + 2(R + 4 R\_[kl]{} - 2 R\_[klkl]{}-p \_k\_l-q \_k \_l) ) d\^3x ,\[eee\] where $2(R + 4 R_{kl} - 2 R_{klkl})$ is the contribution from the Wald construction. Also $\tilde p= 2 \,\gamma\, p$ and $ \quad \tilde q=2\,\gamma\, q.$ The EGB theory belongs to the general Lovelock class of action functions which give rise to quasi linear equation of motions. It has already been shown in [@Sarkar:2013swa] that black holes in all Lovelock theories obey a linearized second law if one uses the JM entropy functional. For EGB gravity, the JM entropy functional is the intrinsic Ricci scalar $(\R)$ of the horizon slice and using the null Gauss-Codazzi equation we can cast $\R$ as,
\[JMGB\] =R + 4R\_[kl]{} - 2 R\_[klkl]{}-[43]{}\_k\_l+2\_k\_l.
We will now explicitly show below that in EGB gravity the entropy functional which obeys the second law is indeed the JM entropy by fixing the coefficients $p$ and $q$ in (\[eee\]).
First, we proceed to evaluate the [*rhs*]{} of eq. (\[Ray3\]) to study the second law. Note that, if we use the metric (\[vd\]) with the choice of $d\Sigma_{3}^2$ to be a flat metric then the shear term in (\[Ray3\]) will vanish identically and $q$ will remain undetermined. This happens because isometries of the metric on a horizon slice essentially coincides with that of a sphere. So we will break the symmetry by adding a cross term and the metric on the horizon slice takes the form, ds\^2=-f(r,v) dv\^2+2 dv dr + r\^2( dx\^2+ dy\^2+ dz\^2+ \_[1]{} h(r,v) dx dy ) . We will assume that this shear mode $h(r,v)$ will be balanced by some matter stress tensor still obeying the NEC. Also we do not require to find the explicit form of $h(r,v)$ for our analysis. We will calculate $E_{kk}$ order by order in $\epsilon_{1}$ and extract the coefficients of the linear order terms in $\e$ from the evolution equation. Setting those terms to zero will satisfy the linearized second law and in the process $p$ and $q$ will be determined. Below we quote expressions for $\theta's$ and $\sigma's$ for this solution. \[exp1\] \_[k]{}=+(\_[1]{}\^2), \[shear1\] \_[l]{}=+(\_[1]{}\^2) and \_k\_l=-+(\_[1]{}\^4).
Now evaluating $E_{kk}$ and extracting the zeroth order terms in $\e_1$ (and linear order in $\e$), we get the following equation, (4-3 p)=0. From $\mathcal{O}(\epsilon_{1}^{2})$ terms (which is when $h(r,v)$ makes its first appearance) we get another equation, 2(-6+6p+q) h(r,v)\^2+3(2-3p-q)h(r,v)r(v)h’(r,v)+(2+q)r(v)\^2h’(r,v)\^2=0. It is evident that, both of these equations are satisfied if $$p=\frac{4}{3}\,\quad q=-2.$$ This shows that linearized second law fixes all the quadratic ambiguities in entropy functional (\[gen1\]) uniquely and it is the JM entropy which obeys the linearized second law for GB black holes! Although we have shown this calculation using the 5-dimensional lagrangian for concreteness, the same result holds in any dimension $D>4$.
$R_{ab}^2$ theory
-----------------
Let us repeat the above analysis for other curvature squared theories. First we take $R_{ab}^2$ theory and we set $\a=\g=0$ and $\beta >0 $ in (\[Flag\]). We will start with the following entropy functional,
\[e1\] §=d\^[3]{} x (1-2(R\_[kl]{}-p\_[k]{}\_[l]{}-q\_k\_l)). When $p$ and $q$ are zero then this reduces to the corresponding Wald entropy for the stationary case. Again proceeding as before we will use the linearized second law to fix these coefficients. Like the Gauss-Bonnet case we get two independent equations and solving them we get, $$p=\frac{1}{2}\,\quad q=0.$$ [As argued in [@Bhattacharjee:2015yaa], the shear part has not contributed to the entropy functional at linear order and the other coefficient of the entropy functional exactly matches with the one which has been shown to obey the linearized second law in [@Bhattacharjee:2015yaa].]{}
$R^2$ theory
------------
We now consider the case $\beta=\g=0$ and $\a >0 $ in (\[Flag\]) and the entropy functional now reads, \[e2\] §=d\^[3]{} x (1+2(R-p\_[k]{}\_[l]{}-q\_k.\_l). It has already been argued in [@Jacobson:1995uq] that $f(R)$ theory obeys a linearized second law and the Wald entropy functional itself does the job. It is clear, in this case setting $p$ and $q$ equal to zero will correspond to the Wald entropy. If we again repeat the linearized second law analysis we find, $$p=0\,\quad q=0.$$ So this is exactly what is expected from the earlier analysis in [@Jacobson:1995uq].
Therefore, considering all the individual terms in (\[Flag\]) we have demonstrated that for any general curvature squared theory we can fix the entropy functional completely using only the linearized second law.
Beyond linearized second law {#beyond}
============================
In this section, we will consider local entropy increase law beyond linear order which means we are not allowed to neglect the terms in (\[Ray3\]) which have been thrown away in the previous section. Now we will be considering all the relevant terms in equation (\[Ray3\]) and will determine the criteria such that second law holds non-perturbatively in the coupling. For this, we will assume a spherically symmetric Vaidya like solution. This means, the metric in (\[vd1\]) will now have the following form, \[vd\] ds\^2=-f(r,v) dv\^2+2 dv dr + r\^2d\_[3]{}\^2. In fact for EGB gravity one can obtain such a solution just setting the mass in the static spherically symmetric Boulware-Deser [@BD] solution to be a function of the advanced time $v$.
Beyond linearized level, one has to evaluate the full $E_{kk}$ in the evolution equation (\[Ray3\]). Since the solution is spherically symmetric, shear is identically zero. In fact, for this particular case the evolution equation will be of the form, -=-8T\_[kk]{}-\^[2]{}\_k, Note that we must have $d\Theta / dt < 0$ to have a local entropy increase law. Also, we have $T_{kk} > 0$ by NEC. Now consider a situation where the stationary black hole is perturbed by some matter flux and we are examining the second law when the matter has already entered into the black hole. In that case, the above evolution equation does not have any contribution from matter stress energy tensor and the evolution will be driven solely by the $\theta_k^2$ term. So we will have a equation of the form, -=-\^[2]{}\_k. In such a situation, if we demand the entropy is increasing, we have to fix the sign of the coefficient of $\theta_k^2$ term. We evaluate the coefficient in the stationary background and impose the condition that overall sign in front of $\theta^{2}_k$ is negative. This will give us a bound on the parameters of the theory under consideration.
Gauss-Bonnet Gravity
--------------------
We start with the EGB gravity. The event horizon is now a null surface whose equation is $r = r(v)$. Calculating the r.h.s. of (\[Ray3\]) for the metric (\[vd\]) we obtain, -=-, where we have identified $$\theta_k^2=\frac{9f(r,v)^2}{4 r(v)^2}$$ and introduced $\zeta$ as the coefficient of the $\theta_k^2$ terms. The expression of $\zeta$ is given by, \[zeta\] =.
For $\g=0$, the coefficient reduces to $1/3$ which matches with GR. $\mathcal{R}$ is the ricci scalar evaluated on the horizon. To satisfy the entropy increase law we now set,
\[cond\] >0.As discussed earlier, we will evaluate $\zeta$ for different stationary backgrounds and determine bounds on the coefficient $\g$ imposing the condition (\[cond\]).
### Asymptotically flat case {#BD-bnd}
Now we consider the Boulware-Deser (BD) [@BD] black hole as the background, for which, f(r)=1+. Also, \[eqn3\] r\_[h]{}\^[2]{}+2=M determines the location of the horizon. Existence of an event horizon demands $r_h^2>0$. In this case horizon topology is a sphere. Now, evaluating $\zeta$ for the above background at the horizon $r =r_{h}$, we get the following inequality <0, \[ineq\] which leads to $$\begin{aligned}
&M>2 |\g|\,\quad if~ M>0\\
&M<-2|\g|\,\quad if~ M<0\,.\end{aligned}$$ To understand this better, note that we require $M>2\g$ to avoid the naked singularity of the black hole solution for $\g>0$. Thus in this case for a spherically symmetric black hole $\zeta$ will be positive and hence second law will be automatically satisfied. The condition of the validity of the second law is same as that for having a regular event horizon.
Also, for $\g > 0$, it is possible to make $r_h$ as small as possible by tuning the mass $M$. But when $\g$ is negative (a situation that appears to be disfavoured by string theory, see [@BD], [@bms] and references therein), $r_h$ cannot be made arbitrarily small and it would suggest that these black holes cannot be formed continuously from a zero temperature set up. Notice that we could have reached the conclusion without the second law if $M$ is considered to be positive–however, our current argument does not need to make this assumption. Due to this pathology, it would appear that the negative GB coupling case would be ruled out in a theory with no cosmological constant.
### AdS case
Next we consider the 5-dimensional AdS black hole solution for EGB gravity as the background. The function $f(r)$ now becomes [@rgcai], f(r)=k+(1-). Here $l$ is the length scale with the cosmological constant ($2\Lambda=-12/l^2$) and $k$ can take values $0,1$ and $-1$ corresponding to planar, spherical or hyperbolic horizons respectively. The intrinsic Ricci scalar on the horizon is $6k/r_h^2$ where, the horizon is at r\_h\^2=. First we will consider black brane solution for which $k=0$ and the horizon is planar. The intrinsic curvature of the codimension two slices of the event horizon vanishes. In this case the coefficient $\zeta$ reads, \[betajm\] = (1-16). Introducing a rescaled coupling $\lambda_{GB} l^2=2\g$ [@Brigante; @Buchel] we get, = (1-8\_[GB]{}). Again demanding positivity of $\zeta$ we get, \_[GB]{}< , so to satisfy the second law using $\S$ at $\mathcal{O}(\epsilon^2)$ order we need to impose a bound on the GB coupling. Incidentally the same has to be imposed to avoid instabilities in the sound channel analysis of the quasinormal mode for dual plasma. It was shown in [@Buchel] that when $\lambda_{GB} >\frac{1}{8}$ the Schroedinger potential develops a well which can support unstable quasinormal modes in the sound channel. It is interesting to see that the second law knows about this instability.
Next we consider $k=-1$ case. This will give us a black hole with a hyperbolic horizon. We will also set $r_{0}=0$, which corresponds to the zero mass solution. In this limit we find , =5-4. Demanding $\zeta>0$ we get, \_[GB]{} < .
If we take the limit $r_0/l\rightarrow \infty$ in the expression of $\zeta$ with $k=\pm 1$ (i.e. the case when the horizon sections becomes planar), we recover the bound on GB coupling $\lambda_{GB}$ for the $k=0$ case. However, this is a weaker bound on $\lambda_{GB}$ and the strongest bound on $\lambda_{GB}$ comes from the hyperbolic black hole in the massless limit. This is illustrated in figures 1a and 1b. We have checked that for the $k=1$ case other thermodynamic considerations (e.g., positive entropy, black hole being the correct phase) do not lead to stronger bounds.
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![(Colour online) (a) Plot of $\zeta$ for different $r_0$ for $k=-1$. (b) Plot of $\zeta$ for different $r_0$ for $k=1$. The black dot denotes $\lambda_{GB}=9/100$. We have set the $l=1$. Note that both the figures shows that $\zeta$ will be positive in all the cases provided $\lambda_ {GB} < 9/100.$[]{data-label="strongbound"}](plot1.pdf "fig:"){height="1.8in" width="3.5in"} ![(Colour online) (a) Plot of $\zeta$ for different $r_0$ for $k=-1$. (b) Plot of $\zeta$ for different $r_0$ for $k=1$. The black dot denotes $\lambda_{GB}=9/100$. We have set the $l=1$. Note that both the figures shows that $\zeta$ will be positive in all the cases provided $\lambda_ {GB} < 9/100.$[]{data-label="strongbound"}](plot2.pdf "fig:"){height="1.8in" width="3.5in"}
(a) (b)
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Quite curiously we have recovered the tensor channel causality constraint [@hm; @Brigante; @holoGB][^4]. The bound on $\lambda_{GB}$ in this channel leads to the lower bound on the ratio of shear viscosity ($\eta$) to entropy density ($s$), $4\pi \eta/s\geq 16/25$ in GB holography [@Brigante] . The vector and scalar channels also lead to bounds on the coupling from causality constraints. The strongest bound in GB arises from the tensor and scalar channels [@holoGB]. Our finding coincides with the tensor channel constraint. It will be interesting to see if the other channels can also be reproduced using the second law analysis.
We end this section with one last comment. Instead of putting a bound on the Gauss-Bonnet coupling, one can try to modify the entropy functional $\S$ itself so that all the $\mathcal{O}(\epsilon^2)$ terms will be canceled. One possible modification is to add a term like $\theta_{k}^2\theta_{l}^2$ which vanishes on the stationary horizon and boost invariant in the null direction. So this type of higher order ambiguous term can occur in the Wald derivation. We modify $\S$ as, S\_[modify]{}=d\^[3]{} x (1+2+ \_[k]{}\^2\_[l]{}\^2). $\zeta$ in (\[betajm\]) will be modified as, \[modeq\] =So we can adjust $\chi$ such that the $\zeta$ vanishes and second law automatically holds up to $\mathcal{O}(\epsilon^2)$. One may also can add a term like $\nabla_{i}\theta_{k}\nabla^{i}\theta_{l}$ at this order where $\nabla_{i}$ is the covariant derivative with respect to the surface index. But these additional terms will generate further higher order terms starting from $\mathcal{O}(\epsilon^3).$ So one will need to add more terms to cancel them. It might possible to do this recursively. However, the coefficients of these terms will also depend on the background data. So it will be prudent not to modify the entropy functional. We have put a bound on the coupling instead of modifying the entropy functional at $\mathcal{O}(\epsilon^2)$ order such that second law continues to hold up to this order. It will be interesting to investigate what happens to this bound on $\lambda_{GB}$ when one goes beyond $\mathcal{O}(\epsilon^2)$ order.
$R_{ab}^2$ theory
-----------------
### Asymptotically flat space
First consider the asymptotically flat black hole with spherically symmetric horizon in 5d. In this case, =[13]{}(1++ f”(r,v)+f’(r,v)- f(r,v)) Now, to extract a bound, we need an explicit form of a spherically symmetric static black hole solution in $Ricci^2$ theory. We are not aware of any such solution except the 5d Schwarzschild solution of GR which is also a solution of $Ricci^2$ theory. So, as an example we take such a solution as the background and write, f(r)=1-. Then the validity of the second law gives us the condition, > . When $\beta<0$ this is automatically satisfied whenever a horizon exist whereas for $\beta>0$, a lower bound on the horizon size ensures the validity of the second law.\
Repeating this same analysis for 4d with the metric function $f(r,v)=1- (r_h/ r)$ we get , >, Therefore, in both cases, there is a minimum horizon radius for the second law to hold when $\beta>0$. Again as discussed in \[BD-bnd\], minimum $r_h$ scenario would appear pathological and may be taken as a reason to disfavour the $\beta>0$ sign.
### AdS case
We repeat the same analysis of the previous section for $R_{ab}^2$ theory but with the background as an asymptotically AdS black brane in 5d. The metric function is then given by f(r)= f\_(1-(r\_[h]{}/[r]{})\^4) where $f_{\infty}=\big (\frac{l}{l_{AdS}}\big)^2$ and it satisfies the equation, 1-f\_+\_2 f\_\^2=0. In this case we obtain, =(+ f”(r,v)+-), Define $\lambda_{2} l^2=2\beta $ and after evaluating on the horizon with the above metric function as background, we get =-. From this, the bound becomes, \_[2]{} <. In this case if we repeat out analysis for a zero mass hyperbolic black hole we obtain a lower bound on $\lambda_{2}$. Combining them we get, - <\_[2]{} <. If we repeat the same analysis for 4d and obtain the following bound, $
\lambda_{2} <\frac{2}{15}.
$ No analog of such a bound in the context of AdS/CFT is known in this case. This analysis also suggests that the second law bound arising from hyperbolic horizons is not necessarily connected to the causality constraints. The reason is that for these theories, in the analysis of [@hm], we only get $c_T>0$.
Critical gravity
----------------
We will now consider bounds arising from the second law in quadratic curvature theories with a negative cosmological constant, which only involve terms like $R_{ab}R^{ab}$ and $R^2$. In particular, we will focus on the case of critical gravity [@pope; @others]. For any theory with only Ricci$^2$ or $R^2$, the positive energy condition of Hofman-Maldacena [@hm] does not yield any constraint (see [@anoms] for general expressions) except for the positivity of the two point function of stress tensors which follows from unitarity. In all the examples below, the positivity of black hole entropy leads to the same condition arising from demanding $c_T>0$ where $c_T$ is the coefficient in the two point function of CFT stress tensors. The second law constraint will lead to further conditions as we will see.
### D=4
The action for the 4d version of critical gravity is given by, S = d\^4x ( R+ +l\^2(R\_[ab]{}R\^[ab]{}-R\^2) ) \[clag\] In general this theory posses a massive spin-2 mode in addition to the usual massless spin-2 degree of freedom. It has been shown in [@pope], at $\alpha=-1/2$, which is called the critical point, the additional massive spin-2 mode becomes massless. Also, at the critical point the entropy of the Schwarzschild-AdS black hole becomes zero.
We will now use the second law to put a bound on the coupling $\alpha$. Proceeding as before at the second order we get, =(-+ l\^2 f”(r,v)- l\^2 r(v) f”’(r,v)). In this case, we take the background as, f(r)= (1-()\^3). From this we get the bound on the coupling as, \[b1\] . Also, the black hole entropy has to be positive–if it was zero then it must remain zero at all times. From this we have the following condition, 1+2>0. So, \[b2\] > -. Combining (\[b1\]) and (\[b2\]) we get, \[b3\] - <. Now from the AdS/CFT perspective it is natural to assume the positivity of the coefficient $c_{T}$ of two point correlator of boundary stress tensor. That will give exactly the same condition as in (\[b2\]) which is weaker than (\[b3\]). Repeating the same analysis for a zero mass hyperbolic black hole we did not any further bounds.
### D=5
Next we consider the critical gravity theory in 5d. The action is given by,
S = d\^5x ( R+ + l\^2(R\_[ab]{}R\^[ab]{}-R\^2) ) \[clag\] The critical point now corresponds to $\alpha=-5/27$ [@pope]. Proceeding as before at the second order we get, =(++ l\^2 f”(r,v)- l\^2 r(v) f”’(r,v)). In this case, the background solution is taken as, f(r)= f\_(1-()\^4), where the quantity $f_{\infty}$ satisfies the equation, $
1-f_{\infty}-\frac{3}{4} \alpha f_{\infty}^2=0.
$ The second law thus leads to, \[b4\] -. Again demanding the entropy to be positive we get, $
1+\frac{9}{2} f_{\infty}\alpha>0,
$ from which $
\alpha > -\frac{5 }{27}.
$ Thus we have, \[b6\] - < . If we repeat our analysis for a zero mass hyperbolic black hole, the resulting bound we obtain is weaker than this.
### D=3
We end this section by exploring the case of NMG theory in 3d. The action is given by,
S = d\^3x ( R+ + l\^2(R\_[ab]{}R\^[ab]{}-R\^2) ) \[clag\] The critical point is now at $\alpha=-3.$ For this case, =(1-+ l\^2 f”(r,v)- l\^2 r(v) f”’(r,v)). The background solution is taken as, f(r)= f\_(1-()\^2), where the quantity $f_{\infty}$ satisfies $
1-f_{\infty}+ \alpha f_{\infty}^2=0.
$ Then we get the following bound, \[b7\] . Demanding the positivity of the black hole entropy we get, $
1+\frac{ f_{\infty} \,\alpha}{2}>0.
$ This gives, $
-3 < \alpha < 1.
$ Thus we have, -3 < . In this case, even if we consider a zero mass hyperbolic black hole we obtain the same bound.
Holographic $c$-functions from the entropy evolution equation
=============================================================
In this section we will construct a holographic $c$-function using the FPS entropy functional for general curvature squared theories. Here we will again use the evolution equation for $\Theta$ (\[Ray3\]) and in the process we will have a geometric derivation for the $c$-function.
For unitary Lorentz invariant theory one can define a function in the space of couplings which decreases monotonically from UV to IR, parametrizing the RG flow. It coincides with the central charges of the corresponding CFTs at the UV and IR fixed point. In $1+1$ and $3+1$ dimensions this has been established rigorously [@Zam; @Kom]. Now from the holographic point view this $c$ function plays a pivotal role in understanding nature of RG flow. A proposal for the $c$ function for arbitrary theories of gravity has been given in [@ctheorems; @ctheorems2; @quasitop] for arbitrary dimensions. In this section we will use the Raychaudhuri equation as discussed in the previous sections and derive a $c$ function for a general curvature squared theory. For general higher curvature theories, we do not expect a c-theorem as there are bound to be problems with unitarity in such theories. Thus some condition on the couplings, which presumably only explore a subset of conditions leading to unitarty as in [@ctheorems], is expected. This is what we will find as well. At the UV and IR fixed points this function will coincide with the A-type anomaly coefficient for the curvature squared theory and it will also have to be a monotonically decreasing quantity along the RG flow. To prove the monotonicity we will use Raychaudhuri equation. The basic setup is similar as before.
Setup
-----
To derive a $c$ function purely in terms of geometric quantities we will first start with a specific example. We will consider domain wall type of geometry in $5$ dimensions, the metric for which is given below, ds\^2=dr\^2+e\^[2 A(r)]{}(-d\^2+dx\^2+dy\^2+ dz\^2). The spacetime can be foliated by surfaces of codimension-2 at every constant time slice. The induced metric on such surfaces is given by, ds\^[2]{}\_[surface]{}=e\^[2 A(r)]{}(dx\^2+dy\^2+dz\^2). Next we will consider a hypersurface orthogonal null congruence. The tangent vector for this congruences is given by, k\^[a]{}=(\_)\^a={-e\^[-A(r)]{},e\^[-2 A(r)]{},0,0,0}. It satisfies $
k^{a}\nabla_{a}k^{b}=0$ and $k_{a}k^{a}=0.
$ We can construct another auxiliary null vector $l^a$ as, l\^[a]{}={,,0,0,0 } such that, $k_{a}l^{a}=-1.$ Note that in this case $k^a$ and $l^a$ are not any kind of horizon null generators as compared to the previous sections. They simply correspond to a null congruence that converges along the light sheet projected out of the codimension-2 surface under consideration. Also in this section we will use affine parametrization.
Holographic c-functions in four derivative gravity
--------------------------------------------------
Next we will start with a guess for the $c$ function for curvature squared theory. The action for this theory is given by, S=d\^[4]{} x. The corresponding entropy functional is the FPS functional. S\_[FPS]{}=d\^[3]{} x (1+)= d\^[3]{} x (1+2). $\mathcal{R}$ is the 3 dimensional Ricci scalar intrinsic to the surface. Now we will start with the following candidate, c()= where as usual, $\Theta= \theta_k(1+\rho)+\frac{d\rho}{d\lambda}$ and $\lambda$ is the affine parameter. This appears to be a natural guess since for the Einstein theory it goes over to the c-function in [@cthor]. Then we have to show two things,
1. $c(\lambda)$ is monotonically decreasing under the flow along the null geodesic congruences. That is we have to check the sign of $\frac{dc(\lambda)}{d\lambda}.$
2. Further it has to coincide with the correct central charge (A-type anomaly in four dimensions) at the fixed points.
Now, $$\begin{aligned}
\begin{split}
\frac{ d c(\lambda)}{d\lambda} &= \frac{1}{\sqrt{h}\theta_k^4}\frac{d \Theta}{d\lambda}-\frac{ \Theta }{(\sqrt{h}\theta_k^4)^2}\frac{d}{d\lambda}(\sqrt{h}\theta_k^4)\\ &=\frac{1}{\sqrt{h}\theta_k^4}\frac{d \Theta}{d\lambda}-\frac{\Theta }{\sqrt{h}\theta_k^5}(\theta_k^2+4 \frac{d\theta_k}{d\lambda})
\end{split}
\end{aligned}$$ using $\frac{d\sqrt{h}}{d\lambda}=\sqrt{h}\theta_k.$ Next we will replace $\frac{d\Theta}{d\lambda}$ by (\[Ray2\]) with $\kappa=0$ and $\frac{d\theta_{k}}{d\lambda}$ by Raychaudhuri equation (\[Ray1\]). We obtain, =+ E\_[kk]{}, where, $$\begin{aligned}
\begin{split}
E_{kk}=&\frac{e^{-A(r)} (16 \alpha +5 \beta)^2}{243 A'(r)^5}\Big((2 A^{(3)}(r)+15 A'(r)^3+16 A'(r) A''(r)) (15 A'(r)^4+2 (A^{(4)}(r)+\\&8 A''(r)^2)+8 A^{(3)}(r) A'(r)+6 A'(r)^2 A''(r))\Big)+\frac{e^{-A(r)} (16 \alpha +5 \beta )}{486 A'(r)^4}\Big(-24 (A^{(4)}(r)\\&+8 A''(r)^2 )+60 A'(r)^4 (4 (10 \alpha +2 \beta +3 \gamma ) A''(r)-3)+A'(r)^2 A''(r)\\& (256 (10 \alpha +2 \beta +3 \gamma ) A''(r)-207 )+8 A^{(3)}(r) A'(r) (4 (10 \alpha +2 \beta +3 \gamma ) A''(r)-15)\Big)\\&-\frac{4 e^{-A(r)} (10 \alpha +2 \beta +3 \gamma ) A''(r)}{27 A'(r)^2}.
\end{split}
\end{aligned}$$ Also, $$\begin{aligned}
\begin{split}\label{dcdt}
\frac{dc(\lambda)}{d\lambda}&=\frac{ e^{-A(r)} (16\alpha+5\beta)}{162 A'(r)^5}\Big(-2 A^{(4)}(r) A'(r)+8 A^{(3)}(r) A''(r)-16 A^{(3)}(r) A'(r)^2\\&+15 A'(r)^3 A''(r)+48 A'(r) A''(r)^2\Big)-\frac{e^{-A(r)} A''(r)}{9 A'(r)^4}
\end{split}
\end{aligned}$$ and c()=. Notice that in (\[dcdt\]) the term proportional to $16\alpha+5\beta$ has no possibility of having a definite sign and it does not cancel with terms in $E_{kk}$ so it is natural to set the coefficient to zero. Now $\sqrt{h}\theta_{k}^4$ is positive and $T_{kk}$ using the null energy condition is positive. With $ 16\alpha+5\beta=0$, we find E\_[kk]{}= (-). where we have used $\frac{d}{d\lambda}= k^{a}\nabla_{a}.$ So we will have now, =+(-), Then we can define a effective quantity, |c()= c()+ such that, = 0, Next evaluating $\bar c(\lambda)$ and multiplying both side by a factor of $-\frac{1}{27}$ we get, () 0. Now in terms of the radial coordinate $r$ this becomes, (k\^[r]{}\_[r]{} + k\^\_ )() 0. From this, e\^[-2 A(r)]{} () 0. So , $$\bar c(r)=\frac{1-4 (10 \alpha +2 \beta +3 \gamma ) A'(r)^2}{A'(r)^3}$$ is a monotonically decreasing quantity along the RG flow, where the RG scale is determined by the radial coordinate $r$. At the fixed point the geometry is AdS and hence $$A(r)\equiv \frac{r}{l_{ads}}.$$ The $\bar c(r)$ coincides (upto an inconsequential overall factor) with the A-type anomaly coefficient at UV and IR (see eg. [@anoms] for expressions in general higher curvature theories). In this construction, the starting point of the entropy functional played an important role in reaching the condition $16\alpha+5 \beta=0$. In [@ctheorems; @ctheorems2], this condition emerged by demanding the existence of a “simple" c-function. Our analysis gives another way of looking at the same condition. One may wonder if it is possible to come up with a scenario where we did not need to absorb $E_{kk}$ into the c-function like what we have done. For example, instead of the HEE functional, perhaps one could try to construct a functional that cancels the $E_{kk}$ on the [*rhs*]{} of the Raychaudhuri equation. This does not appear to be the case–starting with the Wald entropy functional instead of the HEE functional does not naturally lead to the $16\alpha+5 \beta=0$ condition.
Discussion
==========
In this paper, we considered four derivative gravity and subjected it to the condition that the second law of black hole thermodynamics is satisfied at linear and quadratic orders in perturbation. We showed that the entropy functional is uniquely fixed (upto second order in extrinsic curvatures) and coincides with the holographic entropy functionals. We also derived interesting bounds on the coupling which arises at next to leading order in perturbations. We conclude here with a brief discussion on possible future directions.
- One should consider more generic perturbations to see what happens at linear and second order. This is important since the possibility exists that for generic perturbations the second law holds only for very specific cases like the Lovelock theories. In [@camanho], it was argued that causality constraints would require adding an infinite set of higher spin massive modes in a GB theory for consistency. It will be interesting to see if the second law knows about this in the perturbative sense used in the current work.
- By considering asymptotically flat black holes and the Gauss-Bonnet theory, we found that for the negative GB coupling we get a minimum horizon radius for the second law to hold. Taking this to be a pathology, this appears to disfavour this sign of the coupling. We further saw that the second law appears to know about the sound channel instablity in the context of Gauss-Bonnet holography. It will be interesting to see if $\lambda>-1/8$ which corresponds to the scalar channel instability [@Brigante] can also be seen from the second law.
- One should consider how our analysis will change if there is matter coupling to the higher curvature terms. In this case, it is not clear how to define the null energy condition. It seems natural that the matter couplings will get constrained by demanding the second law.
- Our analysis can be extended to general theories such as the quasi-topological theories [@quasitop]. In order to get a sufficient number of conditions to fix all the extrinsic curvature terms, one will need to turn on generic perturbations.
- In case of gravitational Chern-Simons terms, it will be interesting to see if and how the validity of second law can fix the entropy functionals [@loga]. Since the ${\it rhs}$ of the Raychaudhuri equation does not know about topological terms, the linearized analysis will not be able to capture the effect of such terms. One may need to consider topology changing processes like black hole mergers [@Sarkar:2010xp] to see the effect of such terms.
- It will be interesting to compare our analysis with what arises from similar considerations in the fluid-gravity correspondence [@shiraz]. Demanding a positive entropy current for higher curvature theories will lead to constraints which are presumably going to be similar to what we have found.
- The connection between the second law constraints and the positivity of relative entropy [@rel] in the context of holography can also be explored. We found that when we consider the second law for topological black holes with zero mass, we recover the bound from the tensor channel causality constraint. Why is this happening? A plausible explanation is as follows. The topological black holes with zero mass are just AdS spaces written in different co-ordinates and the finite Wald entropy of these black holes can be interpreted as an entanglement entropy across a sphere [@ctheorems; @chm] in the dual field theory. Thus the second law in this context may be related to the positivity of relative entropy due to a time dependent perturbation corresponding to the infalling matter we have considered. Relative entropy is expected to be positive for a unitary field theory. Hence there appears to be an interesting interplay between bulk causality and field theory unitarity. It will be interesting to make this connection more precise. It will also be interesting to understand the bounds in the context of c-theorems using the second law for causal horizons as advocated recently in [@sb].
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Shamik Banerjee, Sayantani Bhattacharyya, Menika Sharma and Aron Wall for discussions. AB thanks IIT-Gandhinagar for hospitality during the course of this work. The research of SS is partially supported by IIT Gandhinagar start up grant No: IP/IITGN/PHY/SS/\
201415-12. AS acknowledges support from a Swarnajayanti fellowship, Govt. of India.
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[^1]: In the context of entanglement entropy, one typically evaluates this formula on a codimension-2 surface with one timelike $n^{(1)a}$ and one spacelike $n^{(2) a}$ normal such that , $n^{(1)a}=\frac{k^a+l^a}{\sqrt{2}}$ and $n^{(2) a}=\frac{k^{a}-l^{a}}{\sqrt{2}}$ and the extrinsic curvature $\mathcal{K}_{ab}^{(i)}$ for that codimension-2 surface with a induced metric $h_{ab}$ defined as, $\mathcal{K}_{ab}^{(i)}=\frac{1}{2}\mathcal{L}_{n^{(i)}} g_{ab}.$ Trace of this is defined as $\mathcal{K}^{(i)}=\mathcal{K}_{ab}^{(i)}h^{ab}$. Also we have $\mathcal{K}_{i}\mathcal{K}^{i}=-2\theta_{k}\theta_{l}$ and $\mathcal{K}_{(i)ab}\mathcal{K}^{(i)ab}= -2\Big(\frac{\theta_{k}\theta_{l}}{D-2}+\sigma_{k}^{ab}\sigma_{l ab}\Big)$, where $D$ is the dimension of the bulk spacetime.
[^2]: The criterion in [@Wall:2015raa] is sufficient but may not be necessary.
[^3]: Interested readers are referred to [@Kar; @Pois] for discussions on the Raychaudhuri equation.
[^4]: We can repeat the same analysis for higher dimensions using the normalizations in [@holoGB]. Curiously, we find the same bound $\lambda_{GB} < \frac{9}{100}$ which is stronger than the causality bounds for $d>5$ in [@holoGB]. For flat case ($k=0$), the bound on GB coupling in $d$ dimensions can be expressed as $\lambda_{GB}<1/(2(d-1))$. This means for any dimension the strongest bound arises when the black holes have hyperbolic horizon ($k=-1$) with $r_0=0$. This would also suggest that in the large $d$ limit, $\eta/s\rightarrow 1/4\pi$ contrary to $1/8\pi$ obtained in [@holoGB].
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abstract: 'In driven-dissipative systems, the presence of a strong symmetry guarantees the existence of several steady states belonging to different symmetry sectors. Here we show that, when a system with a strong symmetry is initialized in a quantum superposition involving several of these sectors, each individual stochastic trajectory will randomly select a single one of them and remain there for the rest of the evolution. Since a strong symmetry implies a conservation law for the corresponding symmetry operator on the ensemble level, this selection of a single sector from an initial superposition entails a breakdown of this conservation law at the level of individual realizations. Given that such a superposition is impossible in a classical, stochastic trajectory, this is a a purely quantum effect with no classical analogue. Our results show that a system with a closed Liouvillian gap may exhibit, when monitored over a single run of an experiment, a behaviour completely opposite to the usual notion of dynamical phase coexistence and intermittency, which are typically considered hallmarks of a dissipative phase transition. We discuss our results with a simple, realistic model of squeezed superradiance.'
author:
- Carlos Sánchez Muñoz
- Berislav Buča
- Joseph Tindall
- 'Alejandro González-Tudela'
- Dieter Jaksch
- Diego Porras
bibliography:
- 'Sci.bib'
- 'arXiv.bib'
- 'books.bib'
title: 'Symmetries and Conservation Laws in Quantum Trajectories: Dissipative Freezing'
---
Driven dissipative systems are ubiquitous in many body physics and cavity QED [@amo09a; @rodriguez17a; @fink18a; @carr13a; @melo16a; @fitzpatrick17a; @baumann10a; @klinder15a; @hamsen18a; @teufel11a; @kolkowitz12a; @pigeau15a]. These systems are typically gapped and feature a unique, non-equilibrium steady state. In the regime of a dissipative phase transition (DPT), however, this gap vanishes and the null-space of the Liouvillian is spanned by several compatible steady-states. [@kessler12a; @minganti18a; @carmichael15a; @weimer15a; @benito16a; @sieberer13a; @sanchezmunoz18b; @biondi17a; @hwang18a; @mendoza16a]. Due to their fundamental interest and practical applications, such as enhanced metrological properties [@macieszczak16b; @fernandezlorenzo17a], DPTs have attracted a significant amount of attention, with much work being devoted to study the associated phenomena of bistability [@fink18a; @carr13a; @melo16a; @mendoza16a; @letscher17a; @muppalla18a; @schuetz13a; @schuetz14a], hysteresis [@rodriguez17a; @hruby18a], intermittency [@lee12a; @fitzpatrick17a; @hruby18a; @malossi14a; @muppalla18a; @ates12a], multimodality [@letscher17a; @malossi14a], metastability [@macieszczak16a] and symmetry breaking [@manzano14a; @wilming17a; @hannukainen18a]. All these effects are understood as different manifestations of the coexistence of several non-equilibrium phases. In particular, many experiments will look for intermittency as the hallmark of such phase coexistence [@lee12a; @fitzpatrick17a; @hruby18a; @malossi14a; @muppalla18a; @ates12a]. Intermittency is a phenomenon defined by a random switching between periods of high and low dynamical activity (for instance in the rate of photon emission). This behaviour, which is observed during a single run of the experiment, is conveniently described using the formalism of quantum jumps in which the system is characterized in terms of a pure wavefunction that undergoes stochastic evolution [@zoller87a; @molmer92a; @plenio98a]. The timescale $\tau$ of this intermittency is given by the inverse of the Liouvillian gap or asymptotic decay rate (ADR), i.e. the eigenvalue $\lambda_2$ of the Liouvillian operator $\mathcal L$ with the second largest real part [@macieszczak16b; @flindt13a; @hickey14a]. Since a DPT is defined by a vanishing Liouvillian gap [@kessler12a; @minganti18a], it will necessarily imply that $\tau$ diverges. In most typical situations, this closing is reached in the thermodynamic limit of a many-body system. Consequently, for any finite system, the long-time limit where intermittency is observable will, at least formally, exist. There are, however, situations in which the Liouvillian gap vanishes exactly and such a long-time limit cannot be taken. This is the case of systems featuring a *strong symmetry* [@buca12a]. Liouvillians $\mathcal L$ with a strong symmetry have a degenerate steady state—implying that $\lambda_2=0$—and an associated conservation law for the symmetry operator, $\dot A=\mathcal L^\dagger A=0$ [@buca12a; @albert14a]. Since the Liouvillian gap is closed exactly for any system size, the long-time limit of intermittency described before does not exist, and the dynamics is split into different, unconnected ergodic symmetry sectors.
In this work, we study the quantum trajectories of open quantum systems with a strong symmetry. We show that, when initialized in a superposition involving different symmetry sectors, the system will evolve towards a single one of them in each individual trajectory, remaining there for the rest of the realization. This non-ergodic phenomenon, that we term *dissipative freezing*, is in stark contrast with the typical looked-for phenomenology of intermittency in a DPT and predicts a completely different dynamical behaviour at the level of individual realizations of the experiment. Related effects have already been discussed in different contexts: in Ref. [@benoist17a] exponential stability of subspaces for quantum trajectories was demonstrated; in Ref. [@benoist14a] it was shown that a quantum stochastic master equation describing non-demolition measurements converges to a pure state; in Ref. [@vanHorssen14a] a similar effect was discussed for quantum Markov chains. An important result of our work is to relate this phenomena to the symmetries of the master equation. Notably, it implies that the conservation law for the strong-symmetry operator is broken at the level of trajectories and can only be recovered under ensemble-averaging. This is a purely quantum phenomenon, since it requires an initial superposition of different symmetry sectors that cannot be implemented classically, i.e. a single classical trajectory is always fully realistic and located in only one of this sectors. Understanding this phenomena is important for the dynamical characterization of dissipative systems with a closed Liouvillian gap. This limit has been proven relevant in quantum metrology, since it yields a Heisenberg scaling (quadratic in time) of the quantum Fisher information [@macieszczak16b].
To discuss the effect of dissipative freezing, we analyse a model that can be solved numerically, yet displays a rich variety of non-ergodic dynamics. This model consists of a coherently-driven spin ensemble with squeezed, collective spin decay, which can be implemented by adiabatic elimination of a cavity mode coupled to a multi-component atomic condensate via cavity-assisted Raman transitions [@jaksch01a; @micheli03a; @dimer07a; @dallatorre13a; @gonzaleztudela13b].
*Model and phase diagram.—* The master equation describing the dynamics of the $N$-spin ensemble is ($\hbar=1$): $$\dot{\rho}=-i\Omega[S_x,\rho]+\frac{\Gamma}{2J}\mathcal L_{D_\theta}[\rho],
\label{eq:master-equation}$$ where $\mathcal{L}_O[\rho]\equiv 2 O\rho O^\dagger - \{O^\dagger O, \rho\} $ is the usual Lindblad superoperator [@carmichael_book02a], and the operator $D_\theta$ describes the quantum jumps undergone by the system, $D_\theta\equiv \cos(\theta) S_- + \sin(\theta) S_+$. In these equations, $S_\pm$ and $S_z$ are collective spin operators obeying angular momentum commutation relations, $\Omega$ is the driving amplitude, $\Gamma$ is the quantum-jump rate, and $J=N/2$ is the total angular momentum, which is conserved in the dynamics. The squeezed decay operator $D_\theta$ includes both $S_-$ and $S_+$, with a weight parametrized by the angle $\theta$ and an associated dark state which is a spin-squeezed state for $\theta\neq (0,\pi/2)$ [@dallatorre13a]. Fig. \[fig:2\] depicts the phase diagram in the $(\Omega,\theta)$ plane in terms of the magnetization and spin-squeezing, featuring two types of non-equilibrium phases (discussed in more detail in Ref [@arXiv_sanchezmunoz19a]) *i)* The *ferromagnetic (F) phase* is characterized by a well-defined magnetization, diverging spin-squeezing at the phase transition, small fluctuations in the counting distributions of quantum jumps, high purity and ergodic dynamics. Any initial state eventually relaxes into a stationary, almost pure gaussian steady-state. In the thermodynamic limit, this phase is well described within a Holstein-Primakoff approximation. *ii)* In the *thermal (T) phase* the steady-state is highly mixed, and close to the infinite-temperature state $\rho \propto \mathbb{1}$. This phase is characterized by zero mean magnetization, small purity, large spin fluctuations, high rate of quantum jumps and large fluctuations in the output field.
![Steady state observables for a finite system size $N=50$. Dashed lines indicate the critical line $\Omega_c(\theta)$. (a) Spin magnetization $M\equiv \langle S_z\rangle/J$, featuring ferromagnetic ($\text F$) and thermal ($\text T$) phases. (b) Spin squeezing $\xi_\bot^2\equiv N \langle\Delta S_x\rangle^2/\langle \mathbf S\rangle^2$.[]{data-label="fig:2"}](fig2-phase-diagram-squeezing.pdf){width="1\columnwidth"}
The transition from the ferromagnetic to the thermal phase occurs at the critical driving $\Omega_c(\theta)=\Gamma(\cos^2\theta-\sin^2\theta)$. All the results presented here apply to the case $\theta\leq \pi/2$, trivially extended to $\theta \geq \pi/2$ by a spin flip. Hence, we have a spin-up and spin-down version of each phase, denoted as $\text F_{\uparrow/\downarrow}$ and $\text T_{\uparrow/\downarrow}$ in Fig. \[fig:2\].
*Liouvillian eigenvalues and symmetries.—* In the large driving limit of the thermal phase the Liouvillian features a particularly interesting spectrum of eigenvalues that can be derived analytically [@arXiv_sanchezmunoz19a; @ribeiro19a]: $$\lambda_{q,k}^\pm=\pm iq\Omega-\frac{\Gamma_\theta}{2J}q^2-\frac{\chi_\theta}{4J}\left[q+k(1+k+2q)\right],
\label{eq:exact-eigenvalues}$$ with $\Gamma_{\theta}\equiv \Gamma(\cos\theta+\sin\theta)^2$, $\chi_{\theta}\equiv \Gamma(\cos\theta-\sin\theta)^2$, $q=0,1,\ldots 2J$, $k = 0,1,\ldots 2J-q$. Equation shows that, besides the steady-state eigenvalue $\lambda_{0,0}=0$, other eigenvalues with zero real part can be obtained in two ways: either reaching the thermodynamic limit $J\rightarrow \infty$, or setting $\theta=\pi/4$ ($\chi_\theta=0$). For any fixed $q$, $\lim_{J\rightarrow\infty}\Re[\lambda_{q,k}^\pm]=0$, which implies eigenstates with finite, purely imaginary eigenvalues and, therefore, the absence of stationarity and emergence of oscillatory dynamics in the long-time limit [@buca19a], which has recently attracted attention in similar models [@iemini18a; @tucker18a]. The focus of this paper is, however, the situation $\theta=\pi/4$, where the Liouvillian gap closes exactly for any system size due to the presence of a *strong symmetry*, i.e., an operator $A$ that satisfies $[H,A]=0$ and $[L_\mu,A]=0$, with $H$ the Hamiltonian and $L_\mu$ the set of quantum-jump operators of the master equation [@buca12a]. In the model considered here, $A=S_x$. All the $\rho^{(m)}_0=|m\rangle\langle m|$ built from eigenstates $|m\rangle$ of $S_x$ are steady states of the dissipative dynamics [^1].
*Dissipative freezing of the dynamics.—* The exact closing of the Liouvillian gap for any system size in the presence of a strong symmetry differs from the usual situation in which this closing, characteristic of a DPT [@kessler12a; @minganti18a; @fink18a; @fitzpatrick17a], occurs in the thermodynamic limit [@fink18a; @carr13a; @melo16a; @mendoza16a; @letscher17a; @lee12a; @fitzpatrick17a; @hruby18a; @muppalla18a; @ates12a; @malossi14a]. In the presence of a strong symmetry, multiple degenerate steady states can exist [@buca12a] and the actual steady state of the system is then composed of a particular superposition of these states, fixed by the initial conditions [@macieszczak16a; @minganti18a]. Since in this case the evolution is not necessarily ergodic, it is not guaranteed that a single trajectory will switch among these states, which is the main assumption behind the notion of intermittency [@lee12a; @fitzpatrick17a; @hruby18a; @malossi14a; @muppalla18a; @ates12a]. More importantly, we want to clarify whether the conservation law $\dot A=0$ will apply at the level of an individual trajectory, since, to the best of our knowledge, this is only guaranteed when $A$ is unitary.
![Three different quantum trajectories at $\theta=\pi/4$ for the same initial state (a superposition of three eigenstates of $S_x$). Panels (a-c) show the three possible types of trajectories that occur. The inset in (a) shows the exponential decrease of the occupation of non-selected states. Parameters: $J=5$, $\Omega=0.8\Gamma$. []{data-label="fig:symmetry"}](fig-ssb.pdf){width="1\columnwidth"}
The model of squeezed superradiance that we consider here represents, in the particular case $\theta=\pi/4$, one of the simplest implementations of a strong symmetry. The unraveling of the evolution in individual trajectories in this model reveals an effect that we term “dissipative freezing” of the dynamics. The phenomenon is depicted in Fig. \[fig:symmetry\] (a–c): after initializing the state in a given superposition—in this example, of the $S_x$ eigenstates $|0\rangle$, $|3\rangle$ and $|5\rangle$—the wavefunction eventually evolves into a single one of the eigenstates, with the occupation of any of the other ones decaying exponentially with time. The evolution is thus effectively frozen in one eigenstate for any individual realization of the dynamics, and the conservation law $\dot S_x=0$ is broken.
An eigenstate of a strong symmetry is stationary under this stochastic evolution. To prove this, we consider the general form of any wavefunction undergoing a stochastic, dissipative evolution described by $\tilde H$ and the set of quantum jump operators $\{L_\mu\}$. Starting from an initial state $|\psi(t_0)\rangle$, the wavefunction evolves for a time $t$ experiencing $n$ quantum jumps at times $(t_1,\ldots,t_n)< t$ with jump operators $(L^{(1)},\ldots,L^{(n)})$, where $L^{(i)}\in \{L_\mu \}$. The form of the wavefunction is then given by a nonunitary evolution $|\psi(t)\rangle = \frac{1}{\mathcal N}\tilde U(t,t_n,\ldots,t_0)|\psi(t_0)\rangle$, where $\mathcal N$ is a normalizing constant, and $\tilde U(t,t_n,\ldots,t_0)$ is an evolution operator given by: $$\tilde U(t,t_n,\ldots,t_0) = e^{-i \tilde H (t-t_n)}\prod_{m=1}^{n} L^{(m)}e^{-i \tilde H (t_{m}-t_{m-1})},$$ with $\prod_{m=0}^n O_m \equiv O_n \cdot O_{n-1}\cdot\ldots \cdot O_0 $. Let us consider a strong symmetry operator $A$, so that $[A,\tilde U]=0$. Therefore, if $|\psi(t_0)\rangle$ is an eigenstate of a strong symmetry $A|\psi(t_0)\rangle=\lambda |\psi(t_0)\rangle$, we obtain $$\begin{gathered}
A|\psi(t)\rangle = A \tilde U(t,t_n,\ldots,t_0) |\psi(t_0)\rangle \\ = \tilde U(t,t_n,\ldots,t_0) A|\psi(t_0)\rangle = \lambda |\psi(t)\rangle,\end{gathered}$$ i.e. an eigenstate of $A$ remains unchanged at the level of individual trajectories. This proof can be easily extended to the eigenstates of any power $A^n$. This fact may suggest that any quantum trajectory could eventually get “trapped” into one of these eigenstates, in a picture somewhat analogue to dark-state cooling [@griessner06a] or population trapping [@aspect88a]. However, for this to happen, the combination of non-Hermitian Hamiltonian evolution and quantum jumps (which have opposing effects on the occupancy of each eigenstate) should bring the system into one of these eigenstates in the first place. It is a priori not certain that this will occur.
Here, we prove that this is indeed the case when $\dot\rho = -i\Omega[A,\rho] + \Gamma/(2J)\mathcal L_A\{\rho\}$; i.e. dynamics with a single quantum jump $L$ and a general, Hermitian strong-symmetry $A\propto H \propto L$. We set $t_0 = 0 $ and consider an initial state $|\psi(0)\rangle = \sum_m c_m(0)|m\rangle$, expanded in the basis of eigenstates of $A$, $|m\rangle$, with eigenvalue $m$. For *any* general quantum trajectory that evolves for a time $t$ undergoing $n$ quantum jumps, the probability for the final state to be in an eigenstate of $|m\rangle$ takes the form [@arXiv_sanchezmunoz19a]: $$p(m; t,n) = \frac{1}{\mathcal N} \left(e^{- |m|^2}|m|^{2\alpha}\right)^{t\Gamma/J}|c_m(0)|^2,
\label{eq:S-freezing}$$ with ${\alpha = nJ/(t\Gamma)}$ and $\mathcal N$ a normalizing constant. The exponent $t\Gamma/J$ in Eq. tends to enhance the maximum of the function in parenthesis as time increases. Hence, after normalization, $p(m;t,n)$ tends to zero for all $m$ except for the optimum value. Since the function $e^{-x}x^{\alpha}$ has a maximum at $x=\alpha$, only the eigenstates $|m\rangle$ from the subspace of $A^\dagger A$ yielding the minimum $|\alpha-|m|^2|$ have a non-zero occupancy in the long-time limit $t\gg J/\Gamma$. Equation thus encapsulates the essence of the dissipative freezing effect and is the main result of this paper: for $t\gg J/\Gamma$, any general trajectory will be trapped in an eigenspace of $A^\dagger A$, consequently breaking the conservation law $\dot A=0$ if initialized in a superposition of different eigenspaces. In the long-time limit, the total number of jumps recorded in a trajectory allows one to unambiguously determine, from those eigenspaces of $A^\dagger A$ having an overlap with the initial state, which one has the system been trapped into.
For the particular case that we study in this paper, $H \propto L\propto A=S_x$, $m=-J,\ldots J$. In this case, the eigenstates of $A^\dagger A=S_x^2$ are doubly degenerate. For $t\gg J/\Gamma$, the probability distribution for any quantum trajectory is $p(m;t,n) \propto\sum_{ m}(\delta_{m,\tilde m}+\delta_{m,-\tilde m})|c_{m}(0)|^2$, with $\tilde m$ the natural number $\leq J$ closest to $\sqrt{nJ/(t\Gamma)}$. The resulting probability distribution versus $n/t$ is plotted in Fig. \[fig:pn\_freezing\](a) for an initial state composed of an equal superposition of all the eigenstates.
The phenomenon can be interpreted in terms of a quantum-measurement description of dissipative dynamics [@haroche_book06a; @wiseman_book10a; @gammelmark14a]. The information provided by the quantum jumps makes the eigenspaces of $A^\dagger A$ with a particular eigenvalue increasingly likely, and continuously updates the state accordingly. We stress that this picture applies to any dissipative dynamics and that this update is different from a projective measurement of $A^\dagger A$ that would collapse the system into one of its eigenstates. In most situations, the update after each jump is not able to freeze the state due to the non-Hermitian evolution between jumps, which also changes the occupancy of these eigenstates. In our case, this is prevented by the strong symmetry, and the effect of the jumps pile up, giving rise to the phenomenon of dissipative freezing.
![(a) Probability distribution for any quantum trajectory at time $t=100 J/\Gamma$, versus the number of jumps $n$, in the model of squeezed superradiance for $\theta=\pi/4$, $J=10$. The initial state is an equal superposition of all the eigenstates $|m\rangle$ of $S_x$. The resulting wavefunction always freezes into an eigenstate of $S_x^2$. Inset shows the probability distribution for the value $n/t = 5$ indicated by the dashed line. (b) $p_T(K)$ at $T=3\cdot 10^3/\Gamma$ in the case of our model. $N=20$, $\Omega = 0.8\Gamma$, $\theta = \pi/4$ (strong symmetry point). (c) $\langle k\rangle_s$ versus $\theta$, featuring the coexistence between a bright and a dark phase in the vicinity of $\theta=\pi/4$. []{data-label="fig:pn_freezing"}](fig-pn-freezing.pdf){width="1\columnwidth"}
*Multimodality*. Having demonstrated the absence of intermittency in the presence of a strong symmetry, it is important to discuss the implications in other phenomena that are usually also linked to DPTs and dynamical phase coexistence, such as the multimodality of the activity distribution [@ates12a]. We analyse the problem from the framework of the thermodynamics of quantum trajectories. The activity is defined as the mean number of quantum jumps undergone by the system per unit time, which can be expressed through the probability distribution $p_T(K)$ of counting $K$ jumps on a time $T$. In the theory of thermodynamics of quantum trajectories [@garrahan10a; @ates12a], the activity in the long time limit is postulated to follow a large deviation principle $p_T(K) \asymp e^{-T\varphi(K/T)}$, where the rate function $\varphi(K/T)=-\ln p_T(K)/T$ has the properties of an entropy density [@touchette09a]. Equivalently, the cumulant generating function also has a large deviation form $Z=\langle e^{sK}\rangle \asymp e^{t\lambda(s)}$. Here, $\lambda(s)$ plays the role of a free energy, related to the entropy by a Legendre transformation $\lambda(s) = \max_k[ks-\varphi(k)]$. $\lambda(s)$ encapsulates the statistical properties of the trajectories, and it allows to write the mean activity as $\langle k \rangle \equiv \langle K\rangle/T= \partial \lambda(s)/\partial s|_{s=0}$. $\lambda(s)$ can be obtained as the largest eigenvalue of the tilted Liouvillian $\mathcal W_s(\rho) = \mathcal L(\rho) - (1-e^s)L\rho L^\dagger$, with $L$ the operator inducing the jumps that we are recording. Once $\lambda(s)$ is found this way, the Legendre inverse transformation $\varphi(k)=\max_s[ks-\lambda(s)]$ allows us to obtain $\varphi(k)$. This relation, however, requires $\lambda(s)$ to be differentiable for all $s\in \mathbb{R}$ or, equivalently, $\varphi(k)$ to be concave for all $k\in\mathbb{R}$ [@touchette09a].
In the presence of a strong symmetry, however, $\varphi(k)$ is non-concave and, therefore, $\lambda(s)$ is non-analytic. To show this, we consider a strong symmetry $A$ with eigenstates $|m\rangle$ and only one quantum jump operator, $L=\sqrt{\Gamma/J}A$. For an initial state $\rho(0) = \sum_m c_m(0) |m\rangle\langle m|$, the quantum-jump probability distribution takes the form [@arXiv_sanchezmunoz19a]: $$p_T(K)= \sum_m \frac{1}{K!}\left(\frac{T\Gamma |m|^2}{J} \right)^K e^{-\Gamma |m|^2 T/J}c_m(0),
\label{eq:pn}$$ which is dependent on the initial state via the coefficientes $c_m(0)$. This equation presents the multimodal structure depicted in Fig. \[fig:pn\_freezing\](b), plotted for the particular case of our model, $A=S_x$. Each steady state $\rho_0^{(m)}=|m\rangle\langle m|$ with non-zero overlap with $\rho(0)$ manifests as a distinct peak in the counting distribution $p_T(K)$, centered at the value $K_m=T|m|^2\Gamma/J $, which is the emission rate expected for that particular state. The multimodal structure of $p_T(K)$—and consequently of $\varphi(k)$—cannot be obtained from $\lambda(s)$ through an inverse Legendre transformation [@touchette09a], therefore, it points towards a non-analicity of $\lambda(s)$. As we prove in the Ref [@arXiv_sanchezmunoz19a], this non-analicity consists of a discontinuity of $\langle k\rangle_s \equiv \partial \lambda(s)/\partial s $ at $s=0$, as it is usually described in the context of dynamical phase transitions [@garrahan07a; @garrahan10a; @ates12a; @flindt13a; @hickey14a]. A first-order phase transition in $\langle k\rangle_s$ is therefore linked to the phenomenon of dissipative freezing.
Our model allows us to explore how this discontinuity turns into a continuous crossover as we depart from the strong-symmetry point $\theta=\pi/4$. This is shown in Fig. \[fig:pn\_freezing\](c), where we plot $\langle k\rangle_s$ versus $\theta$. For $\theta=\pi/4$, the limit $s\rightarrow 0^+$ features a bright phase characterized by a high activity, whereas for $s\rightarrow 0^-$ we find a dark phase with virtually no quantum jumps (see insets). Away from this point, we obtain a crossover consistent with the first-order phase transition smoothed by finite-size effects usually observed in finite many-body systems undergoing a DPT [@ates12a]. It implies that $\lambda(s)$ is analytic, and that, in the long-time limit, $p_T(K)$ is unimodal. Unimodality is a consequence of intermittency: the switching between different dynamical phases destroys the multimodal distribution for times longer than $\tau=-\Re(1/\lambda_2)$ [@macieszczak16b]. Therefore, intermittency is unequivocally connected to a crossover in $\langle k\rangle_s$ at $s=0$, and dissipative freezing, to a discontinuous, first-order transition. Alternatively, dissipative freezing can be described as the survival of multimodality in the long-time limit.
The survival of multimodality is of strong importance in the context of enhanced quantum metrology, where it has been proven that there is a Heisenberg scaling of the quantum Fisher information for times shorter than the correlation time $\tau$ [@macieszczak16b]. Since systems with a strong symmetry will feature an *asymptotic* quadratic scaling of the quantum Fisher information for all times, our results may be of relevance in the design of sensing protocols aimed to exploit this feature on continuous Bayesian parameter estimation from photon counting [@gammelmark14a; @kiilerich14a; @kiilerich16a]. Beyond the model considered here, our results have strong implications for the dynamical characterization of DPTs in more complex systems where the existence of a strong symmetry can provide a way to tune the Liouvillian gap to zero without the need of reaching a thermodynamic limit.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
C.S.M. kindly acknowledges F. Minganti for fruitful and insightful discussions. B.B and C.S.M are grateful to Juan P. Garrahan for very useful comments and insightful discussions. C.S.M. is funded by the Marie Sklodowska-Curie Fellowship QUSON (Project No. 752180). B.B., J.T. and D.J. acknowledge support from the EPSRC grants No. EP/P009565/1 and EP/K038311/1, and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 319286 Q-MAC. AGT and DP acknowledge support from CSIC Research Platform on Quantum Technologies PTI-001 and from Spanish project PGC2018-094792-B-100 (MCIU/AEI/FEDER, EU).
[^1]: Note that our system is fundamentally different from a single spin in a magnetic field pointing along the $x$ axis due to its driven-dissipative character: an initial superposition of different eigenstates of $S_x$ would never reach a stationary state in the purely Hamiltonian case.
|
---
abstract: 'The shape of an object is an important characteristic for many vision problems such as segmentation, detection and tracking. Being independent of appearance, it is possible to generalize to a large range of objects from only small amounts of data. However, shapes represented as silhouette images are challenging to model due to complicated likelihood functions leading to intractable posteriors. In this paper we present a generative model of shapes which provides a low dimensional latent encoding which importantly resides on a smooth manifold with respect to the silhouette images. The proposed model propagates uncertainty in a principled manner allowing it to learn from small amounts of data and providing predictions with associated uncertainty. We provide experiments that show how our proposed model provides favorable quantitative results compared with the state-of-the-art while simultaneously providing a representation that resides on a low-dimensional interpretable manifold.'
author:
- Alessandro Di Martino
- Erik Bodin
- Carl Henrik Ek
- 'Neill D.F. Campbell'
bibliography:
- 'references.bib'
title: |
Gaussian Process Deep Belief Networks:\
A Smooth Generative Model of Shape with Uncertainty Propagation
---
Introduction
============
The space of silhouette images is challenging to work with as it is not smooth in terms of a representation as pixels. A transformation that we would consider semantically smooth might correspond to a drastic change in pixel values. Our goal is to learn a smooth low dimensional representation of silhouette images such that images can be generated in a natural manner. Further, as data is at a premium, we want to learn a fully probabilistic model that allows us to propagate uncertainty throughout the generative process. This will allow us to learn from *smaller amounts of data* and also associate a quantified uncertainty to its predictions. This uncertainty allows the model to be used as a building block in larger models.
The results of our model challenge the current trend in unsupervised learning towards maximum likelihood training of increasingly large parametric models with increasingly large datasets. We demonstrate that by propagating uncertainty throughout the model, our approach outperforms two standard generative deep learning models, a Variational Auto-Encoder (VAE [@Kingma2013]) and a Generative Adversarial Network (InfoGAN [@chen2016infogan]) with comparable architectures and can achieve similar performance with far smaller training datasets.
In our work we revisit a few classic machine learning models with complementary properties. On the one hand, parametric models such as Restricted Boltzmann Machines (RBMs) [@smolensky1986parallel] are particularly interesting as they are stochastic, generative and can be stacked easily into *deeper* models such as deep belief networks (DBNs); these can be trained in a greedy fashion, layer by layer [@hinton2006reducing]. RBMs can approximate a probability distribution on visible units. DBNs, in addition, learn deep representations by composing features learned by the lower layers, yielding progressively more abstract and flexible representations at higher layers and often leading to more expressive and efficient models compared to shallow ones [@bengio2007scaling].
However, DBNs suffer from a number of limitations. Firstly, they do not guarantee a smooth representation in the learned latent space. Secondly, the classic contrastive divergence algorithm used for greedy training is slow and can place limitations on architectures. Finally, a DBN does not provide any explicit generative process from a manifold, as the standard way to sample from a DBN is to start from a training example and perform iterations of Gibbs sampling.
The Gaussian Process Latent Variable Model (GPLVM) [@lawrence2005probabilistic] combines a Gaussian process (GP) prior with a likelihood function in order to learn a representation. By specifying a prior that encourages smooth functions a smooth latent representation can be recovered. However, to make inference tractable the likelihood is also chosen to be Gaussian which does not reflect the statistics of natural images. Further, even though the mapping from the latent space is non-linear the posterior is linear in the observed space. This makes the GPLVM unsuitable for modelling images. To circumvent this one can compose hierarchies of GPs [@damianou2013deep], however, these models are inherently difficult to train.
The characteristics of the DBN and GPLVM can be considered complementary, where the DBN excels the GPLVM fails and vice versa. Unfortunately, combining the two models into a single one by simply stacking a GPLVM on top of a DBN would not preserve uncertainty propagation. Furthermore, this would pose a challenge to training (while the GPLVM is a non-parametric model trained by optimizing an objective function, a DBN is a parametric model, with non-differentiable Bernoulli units, and is trained with contrastive divergence). Another important challenge is learning from very little data. The ability to learn from a small dataset expands the applicability of a model to domains where there is a lack of available data or where collection of data is costly or time-consuming.
In this paper we address these challenges and present the following contributions:
1. A model (which we call GPDBN) that combines the properties of a smooth, interpretable manifold for synthesis with a data specific likelihood function (a deep structure) capable of decomposing images into an efficient representation while propagating uncertainty throughout the model in a principled manner.
2. We train the model end to end using back propagation with the same complexity as a standard feed-forward neural network by minimising a single objective function.
3. We also show that the model is able to learn from very little data, outperforming current generative deep learning models, as well as scaling linearly to larger datasets by the use of mini-batching.
Related Work {#related_work}
============
Modelling of shape is important for many computer vision tasks. It is beyond the scope of this paper to make a complete review of the topic, we refer the reader to the comprehensive work of Taylor [@statModelsOfShape]. In our work we focus on recent unsupervised statistical models that operate directly on the pixel domain. Interest in these models was revived by the Shape Boltzmann Machine (SBM) work of Eslami et al. [@eslami2013shape] and they have been shown to be useful for a variety of vision applications [@eslami2012generative; @deepPartBasedGenerativeShape; @Vedaldi2015semanticpartsegmentation]. These deep models can also be readily extended into the 3D domain, , by recent work on 3D ShapeNets [@shapenets2015]. Detailed analysis of the DBN, GPLVM and SBM is provided in § \[background\].
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Desirable Properties]{} Table \[tab:related\_work\] highlights the desirable properties of the most closely related previous works. We have identified four advantageous properties: (i) It is well known that pixel silhouettes are not well modelled by a Gaussian likelihood. (ii) The utility of an unsupervised shape model is well described by the properties of its latent representation. Ensuring a smooth manifold opens up a number of applications to data in the pixel domain that previously required custom representations, , interactive drawing [@sketchPaper]. (iii) A fully generative model ensures that there is a well defined space that can be sampled as well as interpreted; , dynamics models can be defined in such a space to perform tracking [@Rotopp2016; @victor2012pwp3dsdf]. (iv) Correctly propagating uncertainty is vital to perform data efficient learning, for example when data is scarce or expensive to obtain.
\
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Auto-Encoders]{} The VAE model by Kingma and Welling [@Kingma2013] performs a variational approximation of a generative model with a non-Gaussian likelihood through a feed-forward or Multi-Layer Perceptron (MLP) network. In addition, it uses MLP networks to encode the variational parameters (in a similar manner to [@lawrence2006local]). While this model provides a generative mapping, the feed-forward (decoder) network fails to propagate uncertainty from the latent space. Furthermore, the independent prior on the latent space does not promote a smooth manifold; any smoothness arises as a by-product of the MLP encoding network. This characteristic depends on the MLP architecture and is not directly parametrised. The key limitation of the VAE for our purposes is the lack of uncertainty propagation that results in poor results with limited training data.
The guided, non-parametric autoencoder model of Snoek et al. [@nonparametricGuidanceAutoEncoder] appears similar, however, there are a number of important differences. They use label information (supervision) to guide a latent space learning process for an autoencoder; this is not a pure unsupervised learning task and we do not have label information available to us. Furthermore, as with the VAE, uncertainty is not propagated from the latent manifold to the output space due to the use of the feed-forward network to the output. [startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[InfoGAN]{} Another prominent generative model in unsupervised learning is the Generative Adversarial Network (GAN) [@goodfellow2014gan]. The model learns an implicit generator distribution using a minimax game between a deep generator network, which transforms a noise variable to a sample, and a deep discriminator network, which is used to classify between samples from the generator distribution and the true data distribution. One issue common with GAN models is that they do not provide a smooth latent manifold for synthesis nor uncertainty in their estimates (like the VAE). From the plethora of different variations of GANs models available in the literature we have chosen to include in our comparisons the InfoGAN model [@chen2016infogan], since it also considers the goal of interpretable latent representations (by maximising the mutual information between a subset of GAN’s noise variables and observations).
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[ShapeOdds]{} The recent ShapeOdds work of Elhabian and Whitaker [@ShapeOdds] confers state-of-the-art performance and captures many of the desired properties including a generative probabilistic model that propagates uncertainty. The approach taken is quite different to ours as they specify a detailed probabilistic model including a Gaussian Markov Random Field (MRF) with individual Bernoulli random variables for the pixel lattice. In contrast, our model is more flexible, we allow the network to learn the structure from the data directly but ensure that we still maintain uncertainty quantification throughout. We would also argue that the specific form of the low dimensional manifold we generate is desirable with its guaranteed smoothness that makes the latent space readily interpretable. This provides the tradeoff between the two models. We expect the ShapeOdds model to perform very well at generalisation due to the inclusion of the MRF prior. In contrast, our model will be more data dependent in this respect (weaker prior assumptions on the nature of images), however, it provides a generative space that is highly interpretable and easy to work with. We identify that a topic for further work would be to combine our smooth priors with the likelihood model of ShapeOdds.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[GPLVM Representations]{} A possible workaround to the problem of non-Gaussian likelihoods is to perform a deterministic transformation to a domain where the data is approximately Gaussian. This has been successful for domains where, for example, the shape can be represented in a new geometric representation away from pixels, , parametric curves [@CampbellSIGGRAPH14; @victor2011nonlinearshapemanifolds]. However, this is application dependent and not suitable for arbitrary pixel based silhouettes considered here. A common approach that retains the pixel grid is to transform it into a level-set problem via the distance transform, , [@victor2012pwp3dsdf]. This can improve results in some settings, however, the uncertainty is not correctly preserved and therefore not correctly captured in predictions. We denote this model GPLVMDT in our comparisons.
Background
==========
Deep Belief Networks {#dbn}
--------------------
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[RBM]{} The restricted Boltzmann machine (RBM), or Harmonium, [@smolensky1986parallel] is a generative stochastic neural network that learns a probability distribution over a vector of random variables. The RBM is when stacked the basic the basic component of a deep belief network. The graphical model of the RBM is an undirected bipartite graph, consisting of a set of visible random variables (or units): $\bm{v}$, and a set of hidden units $\bm{h}$ (Fig. \[fig:rbm\]). Typically, all variables are binary (Bernoulli), taking on values from $\{0,1\}$.
The RBM model specifies a probability distribution over both the visible and hidden variables jointly as $$\label{gibbs_distr}
P(\bm{v}, \bm{h}) = \frac{1}{Z}\exp{(-E(\bm{v}, \bm{h}))}$$ which defines a Gibbs distribution with energy function $$\label{rbm-energy}
E(\bm{v}, \bm{h}) = -\bm{v}^\top\bm{W}\bm{h} - \bm{b}^\top\bm{v} - \bm{c}^\top\bm{h}\ ,$$ where $\bm{W}$, $\bm{b}$, $\bm{c}$ are the parameters of the model: $\bm{W}$ as a linear weight matrix and $(\bm{b},\bm{c})$ are bias vectors for the visible and hidden units respectively. The normalising constant $Z$ is the, computationally intractable, sum over all possible random vectors $\bm{v}$ and $\bm{h}$.
The bipartite structure of the model (, the graph has no visible-visible or hidden-hidden connections, as shown in Fig. \[fig:rbm\]), affords efficient Gibbs sampling from the visible units given the hidden variables (or vice versa). The conditional distribution of the hidden units given the visible ones, and vice versa, factorize as each set of variables are conditionally independent given the other: $$P(\bm{h} \,|\, \bm{v}) = \textstyle\prod_{j=1}^H P(h_j \,|\, \bm{v}), \;
P(\bm{v} \,|\, \bm{h}) = \textstyle\prod_{i=1}^V P(v_i \,|\, \bm{h}) \label{rbm-conditional}\ .$$ Replacing binary units with Gaussian units can be performed by modifying the energy function [@hinton2012practical]. Unfortunately, parameter learning is difficult since direct calculation of the gradients of the log likelihood w.r.t. the parameters requires the intractable computation of the normalising constant $Z$. In *current* practice, the approximate maximum-likelihood contrastive divergence algorithm is used [@carreira2005contrastive].
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[DBN]{} When multiple layers of RBMs are stacked on top of each other they form a deep belief network (Fig. \[fig:dbn\]). Hinton [@hinton2006reducing] demonstrated that a DBN can be trained in a greedy fashion, layer by layer. Essentially, the samples (activations) from the hidden units of a trained layer are used as the data to train the next layer in the stack.
\
\
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Sampling]{} Sampling from an RBM proceeds by conditioning on some input data and performing a Gibbs sample for the hidden units. Subsequently, a Gibbs sample can be drawn for the visible units by conditioning the hidden units on this sample. This process is then repeated for a number of cycles. Since a DBN is a stack of RBMs, this process has to be repeated for all layers; the output of one layer becomes the input to condition on for the next layer. In this way, an input data point can be propagated up and down the network.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Limitations]{} Although a DBN is good at learning low-dimensional stochastic representations of high-dimensional data, it has three key drawbacks that we will address by combining the strengths of the DBN with a flexible non-parametric model in § \[model\]:
1. It lacks a directed generative sampling process from a well defined latent representation. In order to generate a sample one must condition on some input data and propagate it through the network back and forth until a sample from the lowest layer is obtained.\
2. There is no explicit representation of the uncertainty, instead this only arises implicitly through the propagation of point estimates (samples) at each layer.\
3. A side effect of the conditional independence assumption of is that the correlations between the hidden units of the top layer of a DBN are not captured because each latent dimension is independent. Most importantly, a DBN does not, therefore, give any guarantee about learning a smooth latent space.
GPLVM {#gp}
-----
The Gaussian Process Latent Variable Model (GPLVM) [@lawrence2005probabilistic] learns a generative representation by placing a Gaussian process (GP) prior over the mapping from the latent to the observed data. This approach has the benefit that it is very easy to ensure a smooth mapping from the latent representations to the observed data. Further, due to the principled uncertainty propagation of the GP, all predictions will have an associated uncertainty.
In specific, each observed datapoint $\bm{y}_{n}$, $n \in [1,N]$, is assumed to be generated by a latent location $\bm{x}_{n}$ through a mapping $f$. Due to the marginalising property of a Gaussian, the predictive posterior over function values $\bm{f}^*$ at a test location $\bm{x}^*$ can be reached in closed form as, $$\begin{gathered}
\label{noise-free-conditional}
p(\bm{f}^* \mid \bm{Y}, \bm{x}^*, \bm{X}) = \mathcal{N}(\bm{m}_{{\mathrm{GP}}}, \sigma^2_{{\mathrm{GP}}})\\
\bm{m}_{{\mathrm{GP}}} = k(\bm{x}^*, \bm{X}) [k(\bm{X}, \bm{X})]^{-1} \bm{Y}\label{gp-pred-mean}\\ \sigma^2_{{\mathrm{GP}}} = k(\bm{x}^*, \bm{x}^*) - k(\bm{x}^*, \bm{X}) [k(\bm{X}, \bm{X})]^{-1} k(\bm{X}, \bm{x}^*)\label{gp-pred-var}\ ,\end{gathered}$$ where $k(\cdot,\cdot)$ is the covariance function specifying the Gaussian process and $\bm{X} = [\bm{x}_1,\dots,\bm{x}_N]^\top$. We used the common *squared exponential* kernel $$\label{squared-exponential}
k(\bm{x}, \bm{x}') = \alpha^2\exp\left(-\frac{1}{2\ell^2}\left\lVert \bm{x} - \bm{x}' \right\rVert^2 \right) \ ,$$ with hyperparameters $\alpha^{2}$ (signal variance) and $\ell$ (lengthscale), to ensure a smooth manifold. Importantly, even though the function $f$ can be non-linear, the relationship between the predicted mean and the training data $\bm{Y}$ is linear. Due to this linearity, a GPLVM is inherently not suitable for modeling image data.
Shape Boltzmann Machine {#sbm}
-----------------------
The Shape Boltzmann Machine (SBM) [@eslami2013shape] is a specific architecture of the Boltzmann machine. It consists of three layers: a rectangular layer of $N \times M$ visible units $\bm{v}$, and two layers of latent variables: $\bm{h}^{1}$ and $\bm{h}^{2}$. Each hidden unit in $\bm{h}^1$ is connected only to one of the four subsets of visible units of $\bm{v}$ (Fig. \[fig:sbm\]). Each subset forms a rectangular patch and the weights of each patch (except the biases) are shared so that a patch effectively behaves as a local receptive field. To avoid boundary inconsistencies, the patches are slightly overlapped (in Fig. \[fig:sbm\], the overlap has size $b$). Layer $\bm{h}^{2}$ is fully connected to $\bm{h}^{1}$.
While the SBM offers improved generalization over a DBN with the same number of parameters, the SBM has a fixed structure which is not easily extended to more layers or patches. In contrast, a DBN, as a stack of simple RBMs, has a more generic and flexible structure which can be adapted easily and combined with other models. Furthermore, like the DBN, the SBM lacks of a proper generative process.
The GPDBN Model {#model}
===============
In our model, we connect a DBN and GPLVM so that the data space of the GPLVM corresponds the latent space of the DBN (Fig. \[fig:model-diagram\]) to obtain a model that can be optimized by minimizing a single objective function.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[New Concrete Layers]{} The uppermost hidden layer of the DBN has Gaussian units to interface with the Gaussian likelihood of the GPLVM. In the lower layers, we replace the standard binary units with a *Concrete distribution* [@maddison2017concrete]. This is a continuous relaxation to discrete random variables, in our case, to the Bernoulli distribution. This allows us to draw low bias samples, in an analogous manner to the reparameterization trick [@Kingma2013], using a function that is differentiable with respect to the model parameters, $$\mathrm{Concrete}\left(p, u\right) = \mathrm{Sigmoid}\!\left(\textstyle\frac{1}{\lambda}\big(\log p -\log(1-p) + \log u - \log(1 - u) \big) \right)\ ,\label{concrete}$$ where $p$ is the parameter of a Bernoulli distribution, $\lambda$ is a scaling factor, which we fix to $0.1$, and $u$ is a uniform sample from $[0,1]$.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Learning]{} Given a dataset $\mathcal{D} = \{\bm{t}_{n}\}_{n=1}^N$, we train the model end-to-end by minimizing the following objective function jointly with respect to all the parameters and the matrix of latent points $\bm{X}$ (omitted from the notation to avoid clutter):$$\label{objective}
{\scriptsize{L = \textstyle\sum_{n=1}^N \underbrace{\left(\bm{t}_{n}\log(\bm{s}_{n}) + (1 - \bm{t}_{n})\log(1 - \bm{s}_{n}) \right)}_{\text{data term}}
+ \frac{1}{2} \underbrace{\mathrm{Tr}\!\left[\bm{K}^{-1}\bm{H} \bm{H}^\top\right]}_{\text{joint term}}
+ \frac{D}{2} \underbrace{\log |\bm{K}|}_{\text{\shortstack{complexity\\term}}} + \underbrace{||\bm{X}||^2}_{\text{\shortstack{prior\\term}}}\ .}}$$ Here, $\bm{t}_{n}$ is a training datapoint, $\bm{s}_{n}$ is a sample from the model, $\bm{K} = k\big(\bm{X}, \bm{X}\big) + \sigma^{2}\bm{I}_N$ is the covariance matrix of the latent points and $D$ is the number of Gaussian units in the uppermost DBN layer (equal to the dimension of the GPLVM output space). We use a standard Gaussian as the prior on $\bm{X}$. The variance of the noise parameter is $\sigma^2$ and $\bm{I}_N$ is an $N \times N$ identity matrix.
To join the two models, the $N \times D$ matrix of activations $\bm{H}$, from the Gaussian units, is defined as: $$\label{gaussian-activations}
\bm{H} = \bm{A} + \bm{\sigma}^{{\mathrm{GP}}} \otimes \bm{\sigma}^{{\mathrm{DBN}}} \odot \bm{\mathcal{E}}\ ,$$ where $\bm{A} = [\bm{m}^{{\mathrm{GP}}}_1, \dots, \bm{m}^{{\mathrm{GP}}}_N]^{\top}$ is a matrix in which each row is the mean output of the Gaussian units corresponding to each input training datapoint. This is combined with $\bm{\sigma}^{{\mathrm{GP}}}$, the $N \times 1$ vector of predictive standard deviations from the GPLVM , and $\bm{\sigma}^{{\mathrm{DBN}}}$, the $1 \times D$ vector of standard deviation parameters of the Gaussian units. Note that $\otimes$ is an outer product, and $\odot$ is an element-wise product.
The $\bm{H}$ matrix represents the observed data for the GPLVM and is updated at each training iteration by sampling $\bm{\mathcal{E}}$ a different $N \times D$ matrix of independent Gaussian noise, $\mathcal{E}_{n,d}\sim\mathcal{N}(0,1)$. This is a second application of the reparameterization trick. At each iteration, $\bm{H}$ is always normalized, to match our zero mean GP assumption, by subtracting its column-wise mean and dividing by $\bm{\sigma}^{{\mathrm{DBN}}}$.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Minibatches]{} The objective can be evaluated on an uniformly drawn subset of data $\{\bm{t}_{b}\}_{b=1}^B$ yielding an estimator for the full objective, $$\begin{aligned}
\label{estimated_objective}\small
L_{\text{batched}} &\simeq \textstyle\frac{N}{B} \textstyle\sum_{b=1}^B \big(\bm{t}_{b}\log(\bm{s}_{b}) + (1 - \bm{t}_{b})\log(1 - \bm{s}_{b})\big)
+ \frac{N}{2B} \mathrm{Tr}\left[\bm{K}_{B}^{-1}\bm{H}_{B} \bm{H}_{B}^\top\right]\nonumber\\[3pt]
&\qquad + \textstyle\frac{ND}{2B}\log |\bm{K}_{B}| + \frac{N}{B}||\bm{X}_{B}||^2 \ ,\end{aligned}$$ where $\bm{H}_{B}$ and $\bm{K}_{B}$ corresponds to $\bm{H}$ and $\bm{K}$ evaluated on the subset $\bm{X}_{B}$ of $\bm{X}$. Using this estimator the model can be optimised using mini-batching to scale linearly to larger datasets. We note that the matrix inversion does introduce bias into the estimator; empirical results suggest this is small and removing it is a topic for future work.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Scaling via Convolutional Architecture]{} When defining the likelihood directly over the pixels, the fully-connected conditional independence of the RBM layers limits scalability in terms of image size. This can be circumvented by adding convolution and deconvolution steps to replace the dense matrix product in in the lower layers.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Sampling]{} A sample $\bm{s}_{n}$ from the model is drawn by first generating a hidden sample $\bm{h}_{n}$ from latent point $\bm{x}_{n}$: $$\label{h_n}
\bm{h}_{n}(\bm{x}) = (\bm{m}^{{\mathrm{GP}}}_{n} + \sigma^{{\mathrm{GP}}}_{n} \times \bm{\epsilon}_{n}) \odot \bm{\sigma}^{{\mathrm{DBN}}} + \bm{h}_{\mu}\ ,$$ using $\bm{m}^{{\mathrm{GP}}}_{n}$ and $\sigma^{{\mathrm{GP}}}_{n}$ as the predictive mean and standard deviation of the GPLVM given latent point $\bm{x}_{n}$. This is combined with a sample $\bm{\epsilon}_{n}$, a $1 \times D$ vector of spherical Gaussian noise. The term $\bm{h}_{\mu}$ is the mean vector that is subtracted from $\bm{H}$ in the normalization step. The sample $\bm{h}_{n}$ is then propagated down through the DBN, sampling layer-by-layer, to give an output sample $\bm{s}_{n}$.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Prediction and Projection]{} Since we have a simple sampling process, we can propagate uncertainty for our predictions by taking the empirical mean of a set of $J$ samples from the model as $\bm{s}_* = \frac{1}{J} \sum_{j}^{J} \bm{s}_j \!\big( \,\bm{h}_j(\bm{x}_*) \big)$ for the latent location $\bm{x}^{*}$. Since we can efficiently take gradients through the sampling process, we can project new observations into the latent space by minimizing the reprojection error w.r.t. the latent locations for predictions from a set of random starting locations in the manifold.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Interpretation]{} We note that the objective consists of terms in contrast with each other. The first encodes a *data* term that ensures the observed data is well represented by the model. The third provides a *complexity* term that encourages a simple (low complexity) latent space $\bm{X}$ through the covariance matrix $\bm{K}$ to prevent overfitting.
The second term “glues” the two models together by ensuring that the covariance matrix $\bm{K}$ is a good model of the covariance of the Gaussian units at the top of the DBN. This in turn, ensures that the DBN learns an appropriate network to give sensible Gaussian activations rather than the unconstrained binary activations from a normal DBN. The last term encodes a *prior* which encourages the latent points to stay close to the origin.
The applications of the reparamerization trick ensures that efficient, low variance samples can be taken during training with gradients propagated throughout all parts of the network. The use of sampling and stochastic networks allows uncertainty to be propagated down through the entire model as well to ensure uncertainty is well quantified both at training and test time.
Experiments
===========
In keeping with previous work, we evaluated our models in terms of four experiments: (i) *Synthesis*, that is, generating samples that are plausible. (ii) *Representation and Generalisation*, demonstrating the ability to capture the variability of the silhouettes away from the training data. (iii) *Smoothness*, evaluating the quality of the learned latent space through interpolation; smooth trajectories in the latent space should produce smooth variations in the silhouette space. (iv) *Scaling*, evaluating how the model performs with respect to the size of the training dataset.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Our Models]{} In the comparisons, our main model (which we will refer to as GPDBN) consists of a three-layer DBN plus a GPLVM layer connected as described in § \[model\]. From the bottom (observed) to the top (hidden) layer the architecture consists of $200$ (Concrete units), $100$ (Concrete) and $50$ (Gaussian). The connected GPLVM layer has only $2$ latent dimensions for easy visualisation. The model is optimized jointly as described in § \[model\]. Our second model, GPSBM, is similar to the GPDBN where the three-layer DBN has been replaced with an SBM architecture of [@eslami2013shape] with hidden Concrete units in the bottom layer and hidden Gaussian units at the top. We implemented all our models in the TensorFlow [@Tensorflow] framework and trained using the Adam optimizer [@AdamOpt].
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Baselines]{} For comparison, we compared our models to size baselines: (i) A vanilla GPLVM with $2$ latent dimensions. (ii) GPLVMDT, a GPLVM operating on a signed distance function representation in a similar manner to [@victor2012pwp3dsdf]; samples are obtained by thresholding through the hyperbolic tangent function. (iii) The state-of-the-art ShapeOdds model [@ShapeOdds]. (iv) A DBN with binary units and the same architecture as our GPDBN. (v) The SBM [@eslami2013shape] model with binary units (trained layer by layer with contrastive divergence like the DBN) with the same architecture as our GPSBM. (vi) The VAE [@Kingma2013] model with the same architecture as our GPDBN (mirrored for the decoder) and $2$ latent dimensions. (vii) An InfoGAN [@chen2016infogan] with same architecture as the VAE and GPDBN (mirrored for the discriminator) and $2$ latent dimensions of structured noise.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Datasets]{} In keeping with previous work, we trained the models on the Weizmann horse dataset [@borenstein2004combining], which consists of $328$ binary silhouettes of horses facing left. The limited number of training samples and the high variability in the position of heads, tails, and legs make this dataset difficult. We also trained the models on $300$ binary images from the Caltech101 dataset of motorbikes facing right [@Caltech101]. All images in both datasets have been cropped and normalized to $32 \times 32$ pixels. The test datasets consisted of the challenging held-out data from [@eslami2013shape]; an additional $14$ horses and $9$ motorbikes not contained in the training datasets.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Synthesis]{} Fig. \[fig:gpdbn\_horse\_manifold\], shows the manifold learned by the GPDBN on the Weizmann horse dataset. Each blue point on the manifold represents the latent location corresponding to a training datapoint. The heat map is given by the log predictive variance that encodes uncertainty in the latent space. The model is more likely to generate valid shapes from any location in the bright regions (, low variance regions).
Unlike GP based models, a standard DBN (or the SBM) does not learn such a generative manifold. This implies, first of all, that a DBN does not allow us to sample “from the top” in a direct manner. Instead we must provide a test image to the visible units and condition on it before propagating it up and down the network for a few iterations to obtain a sample. Secondly, like the VAE and InfoGAN, a DBN does not provide information about how plausible a generated sample is.
\
A smooth generative manifold, such the one learned by our model in Fig. \[fig:gpdbn\_horse\_manifold\] is informative as it gives us an indication about where to sample from to get plausible silhouettes. Fig. \[fig:realism\] compares silhouettes generated by the models that allow sampling from the manifold.[^1] We note that the GPLVM and GPLVMDT produce blurry images since the shapes present interpolation artifacts from the Gaussian likelihood. In contrast, the results from both the GPDBN and GPSBM are sharper.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Representation and Generalisation]{} In the recent literature on shape modelling, quantitative results are reported in terms of the distance between the test data not seen by the model and the most likely prediction under the model. For the models that can be sampled from, this amounts to finding the location on the manifold that most closely represents the test input (discussed for our model in § \[model\]). For the models that learn an explicit manifold we find the closest silhouette to a test silhouette $\bm{t}^{*}$ by minimising the following objective with respect to a latent location $\bm{x}^{*}$ on the manifold: $$\label{generalisation-objective}
L_{\text{proj}}(\bm{x}^{*}) = \textstyle\frac{1}{P}\textstyle\sum_{i=1}^V \left(\bm{t}^{*}\log(\bm{s}_{i}) + (1 - \bm{t}^{*})\log(1 - \bm{s}_{i}) \right) + \gamma\times \log(\sigma^2(\bm{x}^{*}))\ ,$$ where we use $V$ samples to evaluate the cross entropy to the test silhouette. The second term is the log predictive variance of the latent location $\bm{x}^{*}$ (as defined in Eq.), this encourages the model to generate plausible silhouettes from the manifold. The scaling factor $\gamma$ ensures that the two term have approximatively the same scale.
\
\
Samples for a DBN (or SBM) are usually generated by conditioning on an observed sample and propagating it through the network for several cycles, as described in § \[dbn\], with Gibbs samples taken after a burn in period. In our experiments, we fixed the conditioning on the test datapoint and averaged the results of a number of propagated samples through the model to prevent the sample chain from drifting away from the test data.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Projection under Noise]{} To provide a challenging evaluation, we take unseen test data, corrupt it with noise and ask each models to find their most likely silhouette. Simply asking to reconstruct the test data would not be a sufficient evaluation since an identity mapping would be able to perform this task. Instead, we need the model to demonstrate that it can reject data that should not be in the trained model (the noise). In Fig. \[tab:results\_horses\_20\_noise\], we report the results for our proposed model and the baseline methods. We use the Structured Similarity (SSIM) [@ssim] metric (range \[0,1\] with high values better) with a small window size of $3$ to perform quantitative evaluations since it is known to outperform both cross-entropy and MSE as a perceptual metric. A random sample of corresponding silhouettes for the horse dataset are provided in Fig. \[fig:generalisation\_horses\_20\_noise\]. We also test our model in a more challenging environment, Fig. \[fig:generalisation\_noisy\_data\_and\_table\], where test data has been corrupted by significant noise. The quantitative comparisons shown that our GPDBN and GPSBM models have captured a high quality probabilistic estimate of the data manifold while still preserving interpretability.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Interpolation Test]{} We trained a GPDBN, VAE and InfoGAN models on a $30$ image dataset (which we call *stars* dataset) generated from a *known* 1-dimensional manifold using a simple script. The full dataset is displayed in the top row of Fig. \[fig:geodesic\_test\]. The deterministically generated dataset allows us to determine quantitatively whether interpolations in the latent space are representative of the true data distribution. The middle rows of Fig. \[fig:geodesic\_test\] show the model outputs for the interpolation between two latent points corresponding to a four-pointed *star* (leftmost sample) and a *square* (rightmost sample). The uncertainty information of the GPDBN allows us to go from one point to the other passing through low-variance regions by following a geodesic [@Tosi:2014tt]. We can see that the GPDBN produces smoothly varying shapes of high quality that reflect the true manifold. In contrast, the VAE and InfoGAN results do not smoothly follow the true manifold and contain some erroneous interpolants that are not part of the true distribution; this is supported by the quantitative results that measure the quality of the samples to the true data using SSIM. The ability to exploit variance information in the GPDBN is clearly an advantage over the VAE and InfoGAN where the absence of direct access to the latent predictive posterior distribution prevents easy access to geodesics. Further demonstrations of the smoothness are available in supplementary material.
[C[0.12]{}C[0.85]{}]{}Dataset & ![Example results of the interpolation test between two training points from the stars dataset. The top row shows the geodesic interpolation generated by the GPDBN. The middle and the last rows are the linear interpolation generated by the VAE and InfoGAN respectively. The bottom row provides the mean and standard deviation of the SSIM score over $10$ interpolation experiments. (In this picture black and white are inverted respect to the training dataset.)[]{data-label="fig:geodesic_test"}](figures/star-dataset-border.png "fig:"){width="\linewidth"}\
\
GPDBN&![Example results of the interpolation test between two training points from the stars dataset. The top row shows the geodesic interpolation generated by the GPDBN. The middle and the last rows are the linear interpolation generated by the VAE and InfoGAN respectively. The bottom row provides the mean and standard deviation of the SSIM score over $10$ interpolation experiments. (In this picture black and white are inverted respect to the training dataset.)[]{data-label="fig:geodesic_test"}](figures/interpolation/gpdbn_geodesic_interpolation.pdf "fig:"){width="\linewidth"}\
VAE&![Example results of the interpolation test between two training points from the stars dataset. The top row shows the geodesic interpolation generated by the GPDBN. The middle and the last rows are the linear interpolation generated by the VAE and InfoGAN respectively. The bottom row provides the mean and standard deviation of the SSIM score over $10$ interpolation experiments. (In this picture black and white are inverted respect to the training dataset.)[]{data-label="fig:geodesic_test"}](figures/interpolation/vae_linear_interpolation.pdf "fig:"){width="\linewidth"}\
InfoGAN&![Example results of the interpolation test between two training points from the stars dataset. The top row shows the geodesic interpolation generated by the GPDBN. The middle and the last rows are the linear interpolation generated by the VAE and InfoGAN respectively. The bottom row provides the mean and standard deviation of the SSIM score over $10$ interpolation experiments. (In this picture black and white are inverted respect to the training dataset.)[]{data-label="fig:geodesic_test"}](figures/interpolation/infogan_linear_interpolation.pdf "fig:"){width="\linewidth"}
--------------------------- -------------------------- -----------------------------
GPDBN: $0.95 \pm 0.01$ VAE: $0.87 \pm 0.03$ InfoGAN: $0.93 \pm 0.06~$
\[-15pt\]
--------------------------- -------------------------- -----------------------------
![Graph showing the SSIM score of the output of the GPDBN, InfoGAN and VAE models against the test data without noise as the training dataset size increases from $100$ to $10000$ points. A higher score is better.[]{data-label="fig:generalisation_graph"}](figures/generalisation_graph.pdf "fig:"){width="0.5\linewidth"}\
![ The model can be made to scale to a large number of datapoints by optimizing the objective using mini-batching . Furthermore, scalability in the size of images can be obtained by adding upscaling and convolutions in the lowermost layer. Left: GPDBN trained on MNIST comprised of 60,000 28x28 images. Right: GPDBN trained on Weizmann Horses comprised of 328 300x300 images. []{data-label="fig:scalability"}](figures/batched_mnist_full.png "fig:"){width="0.4\linewidth"} ![ The model can be made to scale to a large number of datapoints by optimizing the objective using mini-batching . Furthermore, scalability in the size of images can be obtained by adding upscaling and convolutions in the lowermost layer. Left: GPDBN trained on MNIST comprised of 60,000 28x28 images. Right: GPDBN trained on Weizmann Horses comprised of 328 300x300 images. []{data-label="fig:scalability"}](figures/cnn_weizmann_300.png "fig:"){width="0.4\linewidth"}
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Scaling Experiments]{} In Fig. \[fig:generalisation\_graph\] we compare the performance of the GPDBN, InfoGAN and VAE models as the size of the training dataset increases; here we use the standard MNIST digit dataset. We used a 10-dimensional latent space for all of the three models to account for the larger quantity of data. Similarly to the experiments in Figs. \[fig:generalisation\_horses\_20\_noise\_and\_table\] and \[fig:generalisation\_noisy\_data\_and\_table\], we took 30 random images from the MNIST test data, add 20% salt-and-pepper noise, and calculated the SSIM score between the output of the models and the test data without noise. We plotted the score against dataset size (in log scale). We can see that the GPDBN model is able to capture a high quality model of the data manifold even from small datasets; for example, it achieves the same quality as a VAE trained on 10,000 images using only 100. We argue that the propagation of uncertainty throughout the model provides the advantage over both the VAE and InfoGAN which are both trained with only maximum likelihood approaches.
In Fig. \[fig:scalability\] we provide results that demonstrate that our approach also overcomes scaling issues normally present in GP models and DBNs. Firstly, we show training on the 60,000 MNIST images via our proposed mini-batching approach. In addition, we also show the manifold for higher resolution images from the horse dataset ($300\times300$). By using convolutional architectures, we can scale the number of parameters in an identical manner to convolutional feed-forward networks and our concrete layers allow us to train from random weight initialisation using back propagation without the need to use slow contrastive divergence. With both these approaches we still maintain our full uncertainty model so the same model can perform well with small and large datasets.
Conclusion
==========
We have presented the GPDBN, a model that combines the properties of a smooth, interpretable low-dimensional latent representation with a data specific non-Gaussian likelihood function (for silhouette images). The model fully propagates and captures uncertainty in its estimates, it is trained end to end with the same complexity as a standard feed-forward neural network by minimising a single objective function, and is able to learn from very little data as well as scaling to larger datasets linearly by using mini-batching. We have shown both quantitatively and qualitatively that our model performs on par with the best shape models while at the same time introducing a smooth and low-dimensional latent representation with associated uncertainty that facilitates easy synthesis of data.
[startsection[paragraph]{}[4]{}[@]{} [4@ plus 2@ minus 2@]{} [-0.5em plus -0.22em minus -0.1em]{} [****]{}]{}[Acknowledgements]{} This work was supported by the EPSRC CAMERA (EP/M023281/1) grant and the Royal Society.
Interactive Manifold Demonstration
==================================
Please see the interactive manifold demonstration included with the supplemental material. The demonstration is a standalone application that can be viewed with a recent javascript, HTML5 compliant browser by loading the file “*interactive\_demo.html*”. The data and rendering code is contained in the “*data*” directory is accessed through the standalone webpage. Figure \[fig:demo\] provides an illustration of what the page should look like with instructions for use in blue text. Figures \[fig:demo\_model\_select\] and \[fig:demo\_zoom\_select\] show the options to select different models, datasets and zoom levels.
![The interactive manifold demonstration included as “*interactive\_demo.html*”. Please see the instructions for use shown in blue. While both the GPLVM and our GPDBN models provide interpretable, smooth manifolds, the Gaussian likelihood assumptions of the GPLVM lead to poor shape interpolations that are blurred and contain artifacts. Importantly, the heat map encodes the confidence in the estimated shapes via the predictive variance of the model.[]{data-label="fig:demo"}](figures/demo/demo_annotated.png){width="0.9\linewidth"}
![Selecting the zoom combo box switches between different scales of the output image from $1$ (actual size) to $8$ times the actual size.[]{data-label="fig:demo_zoom_select"}](figures/demo/demo_model_select.png){width="0.7\linewidth"}
![Selecting the zoom combo box switches between different scales of the output image from $1$ (actual size) to $8$ times the actual size.[]{data-label="fig:demo_zoom_select"}](figures/demo/demo_zoom_select.png){width="0.7\linewidth"}
Additional Results
==================
In Figure \[fig:gpdbn\_motorbike\_manifold\] we provide an illustration of the learned manifold for our GPDBN model on the Caltech101 dataset of motorbikes facing right [@Caltech101]. We also trained a GPDBN on two types of data at the same time (horses and motorbikes). Fig. \[fig:gpdbn\_horses\_and\_motorbikes\_manifold\] shows what the manifold looks like (two separate clusters are clearly distinguishable).
![Manifold learned by the GPDBN model on the motorbikes dataset. Moving over the manifold changes the shape of the motorbike producing smooth silhouette transitions. The heat map encodes the predictive variance of the model with darker regions indicating higher uncertainty and lower confidence in the silhouette estimates.[]{data-label="fig:gpdbn_motorbike_manifold"}](figures/gpdbn_motorbikes_manifold.pdf){width="0.7\linewidth"}
![Manifold learned by the GPDBN model trained on 250 horses + 250 bikes.[]{data-label="fig:gpdbn_horses_and_motorbikes_manifold"}](figures/gpdbn_horses_and_motorbikes_manifold.pdf){width="0.8\linewidth"}
Additional Representation and Generalisation Results
----------------------------------------------------
A fundamental ML problem is that there is no objective quantitative method of assessing unsupervised learning models. An identity function that simply outputs the test input would be able to achieve perfect reconstruction, however, under such a model all silhouettes would be equally likely (even implausible ones); generalising to implausible shapes is not a property that we want. In contrast, the predictive uncertainty of our GPDBN model tells us how plausible a generated silhouette is (a key feature).
Our proposal for a good quantitative assessment is to measure the reconstruction of the generated output corresponding to a noisy version of the input. An identity function would return the noisy version and be penalised for producing an implausible shape. In contrast, a good model should return a projection to a plausible shape. By using a noisy version the closest plausible shape should be the uncorrupted test image so we can compare to this to quantify the performance.
In addition to the results in the paper we report in Tab. \[tab:results\_horses\_10\_noise\] the SSIM reconstruction score for 10% salt-and-pepper noisy horses to demonstrates that our models consistently outperform the competition even with very little noise.
Fig. \[fig:generalisation\_bikes\_20\_noise\_and\_table\], Fig. \[fig:generalisation\_bikes\_40\_noise\_and\_table\] and Fig. \[fig:generalisation\_bikes\_60\_noise\_and\_table\] show the results of an experiment where test data has been corrupted by significant noise (20%, 40% and 60% respectively) and we wish to project onto the manifold of valid silhouettes. The quantitative comparisons indicate that our models have managed to capture a good probabilistic estimate of the data manifold while still preserving interpretability.
\
\
\
\
{width="0.85\linewidth"}\
[Ablation comparisons of GPDBN models variants using sigmoidal units plus Dropout, showing the importance of Concrete units for proper uncertainty propagation. (A *GPNN* architecture is exactly like a GPDBN except that it uses sigmoidal units.)]{} \[fig:gpdbn\_fixed\_dropout\]
Method SSIM
------------------------ -----------------
Large net (x3 units) $0.52 \pm 0.12$
Narrow net (1/3 units) $0.47 \pm 0.15$
Deep net (+1 layer) $0.45 \pm 0.20$
Shallow net (-1 layer) $0.58 \pm 0.08$
: SSIM of the output (without 20% salt-and-pepper noise) of the GPDBN using different network architectures.[]{data-label="tab:different_architectures"}
Robustness and Ablation
-----------------------
Given the space constraints in the paper we have given priority to what we believe is the most important; we have provided extensive results and comparisons with recents models (plus an interactive demo). In the following paragraphs we address feature ablation and model robustness.
We note that the two components of our GPDBN model, that is the GPLVM and DBN, are both essential, none of them can be ablated because the former provides the smooth manifold and predictive uncertainty while the latter increases the capability of generating image data. Ablation of our important Concrete units to dropout units impedes uncertainty propagation and reduces performance (Fig. \[fig:gpdbn\_fixed\_dropout\]). Moreover, our comparisons show that both GPLVM and DBN are weaker as standalone models.
The architectures used were based on previous work to enable fair comparison. In addition, we have trained a GPDBN on the horse dataset experimenting with four different networks (increasing and decreasing the number of units and layers). By removing one layer (Tab. \[tab:different\_architectures\]) we got slightly better performance to the more generic architecture proposed in the paper. It might be that a shallow network is better suited for this data given the small number of training examples. We see this positively as the model can be fine-tuned to achieve even higher performance depending on the specific data. Finding the optimal number of layers and weights (parameters) is an open issue common to many deep learning methods. We think that replacing the GPLVM part with a Bayesian one ([@titsias2010bayesian]) would solve this problem for the GPDBN allowing it to use the optimal number of parameters automatically.
DBNs and Mini-batching
----------------------
Traditionally, DBNs do not train on large images, this is because of the high number of parameters therefore, the use of convolutions is necessary. In contrast, GPDBN mini-batching allows us to deal with a large number of images and empirically any introduced bias does not really reduce the performance. For example, we trained a GPDBN on $1000$ MNIST digits (as in Fig 6 in the paper) with mini-batch size $100$, we obtained SSIM: $0.63 \pm 0.09$, which is even higher than the non mini-batched equivalent (SSIM: $0.50 \pm 0.08$).
Smoothness Experiment
---------------------
{width="0.9\linewidth"}\
[cccc]{} GPDBN & GPSBM & GPLVM & GPLVMDT\
{width="0.23\linewidth"} & {width="0.23\linewidth"} & {width="0.23\linewidth"} & {width="0.23\linewidth"}\
{width="0.24\linewidth"} & {width="0.24\linewidth"} & {width="0.24\linewidth"} & {width="0.24\linewidth"}\
------------------------------------------------------------------------
{width="0.1\linewidth"} & {width="0.1\linewidth"} & {width="0.1\linewidth"} & {width="0.1\linewidth"}
\
This additional experiment also highlights the benefit of the uncertainty associated with our model and how it manifests itself in the proposed GPDBN in contrast with the other methods. There is an inherent trade-off between a simple topology of the manifold and the smoothness of the mapping. To exemplify this we generated a dataset of a shape deformed in a cyclic manner Fig. \[fig:manifolds\]. The resulting latent space is structured as a circle clearly reflecting the topology of the deformation. Importantly, if the uncertainty in the model reflects that of the data we should move along ridges of high probability (manifold geodesics) to generate realistic data. Our proposed model is directly applicable to such approaches as described in [@Tosi:2014tt]. Further, the experiment highlights how the uncertainty effects the prediction. When generating shapes corresponding to a region of the manifold where the model is highly uncertain we would, if the model have captured the characteristics of the data well, expect images corresponding to the *average* shape. As can be seen the GPDBN clearly generates the average shape while the other methods fail to capture this characteristic in the data making it challenging to interpret the uncertainty.
#### Structure
The blue points on the manifolds in Fig. \[fig:manifolds\] show the results for each of the smooth manifold based models. We note that all the manifolds have correctly identified a smooth trajectory for the training data. In addition, all but the GPSBM have captured the periodic repetition by closing the path; this is possibly due to the symmetry in the dataset not reflecting the shared architecture of the SBM.
The red points represent test locations corresponding to the samples in the third row of Fig. \[fig:manifolds\]. Here we see that all models are correctly interpolating the overall pattern, however, the Gaussian likelihood of the GPLVM introduces artefacts in the silhouettes that are not found in the results from the GPDBN and GPSBM. The GPLVMDT improves over the GPLVM but still produces blurred results.
#### Uncertainty
Finally, the real power of the GPDBN model is captured by looking at what happens when you leave the manifold. The final row of silhouettes are samples from the orange points that are in regions of high predictive variance (low confidence). Both the GPLVM and the GPLVMDT produce completely unreasonable results. Whereas the GPDBN *captures the uncertainty in the manifold perfectly*; we see the average probability of the entire dataset with the predictive probability correctly captured. The GPDBN results are the mean of a set of samples from the model and away from the manifold these results are correctly approaching the mean of the training data. Interestingly, the results for the GPSBM show the asymmetry in the shared weights; leaving the manifold in two different directions averages different regions of the training data.
[^1]: When we show generated silhouettes from any model, we actually show grayscale images denoting pixel-wise probabilities of turning white rather than binary samples.
|
---
abstract: '[We present [*Chandra*]{} observations of the galaxy cluster A4059. We find strong evidence that the FR-I radio galaxy PKS 2354–35 at the center of A4059 is inflating cavities with radii $\sim 20\kpc$ in the intracluster medium (ICM), similar to the situation seen in Perseus A and Hydra A. We also find evidence for interaction between the ICM and PKS 2354–35 on small scales in the very center of the cluster. Arguments are presented suggesting that this radio galaxy has faded significantly in radio power (possibly from an FR-II state) over the past $10^8\yr$.]{}'
author:
- 'Sebastian Heinz, Yun-Young Choi, Christopher S. Reynolds, and Mitchell C. Begelman'
title: '[*Chandra*]{} ACIS-S observations of Abell 4059: signs of dramatic interaction between a radio galaxy and a galaxy cluster.'
---
introduction
============
Clusters of galaxies are complex dynamical structures and their cores are subject to an array of interesting physical processes. Constraints from imaging X-ray observations suggest that the hot X-ray emitting intracluster medium (ICM) in the core regions of rich clusters is radiatively cooling on timescales shorter than the life of the cluster, giving rise to cooling flows [@fabian:94 and references therein]. The central dominant galaxy present in many clusters often hosts a radio loud active galactic nucleus (AGN). It has been suggested [e.g., @binney:95] that cooling flows and central cluster radio galaxies are intimately related via complex feedback processes. It is easy to see how radio galaxy activity resulting from black hole accretion can be associated with a cooling flow. However, the impact of a radio galaxy on its environment is much less clear.
Theoretically, we expect radio jets to inflate cocoons of relativistic plasma that expand into the surrounding ICM [e.g. @begelman:89; @kaiser:97; @reynolds:01, hereafter RHB]. The energy input by this process has recently come under investigation for its potential role in heating cluster cores [@bruggen:01; @quilis:01; @reynolds:01b]. However, while our simulations suggest that about half of the energy injected by the jets can be thermalized in the cluster center, numerical simulations of this process still carry a large degree of uncertainty, since limited computational resources require significant simplications. In order to verify the validity of the assumptions and to design future models, we require guidance from observations of radio-galaxy/cluster interactions.
Imaging X-ray observatories, such as the [*Chandra X-ray Observatory*]{} (CXO), provide a direct probe of this interaction. Both [*ROSAT*]{} and CXO observations of Perseus A have found X-ray cavities coincident with the radio lobes [@boehringer:93; @fabian:00], surrounded by X-ray shells which appear to be slightly cooler than the unperturbed ICM (see, for example, RHB for a possible explanation). Similar features are seen in CXO observations of Hydra A [@mcnamara:00; @david:01] and Abell 2052 [@blanton:01].
In this [*Letter*]{}, we present CXO observations of the rich galaxy cluster Abell 4059 ($z=0.049$). The cD galaxy of A4059 hosts the FR-I radio galaxy PKS 2354–35. A short [*ROSAT*]{} High Resolution Imager (HRI) observation of this source suggested the presence of two ICM cavities at the same position angle as the radio lobes [@huang:98, hereafter HS]. In § 2 we discuss our observations, confirming the presence of these cavities, and show that A4059 displays significant additional morphological complexity. Constraints on models for this source are discussed in § 3, § 4 presents our conclusions. We assume a Hubble constant of $H_0=65 \kmpspMpc$ and $q_0 = 1/2$, giving a linear scale of $1\, {\rm kpc\,arcsec^{-1}} = 0.492\, {\rm kpc\,pixel^{-1}}$.
Observations
============
Data Reduction
--------------
The galaxy cluster Abell 4059 was observed with the [*Advanced CCD Imaging Spectrometer*]{} (ACIS) on 24-Sept-2000 (22.3ksec exposure) and on 4-Jan-2001 (18.4ksec exposure). The radio nucleus of PKS 2354-35 was placed 1arcmin from the nominal aim point of the back-illuminated S3 chip, placing the bulk of the emission on chip-S3. In this [*Letter*]{}, we only use data collected on this chip. The data was read out at the standard 3.2 sec frame rate, telemetered using faint mode, and filtered on ASCA event grades. The energy range was restricted to the 0.3-10 keV band and corrected for exposure and vignetting, spectral fitting was restricted to 0.3-8.0 keV.
Some of the Sept-2000 data were affected by a period of relatively high background, with up to 10 counts per second per chip for the back-illuminated chips. However, most of the cluster emission is well above this rate and we decided to use the entire data set for imaging. For the spectral analysis, we filtered the data on the counts from chip-S1 to fall within a factor of 1.2 of the quiescent background rate, which rejected 33% of the observing window. We then used the quiescent background files by Markevitch[^1] for background subtraction. The background subtracted total count rate on chip-S3 is 5.1 counts/second.
Morphological Appearance {#sec:morphology}
------------------------
Figure 1a shows an adaptively smoothed image of the central 3 arcmin of A4059. Overlayed is the 8GHz image by [@taylor:94, hereafter T94]. While the large scale emission appears smooth (probably due to the larger smoothing length used by the [csmooth]{} algorithm), the core of A4059 is not relaxed: it is double peaked with one peak at the cluster center and the other 15arcsec south-west of the center. The smoothed image reveals a distorted hour-glass like structure (yellow regions in Fig. 1a), centered on the nucleus and oriented perpendicular to the radio axis.
The most interesting detail of the X-ray image are the two X-ray holes already noticed by HS, clearly visible in the 5-sigma smoothed image. The higher quality CXO image confirms that these are real local brightness minima, not just a visual effect caused by a bright central bar perpendicular to the cluster elongation. A noticeable offset exists between the nucleus and the axis between the cavity centers, with the nucleus being shifted to the south-west by $\sim 12\arcsec$ from this axis. The cavity centers are roughly $50\arcsec$ apart.
To assess the significance of the cavities we extracted an azimuthally averaged radial surface brightness profile of the central region, excluding the cavity regions shown in Fig. 1a (a by-eye approximation) and calculated the expected surface brightness in the two cavities. Based on this estimate, the NW and SE cavities are significant to 26 and 5.3 sigma, respectively. However, because the central region of A4059 is strongly perturbed (roughly inward of 1 arcmin from the center), it is difficult to make a rigorous statement concerning the significance of the cavities this way.
However, the outer regions of the cluster appear relaxed. As an alternative method, we took radial surface brightness profiles from the combined SE- and NW quadrants and the combined SW-NE quadrants. As a first order correction for cluster ellipticity we shifted the radial SW-NE profile outward by a factor of 1.13, producing a good match at large radii with the SE-NW profile. Because the temperature at large $r$ is relatively uniform, we used an isothermal $\beta$-model to fit the surface brightness, which represents the outer cluster very well. Because the inner cluster regions are not well represented by a $\beta$-model, we only used points further than 60 arcsec from the center for the fit ($r_{\rm c} \sim
50\arcsec$, $\beta=0.52$, $\chi^{2}/dof=59/43=1.4$).
We then compared the flux measured inside the cavity contours in Fig. 1a to that expected from the $\beta$-model. The NW cavity has 33 sigma significance compared to the best fit $\beta$-model, while the less pronounced SE cavity has 13 sigma significance. Forcing the $\beta$-model normalization to be consistent with the observed cavity flux increases the reduced chisquare of the fit by a factor of 2.7 (NW cavity) and 1.9 (SE cavity), corresponding to a significance of 8.5 and 6 respectively.
The radio overlay in Fig. 1a shows that the 8GHz radio lobes are only partly coincident with the cavities. The NW lobe covers a good fraction of the NW cavity. The SE lobe is much smaller than the northern lobe and not spatially coincident with the SE cavity. We note, however, that low-frequency radio observations sensitive to spatial scales of $0.1-1$arcmin, which might reveal the full extent of the radio lobes, do not yet exist.
The central peak contains interesting sub-structure (Fig. 1b., smoothed to a S/N of 3-4). The brightest sub peak, which is well resolved by CXO and has a diameter of 3–4$\sim$ 3–4kpc, is coincident with the core of PKS 2354–35 (within the CXO pointing accuracy) and could be emission from the hot interstellar medium of the central galaxy. The three sub-peaks are located around a local brightness minimum. Our hardness ratio analysis (§\[sec:spec\]) suggests that this minimum is due to lack of emission rather than intervening absorption. The close correspondence between these peaks and the SE radio lobe suggests that this substructure might be caused by on-going interaction.
Spectroscopic Properties {#sec:spec}
------------------------
We used the adaptive binning routine of @sanders:01 to produce hardness ratio maps. We define the hardness ratios $h_{1} = (1-2{\rm \,
keV})/(0.3-1{\rm \, keV})$ and $h_{2} = (2-10{\rm \, keV})/(1-2{\rm \,
keV})$ as the ratio of the counts in the respective energy bands and $c_{1}$ as the surface brightness in the 0.3-10 keV band in ${\rm cts\
cm^{-2}\ s^{-1}\ pixel^{-1}}$. For ICM observations, $h_1$ is mainly sensitive to absorption variations, whereas $h_2$ is a temperature diagnostic. While there is no sign of varying absorption across the field (as seen in $h_1$), the $h_2$ map clearly shows a radial temperature gradient, as would be expected in a cooling flow cluster (Fig. 2). It also shows a global temperature gradient, with the SW half of the image appearing hotter than the NE half.
We have extracted a global spectrum of A4059 from the central 90arcsec. Fitting this spectrum with a two temperature [wabs\*zwabs\*(mekal+mekal)]{} thermal plasma model (Galactic neutral hydrogen column fixed to $N_{\rm
H,G} = 1.45\times 10^{20}\,{\rm cm^{-2}}$) results in the best fit parameters $kT_1=1.34^{+0.53}_{-0.19} \keV$, $kT_2=3.90^{+1.19}_{-0.36}
\keV$, $Z=0.60^{+0.16}_{-0.11}$, $N_{\rm H,z}=5.36^{+0.48}_{-0.54}\times
10^{20}\,{\rm cm^{-2}}$, $\chi^2/dof=1.29$ (3 sigma error bars). A [wabs\*zwabs\*(mkcflow + mekal)]{} cooling flow model provides a similarly reasonable fit ($kT_{\rm 1} = 0.1^{+0.75}_{-0.1}\,{\rm keV}$, $kT_{\rm
2}=kT_{\rm {\tt mekal}}=3.80^{+0.16}_{-0.13}\,{\rm keV}$, $Z=0.71^{+0.09}_{-0.09}$, $\dot{M} = 27.6^{+6.0}_{-5.9}\, M_{\sun}{\rm \,
yr^{-1}}$, $N_{\rm H,z}=5.54_{-0.40}^{+0.25} \times 10^{20}\,{\rm
cm^{-2}}$, $\chi^2/dof = 1.26$).
Discussion
==========
Like Perseus A and Hydra A, A4059 shows clear signs of interaction between the radio galaxy and the ICM. However, unlike in these systems, there is no one-to-one correspondence between the radio lobes and the X-ray holes. We argue below that the cavities were indeed created by the radio activity (given the similarity in position angle and the fact that there are no other viable models for producing large ICM cavities), but that the radio galaxy has faded since they were formed. This is particularly interesting since CXO has recently found clusters with cavities but without a radio source [e.g., @mcnamara:01]: A4059 could be a ‘missing link’ between cavities with and without detectable radio lobes, supporting the notion that cluster cavities can be created by radio galaxies without having to show current radio activity.
Evolutionary state
------------------
In the hydrodynamic simulations of RHB, we identified three evolutionary phases. Early on, the cocoon is highly overpressured, driving a strong shock into the ICM. Simulated X-ray maps show a cavity surrounded by a thin, hot shell. Once the cocoon comes into pressure equilibrium with the ICM, the sideways expansion of the cocoon becomes sub-sonic and the shock becomes a compression wave, though active jets can keep the advance speed of the jet heads supersonic for much longer, driving “sonic booms” into the ICM. For roughly a sound crossing time, the well defined cavity created during the supersonic phase will survive and be observable. Then, hydrodynamic instabilities will destroy the cavity altogether.
A strong shock would show up as a sharp feature in our X-ray images, which have an effective resolution[^2] of 4–5arcsec at the edges of the cavities. The absence of such a shock indicates that the source expansion is no longer highly supersonic. We identify the hourglass-like feature in the core with the compression wave (“sonic boom”) found in our hydrodynamic simulations (Fig. 1, RHB; in our simulated X-ray maps the brightest emission also tends to be in the equatorial plane).
Source Power
------------
We can estimate the radio source parameters based on the presence of the X-ray cavities (@heinz:98, RHB, @churazov:00). We use the two-temperature fit of § 2 to estimate the physical parameters of the ICM. Taking the hot emission to arise uniformly in a sphere of 90radius yields an electron density of $n_{\rm hot}\gtrsim 0.009\,{\rm
cm^{-3}}$. Assuming the cold gas is in pressure equilibrium with the hot gas gives an electron density of $n_{\rm cold} \gtrsim 0.031\, {\rm
cm^{-3}}$ and a volume filling factor of $5\times 10^{-3}$.
We assume that both X-ray cavities are completely evacuated by the lobes and estimate their size from the smoothed images by approximating them as spheres. While this is clearly a simplification, it will be sufficient for this order of magnitude estimate. A ‘by eye’ fit of the cavities gives bubble radii of $r_{\rm bub} \sim 20\arcsec$ (20 kpc). The pressure in the hot phase is $p_{\rm ICM} \gtrsim 1.1\times 10^{-10}\,{\rm erg\,cm^{-3}}$, relatively close to the minimum energy pressure in the lobes of $p_{\rm ME}
\sim 3 - 5\times 10^{-11}{\rm ergs\ cm^{-3}}$ (T94). At a minimum, the radio galaxy has to perform “pdV” work against the ICM. Including the internal energy of the plasma within the cavities, this gives an integrated energy output of $E_{\rm tot} \gtrsim 8\times 10^{59}\, {\rm ergs}$.
The radio galaxy had to inject this energy into the bubbles before they floated out of the cluster core. This buoyancy timescale is approximately twice the sound crossing time of the relevant region of the cluster, $\tau\sim 4r_{\rm bub}/c_{\rm s}\sim 8\times 10^{7}\yr$, where we have used the sound speed for a $4\keV$ gas, $c_{\rm s}\sim 1000\kmps$. The time-averaged source power needed to produce the cavities is then $L_{\rm
kin}\sim E_{\rm b}/\tau\gtrsim 3\times 10^{44}\ergps$.
Alternatively, we can estimate the source age from the “sonic boom” arguments of RHB. The hour-glass structure through the cluster center is roughly 50 long (i.e., $25\arcsec \sim 25\,{\rm kpc}$ on either side of the center). Following RHB, we equate this to the distance traveled by a shock/compression wave which moves at least at the sound speed of the hot ICM. This gives a source age of $\tau \lesssim 2.4\times
10^{7}\,{\rm yrs}$ and a time-averaged power of $L_{\rm kin}\gtrsim E_{\rm
b}/\tau\sim 10^{45}\ergps$.
A third, X-ray independent way to estimate the source power is based on the radio flux. The flux densities at 5GHz and 8GHz are given by T94 as 76mJy and 34mJy, while the 1.4GHz NVSS flux @condon:98 is 1.3Jy. The NVSS flux lies a factor of 2 above the extrapolation of the 5–8GHz flux, and the NVSS image suggests spatial extension on arcmin scales (a factor of $\sim 2$ larger than seen at 5–8GHz). This suggests that NVSS is detecting low frequency emission from plasma that is emitting a steep radio spectrum, possibly indicating that it has suffered synchrotron aging. A reasonable upper limit on the current radio power can be derived by taking the 1.4GHz luminosity, and using the arguments of @bicknell:98 to convert it into a kinetic luminosity. Taking the smallest reasonable value of their conversion parameter, $\kappa_{1.4} >
10^{-12}$, we estimate an upper limit on the current kinetic power of $L_{\rm kin} < 7\times 10^{43}\ergps$.
Comparing the time-averaged source power (derived the X-ray cavities) to that derived from the radio luminosity (which is equivalent to the source power averaged over the synchrotron cooling time of the 1.4GHz electrons, $\tau_{\rm cool} \lesssim 10^7\,{\rm yrs}$), one infers that either this source has faded in kinetic luminosity by an order of magnitude or more, or that the magnetic field in the lobes is considerably out of equipartition. Since the thermal pressure is close to the equipartition pressure estimated by T94, we favor the first possibility. Given the uncertainties in these arguments (especially in $\kappa_{1.4}$, for which we chose a conservative value), the source could easily have faded by more than an order of magnitude. Indeed, the fact that the average power is in the realm of FR-II radio galaxies, while morphology and current radio luminosity qualify it as an FR-I, leads us to speculate that PKS2354–35 is an example of an FR-II source that has faded into an FR-I source on a timescale of less than $10^8\yr$.
The apparent offset between the cluster center and the center of the cavities and the asymmetric brightness distribution through the equatorial regions of the hour-glass structure may be evidence for bulk ICM motions. In particular, the morphology suggests a bulk flow in a NE direction which might further squeeze the outward moving compression wave from the radio galaxy, and sweep back the cavity structure. We note that the SW ridge is rather cool and thus cannot be a strong shock resulting from the interaction of a bulk flow with the radio galaxy. Hydrodynamic simulations are required to investigate this system further.
Conclusions
===========
We have presented CXO observations of Abell 4059. While the ICM appears smooth and relaxed on large scales, it shows complex morphology in the core region which is likely the result of interaction between the ICM and the central FR-I radio galaxy PKS2354–35. As was suggested by HS, PKS 2354–35 appears to have inflated two large cavities in the ICM. Together with a central bar-like structure, these cavities produce an hour-glass like morphology which can be readily understood as being due to a radio cocoon expanding into the ICM. While clear correspondence exists between the NW cavity and the NW radio lobe, the SE cavity is much larger than the SE lobe, suggesting that this could be a ‘missing link’ between cavities with and without visible radio lobes.
The absence of sharp edges in the brightness images and of large temperature jumps implies that PKS2354–35 is [*not*]{} driving a strong shock into the ICM. We suggest that it is in the weak-shock/compression-wave phase identified in the hydrodynamic simulations of RHB. Dynamical estimates give a time averaged kinetic source power of at least $L_{\rm kin} \gtrsim 3\times 10^{44}\ergps$, while estimates based on the current radio luminosity indicate a source power of $L_{\rm kin} \lesssim 7\times 10^{43}\ergps$. We suggest that this source has faded by a significant amount (and possibly from an FR-II phase) during the past $10^8\yr$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Eugene Churazov and Torsten Ensslin for helpful discussions. We acknowledge support from SAO grant GO0-1129X, the National Science Foundation grants AST 9529170 and AST 9876887, and NASA under grant NAG5-6337. YC wishes to thank support from the MOST through the National R & D program for women’s universities. CSR thanks support from Hubble Fellowship Grant HF 01113.01-98A.
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[^1]: See
[^2]: After adaptive smoothing to a signal to noise of 5.
|
---
author:
- 'V. Timorin [^1]'
title: Rectification of circles and quaternions
---
[Consider a bundle of circles passing through 0 in 4-dimensional space. It is said to be [*rectifiable*]{} if there is a germ of diffeomorphism at 0 that takes all circles from our bundle to straight lines. We will give a classification of all rectifiable bundles of circles containing sufficiently many circles in general position. This result is surprisingly different from those in dimensions 2 and 3 (Khovanskii and Izadi) due to a connection with the quaternionic algebra.]{}
Introduction {#introduction .unnumbered}
============
Throughout this paper, the word “circle” means a circle or a straight line. We are always assuming that the space $\R^n$ is equipped with a fixed “standard” Euclidean inner product.
A collection of curves in $\R^n$ passing through 0 is said to be a [*simple bundle of curves*]{} if no two of them are tangent at 0. A simple bundle of curves is called [*rectifiable*]{} if there exists a germ of diffeomorphism in a neighborhood of the origin that sends all curves from this bundle to straight lines. Rectifiable bundles of curves appear, for example, in Riemannian geometry — the set of geodesics passing through a given point is rectifiable.
A. G. Khovanskii proved in [@Kh] that a rectifiable simple bundle of more than 6 circles on plane necessarily pass through some point different from the origin. F. A. Izadi [@Iz] generalized Khovanskii’s arguments to dimension 3. A rectifiable simple bundle of circles in $\R^3$ containing sufficiently many circles in general position must pass through some other common point.
In dimension 4, this is not true. The simplest counterexample is a family of circles that are obtained from straight lines by some complex projective transformation (with respect to some identification $\R^4=\C^2$ such that the multiplication by $i$ is an orthogonal operator).
It turns out that in dimension 4 there is a large family of transformations that round lines (i.e., take them to circles). To construct such a family, fix a quaternionic structure on $\R^4$ compatible with the Euclidean structure. If $A$ and $B$ are some affine maps, then the map $x\mapsto A(x)^{-1}B(x)$ rounds lines (the multiplication and the inverse are in the sense of quaternions). Such transformations will be called (left) [*quaternionic fractional transformations*]{}. Right quaternionic fractional transformations $AB^{-1}$ also round lines. Any real projective, complex projective or quaternionic projective transformation is quaternionic fractional.
In this paper, we will prove that a rectifiable simple bundle of circles containing sufficiently many circles in general position is the image of a bundle of straight lines under some left or right quaternionic fractional transformation.
In arbitrary dimension, we have a purely algebraic description of rectifiable simple bundles of circles. So the analytic problem of classification of such bundles is reduced to an algebraic problem.
The paper is organized as follows. In Section 1, for a simple rectifiable bundle of circles we establish an algebraic condition on the second derivative of a rectifying map. This condition is formulated on the asymptotic cone $\{(x,x)=0\}\subseteq\C^2$ where $(\cdot,\cdot)$ is the complexification of the usual inner product. This provides a simple proof of Izadi’s theorem [@Iz]. In Section 2, we show that this algebraic condition is not only necessary but also sufficient in a sense. Thus we obtain an algebraic description of rectifiable simple bundles of circles. In Section 3, we review some important properties of complex and quaternionic structures and relate them to the geometry of the asymptotic cone. In Section 4, we define quaternionic fractional transformations and list some of their properties. Section 5 contains the main rectification result and some its geometrical consequences.
I am grateful to A. G. Khovanskii for useful discussions.
Rectifiable collections of circles
==================================
The following result is true in dimensions 2 [@Kh] and 3 [@Iz].
\[rect23\] Consider a simple bundle of circles in $\R^2$ or $\R^3$ containing sufficiently many circles in general position. If this bundle is rectifiable, then all its circles pass through a common point different from the origin.
On plane, it is enough to take 7 circles. Theorem \[rect23\] means, in particular, that if a generic family of circles can be rectified anyhow, then it can be rectified by means of some inversion. As we will see later, this violates in dimension 4.
We need the following very simple lemma:
\[pr\] Consider a polynomial map $F:\R^n\to\R^n$ such that $F(x)$ is everywhere proportional to $x$. Then $F(x)=G(x)x$ for some polynomial function $G:\R^n\to\R$. If $F$ is homogeneous, then so is $G$.
Introduce a coordinate system $(x_0,\dots,x_{n-1})$. Denote by $F_i$ the $i$-th component of $F$. Then the proportionality condition reads as $x_iF_0-x_0F_i=0$. In particular, $F_0$ is divisible by $x_0$. Denote the quotient by $G$. Then from our equation we see that $F_i=Gx_i$. The last statement of the lemma is obvious. $\Box$
Extend the standard inner product $(\cdot,\cdot)$ from $\R^n$ to $\C^n$ by complex bilinearity. The locus $(x,x)=0$ is called the [*asymptotic cone*]{}. Denote this cone by $C$. The asymptotic cone describes the behavior of circles at infinity. Namely, any nondegenerate circle (not a line) is asymptotic to $C$.
Let $\Phi:(\R^n,0)\to (\R^n,0)$ be a germ of diffeomorphism at 0 that sends several lines passing through the origin to circles. Suppose that the number of lines is big enough and that they are in general position. Denote this set of lines by $\Lc$. We can assume without loss of generality that $d_0\Phi=id$. To arrange this it is enough to compose $\Phi$ with some linear transformation (namely, the inverse of $d_0\Phi$) which certainly takes lines to lines. Let $\Phi=x+\Phi_2(x)+\cdots$ near 0 where $\Phi_2$ denotes the second order terms.
\[cond\] The quadratic map $\Phi_2$ satisfies the following relations on the asymptotic cone: $$(\Phi_2(x),\Phi_2(x))=0,\quad (\Phi_2(x),x)=0.$$
This proposition means that $\Phi_2$ preserves the asymptotic cone and takes each vector $x\in C$ to a vector $y\in C$ such that $x$ and $y$ span a subspace lying entirely in $C$. To give an informal explanation of this result let us assume the following:
- The diffeomorphism $\Phi$ takes germs of [*all*]{} lines passing through 0 to germs of circles.
- Our diffeomorphism can be extended to a neighborhood of the origin in $\C^n$ as a local holomorphic map.
Then $\Phi$ takes germs of complex lines to germs of [*complex circles*]{}. By a complex circle we mean an algebraic curve that is given by equations of the same form as for real circles but in complex variables and with complex coefficients. In particular, we can define a complex circle as the intersection of a complex 2-dimensional plane in $\C^n$ and a complex sphere $(x-a,x-a)=R^2$ where $a\in\C^n$ and $R\in\C$. Thus any complex circle is a plane curve of degree 2.
Take a complex line $L$ from the cone $C$. Then $\Phi(L)$ is a complex circle. We know that this circle is tangent to $L$ at 0 and asymptotic to $C$ at infinity. Denote by $M$ the plane where $\Phi(L)$ lies. Then either $M$ is contained in $C$ or $M\cap C$ is a pair of intersecting lines in $M$. In the latter case $\Phi(L)$ must coincide with one of these lines. Indeed, $\Phi(L)$ intersects both lines at the origin and is asymptotic to one of them. But a plane curve of degree 2 cannot intersect its own asymptotic line. Note that $L$ is clearly in $M\cap C$, so $\Phi(L)=L$.
In any case, $L$ and $\Phi(L)$ span a vector subspace lying entirely in $C$. Hence $\Phi_2(L)$ lies in this subspace. From this the proposition follows.
The above argument can be extended to a rigorous proof but, to give a shorter proof, we will use another idea.
Make the inversion $I$ with respect to the origin and consider the composition $I\circ\Phi$. The diffeomorphism $\Phi$ takes a line from $\Lc$ to a tangent circle (due to the condition $d_0\Phi=id$) and $I$ sends circles or lines tangent at 0 to parallel lines. Therefore, $I\circ\Phi$ maps each line from $\Lc$ to a parallel line.
Consider the Taylor series for $\Phi$ at the origin: $$\Phi(x)=x+\Phi_2(x)+\Phi_3(x)+\cdots,$$ where $\Phi_k(x)$ denotes the order-$k$ terms. Fix some nonzero vector $x$ that spans a line from $\Lc$. This line can be parameterized as $\{xt\}$ where $t$ is a parameter. Hence $I\circ\Phi(xt)$ runs over some line parallel to $x$ as $t$ runs over real numbers. This means that in the expansion of $I\circ\Phi(xt)$ all terms with nonzero powers of $t$ are proportional (parallel) to $x$. We will write down some initial terms of this expansion dropping the terms with zero power of $t$ and those obviously parallel to $x$: $$I\circ\Phi(xt)=
\left(\frac{\Phi_3}{(x,x)}-\frac{2(\Phi_2,x)\Phi_2}{(x,x)^2}\right)t+$$ $$+\left(\frac{\Phi_4}{(x,x)}-\frac{2(\Phi_2,x)\Phi_3}{(x,x)^2}
-\frac{(\Phi_2,\Phi_2)\Phi_2+2(\Phi_3,x)\Phi_2}{(x,x)^2}+
\frac{4\Phi_2(x,\Phi_2)^2}{(x,x)^3}\right)t^2+\cdots$$
The terms with $t$ and $t^2$ must be proportional to $x$. The proportionality conditions are polynomial relations in $x$. If they hold for sufficiently many $x$’s in general position, then they hold everywhere.
The coefficient with $t$ is equal to $$\frac{\Phi_3}{(x,x)}-\frac{2(\Phi_2,x)\Phi_2}{(x,x)^2}.$$ Therefore, the map $\Phi_3(x,x)-2(\Phi_2,x)\Phi_2$ is everywhere proportional to $x$. In particular, the inner product of this map with $x$ is identically zero on the asymptotic cone $\{(x,x)=0\}$. This implies that $(\Phi_2,x)=0$ on $C$. Hence $(\Phi_2,x)$ is divisible by $(x,x)$, and so the map $$\Phi_3-\frac{2(\Phi_2,x)\Phi_2}{(x,x)}$$ is a polynomial proportional to $x$. By Lemma \[pr\] this polynomial is divisible by $x$ in the class of polynomials. Therefore, $\Phi_3$ is a linear combination with polynomial coefficients of $\Phi_2$ and $x$. So it always lies in the linear span of $\Phi_2$ and $x$. In particular, $(\Phi_3,x)=0$ on $C$.
The term with $t^2$ is $$\frac{\Phi_4}{(x,x)}-\frac{2(\Phi_2,x)\Phi_3}{(x,x)^2}
-\frac{(\Phi_2,\Phi_2)\Phi_2+2(\Phi_3,x)\Phi_2}{(x,x)^2}+
\frac{4\Phi_2(x,\Phi_2)^2}{(x,x)^3}.$$ Multiply this expression by $(x,x)^2$ and restrict it to the asymptotic cone. We obtain that $\Phi_2(\Phi_2,\Phi_2)$ is parallel to $x$ on $C$ (note that all other terms are zero on the asymptotic cone). This means that either $\Phi_2$ is parallel to $x$ on $C$ or the coefficient is zero. In both cases we have $(\Phi_2,\Phi_2)=0$ on $C$. $\Box$
[Example.]{} Let us construct an example of transformation that takes all lines to circles and has the identical differential at 0. Pick up a point $a\in\R^n$ and consider the composition of the mirror reflection $$x\mapsto x-2\frac{(a,x)a}{(a,a)}$$ with respect to the orthogonal complement to $a$ and the inversion $$x\mapsto a+\frac{(a,a)(x-a)}{(x-a,x-a)}$$ with center $a$ and radius $|a|$ (so that 0 is fixed). Denote the resulting local diffeomorphism by $T^a$. We have $$T^a(x)=\frac{(a,a)x+(x,x)a}{(a,a)+2(a,x)+(x,x)}=x+\frac{(x,x)a-2(a,x)x}{(a,a)}+\cdots.$$ In particular, the quadratic term of $T^a$ has the form $$T^a_2(x)=\frac{(x,x)a-2(a,x)x}{(a,a)}$$ which is obviously parallel to $x$ on the asymptotic cone.
Now let us return to the general situation: we have a local diffeomorphism $\Phi$ which rounds a sufficiently big and sufficiently general collection $\Lc$ of lines passing through 0. Denote by $\Sc$ the corresponding set of circles.
\[paral\] Suppose that $\Phi_2$ is parallel to $x$ on the asymptotic cone. Then all the circles from $\Sc$ pass through another common point different from the origin.
To prove this, we need 2 very simple algebraic lemmas.
\[w\] Assume that a linear map $\Lambda:\R^n\to\Lambda^2\R^n$ satisfies the condition $\Lambda(x)\wedge x=0$ everywhere. Then there is a vector $b\in\R^n$ such that $\Lambda(x)=b\wedge x$.
Introduce a coordinate system $(x_0,\dots,x_{n-1})$ in $\R^n$. Let $\Lambda_{ij}(x)$ be the coordinates of $\Lambda(x)$ in the standard basis of $\Lambda^2\R^n$. These are linear functions in $x$. The condition $\Lambda\wedge x=0$ can be written in coordinates as follows: $$\Lambda_{ij}x_k+\Lambda_{jk}x_i+\Lambda_{ki}x_j=0.\eqno{(*)}$$ The above formula implies that $\Lambda_{ij}$ vanishes on the subspace $x_i=x_j=0$. Therefore, $\Lambda_{ij}=b_{ij}x_j-c_{ij}x_i$ where $b_{ij}$ and $c_{ij}$ are some numbers. Substitute this expression to $(*)$: $$(b_{ij}x_j-c_{ij}x_i)x_k+(b_{jk}x_k-c_{jk}x_j)x_i+
(b_{ki}x_i-c_{ki}x_k)x_j=0.$$ Equating the coefficient with $x_ix_j$ to zero we obtain $b_{ki}=c_{jk}$. This implies that:
- the coefficient $b_{ki}$ is independent of $i$, denote it by $b_k$;
- the coefficient $c_{jk}$ is independent of $j$, denote it by $c_k$;
- $b_k=c_k$.
Now we have $\Lambda_{ij}=b_ix_j-b_jx_i$ which means that $\Lambda(x)=b\wedge x$ where $b$ is the vector with coordinates $(b_0,\dots,b_{n-1})$. $\Box$
Recall that a map $\Gamma:\C^n\to\C^n$ is [*defined over reals*]{} if it takes $\R^n\subset\C^n$ to $\R^n$.
\[G\] Let $\Gamma:\C^n\to\C^n$ be a vector-valued quadratic form (i.e., a homogeneous polynomial map of second degree) defined over reals and such that $\Gamma(x)$ is everywhere parallel to $x$ on $C$. Then $\Gamma$ has the form $\Gamma(x)=b(x,x)+\lambda(x)x$ where $b\in\R^n$ and $\lambda$ is a linear functional.
Since $\Gamma$ is everywhere parallel to $x$ on the cone $C$, we have $\Gamma(x)\wedge x=0$ there. Therefore, $\Gamma\wedge x$ is divisible by $(x,x)$. Denote the quotient by $\Lambda$. It is a linear map from $\R^n$ to $\Lambda^2\R^n$. Moreover, we have $\Lambda\wedge x=0$ because $(\Gamma\wedge x)\wedge x=0$. By Lemma \[w\] it follows that $\Lambda=b\wedge x$ and hence $(\Gamma-b(x,x))\wedge x$ vanishes everywhere. This means that the polynomial map $\Gamma-b(x,x)$ is proportional to $x$. By Lemma \[pr\] we have $\Gamma-b(x,x)=\lambda(x)x$ where $\lambda$ is some linear function. $\Box$
[Proof of Proposition \[paral\].]{} By Lemma \[G\] the second-order part $\Phi_2$ of a rectifying diffeomorphism $\Phi$ has the form $\Phi_2(x)=b(x,x)+\lambda(x)x$ where $b$ is some vector from $\R^n$ and $\lambda$ is a linear functional.
Consider a circle from $\Sc$ with the tangent vector $x$ at 0. The acceleration with respect to the natural parameter is $$2\frac{\Phi_2-\frac{(\Phi_2,x)x}{(x,x)}}{(x,x)}=
2\frac{\Phi_2-\lambda(x)x-(b,x)x}{(x,x)}=2\left(b-\frac{(b,x)x}{(x,x)}\right)$$ that is the same as for the circle passing through $b/(b,b)$. But the circle is determined by its velocity $x/|x|$ and acceleration (both with respect to the natural parameter). It follows that all the circles from $\Sc$ pass through $b/(b,b)$. $\Box$
Now we can give a simple proof of Theorem \[rect23\].
[Proof of Theorem \[rect23\].]{} In dimensions 2 and 3 the asymptotic cone does not contain any plane. Therefore, $\Phi_2$ must be parallel to $x$ everywhere on the cone. Now Proposition \[paral\] is applicable. $\Box$
[Example.]{} In dimension 4, the statement of Theorem \[rect23\] does not hold. To construct a counterexample, introduce a complex structure on $\R^4$ and identify $\R^4$ with $\C^2$ by means of this complex structure. Consider any complex projective transformation $\Phi$ preserving the origin. It takes complex lines to complex lines, and on each line it induces a projective transformation. On the other hand, a complex projective transformation of a complex line takes real lines to circles. Therefore $\Phi$ takes real lines to circles (note that each real line belongs to exactly one complex line). Thus we get a rectifiable family of circles (through 0). But these circles do not pass through a common point different from the origin since different complex lines meet only at the origin.
Theorem \[rect23\] fails in dimension 4 by the following simple reason. The asymptotic cone now contains many planes, so there is no reason anymore for $\Phi_2(x)$ to be everywhere parallel to $x$ on $C$.
Algebraic criteria for rectification
====================================
We are going to prove now that the conditions on $\Phi_2$ stated in Proposition \[cond\] are not only necessary but also sufficient in a sense.
\[suff\] If a vector-valued quadratic form $\Gamma:\C^n\to\C^n$ defined over reals satisfies the conditions $(x,\Gamma(x))=(\Gamma(x),\Gamma(x))=0$ on the asymptotic cone, then there exists a germ of diffeomorphism $\Phi:(\R^n,0)\to(\R^n,0)$ that rounds lines passing through the origin and such that $d_0\Phi=id$, $\Phi_2=\Gamma$, i.e., $\Phi=x+\Gamma$ up to third-order terms.
Let us introduce the following notation: $$\lambda=\frac{(\Gamma,x)}{(x,x)},\quad\mu=\frac{(\Gamma,\Gamma)}{(x,x)}.$$ We know that $\lambda$ and $\mu$ are polynomials in $x$ ($\lambda$ is a linear functional and $\mu$ is a quadratic form).
First assume that $\lambda=0$ (i.e., $(\Gamma,x)=0$ everywhere). Let us look for a diffeomorphism $\Phi$ of the form $\Phi(x)=x+\Gamma(x)f(x)$ where $f$ is some smooth function that is equal to 1 at 0. We want $\Phi$ to take all lines (passing through 0) to circles. Denote by $I$ the inversion with center at 0 and radius 1. Then the germ of diffeomorphism $$I\circ\Phi=\frac{x+\Gamma f}{(x+\Gamma f,x+\Gamma f)}=
\frac 1{(x,x)}\frac{x+\Gamma f}{1+\mu f^2}$$ sends a neighborhood of 0 to a neighborhood of $\infty$ and is supposed to take each line (passing through 0) to a parallel line. For that it suffices to require that $f/(1+\mu f^2)=1$. Indeed, under the latter requirement we have $$I\circ\Phi(xt)=t^{-1}\frac x{(x,x)(1+\mu f(xt)^2)}+\frac{\Gamma}{(x,x)},$$ and the right-hand side has the form “something parallel to $x$ plus a term independent of $t$” which means that $I\circ\Phi(xt)$ runs over a line parallel to $x$ as $t$ runs over reals. Solving the corresponding quadratic equation on $f$, we obtain $$f=\frac{1-\sqrt{1-4\mu}}{2\mu}.$$ We see that $f$ is a smooth analytic function near 0 such that $f(0)=1$ as we wanted.
Now suppose that $\lambda\ne 0$. Let us look for a diffeomorphism $\Phi$ of the form $\Phi=T^a\circ\Psi$ where $\Psi$ is some other local diffeomorphism at 0. If $\Psi$ takes all lines passing through 0 to circles, then the same is true for $\Phi$. We will try to kill $\lambda$ by choosing an appropriate center $a$. For the second-order terms we have $\Phi_2=\Psi_2+T^a_2$. So it suffices to take $a$ such that $\lambda(x)=-(a,x)/(a,a)$. Now $(\Psi_2,x)=0$ everywhere, so we reduced our problem to the previous case ($\lambda=0$) which is done. $\Box$
Consider a simple bundle $\Sc$ of circles passing through 0 such that in each direction there goes a unique circle from $\Sc$. Such bundle is called [*complete*]{}. Now we can give a description of complete rectifiable bundles of circles in pure algebraic terms.
\[class\] Complete rectifiable bundles of circles in $\R^n$ are in one-to-one correspondence with quadratic homogeneous maps $\Gamma:\C^n\to\C^n$ defined over reals and satisfying the conditions $(x,\Gamma(x))=(\Gamma(x),\Gamma(x))=0$ on the asymptotic cone, modulo maps of the form $x\mapsto \lambda(x)x$ where $\lambda$ are linear functionals.
To each complete rectifiable bundle $\Sc$ of circles assign the quadratic part $\Phi_2$ of any rectifying diffeomorphism $\Phi$. We know that any quadratic homogeneous map $\Phi_2$ defined over reals and satisfying Proposition \[cond\] can be obtained in this way. Let us see to what extend the quadratic map $\Phi_2$ is unique. We saw already that for each circle from $\Sc$ it is enough to know the acceleration at 0 with respect to the natural parameter. The acceleration of the circle with the tangent vector $x$ is equal to $$w(\Phi_2)=2\frac{\Phi_2-\frac{(\Phi_2,x)x}{(x,x)}}{(x,x)}.$$ But the above expression does not determine $\Phi_2$. It is easy to see that if $\Phi_2$ and $\Phi'_2$ differ by $\lambda(x)x$ where $\lambda$ is a linear functional, then $w(\Phi_2)=w(\Phi'_2)$ so the corresponding families are the same. Indeed, it follows from the observation that $\Phi_2-(\Phi_2,x)x/(x,x)$ is just the projection of $\Phi_2$ to the orthogonal complement of $x$. Vice versa, if $w(\Phi_2)=w(\Phi'_2)$, then $\Phi_2-\Phi'_2$ is everywhere parallel to $x$ (since the projections to the orthogonal complement coincide). Hence $\Phi_2-\Phi'_2=\lambda(x)x$ where $\lambda$ is a linear functional. $\Box$
[Example.]{} In dimension 4, the condition $(x,\Gamma(x))=
(\Gamma(x),\Gamma(x))=0$ on $C$ can be interpreted in terms of algebraic geometry as follows. Denote by $Q$ the projectivization of the asymptotic cone. This is a nondegenerate quadratic surface in $\CP^3$. Each point of $Q$ belongs to 2 straight lines lying entirely in $Q$.
To describe all lines in $Q$ it is convenient to identify $Q$ with the image of the Segre embedding $$\CP^1\times\CP^1\to\CP^3,\quad ([u_0:u_1],[v_0:v_1])\mapsto
[u_0v_0:u_0v_1:v_0u_1:u_1v_1]$$ (recall that any nondegenerate quadratic surface in $\CP^3$ can be mapped to any other by a complex projective transformation). Under this embedding, each horizontal line $\CP^1\times\{p\}$ and each vertical line $\{p\}\times\CP^1$ get mapped to straight lines. Hence we have 2 families of lines in $Q$ such that every point of $Q$ belongs to a unique line from each family. These families of lines are called [*generating families of lines*]{}. For each generating family of lines in $Q$ there is the corresponding [*generating family of planes*]{} in $C$. So the cone $C$ is covered by 2 generating families of planes, and every line in $C$ belongs to exactly one plane from each generating family.
The conditions $(x,\Gamma(x))=(\Gamma(x),\Gamma(x))=0$ on the asymptotic cone are equivalent to the following statement: the subspace spanned by $x$ and $\Gamma(x)$ lies entirely in $C$. This means that $\Gamma$ takes $x$ to another point of some line or plane lying entirely in $C$. The map $\Gamma$ is homogeneous. Therefore, it gives rise to a map from $\gamma:\CP^3\to\CP^3$ preserving the projectivization $Q$ of the asymptotic cone $C$. We know that for each point $q\in Q$ there is a line lying entirely in $Q$ and containing both $q$ and its image $\gamma(q)$. We will deduce from this that $\Gamma$ preserves at least one of the generating families of lines in $Q$ (maybe both), i.e., takes each line from some generating family to itself. Indeed, being an algebraic map, $\gamma$ cannot “switch” from one generating family to the other. Below is a formal proof of this statement.
\[gen\] The map $\gamma$ preserves at least one generating family of lines in $Q$.
The surface $Q$ is isomorphic to $\CP^1\times\CP^1$ via the Segre map. Hence $\gamma$ can be given by 2 algebraic maps $$X:(x,y)\in\CP^1\times\CP^1\mapsto X(x,y)\in\CP^1,$$ $$Y:(x,y)\in\CP^1\times\CP^1\mapsto Y(x,y)\in\CP^1.$$ We know that for each point $(x,y)\in\CP^1\times\CP^1$ we have $X(x,y)=x$ or $Y(x,y)=y$. Therefore, $Q$ is the union of 2 algebraic subsets defined by the equations $X(x,y)=x$ and $Y(x,y)=y$. Since $Q$ is irreducible, at least one of our equations is satisfied identically, which means that $\gamma$ preserves at least one of the generating families of lines in $Q$. $\Box$
Now we can deduce the following:
Polynomial homogeneous maps $\Gamma:\C^4\to\C^4$ satisfying the conditions $(x,\Gamma(x))=(\Gamma(x),\Gamma(x))=0$ on the asymptotic cone preserve some generating family of planes in $C$.
Complex and quaternionic structures
===================================
From now on we will work in 4-dimensional space $\R^4$. This section reviews not only well-known classical facts about complex and quaternionic structures, but also their relation to the geometry of the asymptotic cone $C$.
Recall that a [*complex structure*]{} in $\R^4$ is a linear operator $I:\R^4\to\R^4$ such that $I^2=-1$. We will always assume that the complex structure $I$ is compatible with the Euclidean structure, i.e., $I$ preserves the inner product. A complex structure clearly defines an action of $\C$ on $\R^4$ via linear conformal maps. From the definition it follows immediately that $I$ must be skew-symmetric, i.e., $(x,Iy)=-(Ix,y)$ for all $x,y\in\R^4$. In particular, $(Ix,x)=0$. Since the operator $I$ is defined over reals and $I^2=-1$, it should have eigenvalues $i$ and $-i$, both with multiplicity 2.
Note that $I$ preserves the asymptotic cone $C$ (being an orthogonal operator). In particular, all eigenvectors of $I$ belong to $C$. We know that $(Ix,x)=0$ everywhere and in particular on $C$. From the conditions $(x,x)=(Ix,Ix)=(Ix,x)=0$ on $C$ it follows that the subspace spanned by $x$ and $Ix$ lies entirely in $C$. Hence $I$ preserves one of the generating families of planes in $C$.
On the other hand, the complex structure $I$ defines a canonical orientation on $\R^4$. Let us recall the definition. Take 2 vectors $x,y\in\R^4$ in general position. By definition, the canonical orientation is the orientation of the basis $x,y,Ix,Iy$. This orientation is well-defined (i.e., independent of the choice of $x$ and $y$) because the set of degenerate pairs $(x,y)$ (such that $x,y,Ix,Iy$ are linearly dependent) has real codimension 2 in the space $\R^8$ of all pairs. So we can always avoid this set going from any nondegenerate pair to any other. In fact, the degeneracy locus consists of all pairs $x,y$ that are linearly dependent over $\C$, so it is a complex hypersurface.
The space of all complex structures on $\R^4$ has 2 connected components. Complex structures from the same component preserve the same generating family of planes in $C$ and provide the same canonical orientation.
A connected component to which a complex structure $I$ belongs will be called the [*orientation*]{} of $I$. Note that the orientation of $I$ has nothing to do with $\det(I)$ which is always equal to 1 — any complex structure preserves orientation of the ambient space.
Now let us pass to quaternionic structures. A [*quaternionic structure*]{} on $\R^4$ is a choice of 3 linear operators $I,J,K:\R^4\to\R^4$ such that $$I^2=J^2=K^2=-1,$$ $$IJ=-JI=K,\quad JK=-KJ=I,\quad KI=-IK=J.$$ In particular, the operators $I,J,K$ are complex structures. We will assume that they are compatible with the inner product. A quaternionic structure gives rise to an action of the skew-field $\H$ of quaternions on $\R^4$ via linear conformal maps. This action is called the [*quaternionic multiplication*]{}.
Let $(I,J,K)$ be any quaternionic structure on $\R^4$. Then all 3 complex structures $I,J,K$ have the same orientation. Therefore, quaternionic multiplication preserves one of the generating families of planes in the asymptotic cone.
Let us prove, for example, that $I$ and $J$ provide the same canonical orientation. Take any vector $e\in\R^4$. It is enough to show that the bases $(e,Ke,Ie,IKe)$ and $(e,Ke,Je,JKe)$ have the same orientation. But $IKe=-Je$ and $JKe=Ie$, so the statement becomes obvious. $\Box$
Let $a\in\H$ be a quaternion. It gives rise to the operator of multiplication $A:x\mapsto ax$. If $a=a_0+a_1i+a_2j+a_3k$, then the corresponding operator is $A=a_0+a_1I+a_2J+a_3K$. We know that the operator $A$ satisfies the conditions $(x,Ax)=(Ax,Ax)$ on $C$. In particular, both forms $(Ax,Ax)$ and $(Ax,x)$ are divisible by $(x,x)$. We can write down the quotients explicitly.
\[quot\] If $A$ is the operator of multiplication by a quaternion $a\in\H$ (with respect to some quaternionic structure on $\R^4$), then $$(Ax,Ax)=(a,a)(x,x),\quad (Ax,x)=\Re(a)(x,x).$$ In particular, these forms are independent of the choice of a quaternionic structure.
This is a very simple computation based on the fact that $(x,Ix)=(x,Jx)=(x,Kx)=0$ for all $x\in\R^4$. $\Box$
Let us summarize some properties of quaternionic structures that are of particular importance for us. These properties follow directly from what we saw already.
\[quatern\] The set of all quaternionic structures in $\R^4$ has 2 connected components. Each component corresponds to a certain orientation of 3 complex structures involved. Quaternionic multiplications with respect to quaternionic structures from the same component preserve the same generating family of planes in $C$. Different components correspond to different families of generating planes.
We will say that quaternionic structures from the same connected component have the same [*orientation*]{}. Note that the orientation of a quaternionic structure has nothing to do with determinants of quaternionic multiplications. Quaternionic multiplications (with respect to any quaternionic structure) always preserve the orientation of the ambient space.
[Example.]{} Identify $\R^4$ with $\H$. Denote by $I,J,K$ the operators of left multiplication by $i,j,k$ respectively. The structure $(I,J,K)$ is called the [*left quaternionic structure*]{} on $\H$. If we take right multiplication instead of left multiplication, then we get the [*right quaternionic structure*]{}. Left and right quaternionic structures on $\H$ have different orientations.
Let us introduce some notions. We say that a linear operator is [*almost orthogonal*]{} if it has the form $const\cdot A$ where $A$ is an orthogonal operator. Analogously, an operator is [*almost skew-symmetric*]{} if it has the form $const+A$ where $A$ is skew-symmetric.
A linear operator $A:\R^4\to\R^4$ is the multiplication by a quaternion (with respect to some quaternionic structure on $\R^4$) if and only if it is almost orthogonal and almost skew-symmetric. The property of being a quaternionic multiplication depends only on the orientation of a quaternionic structure, not on a structure itself.
A quaternionic multiplication is clearly almost orthogonal and almost skew-symmetric. If follows from Lemma \[quot\]. Now consider an almost orthogonal and almost skew-symmetric operator $A$ and present it by a matrix in some orthonormal basis. Denote by $a_0,a_1,a_2,a_3$ the entries of the first column of $A$. Since $A$ is almost skew-symmetric, it has the form $$\left(\begin{array}{cccc}
a_0 & -a_1 & -a_2 & -a_3 \\
a_1 & a_0 & \alpha & \beta \\
a_2 & -\alpha & a_0 & \gamma \\
a_3 & -\beta & -\gamma & a_0
\end{array}\right).$$ The columns must be orthogonal and have the same length. From the corresponding equations we obtain that either $\alpha=a_3,\beta=-a_2,\gamma=a_1$ or $\alpha=-a_3,\beta=a_2,\gamma=-a_1$. The first case corresponds to the left multiplication by $a=a_0+a_1i+a_2j+a_3k$ with respect to the standard quaternionic structure (assigned to the given basis). The second case corresponds to the right multiplication by $a$. No matter what orthonormal basis we chose. Thus the second statement follows. $\Box$
Quaternionic fractional transformations
=======================================
Let us identify $\R^4$ with the skew-field $\H$ of quaternions. Consider 2 affine maps $A,B:\R^4\to\R^4$. The map $B^{-1}A$ (the multiplication and the inverse are in the sense of quaternions) is called a (left) [*fractional quaternionic transformation*]{} provided that it is one-to-one at least in some open subset of $\R^4$. A [*right quaternionic fractional transformation*]{} is a local transformation of the form $AB^{-1}$ where $A$ and $B$ are some affine maps.
[Example 1.]{} Any real projective transformation is quaternionic fractional. This corresponds to the case when $B$ takes real values only.
[Example 2.]{} Any complex projective transformation is quaternionic fractional. This happens if $B$ takes complex values only and both $A$ and $B$ are complex linear (i.e., commute with the multiplication by $i$).
[Example 3.]{} Consider a map of the form $x\mapsto (xa+b)^{-1}(xc+d)$ where $a,b,c,d$ are quaternions. We are assuming that the denominator is not proportional to the numerator (in particular, the denominator is not identically zero). This map is called a (left) [*quaternionic projective transformation*]{}. Any quaternionic projective transformation is clearly quaternionic fractional. Note that each quaternionic projective transformation takes all lines to circles. Indeed, we have $$(xa+b)^{-1}(xc+d)=(xa+b)^{-1}((xa+b)\alpha+\beta)=\alpha+(xa+b)^{-1}\beta$$ where $\alpha=a^{-1}c$, $\beta=d-b\alpha$. Hence a quaternionic projective transformation is a composition of a dilatation, reflected inversion and a translation. This composition obviously rounds lines.
Any quaternionic fractional transformation rounds lines (to be more precise: it takes germs of lines to germs of circles).
Consider a line $L$ in $\R^4$. Let $t$ be a linear parameter on $L$. If $A$ and $B$ are some affine maps, then their restrictions to $L$ are $at+b$ and $ct+d$ respectively. So on the line $L$ the transformation $A^{-1}B$ coincides with the quaternionic projective transformation $x\mapsto (ax+b)^{-1}(cx+d)$. But the latter rounds lines. $\Box$
Rectification at a point
========================
In this section, we will prove the following theorem:
\[rect4\] Consider a simple bundle of circles in $\R^4$ containing sufficiently many circles in general position. If this bundle is rectifiable, then there exists a left or right quaternionic fractional transformation $T$ such that $T^{-1}$ sends all these circles to straight lines.
Denote the given set of circles by $\Sc$. Let $\Phi$ be a local diffeomorphism such that $d_0\Phi=id$ and $\Phi^{-1}$ rectifies all circles from $\Sc$. Then by Proposition \[cond\] the quadratic term $\Phi_2$ satisfies the relations $(\Phi_2,x)=(\Phi_2,\Phi_2)=0$ on the asymptotic cone. This means that $\Phi_2$ preserves one of the generating families of planes in $C$.
\[Phi\_2\] There exists a linear operator $A:\R^4\to\R^4$ such that $\Phi_2(x)=A(x)x$ or $\Phi_2(x)=xA(x)$ where the product is in the sense of quaternions.
Fix an identification $\R^4=\H$. Extend the operators $I$, $J$ and $K$ of left multiplication by $i$, $j$ and $k$ respectively to $\C^4$ by complex linearity. Note that the operator $I$ is quite different from the multiplication by $\sqrt{-1}$ in $\C^4$. By Proposition \[quatern\] the left quaternionic multiplication preserves one of the generating families of planes in $C$. Assume that $\Phi_2$ preserves the same family. Otherwise we should consider the right multiplication instead of the left multiplication.
Recall that the [*quaternionic conjugation*]{} is the map $$x=x_0+x_1i+x_2j+x_3k\mapsto \bar x=x_0-x_1i-x_2j-x_3k.$$ We can extend this map to $\C^4$ by complex linearity. Note that $i$ is now a vector from $\R^4$, not a complex number. Let us multiply $\Phi_2$ by $\bar x$ in the sense of quaternions. Note that $$\Phi_2\bar x=(\Phi_2,x)+(\Phi_2,Ix)i+(\Phi_2,Jx)j+(\Phi_2,Kx)k.$$ But this expression is zero on the cone $C$ since $\Phi_2$, $x$, $Ix$, $Jx$ and $Kx$ lie on the same plane belonging to $C$. Therefore, $\Phi_2\bar x$ is divisible by $(x,x)$. The quotient is a linear map $A$. Since $\bar x/(x,x)=x^{-1}$, we have $\Phi_2x^{-1}=A(x)$, i.e., $\Phi_2(x)=A(x)x$. $\Box$
Now we can prove Theorem \[rect4\] and even more precise statement:
Under assumptions of Theorem \[rect4\] the family of circles can be obtained from the family of their tangent lines by one of the transformations $x\mapsto (1-A(x))^{-1}x$ or $x\mapsto x(1-A(x))^{-1}$ where $A$ is some linear operator. This answer does not depend on the choice of a quaternionic structure.
Note that both transformations have the identical derivative at 0 and their second-order terms are $A(x)x$ and $xA(x)$ respectively. These transformations are quaternionic fractional so they round lines. The corresponding families of circles passing through 0 are determined by the second-order terms. But by Lemma \[Phi\_2\] the quadratic maps $A(x)x$ and $xA(x)$ are the only possible second-order terms of transformations that round lines. $\Box$
For a complete rectifiable bundle $\Sc$ of circles there is a transformation of the form $x\mapsto (1-A(x))^{-1}x$ or $x\mapsto x(1-A(x))^{-1}$ that takes the family of all lines passing through 0 to $\Sc$. To fix the idea, assume that this is the left transformation $\Phi:x\mapsto (1-A(x))^{-1}x$.
The center of the circle from $\Sc$ with the tangent vector $x$ at 0 is $-\frac 12(\Im A(x))^{-1}x$. This point can be infinite which means that the corresponding circle is a straight line.
We know that the acceleration with respect to the natural parameter is $$w(x)=2\frac{\Phi_2-\frac{(\Phi_2,x)x}{(x,x)}}{(x,x)}.$$ Therefore the center is located in the point $$\frac w{(w,w)}=
\frac 12\frac{ \frac{\Phi_2}{(x,x)} - \frac{(\Phi_2,x)x}{(x,x)^2} }
{ \frac{(\Phi_2,\Phi_2)}{(x,x)^2} - \frac{(\Phi_2,x)^2}{(x,x)^3} }.$$ By Lemma \[quot\] we have $(\Phi_2,\Phi_2)=(A,A)(x,x)$ and $(\Phi_2,x)=(\Re A)(x,x)$. Simplifying the above expression we get the following formula for the center: $$\frac 12\left(\frac{A-\Re A}{(A,A)-(\Re A)^2}\right)x=
\frac 12\frac{\Im(A)}{(\Im A,\Im A)}x=-\frac 12(\Im A)^{-1}x.$$ $\Box$
The previous proposition has the following geometric corollary:
The family $\Sc$ contains at least one line. The union of all straight lines from $\Sc$ is a vector subspace of $\R^4$.
[Remark.]{} We see that the set of all complete rectifiable families of circles passing through 0 is naturally identified with the union of 2 affine spaces of dimension 12 (=dimension of all possible $\Im A(x)$). The intersection of these components has dimension 4 and consists of all families rectifiable by means of inversions (i.e., of families whose circles meet at a point different from 0 — this happens if $\Im A$ is independent of $x$). The two components can be distinguished by the “orientation”.
We can describe an affine structure on each component in geometric terms. Namely, take any 2 circles $S_1$ and $S_2$ tangent at 0. After an inversion, they become parallel lines. For two parallel lines $L_1$ and $L_2$ we can take their barycentric combination $$L=\lambda L_1+(1-\lambda)L_2=
\{\lambda x+(1-\lambda)y|\ x\in L_1,y\in L_2\},\quad \lambda\in\R.$$ Make the inversion again. The line $L$ goes to a circle $S$. Put by definition $S=\lambda S_1+(1-\lambda)S_2$. Now we can take barycentric combinations of complete bundles of circles. Namely, let the circle of the new bundle passing through 0 in direction $x$ be $S=\lambda S_1+(1-\lambda)S_2$ where $S_1$ and $S_2$ are circles from the old bundles going from 0 in direction $x$. It turns out that if two rectifiable bundles have the same “orientation”, then their barycentric combinations are also rectifiable.
[Open question.]{} How many complete rectifiable simple bundles of circles are there? We saw that in $\R^n$ the space of all complete rectifiable bundles of circles passing through 0 is finite-dimensional. What is its dimension (as a function of $n$)? Is there an explicit geometric description of such bundles in dimensions $>4$?
[9]{} Khovanskii A. G., [*Rectification of circles*]{}, Sib. Mat. Zh., [**21**]{} (1980), 221–226
Izadi F. A., [*Rectification of circles, spheres, and classical geometries*]{}, PhD thesis, University of Toronto, (2001)
[^1]: Partially supported by RFBR 99-01-00245 and CRDF RM1-2086
|
---
author:
- 'G. Carinci, A. De Masi, C. Giardinà, E. Presutti'
title: |
Free boundary problems\
in PDEs and particle systems
---
[G. Carinci, A. De Masi, C. Giardinà, E. Presutti]{} [Free boundary problems]{}\
[in PDEs and particle systems]{}
|
---
author:
- |
L. Andrianopoli$^a$, M. Derix$^a$, G.W. Gibbons$^b$, C. Herdeiro$^b$, A. Santambrogio$^a$, A. Van Proeyen$^{a}$ [^1]\
$^a$ Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,\
Celestijnenlaan 200D B-3001 Leuven, Belgium\
$^b$ D.A.M.T.P., University of Cambridge,\
Silver Street, Cambridge CB3 9EW, U.K.
title: 'Embedding Branes in Flat Two-time Spaces'
---
[Embedding]{} geometric manifolds in higher dimensional flat spaces can be a useful tool for studying global properties of these manifolds. Typical examples of extending the dimension for a better understanding of the geometry are the description of the $d$-dimensional sphere $S^d$ by embedding it in $(d+1)$-dimensional Euclidean space and the description of $AdS_d$ as a hyperboloid in flat $(d+1)$-dimensional space with two timelike directions. The embedding is encoded, in both cases, in one constraint on the embedding coordinates and makes the $SO(p,q)$ global symmetry [^2] of these geometries manifest. Any choice of coordinates on the $d$-dimensional manifold will break this manifest symmetry.
The two examples above are combined in [@conffads], where the $AdS_{p+2}\times S^{n-1}$ near-horizon geometry of $p$ branes in $D=p+n+1$ dimensions is described starting from a flat $(D+2)$-dimensional space. Two constraints are imposed, which respectively reduce $p+3$ dimensions to the $AdS_{p+2}$ manifold and $n$ dimensions to the $S^{n-1}$ sphere. The Born–Infeld action for the near-horizon theories of these branes can then be expressed in terms of $(D+2)$-dimensional fields, while the embedding constraints are imposed by means of Lagrange multipliers. The Wess–Zumino (WZ) terms in these actions can be obtained from a constant $(p+2)$ form in $D+2$ dimensions, which must be integrated over a $(p+2)$-dimensional manifold which has the worldvolume as its boundary.
This talk is based on the paper [@noi]. We will show, following [@noi], how the full spacetime metric of a brane can be embedded isometrically in ${\IE}^{D,2}$. In this way we generalize the construction of [@conffads] to not just near-horizon, but to the full brane geometry. We will show that even if the geometry is not a product of $AdS$ times a sphere, the brane geometry can still be embedded in flat $(D+2)$-dimensional space with signature $(D,2)$. The two constraints are no longer independent in the sense that they do not constrain separate coordinates of the embedding space, but instead a non-trivial mixing of the coordinates is involved.
As in [@conffads], also the forms for the WZ terms are obtained in this picture. For that construction, we will assume [@hew] that a $p$ brane evolving in a space with two times couples to a $(p+3)$-form field strength. The field strength is then contracted to a $(p+2)$-form which can be used for the WZ term. To make this contraction we will have to introduce an extra vector field which only in the near-horizon limit will have an elegant form.
One may wonder whether the whole geometry cannot be embedded with just one extra dimension and why we need two timelike directions for the embedding space. First of all, it has been shown [@eise] that the embedding of a surface in a flat space of co-dimension 1 imposes, by use of the Einstein equations, that the surface has constant curvature (if the surface has dimensionality $d>2$). This corresponds to familiar cases as the embedding of spheres and (anti–)de Sitter in flat spaces. Therefore, in order to embed brane backgrounds, which do not have constant curvature, we need at least two extra dimensions.
To determine the signature of the embedding space we use the following argument. An interesting aspect of brane spacetimes is that they are not globally hyperbolic[^3]. According to Penrose [@Penrose], a global isometric embedding in normal flat Minkowski space, i.e. in ${\IE}^{n,1}$, is only possible for a spacetime which is globally hyperbolic. Penrose’s argument is essentially that the restriction of the time coordinate $X^0$ of ${\IE}^{n,1}$ to the embedded spacetime $M$ would serve as a time-function on $M$, i.e., a function which increases along every future directed timelike curve. Moreover, if the embedding is suitably regular, the level sets (constant time slices) would actually serve as Cauchy surfaces on $M$, implying global hyperbolicity. In a sense any embedded surface inherits global hyperbolicity from the ambient space. No such obstruction arises for embeddings in flat space with more than one timelike direction. We will describe therefore a minimal embedding of general brane backgrounds in flat spaces with two extra dimensions and $(D,2)$ signature.
In section \[ss:embedding\] we give the embedding of the geometry, commenting on global properties and on the near-horizon approximation. The worldvolume actions will be constructed in section \[ss:braneaction\]. The essential step in that section is the construction of the forms. General results for the electric field strengths are given, before completing the construction for the example of the D3-brane.
Embedding: The geometry {#ss:embedding}
=======================
In this section we describe the embedding of a $SO(n)$ invariant $p$-brane in a $(D+2)=(n+p+3)$-dimensional spacetime. We will obtain the embedding by requiring that the known metric of the brane is obtained from a flat $(D,2)$ metric, i.e. we demand that the embedding is isometric [@goenner; @friedman]. The $D$-dimensional $p$-brane geometry can generally be described by a metric of the form ds\^[2]{}&=&A(r)\^[2]{}+\
&&+B(r)\^[2]{}dr\^[2]{}+ C(r)\^[2]{}d\_[n-1]{}\^[2]{}, \[metricbra\] where $dx_p\cdot dx_p$ is the $p$-dimensional spacelike part of the worldvolume, and $d\Omega_{n-1}^{2}$ is the metric for the $n$-sphere.
On the embedding space $\IE^{(D,2)}$ we take cartesian coordinates $X^{M}$, with $M=0,\ldots ,D+1$, which we divide as $X^M=(X^\mu, X^{p+1}, X^{p+2}, X^\alpha)$ (with $\mu =0,\ldots
,p$, $\alpha=p+3,\ldots ,D+1$). Using these coordinates, the metric looks as follows ds\^[2]{}&=&-(dX\^[0]{})\^[2]{}+(dX\^[1]{})\^[2]{}+…+(dX\^[p+1]{})\^[2]{}+\
&&-(dX\^[p+2]{})\^[2]{} +…+(dX\^[D+1]{})\^[2]{}.
To obtain the two embedding constraints, we start by making a change of the $(D+2)$-dimensional coordinates, such that we make a subgroup $SO(p,1)\times SO(n)\subset
SO(p+n+1,2)$ manifest. This is achieved by using a mixture of hyperspherical and horospherical coordinates $\{\rho ,z,x^\mu ,\beta ,n^\alpha \}$ $$\begin{aligned}
&& X^-\equiv X^{p+2}-X^{p+1}=\frac{\rho}{z},\nonumber\\
&&X^+ \equiv X^{p+2}+X^{p+1}= \rho z+\frac{\rho}{z}x^{\mu}x_{\mu},\nonumber\\
&& X^{\mu}=\rho \frac{x^{\mu}}{z}\,,\quad X^{\alpha}=\beta n^{\alpha}\,,
\label{embori}\end{aligned}$$ where $n^{\alpha}$ ($\sum_{\alpha}(n^{\alpha})^{2}=1$) parametrise the sphere $S^{n-1}$. In these new coordinates, the metric reads ds\^[2]{}=-d\^[2]{} +d\^[2]{}+\^[2]{}dn \^dn\^. \[metricemb\]
Comparing (\[metricemb\]) and (\[metricbra\]) we identify $dx^\mu dx_\mu$ with $-dt^2+dx_p.dx_p$ and $dn^\alpha dn^\alpha
$ with $d\Omega _{n-1}^2$. Then $\beta $, $\rho $ and $z$ are functions of $r$ and are still to be determined. The comparison gives $$\begin{aligned}
&& \beta=C(r) \, ,\quad \frac{\rho}{z}=A(r) \, ,\nonumber\\
&&
-d\rho^{2}+\frac{\rho^{2}}{z^{2}}dz^{2}+d\beta^{2}=B(r)^{2}dr^{2}\,.
\label{ident}\end{aligned}$$ The differential equation can be rewritten to give =(z)’ F’. \[df\]
From all this we can derive the following embedding functions $$\begin{aligned}
X^- &=& A(r) \, , \quad
X^+ = F(r)+A(r)x^{\mu}x_{\mu}\nonumber\\
X^{\mu}&=&A(r)x^{\mu} \, , \quad
X^{\alpha}=C(r) n^{\alpha}\,.
\label{embedfun}\end{aligned}$$ We can, furthermore, express the constraints in terms of the $X^M$ coordinates only. Denoting the inverse function with an overbar, i.e., $\bar{f}(f)=f(\bar{f})={\mathord{\!\usebox{\uuunit}}}$, we can write $r=\bar{A}(X^-)$. Thus, our two constraints are $$\begin{aligned}
\phi_1 &=& X^-X^+-X^{\mu}X_{\mu}-
X^- F(\bar{A}(X^-))=0\nonumber\\
\phi_2 &=& \sum_{\alpha}(X^{\alpha})^{2}-
\left[ C(\bar{A}(X^-))\right]^{2}=0\,.
\label{constraints}\end{aligned}$$ These constraints are therefore determined by the functions $A$, $C$ and $F$. The latter is determined up to a constant by (\[df\]) in terms of $A$, $B$ and $C$.
Note that so far there is no definition of the radial variable $r$. We can use different parametrizations, e.g. it will turn out that in some cases it is useful to take $A$ or $C$ itself as the radial variable. In the standard brane cases, the functions $A$, $B$ and $C$ will take the form of some harmonic function to some power in the transverse space of the brane. We will further adopt the name $r$ for that transverse coordinate, use just the name $A$ for the parameter in the first mentioned parametrization, and use $R$ for the radial coordinate such that $C(R)=R$.
From now on we will assume that the functions $A$, $B$ and $C$ are indeed harmonic functions in $n$ dimensions with a flat limit at $r\rightarrow \infty$. For non-dilatonic D- and M-branes, they are of the following form $$\begin{aligned}
&H = \left(1+\frac{1}{r^{\kappa}}\right)\,,\quad
A(r) = H^{-\frac{1}{p+1}}&\nonumber\\
&B(r) = H^{\frac{1}{\kappa}}\,,\quad
C(r)= r H^{\frac{1}{\kappa}}&
\label{harmonic}\end{aligned}$$ where $\kappa \equiv n-2=D-p-3$. Here a priori $r>0$ and $r=0$ corresponds to the horizon, but we will come back to this later.
With this explicit form for the functions $A$, $B$ and $C$ we can evaluate the function $F$. Using (\[df\]) we get F’(r) = - w r\^[1-]{} (1+r\^[-]{})\^[ + -1]{} (1+2r\^) , (where $w=\frac{p+1}{\kappa}$), which can be integrated to give (up to a constant) F(r) &=& - . \[fofr\] Here we used the incomplete Beta function B\_x(a,b) &=& \_0\^x t\^[a-1]{} (1-t)\^[b-1]{} dt=\
&=& a\^[-1]{}x\^a \_2F\_1(a,1-b;a+1;x), which is defined for $0 < x \leq 1$. This means that $F(r)$ is well defined in the region $r>0$, which is what we were looking for.
Note that near the horizon ($r\to 0$), where the brane geometry is well described by $AdS_{p+2}\times S_{D-p-2}$ [@interp], we get $ F\sim w^2 r^{-\frac{1}{w}}$ and the embedding functions (\[embedfun\]) reduce to those used in [@conffads].
Using the embedding we can now study the global properties of the brane geometries. Before considering the higher dimensional D- and M-branes, let us first look at the simpler example of the extreme Reissner–Nordstr[ø]{}m (RN) black hole. (A large list of embedding functions for other solutions of General Relativity is given in [@rosen]). The RN black hole fits our general embedding scheme with $D=4$ and $p=0$, $\kappa =1$, $w=1$. Here, rather than working with the radial variable $r$ as in (\[harmonic\]), we use the variable $R\equiv r+1$, which has the property $C(R)=R$. Then the functions $A$ and $B$ are given by $A(R) = B(R)^{-1} = 1-1/R$. The horizon is now at $R=1$ and $R=0$ corresponds to the singularity. Using (\[df\]), we then find $$F_{RN}(R) = \frac{1}{R-1} -3R - R^2 - 4 \log |R-1| \,.$$
The entire Reissner–Nordstr[ø]{}m black hole geometry can be drawn using parametrization (\[embedfun\]) as is shown in figure \[rn\]. We only draw the relevant directions ($X^-$, $X^+$, and $X^0$), which basically means we only draw the $R$ and $t$ coordinates of the black hole (every point in the graph should be thought of as a 2-sphere). The lines in the graph are therefore constant $t$ and constant $R$ lines.
We can read off the following global features from the picture. The geometry consists of 2 distinct regions: region I, the asymptotically flat region for $R>1$ which corresponds to $X^->0$. For big $R$ the surface flattens and $X^- \to 1$, which is the flat limit. Region II is the region inside the horizon ($X^- <0$), the singularity ($R=0$) corresponds to $X^- \to -\infty$. This is the global picture we recognize from the familiar Penrose diagram for extreme RN black hole as can be found in [@noi].
The two regions are connected in an [*$AdS$-throat*]{}. It seems that these two regions are disconnected, the constant time lines all diverge near $X^-=0$ and never cross the horizon, but this is just an artifact of the parametrization. Actually, we know that the near-horizon geometry is equivalent to $AdS_2$, which is known to have no problems at its ’horizon’. Indeed, a different parametrization exists (the advanced or retarded Finkelstein coordinates) in which lightlike geodesics pass smoothly through the horizon into the interior region, as is depicted in figure \[rnv\].
One of the features of $AdS$ spaces is that they admit closed timelike curves. The usual remedy for this is to consider the covering space $CAdS$ instead of $AdS$ itself. Looking at figures \[rn\] and \[rnv\] we see that the RN black hole geometry suffers from the same problem, it admits closed timelike curves. Again this is remedied by considering the covering space. The result of this of course is that the space then consists of multiple universes.
Let us now move to the non-dilatonic branes. As discussed in [@GHT], the general brane solution case (\[harmonic\]) can be divided in two classes: $p$ odd or $p$ even, with quite different global properties.
Let us first consider the $p$ *odd* case. In the exterior region ($r>0$), the function $A(r)$ is analytic and positive and vanishes as $r\to 0$. If we take $A$ to be our new radial variable instead of $r$, we see that $A$ can be continued through the horizon to negative $A$ [@GHT]. The range of $A$ is from -1 to 1. The analytic extension of the metric is && ds\^2= A\^2 dx\_dx\^+ (1- A\^[p+1]{}) \^[-]{}\
&& , \[dsA\] which is even in $A$. This leads to F(A)&=& - (A) . The embedding functions (\[embedfun\]) are then odd in $A$. This means that the embedded space is symmetric around the horizon and completely nonsingular. For the non-dilatonic branes, the D3 and M5 fit this picture. The embedding, depicted in figure \[d3m5\] for the D3-brane case, nicely shows these features (an analogous picture for the M5-brane can be found in [@noi]). It is clearly visible there is no interior region, just two symmetric ’exterior’ regions connected in the AdS-throat as was expected from the Penrose diagram [@GHT] [@noi].
In the $p$ *even* case, the metric and embedding functions are neither even nor odd. It is useful in this case to adopt so-called Schwarzschild coordinates, defined by $R^\kappa =r^\kappa+1$. In these coordinates the horizon (which is still a coordinate singularity) is at $R=1$. At $R=0$ there is a true curvature singularity. Expressed in this coordinate, $A(R)$ can be continued through the horizon into negative $A$ and its range is $\{-\infty,1\}$. As already stated in [@GHT], the Penrose diagram for these spaces is equivalent to the extreme Reissner–Nordstr[ø]{}m diagram.
The embedding of the M2-brane metric illustrates these features. The expression (\[fofr\]) of $F$ is only well defined in the region $R>1$. It is not possible to find a continuous expression for $F$ valid in both regions ($0<R\leq 1$ and $R>1$). But, nevertheless, a continuous embedding is obtained using in the interior region F(R<1) &=& . The global properties of the M2-brane are qualitatively the same as those of the RN black hole depicted in figure \[rn\]. We refer to [@noi] for the M2-picture.
The brane action {#ss:braneaction}
================
We would like to write the action of a brane placed in the background of other branes using the embedding of the previous section. A typical (schematic) form of the action is $$\begin{aligned}
S_{p+1}&=& \int_W d^{p+1}\xi \sqrt{-\det {\cal G}_{\mu \nu }} +
\int_{B} \Omega _{(p+2)} +\nonumber\\
&& + \int_W d^{p+1}\xi [\lambda_1 \phi_1 +
\lambda_2 \phi_2]\,,
\label{action}\end{aligned}$$ where $W=\partial B$ is the $(p+1)$-dimensional world volume of the brane. The expression for ${\cal G}_{\mu \nu }$ differs in each case. For example, for Dp-branes ${\cal G}_{\mu\nu} \equiv \partial _\mu X^M \partial _\nu
X^N \eta_{MN}+{\cal F}_{\mu\nu}$, with ${\cal F}_{\mu\nu}$ the field strength of the gauge field living on the world volume of the brane. The fields $\lambda_1,
\lambda_2$ are two Lagrange multipliers implementing the constraints (\[constraints\]). $\Omega _{(p+2)}(X^M)$ is a function of the forms coupling to the brane, such that it reduces to the appropriate Wess–Zumino term when projected onto the physical hypersurface. Its explicit form will be determined for the D3-brane case in section \[ss:d3fs\]. An analogous treatment for M2 and M5 is given in [@noi].
Embedding the field strength
----------------------------
Let us now try to embed the field strengths appearing in the Wess–Zumino term. We will assume [@hew] that a brane (extended in $p$ spatial directions) fluctuating in a spacetime with two times should evolve in both time directions, and therefore couple to a (p+3)-form field strength. We assume therefore that the $(D+2)$-dimensional theory can be coupled to a rank $p+3$ electric field strength $K_e$, and to a rank $n$ magnetic field strength $K_m$.
This ansatz is the most natural one for the D3-brane, because in this case the $10$-dimensional self-dual field strength is extended to a self-dual field strength in $12$ dimensions. If there would be a supergravity theory in $D=12$, the bosonic configuration with flat $(10,2)$ space and a constant self-dual field strength would solve the equations of motion. This is obvious for the Maxwell equation (there can be no Chern–Simons terms built from a 5 form potential in 12 dimensions and so the Maxwell equation would take the standard form), but for the Einstein equations it is only true because the field strength is self-dual. In a $D$-dimensional spacetime with zero or two times, a self-dual field strength has a vanishing energy momentum tensor for $D= 4$ mod 4. (For Lorentzian signature it is $D=2$ mod 4). What this would mean is that the ten-dimensional D3-brane solution would just be the projection to a complicated hypersurface of an almost trivial 12 dimensional supergravity solution.
Let us start by analysing how an electric $(p+2)$-form field strength $F^{(p+2)}$ gets embedded in the $(D+2)$-dimensional space. Our aim is to obtain $F$ as a restriction of a $p+3$-form $K^{(p+3)}$ to the $D$-dimensional hypersurface $\Sigma$. A general non-dilatonic brane is described in $D$ dimensions by the fields [@d3brane] $$\begin{aligned}
ds^{2}&=&H^{-\frac{2}{p+1}}\left[ -dt^{2}+dx_1^{2}+ \cdots +dx_p^{2} \right]
+\nonumber\\
&&+H^{\frac{2}{\kappa}}\left[ dr^{2} +r^2 d\Omega_{D-p-2}^{2}\right],
\nonumber\\
G_{01...p}&=&-H^{-1}=-A^{p+1},
\nonumber\\
\Phi&=&0\,,
\label{fields}\end{aligned}$$ using the notation of section \[ss:embedding\]. We can write the (electric) field strength as ($F\equiv dG$) F = -(p+1) A\^p A’ drdtdx\_1dx\_p, \[fieldstrength\] where the prime denotes differentiation with respect to $r$.
To find the embedding, we start by considering a constant $(p+3)$-form in $D+2$ dimensions $$K_e=\frac{p+1}{(p+3)!}\epsilon_{\mu'_{0}...\mu'_{p+2}}
dX^{\mu'_{0}}\wedge dX^{\mu'_{1}}...\wedge dX^{\mu'_{p+2}}
\label{p+3form}$$ ($\mu'=0,\ldots,p+2$). In order to get a rank $(p+2)$ field strength, we contract $K_e$ with a vector field $V$, with components $V=V^M(\frac{\partial}{\partial X^M}$), which so far remains arbitrary. (There is a sign ambiguity in this contraction; we chose to make it on the left, i.e. $K(V)_{\mu'_{1}...\mu'_{p+2}}\equiv
V^{\mu'_{0}}K_{\mu'_{0}...\mu'_{p+2}}$). Such a contraction yields $$K_e(V)=\frac{p+1}{(p+2)!}\epsilon_{\mu'_{0}...\mu'_{p+2}}V^{\mu'_{0}}
dX^{\mu'_{1}}\wedge ... dX^{\mu'_{p+2}}\,.
\label{p+2form}$$ Then we reduce the resulting $(p+2)$-form to the $D$ dimensional hypersurface by using the embedding functions (\[embedfun\]), $$\begin{aligned}
&K_e(V)|_{\Sigma} =
\frac{p+1}{2}A'A^{p+1} dr\wedge dt \wedge dx_1 \wedge \cdots dx_p \times
\nonumber\\
&\times
\left[ 2V^{\mu}x_{\mu}+ V^{+}(\frac{F'}{A'}-x^{\mu}x_{\mu})
-V^{-}\right]\,,
\label{hev}\end{aligned}$$ where we defined $V^{\pm} \equiv V^{p+2}\pm V^{p+1}$. Next we impose that $K_e(V)|_\Sigma = F$. From this we can determine $V^{M}$, using the ansatz $V^{\mu'}=\alpha(r)X^{\mu'}$. Because $K_e$ only has components in the longitudinal directions, $V^\alpha$ stays undetermined. When the field strength also includes a magnetic part, this $V^\alpha$ comes into play, as we will see in the next subsection. It follows that, in order for (\[hev\]) to match with (\[fieldstrength\]), $\alpha (r)$ has to obey (r)(-F)=- \[alpha\] which gives, using (\[df\]) (r)=. or, in terms of the embedding coordinates ($\alpha(r)\to \alpha(r(X))\equiv
\alpha(X)$) (X)= and $V^{\mu'}(X)=\alpha (X)X^{\mu'}$.
D3-brane embedding {#ss:d3fs}
------------------
Let us now discuss, as an example, how this construction works for the D3-brane in the background produced by other D3-branes. We refer to [@noi] for a discussion of the embedding of the M2- and M5-brane.
The 10-dimensional Wess–Zumino term is the integral of the self-dual field strength $F$ that couples to the D3-branes solution of the type IIB supergravity theory. For the 12-dimensional theory we construct a self-dual 6 form $K$, i.e. $$\star{K}\wedge K = \eta_{12}|K|^{2},$$ where $\eta_{12}$ is the volume form on $\IE^{(D,2)}$. Our aim is to obtain $F$ as a restriction of $K$ to the 10 dimensional surface $\Sigma$. The D3-brane is described by the fields (\[fields\]) with $p+1=\kappa=4$, $D=10$. We can therefore write the self-dual field strength in terms of the embedding functions as $$F =- 4 A'A^{3}dt\wedge dx\wedge dy \wedge dz \wedge dr
+ 4 \omega_{(5)} \, ,
\label{fieldstr}$$ where $\omega_{(5)} \equiv
\sin(\theta)^{4}\sin(\phi_{1})^{3}\sin(\phi_{2})^{2}\sin(\phi_{3}) d\theta \wedge d\phi_1 \wedge ... \wedge d\phi_4$ is the volume form on the unit 5-sphere.
To find the embedding, we again start by considering a constant form in the embedding space, which in this case we take to be a self-dual six-form K&=&(\_[’\_[0]{}...’\_[5]{}]{}dX\^[’\_[0]{}]{}dX\^[’\_[1]{}]{}...dX\^[’\_[5]{}]{} +\
&+& \_[\_[1]{}...\_[6]{}]{}dX\^[\_[1]{}]{} dX\^[\_[1]{}]{}...dX\^[\_[6]{}]{}), \[sixform\] In order to get a rank 5 field strength, we contract, as we have done for the general electric case, $K$ with a vector field $V$. Such a contraction yields K(V)&=&(\_[’\_[0]{}...’\_[5]{}]{}V\^[’\_[0]{}]{} dX\^[’\_[1]{}]{}...dX\^[’\_[5]{}]{} +\
&+& \_[\_[1]{}...\_[6]{}]{} V\^[\_[1]{}]{} dX\^[\_[2]{}]{}...dX\^[\_[6]{}]{}). \[fiveform\] Again we reduce the resulting 5 form to the 10-dimensional hypersurface by using the embedding functions (\[embedfun\]). By requiring the matching $K(V)|_{\Sigma}=F$, we get the constraints on our vector field $V$. The resulting 5 form $K(V)|_\Sigma$ is precisely the Wess–Zumino term $\Omega_5$ we were looking for.
Let us analyse separately the two terms in the right hand side of (\[fieldstr\]), (\[sixform\]) and (\[fiveform\]). The electric part has already been studied in the general case in the previous subsection. In this case it gives $V^{\mu'}=\alpha(r) X^{\mu'}$ with $$\alpha (X)=\frac{2}{2 (X^{\alpha}X_{\alpha})^3 -X^{M}X_{M}}\,.$$ The magnetic part in (\[sixform\]) can be rewritten in terms of the radial coordinate $r$ and the angular coordinates $\theta$,$\phi_i$ ($i=1,\cdots
,4$) $$\begin{aligned}
\frac{1}{6!} \epsilon_{\alpha_{1}...\alpha_{6}}dX^{\alpha_{1}} \wedge \cdots
\wedge dX^{\alpha_{6}} =\nonumber\\
\quad\quad = C' C^5 dr \wedge \omega_5\,.\end{aligned}$$ For the second term in (\[fiveform\]) to match with the second term in (\[fieldstr\]), we have to require that the vector $V^\alpha$ points in the radial direction when decomposed in the $r,\theta , \phi_i$ basis, that is V\^ V\^ . \[radialv\] This gives \_[\_[1]{}...\_[6]{}]{}V\^[\_[1]{}]{} dX\^[\_[2]{}]{} ... dX\^[\_[6]{}]{} =\
= C’ C\^5 V\^ \_5. Matching this with (\[fieldstr\]) requires V\^ = (C’ C\^5)\^[-1]{}, which, using the ansatz $V^{\alpha}=\epsilon(r) X^{\alpha}$, is solved by[^4] V\^= C\^[-6]{} X\^= (1+r\^4)\^[-]{}X\^.
We notice that $\epsilon(r \to 0)=\alpha(r \to 0) \to 1$, so that in the near-horizon approximation we have $V^{M}=X^{M}$ as was already found in [@conffads].
The general form of the vector field in terms of the 12-dimensional coordinates is V\^[’]{}=, V\^=.
Discussion
==========
The aim of this talk has been to report on a global description [@noi] of non-dilatonic branes by isometrically embedding them in flat space with two extra dimensions and two times, thus extending the ideas of [@conffads]. We have gained a rather clear global picture of the geometry, giving insight in the structure around coordinate singularities and in the symmetries. In particular, the differences between $p$-branes with $p$ even and $p$ odd, previously pointed out in [@GHT], are clearly apparent. Like the familiar embedding of anti-de Sitter spacetime as a quadric, our embeddings are periodic in time. This is consistent with some suggestions in [@GaryWrap], but one may of course always pass to the covering space.
In the context of supergravity and string theory, $p$-branes are coupled to $(p+2)$-form field strengths. An embedding of the brane thus has to include, besides the embedding of the geometry, a prescription for the forms in the higher dimensional space. This is obtained by defining constant $(p+3)$-forms in $D+2$ dimensions, and contracting them using a vector $V$. The form of $V$ is determined by matching the projection on the surface with the known forms for the branes field strengths.
Unfortunately, the geometric significance of the vector field $V$ remains unclear. In the case of the M2-brane it is not even unique, since the $V^\alpha $ components are arbitrary. A co-dimension 2 surface has a 2-dimensional normal plane. In the D3 and M5 cases, the vector $V$ does not lie in this 2-plane, except in the near-horizon limit. Specifically, the normal 2-plane is spanned by $\partial _\mu \phi _1$ and $\partial _\mu \phi _2$. One may check that $V$ is not a linear combination of $\partial _\mu \phi _1$ and $\partial _\mu \phi _2$. The bosonic action for probe branes in the embedded background (\[action\]) is completely determined after the construction of $V$. It could be interesting to investigate if the vector $V$ can have some role in the context of F-theory [@vafa].
Finally it is possible that the methods developed in this paper may be applicable to scenarios in which one regards the universe as a brane embedded in a higher-dimensional spacetime.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the European Commission TMR programme ERBFMRX - CT96 - 0045. C.H is funded by FCT (Portugal) through grant no. PRAXIS XXI/BD/13384/97.
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[^1]: Onderzoeksdirecteur, FWO, Belgium
[^2]: $SO(d+1)$ for the sphere, $SO(d-1,2)$ for AdS
[^3]: A space is called globally hyperbolic if it possesses a Cauchy surface
[^4]: We used the relation $C^2(r)=X^\alpha X_\alpha$, from which $\frac{\del r }{\del X^\alpha}=\frac{\del r }{\del C^2(r)}\frac{\del C^2 }{\del X^\alpha}=(CC')^{-1}X^\alpha$.
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abstract: 'The design of electrically driven quantum dot devices for quantum optical applications asks for modeling approaches combining classical device physics with quantum mechanics. We connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems to meet this requirement. By coupling the van Roosbroeck system with a quantum master equation in Lindblad form, we introduce a new hybrid quantum-classical modeling approach, which provides a comprehensive description of quantum dot devices on multiple scales: It enables the calculation of quantum optical figures of merit and the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way. We construct the interface between both theories in such a way, that the resulting hybrid system obeys the fundamental axioms of (non-)equilibrium thermodynamics. We show that our approach guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics. The feasibility of the approach is demonstrated by numerical simulations of an electrically driven single-photon source based on a single quantum dot in the stationary and transient operation regime.'
author:
- Markus Kantner
- Markus Mittnenzweig
- Thomas Koprucki
title: 'Hybrid quantum-classical modeling of quantum dot devices'
---
Introduction
============
Semiconductor quantum dots (QDs) are zero-dimensional nanostructures which provide a discrete spectrum of electronic states due to the confinement of charge carriers in all spatial dimensions. Because of their tunable electro-optical properties and their easy integration into dielectric microcavities, QDs have attracted considerable attention in particular for applications in solid-state based optoelectronic devices [@Bimberg1999; @Michler2003; @Bhattacharya2007; @Michler2009; @Bimberg2011]. These include e.g. highly efficient semiconductor micro- and nanolasers with a few or even a single QD as gain medium [@Noda2006; @Gies2011; @Strauf2011; @Chow2013; @Schneider2013], semiconductor optical amplifiers [@Akiyama2007], and quantum light sources such as single-photon emitters and sources of entangled photon pairs [@Michler2000; @Santori2010; @Buckley2012; @Lodahl2015]. Applications comprise optical communication and quantum information processing [@Kimble2008; @Santori2010; @Buckley2012], quantum cryptography [@Gisin2002], optical computing [@Knill2001] and bio-chemical sensing [@Kairdolf2013].
Currently, quantum optics is making the leap from the lab to commercial applications. On this way, device engineers will need simulation tools, which combine classical device physics with models from quantum mechanics. The modeling and simulation of electrically driven semiconductor devices containing QDs constitutes a considerable challenge. On the one hand, modern optoelectronic devices increasingly employ quantum optical effects based on coherent light matter interaction, entanglement, photon counting statistics and non-classical correlations, which require a quantum mechanical description of the charge carriers and the optical field. In the last decades, light emitting devices based on a single or a few QDs have been successfully described by quantum master equations (QMEs) for the density matrix [@Gies2011; @Steinhoff2012; @Chow2013], which enable a detailed description of the dynamics of open quantum systems. On the other hand, the simulation of electrically driven devices requires a spatially resolved description of the current injection from the highly doped barriers and metal contacts into the optically active region containing the semiconductor QDs. The carrier transport problem is well described by semi-classical transport models such as the van Roosbroeck system [@VanRoosbroeck1950], which describes the drift and diffusion of carriers within their self-consistently generated electric field. The van Roosbroeck system has been applied previously to QD devices, in particular to QD-based intermediate band solar cells [@Marti2002; @Gioannini2013] and for the optimization of the current injection in single-photon sources [@Kantner2016a].
Both fields, the theory of open quantum systems and the semi-classical semiconductor transport theory, are well developed and established for several decades. The scope of this paper is the self-consistent coupling of both theories in order to obtain a comprehensive description of QD-based optoelectronic devices on multiple scales. Therefore, the interface connecting both systems will be constructed in such a way, that the resulting hybrid quantum-classical model guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics.
The paper is organized as follows: In Sec. \[sec:Model-equations\] the model equations are introduced and the physical properties of the hybrid quantum-classical model are discussed. We present the structure of the coupling terms between both systems and investigate important features such as the conservation of charge. In Sec. \[sec:Thermodynamics\] the consistency of the model equations with fundamental axioms of $\text{(non-)}$equilibrium thermodynamics is investigated. In particular, we construct the thermodynamic equilibrium solution by minimizing the grand potential of the coupled system and show that the hybrid model obeys the second law of thermodynamics. In Sec. \[sec:Application\] the approach is applied to the simulation of an electrically driven single-photon source based on a single QD. We study the stationary and transient excitation regime by numerical simulations and show how the model allows to compute the decisive quantum optical figures of merit along with the spatially resolved carrier transport characteristics. Finally, in Sec. \[sec:Outlook\] we give an outlook on extensions of the approach.
Model equations\[sec:Model-equations\]
======================================
We consider a hybrid quantum-classical model that self-consistently couples semi-classical transport theory to a kinetic equation for the quantum mechanical density matrix. The latter one is a QME in a Born-Markov and secular (rotating wave) approximation that describes the evolution of an open quantum system which interacts with its macroscopic environment [@Davies1974; @Lindblad1976; @Gorini1976; @Breuer2002]. In the following, the open quantum system is given by a single or a few QDs. Our approach is based on the assumption that the charge carriers can be separated into (free) continuum carriers and (bound) carriers confined to QDs, which is typically met for optoelectronic devices operating close to flat band conditions (weak electric fields) [@Grupen1998; @Steiger2008; @Koprucki2011]. The model equations read $$\begin{aligned}
-\nabla\cdot\varepsilon\nabla\psi & =q\left(p-n+C+Q\left(\rho\right)\right),\label{eq: Poisson equation}\\
\partial_{t}n-\frac{1}{q}\nabla\cdot\mathbf{j}_{n} & =-R-S_{n}\left(\rho;n,p,\psi\right),\label{eq: electron transport}\\
\partial_{t}p+\frac{1}{q}\nabla\cdot\mathbf{j}_{p} & =-R-S_{p}\left(\rho;n,p,\psi\right),\label{eq: hole transport}\\
\frac{\mathrm{d}}{\mathrm{d}t}\rho = \mathcal{L}\left(\rho;n,p,\psi\right) &=-\frac{i}{\hbar}\left[H,\rho\right]+\mathcal{D}\left(\rho;n,p,\psi\right)\label{eq: quantum master equation}\end{aligned}$$ on the domain $\Omega\subset\mathbb{R}^{3}$. The system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) is subject to initial conditions and boundary conditions modeling electrical contacts and other interfaces [@Selberherr1984]. See Appendix \[sec:Boundary-conditions\] for the boundary conditions considered throughout this paper. A schematic illustration of the modeling approach is shown in Fig. \[fig: model scheme\].
The model (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) differs from the typical quantum optical setting by explicitly considering the spatially resolved semi-classical carrier transport equations (\[eq: Poisson equation\])–(\[eq: hole transport\]) as a part of the system under investigation. As a consequence, here the notion “reservoir” is employed differently from the standard quantum optics literature. In the following, the term *reservoir* refers to the electrical contacts connected to the semiconductor device and the surrounding heat bath, which must be distinguished from the *classical* or *macroscopic environment* of the quantum system, see Fig. \[fig: system-reservoir\]. The continuum carriers, which represent the electronic part of the classical environment of the quantum system, evolve according to the van Roosbroeck system.
![(a) Illustration of the hybrid quantum-classical modeling approach. A quantum system described by a QME is self-consistently coupled to the semi-classical transport equations for the freely roaming continuum carriers. Both systems exchange charge by capture and escape of carriers and interact via their self-consistently generated electric field. (b) Schematic band diagram of the hybrid system in a 1D cross-section of a p-i-n diode with a single QD embedded in the intrinsic zone. The dissipative interactions of the quantum system with its classical environment are described by dissipation superoperators of Lindblad type $\mathcal{D}_e$, $\mathcal{D}_h$ and $\mathcal{D}_0$ for carrier capture and escape, recombination etc. The interaction domain is determined by the spatial profile $w$. []{data-label="fig: model scheme"}](fig1){width="1\columnwidth"}
Van Roosbroeck system
---------------------
Eqns. (\[eq: Poisson equation\])–(\[eq: hole transport\]) represent the standard van Roosbroeck system, extended by additional terms that constitute the coupling to the quantum system. Poisson’s Eq. (\[eq: Poisson equation\]) describes the electrostatic potential $\psi$ generated by the free electron and hole densities $n$ and $p$, the (stationary) built-in doping profile $C$ and the expectation value of the charge density $Q\left(\rho\right)$ of the carriers confined to the QDs. The dielectric permittivity of the semiconductor material is given by $\varepsilon=\varepsilon_{0}\varepsilon_{r}$ and $q$ denotes the elementary charge. The continuity equations (\[eq: electron transport\])–(\[eq: hole transport\]) describe the flux of free electrons and holes in the presence of recombination and transitions between free and bound states. The (net-)recombination rate $R$ includes several recombination channels such as Shockley-Read-Hall recombination, spontaneous emission and Auger recombination. Moreover, carriers can be scattered from the continuum to the QDs which is described by the ${\text{(net-)}}$capture rates $S_{n}$ and $S_{p}$. The van Roosbroeck system must be augmented with additional state equations for the free carrier densities
\[eq: carrier densities\] $$\begin{aligned}
n & =N_{c}F_{1/2}\left(\beta\left(\mu_{c}-E_{c}+q\psi\right)\right),\label{eq: electron density}\\
p & =N_{v}F_{1/2}\left(\beta\left(E_{v}-q\psi-\mu_{v}\right)\right)\label{eq: hole density}\end{aligned}$$
and the electrical current densities
\[eq: current densities\] $$\begin{aligned}
\mathbf{j}_{n} & =\frac{1}{q}\sigma_{n}\nabla\mu_{c},\label{eq: electron current density}\\
\mathbf{j}_{p} & =\frac{1}{q}\sigma_{p}\nabla\mu_{v}.\label{eq: hole current density}\end{aligned}$$
Here, $N_{c}$ and $N_{v}$ denote the effective density of states of the conduction and valence band and $E_{c}$ and $E_{v}$ are the respective band edge energies. The inverse temperature $\beta=\left(k_{B}T\right)^{-1}$ is considered as a fixed parameter and $$F_{\nu}\left(\eta\right)=\frac{1}{\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\mathrm{d}\xi\,\frac{\xi^{\nu}}{e^{\xi-\eta}+1}$$ is the Fermi–Dirac integral of order $\nu$. The state equations (\[eq: carrier densities\]) describe thermalized carrier ensembles in a quasi-equilibrium distribution, where the quasi-Fermi energies of the conduction band $\mu_{c}$ and the valence band $\mu_{v}$ parametrize the deviation from the thermodynamic equilibrium. In accordance with linear irreversible thermodynamics, the current densities are driven by the gradients of the quasi-Fermi energies [@DeGroot1984]. The electrical conductivities $\sigma_{n}=qM_{n}n$, $\sigma_{p}=qM_{p}p$ are products of the carrier densities and the carrier mobilities $M_{n/p}$.
Quantum master equation
-----------------------
The state of the quantum system is described by the density matrix $\rho$, which is subject to the QME (\[eq: quantum master equation\]). Here, the quantum system represents a many-body problem describing the charge carriers confined to QDs and possibly further quasi-particles, e.g. cavity photons, phonons or exciton-polaritons (dressed states).
The Hamiltonian in Eq. (\[eq: quantum master equation\]) takes the form $$H=H_{0}+H_{I},$$ where $H_{0}$ describes the single-particle energies of the confined electrons and holes (and possibly additional particle species). The interaction Hamiltonian $H_{I}$ is assumed to commute with the charge number operator of the quantum system $$N=n_{e}-n_{h}\label{eq: net charge operator}$$ ($n_{e}$ and $n_{h}$ are the number operators of the bound electrons and holes) such that the Hamiltonian part of the evolution conserves the net charge $$\left[H,N\right]=0.\label{eq: commutator N and H}$$ This imposes only a weak restriction on $H_{I}$ and allows e.g. for Coulomb interaction between the confined carriers (configuration interaction) as well as coherent light-matter interaction.
We assume the quantum system to be embedded in a semiconductor device, which represents a macroscopic environment with an infinitely large number of degrees of freedom. The interactions of the quantum system with its environment, e.g. the exchange of energy and charge via recombination and capture or escape of carriers, represent dissipative processes that are described by the dissipation superoperator $\mathcal{D}$. Within the limit of weak system-reservoir coupling one obtains by using the Born-Markov and secular (rotating wave) approximation a dissipation superoperator in Lindblad form [@Breuer2002; @Schaller2014] $$\begin{aligned}
\begin{aligned}
\mathcal{D}\left(\rho;\chi\right) &= \sum_{\alpha\in I_{\alpha}} \mathcal{D}_{\alpha }\left(\rho;\chi\right) \\
& =\sum_{\alpha\in I_{\alpha}}\big(\gamma_{\alpha}(\chi)L_{A_{\alpha}}(\rho)+\hat{\gamma}_{\alpha}(\chi) L_{A^{\dagger}_{\alpha}}(\rho)\big)
\end{aligned}
\label{eq: Dissipator (quantum detailed balance)}\end{aligned}$$ with the *Lindblad superoperator* $$\begin{aligned}
L_{A}\left(\rho\right)=A\rho A^{\dagger}-\frac{1}{2}\left\{ A^{\dagger}A,\rho\right\}.\end{aligned}$$ The admitted irreversible interactions between the quantum system and its environment are indexed by $\alpha\in I_{\alpha}$. The environment considered in this paper is a tensor product of multiple thermal states. This comprises a bosonic heat bath (lattice phonons, thermal radiation) and the thermalized carrier ensembles, which are subject to the van Roosbroeck system (\[eq: Poisson equation\])–(\[eq: hole transport\]). In the hybrid model, the forward and backward transition rates $\gamma_{\alpha}$ and $\hat{\gamma}_{\alpha}$ depend on the state of the macroscopic environment, which is indicated here by the state vector $\chi$. Under the assumptions and approximations outlined above, the dissipation superoperator can be additively decomposed into various channels as given in Eq. (\[eq: Dissipator (quantum detailed balance)\]) [@Schaller2014]. A QME in Lindblad form ensures the preservation of trace, hermiticity and (complete) positivity of the density matrix [@Lindblad1976; @Gorini1976]. The symbol ${\left\{ A,B\right\} =AB+BA}$ denotes the anti-commutator. The operators $A_{\alpha}$ represent the *quantum jump operators*, which are projectors between different eigenstates of $H$. Following the standard construction of a Lindblad-QME for a weak system-reservoir interaction [@Breuer2002] (extended to the case of variable charge number here), we require the jump operators to satisfy
\[eq: jump operator requirements\] $$\begin{aligned}
\left[H,A_{\alpha}\right] & =-\hbar\omega_{\alpha}A_{\alpha},\label{eq: jump operator requirement H}\\
\left[N,A_{\alpha}\right] & =-\ell_{\alpha}A_{\alpha},\label{eq: jump operator requirement N}\end{aligned}$$
where $\hbar\omega_{\alpha}$ denotes the transition energy and ${\ell_{\alpha}\in\mathbb{Z}}$ quantifies the charge transfer of the interaction described by $A_{\alpha}$. In order to classify the dissipation superoperators with respect to their effect on the charge of the quantum system, we collect the dissipators belonging to equal values of $\ell_{\alpha}$ and introduce the notation $$\mathcal{D}\left(\rho;\chi\right)=\mathcal{D}_{e}\left(\rho;\chi\right)+\mathcal{D}_{h}\left(\rho;\chi\right)+\mathcal{D}_{0}\left(\rho;\chi\right), \label{eq: dissipator decomposition}$$ where we have split the index set $I_{\alpha}$ into three disjoint subsets $I_{\alpha} = I_{e} \cup I_{h} \cup I_{0}$. With $\ell_{\alpha\in I_e}=-1$ and $\ell_{\alpha\in I_h}=+1$, the dissipators $\mathcal{D}_{e}$ and $\mathcal{D}_{h}$ can change the charge of the quantum system (by capture and escape of electrons and holes), whereas the processes described by $\mathcal{D}_{0}$ with $\ell_{\alpha\in I_0}=0$ leave the charge invariant (e.g. spontaneous emission, photon absorption, intraband carrier relaxation, outcoupling of cavity photons). Simultaneous capture of multiple carriers with $\vert\ell_{\alpha}\vert \geq 2$ is neglected here. From Eq. (\[eq: jump operator requirement N\]) and $\ell_{\alpha\in I_0} = 0$ one easily obtains $$\mathrm{tr}\left(N\mathcal{D}_{0}\left(\rho;\chi\right)\right)=0.\label{eq: charge conservation of D0}$$ Throughout this paper, we restrict ourselves to dissipation superoperators which satisfy the quantum detailed balance condition with respect to the thermodynamic equilibrium [@Alicki1976; @Kossakowski1977]. This requires a certain relationship between the forward and backward transition rates $\gamma_{\alpha}$ and $\hat{\gamma}_{\alpha}$, which will be discussed in Sec. \[subsec:Microscopic-transition-rates\]. In the case of degenerate energy spectra, the traditional secular approximation must be modified to properly account for degenerate eigenstate coherences. As shown in [@Cuetara2016], this can be done in a thermodynamically consistent way. Finally, we remark that the Lamb-Shift is neglected throughout this paper.
Macroscopic coupling terms and charge conservation\[subsec:Coupling-terms-and\]
-------------------------------------------------------------------------------
By taking the time derivative of Poisson’s Eq. (\[eq: Poisson equation\]) and using Eq. (\[eq: electron transport\])–(\[eq: hole transport\]), we obtain the continuity equation $$\begin{aligned}
\nabla\cdot\mathbf{j}_{\text{tot}} & =q\left(\partial_{t}Q-S_{p}+S_{n}\right)\end{aligned}$$ for the total current density $\mathbf{j}_{\text{tot}}=\mathbf{j}_{n}+\mathbf{j}_{p}+\partial_{t}\mathbf{D}$. Besides the flux of charge carriers, it also includes the displacement current density $\partial_{t}\mathbf{D}=-\varepsilon\partial_{t}\nabla\psi$. For the sake of simplicity, we consider a quantum system comprising only a single QD. The generalization of the approach outlined below to the case of multiple QDs is straightforward. We approximate the electric charge density of the QD by the expectation value of the (net-)charge operator $$Q\left(\rho\right)=-w\left(\mathbf{r}\right)\mathrm{tr}\left(N\rho\right),\label{eq: charge density of the quantum system}$$ where $w$ models the spatial profile of the captured carriers, which is assumed to be identical for all carriers. The function $w$ is normalized such that $\int_{\Omega}\mathrm{d}^{3}r\,w\left(\mathbf{r}\right)=1$. The spatial profile $w$ replaces the absolute squares of the many-body wave functions of the bound carriers. The actual spatial distributions of the confined carriers differ only on a small length scale, which can be safely neglected in the simulation of macroscopic charge transport. In the form of Eq. (\[eq: charge density of the quantum system\]), the model accounts for long range electrostatic correlations induced by the confined carriers.
Using Eqns. (\[eq: quantum master equation\]), (\[eq: commutator N and H\]), (\[eq: dissipator decomposition\]) and (\[eq: charge conservation of D0\]), the time derivative of Eq. (\[eq: charge density of the quantum system\]) is obtained as $$\begin{aligned}
\partial_{t}Q &= -w\left(\mathbf{r}\right) \mathrm{tr}\left(N\mathcal{D}_{e}\left(\rho;n,p,\psi\right)\right)\\
&\phantom{=}\; -w\left(\mathbf{r}\right) \mathrm{tr}\left(N\mathcal{D}_{h}\left(\rho;n,p,\psi\right)\right).\end{aligned}$$ In order to ensure local charge conservation $\nabla\cdot\mathbf{j}_{\text{tot}}=0$, the (net-)capture rates appearing in the carrier transport equations (\[eq: electron transport\]) and (\[eq: hole transport\]) are identified as
\[eq: loss terms\] $$\begin{aligned}
S_{n} & =+w\left(\mathbf{r}\right)\mathrm{tr}\left(N\mathcal{D}_{e}\left(\rho;n,p,\psi\right)\right),\label{eq: electron loss term}\\
S_{p} & =-w\left(\mathbf{r}\right)\mathrm{tr}\left(N\mathcal{D}_{h}\left(\rho;n,p,\psi\right)\right).\label{eq: hole loss term}\end{aligned}$$
The (net-)capture rates $S_{n/p}$ contain all microscopic capture processes connected with transitions between the various multi-particle configurations of the QD.
For different choices of $Q\left(\rho\right)$, e.g. different localization profiles of captured electrons and holes $Q\left(\rho\right)=w_{h}\left(\mathbf{r}\right)\mathrm{tr}\left(n_{h}\rho\right)-w_{e}\left(\mathbf{r}\right)\mathrm{tr}\left(n_{e}\rho\right)$ (with $w_{e/h}$ normalized), the property of local charge conservation is lost in general. However, the violation of local charge conservation is restricted to a small region $\nabla\cdot\mathbf{j}_{\text{tot}}\propto\left(w_{e}\left(\mathbf{r}\right)-w_{h}\left(\mathbf{r}\right)\right)$ and is preserved globally, i.e. it holds $\int_{\Omega}\mathrm{d}^{3}r\,\nabla\cdot\mathbf{j}_{\text{tot}}=0$.
The thermodynamic consistency discussed in the subsequent sections does not crucially rely on the property of *local* charge conservation as enforced by Eq. (\[eq: loss terms\]). With some minor modifications, the approach can be generalized to cases where only the weaker condition of global charge conservation is fulfilled. This allows e.g. for capture rates with a more complicated spatial dependency than the one stated in Eq. (\[eq: loss terms\]). Since the discussion of thermodynamic consistency is least technical in the case of local charge conservation, we assume Eq. (\[eq: loss terms\]) in the following. Other cases can be treated analogously.
![ (a) Spatial arrangement of the system $\mathcal{S}$ and the electrical contacts (reservoirs). The system $\mathcal{S}$ consists of a classical subsystem $\mathcal{S}_{\text{cl}}$ and a quantum mechanical subsystem $\mathcal{S}_{\text{qm}}$ in the interior of the classical domain. The classical system is in contact with several electric contacts, which act as charge reservoirs and are characterized by their chemical potentials $\mu_i$ and a common inverse temperature $\beta$. The reservoirs enter the equations via Dirichlet boundary conditions. (b) Illustration of the coupling scheme. The quantum system is coupled to the van Roosbroeck system via charge transfer mediated by dissipation superoperators $\mathcal{D}_e$ and $\mathcal{D}_h$. Further decay processes, which keep the charge of the quantum system invariant, are described by $\mathcal{D}_0$. Besides the charge transfer, the model system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) accounts for electrostatic interaction between charges in $\mathcal{S}_{\text{cl}}$ and $\mathcal{S}_{\text{qm}}$. []{data-label="fig: system-reservoir"}](fig2){width="1.0\columnwidth"}
Thermodynamics\[sec:Thermodynamics\]
====================================
In the recent years, the on-going miniaturization of (quantum) electronic devices has enabled the investigation of thermodynamical laws on the nanoscale. This has lead to the emergence of the novel field of *quantum thermodynamics* [@Gemmer2004; @Kosloff2013; @Esposito2015; @Goold2016]. Experiments and theory indicate that the fundamental thermodynamical laws also hold in the quantum regime [@Pekola2015; @Strasberg2017] and therefore we view thermodynamic consistency as a crucial feature for any hybrid quantum-classical model.
In this section we discuss the thermodynamic properties of the hybrid model system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]). At first, this concerns a consideration of the energy, charge and entropy balance between the system and its reservoirs. Second, the thermodynamic equilibrium solution of the hybrid system will be constructed by minimizing its grand potential. Moreover, we formulate a relation between the microscopic transition rates satisfying the quantum detailed balance condition. Finally, the hybrid quantum-classical model (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) is shown to have a non-negative entropy production rate, which we interpret as consistency with the second law of thermodynamics.
Energy, charge and entropy balance
----------------------------------
We consider an open system $\mathcal{S}$, which itself consists of a classical subsystem $\mathcal{S}_{\text{cl}}$ and quantum-mechanical subsystem $\mathcal{S}_{\text{qm}}$. The system $\mathcal{S}$ is in contact with several reservoirs $\mathcal{R}_{i}$ as illustrated in Fig. \[fig: system-reservoir\](a). The system $\mathcal{S}$ can exchange energy and charge carriers with the reservoirs. The combined system is assumed to be isolated. The reservoir $\mathcal{R}_{0}$ is a heat bath with fixed background temperature $T$, which comprises the crystal lattice as well as the surrounding radiation field. The reservoirs $\mathcal{R}_{i\geq1}$ model the electrical contacts at the boundary of the device. They are characterized by a common temperature and their chemical potentials $\mu_{i}$ (or applied voltages), which enter the system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) via boundary conditions (cf. Appendix \[sec:Boundary-conditions\]).
The total change of entropy is given by $$\Delta S_{\text{tot}}=\Delta S_{\mathcal{S}}+\Delta S_{\mathcal{R}}\geq0,$$ and the conservation of the total internal energy and charge is expressed as $$\begin{aligned}
\Delta U & =\Delta U_{\mathcal{S}}+\Delta U_{\mathcal{R}}=0,\\
\Delta N & =\Delta N_{\mathcal{S}}+\Delta N_{\mathcal{R}}=0.\end{aligned}$$ The reservoir $\mathcal{R}_{0}$ can exchange only energy with $\mathcal{S}$, hence its change of entropy is given by $\Delta S_{\mathcal{R}_{0}}=\frac{1}{T}\Delta U_{\mathcal{R}_{0}}$. For the contacts $\mathcal{R}_{i\geq1}$, also charge transfer is possible such that $\Delta S_{\mathcal{R}_{i\geq 1}}=\frac{1}{T}\Delta U_{\mathcal{R}_{i}}-\frac{\mu_{i}}{T}\Delta N_{\mathcal{R}_{i}}$. Using the conservation laws state above and $\Delta U_{\mathcal{R}} = \sum_{i\geq 0}\Delta U_{\mathcal{R}_i}$, we obtain $$\Delta S_{\text{tot}}=\Delta S_{\mathcal{S}}-\frac{1}{T}\Delta U_{\mathcal{S}}-\sum_{i\geq1}\frac{\mu_{i}}{T}\Delta N_{\mathcal{R}_{i}},$$ where $\Delta N_{\mathcal{R}_{i}}$ is just the (negative) charge flow across the boundary $\Gamma_{i}$. Using $$\lim_{\Delta t\to0}\frac{\mathrm{\Delta}N_{\mathcal{R}_{i}}}{\Delta t}=\frac{\mathrm{d}N_{\mathcal{R}_{i}}}{\mathrm{d}t}=-\frac{1}{q}\int_{\Gamma_{i}}\mathrm{d}\mathbf{A}\cdot\left(\mathbf{j}_{n}+\mathbf{j}_{p}\right),$$ we obtain the entropy production rate $$\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t}=-\frac{1}{T}\frac{\mathrm{d}F_{\mathcal{S}}}{\mathrm{d}t}+\sum_{i\geq1}\frac{\mu_{i}}{qT}\int_{\Gamma_{i}}\mathrm{d}\mathbf{A}\cdot\left(\mathbf{j}_{n}+\mathbf{j}_{p}\right),\label{eq: entropy production (general)}$$ where $F_{\mathcal{S}}=U_{\mathcal{S}}-TS_{\mathcal{S}}$ denotes the free energy of the system $\mathcal{S}$. In Sec. \[subsec:Entropy-production-and\] it will be shown, that the entropy production rate is indeed always positive for the hybrid model (\[eq: Poisson equation\])–(\[eq: quantum master equation\]). Under chemical equilibrium boundary conditions (all reservoirs $\mathcal{R}_{i\geq1}$ have the chemical potential $\mu_{i}=\mu_{\text{eq}}$), the above expression simplifies further. Exploiting the conservation of total charge, one obtains $$\left.\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t}\right\vert _{\text{eq}}=-\frac{1}{T}\frac{\mathrm{d}\Omega_{\mathcal{S}}}{\mathrm{d}t}\label{eq: entropy production (equilibrium conditions)}$$ with the grand potential $\Omega_{\mathcal{S}}=U_{\mathcal{S}}-TS_{\mathcal{S}}-\mu_{\text{eq}}N_{\mathcal{S}}$. Thus, $\Omega_{\mathcal{S}}$ is a Lyapunov function for the irreversible relaxation of $\mathcal{S}$ into the thermodynamic equilibrium.
Thermodynamic equilibrium\[subsec:Thermodynamic-equilibrium\]
--------------------------------------------------------------
According to Eq. (\[eq: entropy production (equilibrium conditions)\]), the thermodynamic equilibrium solution of (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) can be constructed by minimizing the grand potential $\Omega_{\mathcal{S}}$. Since we assume only a weak coupling between the quantum system and its macroscopic environment, the total entropy, total internal energy and total charge number are given by sums of the classical and the quantum mechanical contribution
\[eq: total thermodynamic potentials\] $$\begin{aligned}
S\left(n,p,\rho\right) & =S_{\text{cl}}\left(n,p\right)+S_{\text{qm}}\left(\rho\right),\label{eq: total entropy}\\
U\left(n,p,\rho\right) & =U_{\text{cl}}\left(n,p\right)+U_{\text{qm}}\left(\rho\right)\nonumber \\
& \phantom{=}+U_{\psi}\left(p-n+Q\left(\rho\right)\right),\label{eq: total internal energy}\\
N\left(n,p,\rho\right) & =N_{\text{cl}}\left(n,p\right)+N_{\text{qm}}\left(\rho\right).\label{eq: total charge number}\end{aligned}$$
Here also the energy contribution $U_{\psi}$ of the electrostatic field is taken into account. The extensive thermodynamic quantities of the macroscopic system are expressed via volume densities $$\begin{aligned}
S_{\text{cl}}\left(n,p\right) & =\int_{\Omega}\mathrm{d}^{3}r\,s_{\text{cl}}\left(n,p\right),\\
U_{\text{cl}}\left(n,p\right) & =\int_{\Omega}\mathrm{d}^{3}r\,u_{\text{cl}}\left(n,p\right),\\
N_{\text{cl}}\left(n,p\right) & =\int_{\Omega}\mathrm{d}^{3}r\,\left(n-p\right)\end{aligned}$$ with the entropy density $s_{\text{cl}}$ and the internal energy density $u_{\text{cl}}$. We consider the continuum carriers to be in a *local thermodynamic equilibrium* [@DeGroot1984]. Hence, the internal energy density and the entropy density can be expressed as functions of the local carrier density
\[eq: classical entropy and energy density\] $$\begin{aligned}
s_{\text{cl}} & =-k_{B}\left(nF_{1/2}^{-1}\left(\frac{n}{N_{c}}\right)-\frac{5}{2}N_{c}F_{3/2}\left(F_{1/2}^{-1}\left(\frac{n}{N_{c}}\right)\right)\right)\nonumber \\
& \phantom{=}-k_{B}\left(pF_{1/2}^{-1}\left(\frac{p}{N_{v}}\right)-\frac{5}{2}N_{v}F_{3/2}\left(F_{1/2}^{-1}\left(\frac{p}{N_{v}}\right)\right)\right),\label{eq: classical entropy density}\\
u_{\text{cl}} & =\frac{3}{2}k_{B}TN_{c}F_{3/2}\left(F_{1/2}^{-1}\left(\frac{n}{N_{c}}\right)\right)+E_{c}n\nonumber \\
& \phantom{=}+\frac{3}{2}k_{B}TN_{v}F_{3/2}\left(F_{1/2}^{-1}\left(\frac{p}{N_{v}}\right)\right)-E_{v}p.\label{eq: classical energy density}\end{aligned}$$
The above relations are obtained for the quasi-free electron and hole gas with parabolic energy dispersion and Fermi–Dirac statistics in three dimensions [@Albinus2002]. The contributions of the quantum system are given by the von Neumann entropy and the expectation values of the Hamiltonian $H$ and the charge number operator $N$
\[eq: quantum mechanical entropy and energy density\] $$\begin{aligned}
S_{\text{qm}} & =-k_{B}\mathrm{tr}\left(\rho\log{\rho}\right),\label{eq: quantum entropy}\\
U_{\text{qm}} & =\mathrm{tr}\left(H\rho\right),\label{eq: quantum internal energy}\\
N_{\text{qm}} & =\mathrm{tr}\left(N\rho\right).\label{eq: quantum charge}\end{aligned}$$
The carriers interact via their self-consistently generated electrostatic field, which yields the contribution $U_{\psi}$ to the internal energy. It is convenient to decompose the total electrostatic potential into $\psi=\psi_{\text{int}}+\psi_{\text{ext}}$, where the internal field $\psi_{\text{int}}=\psi_{\text{int}}\left(\rho_{\text{int}}\right)$ is generated by the total internal carrier density $$\rho_{\text{int}}=p-n+Q\left(\rho\right),$$ whereas the external field $\psi_{\text{ext}}$ arises from the built-in doping profile and voltages applied at the electric contacts. Then, the field energy can be written as [@Albinus1996] $$\begin{aligned}
U_{\psi}\left(\rho_{\text{int}}\right) = \frac{1}{2}\int_{\Omega}\mathrm{d}^{3}r\,\varepsilon\left|\nabla\psi_{\text{int}}\left(\rho_{\text{int}}\right)\right|^{2}+q\int_{\Omega}\mathrm{d}^{3}r\,\rho_{\text{int}}\psi_{\text{ext}}.
\label{eq: internal energy electric field}\end{aligned}$$
Assuming the charge density of the quantum system as stated in Eq. (\[eq: charge density of the quantum system\]), and finally minimizing the grand potential $\Omega_{\mathcal{S}}$ under the constraint $\mathrm{tr}\left(\rho\right)=1$, we obtain the equilibrium free carrier densities as $$\begin{aligned}
n_{\text{eq}} & =N_{c}F_{1/2}\left(\beta\left(\mu_{\text{eq}}-E_{c}+q\psi_{\text{eq}}\right)\right),\\
p_{\text{eq}} & =N_{v}F_{1/2}\left(\beta\left(E_{v}-q\psi_{\text{eq}}-\mu_{\text{eq}}\right)\right)\end{aligned}$$ and the equilibrium density matrix $$\rho_{\text{eq}}=\frac{1}{Z}e^{-\beta\left(H-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)N\right)}.\label{eq: thermal equilibrium density matrix}$$ Here, ${Z=\mathrm{tr}\left(\exp{\left(-\beta\left(H-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)N\right)\right)}\right)}$ represents the grand canonical partition function, $$\left\langle \psi\right\rangle _{w}=\int_{\Omega}\mathrm{d}^{3}r\,w\left(\mathbf{r}\right)\psi\left(\mathbf{r}\right)\label{eq: spatial average}$$ is the averaged electrostatic potential in the vicinity of the QD and the *built-in potential* $\psi_{\text{eq}}$ solves Eq. (\[eq: Poisson equation\]) with the right hand side $q\left(p_{\text{\text{eq}}}-n_{\text{\text{eq}}}+C+Q\left(\rho_{\text{\text{eq}}}\right)\right)$ at equilibrium boundary conditions. The equilibrium density matrix is a grand canonical ensemble, which contains a contribution from the electrostatic potential due to the electrostatic interaction with the macroscopic environment. The latter appears in Eq. (\[eq: thermal equilibrium density matrix\]) as a spatial average using the localization profile $w$ of the confined carriers as a weighting function, see Eq. (\[eq: spatial average\]). This is a remarkable result, which indicates that the quantum system interacts only with its spatially averaged macroscopic environment. We emphasize that this is a direct consequence of the ansatz Eq. (\[eq: charge density of the quantum system\]) and the variation of Eq. (\[eq: internal energy electric field\]) with respect to $n$, $p$ and $\rho$. See Appendix \[sec:Electrostatic-field-energy\] for details.
In the following, the concept of a non-local interaction of the quantum system with its spatially averaged macroscopic environment will be extended to non-equilibrium situations.
Microscopic transition rates and the quantum detailed balance condition\[subsec:Microscopic-transition-rates\]
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We assume the microscopic transition rates in the dissipator (\[eq: Dissipator (quantum detailed balance)\]) to be functions of the spatially averaged macroscopic potentials $$\begin{aligned}
\gamma_{\alpha} & =\gamma_{\alpha}\left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right),\\
\hat{\gamma}_{\alpha} & =\hat{\gamma}_{\alpha}\left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right),\end{aligned}$$ where $\left\langle \cdot\right\rangle _{w}$ denotes the spatial average according to Eq. (\[eq: spatial average\]). The quantum detailed balance condition requires the dissipator to vanish in equilibrium. Hence, the condition $$\begin{aligned}
0&\stackrel{!}{=}\mathcal{D}_{\alpha}\left(\rho_{\text{eq}};n_{\text{eq}},p_{\text{eq}},\psi_{\text{eq}}\right) =\gamma_{\alpha}^{\text{eq}}L_{A_{\alpha}}(\rho_{\text{eq}}) + \hat{\gamma}_{\alpha}^{\text{eq}}L_{A^{\dagger}_{\alpha}}(\rho_{\text{eq}})\end{aligned}$$ can be used to derive a relation between the equilibrium transition rates $\gamma_{\alpha}^{\text{eq}}=\gamma_{\alpha}\left(\mu_{\text{eq}},\mu_{\text{eq}},\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)$ and $\hat{\gamma}_{\alpha}^{\text{eq}}$. From Eq. (\[eq: jump operator requirements\]), one obtains for any $\lambda\in\mathbb{R}$ $$\begin{aligned}
e^{\lambda H}A_{\alpha}e^{-\lambda H} & =e^{-\lambda\hbar\omega_{\alpha}}A_{\alpha},\\
e^{\lambda N}A_{\alpha}e^{-\lambda N} & =e^{-\lambda\ell_{\alpha}}A_{\alpha},\end{aligned}$$ which implies $$\begin{aligned}
A_{\alpha}\rho_{\text{eq}} & =e^{-\beta\left(\hbar\omega_{\alpha}-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)\ell_{\alpha}\right)}\rho_{\text{eq}}A_{\alpha},\\
A_{\alpha}^{\dagger}\rho_{\text{eq}} & =e^{+\beta\left(\hbar\omega_{\alpha}-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)\ell_{\alpha}\right)}\rho_{\text{eq}}A_{\alpha}^{\dagger}.\end{aligned}$$ Subsequently, one obtains $$\begin{aligned}
0 &\stackrel{!}{=}\left(\gamma_{\alpha}^{\text{eq}}-\hat{\gamma}_{\alpha}^{\text{eq}}e^{+\beta\left(\hbar\omega_{\alpha}-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)\ell_{\alpha}\right)}\right)\times\\
& \phantom{=}\times\Big(A_{\alpha}\rho_{\text{eq}}A_{\alpha}^{\dagger}-e^{-\beta\left(\hbar\omega_{\alpha}-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)\ell_{\alpha}\right)}A_{\alpha}^{\dagger}\rho_{\text{eq}}A_{\alpha}\Big),\end{aligned}$$ which yields the desired relation between $\gamma_{\alpha}^{\text{eq}}$ and $\hat{\gamma}_{\alpha}^{\text{eq}}$: $$\hat{\gamma}_{\alpha}^{\text{eq}}=\gamma_{\alpha}^{\text{eq}}e^{-\beta\left(\hbar\omega_{\alpha}-\left(\mu_{\text{eq}}+q\left\langle \psi_{\text{eq}}\right\rangle _{w}\right)\ell_{\alpha}\right)}.$$ This agrees with the relation imposed by the Kubo-Martin-Schwinger (KMS) condition on the equilibrium reservoir correlation functions [@Kossakowski1977; @Breuer2002]. Since throughout this paper we consider only thermalized environments, we extend the above relation to non-equilibrium situations $$\begin{aligned}
\hat{\gamma}_{\alpha} & \left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right)=\label{eq: quantum detailed balance rates}\\
& =e^{-\beta\left(\hbar\omega_{\alpha}-\left(\left\langle \mu_{\alpha}\right\rangle _{w}+q\left\langle \psi\right\rangle _{w}\right)\ell_{\alpha}\right)}\gamma_{\alpha}\left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right)\nonumber \end{aligned}$$ with $\mu_{\alpha\in I_{e}}=\mu_{c}$ and $\mu_{\alpha\in I_{h}}=\mu_{v}$. For charge-conserving processes we require $\ell_{\alpha\in I_{0}}=0$, single electron-capture processes are described by $\ell_{\alpha\in I_{e}}=-1$ and for single hole-capture processes it holds $\ell_{\alpha\in I_{h}}=+1$.
Thus, supposing Eq. (\[eq: quantum detailed balance rates\]), the hybrid model obeys the quantum detailed balance condition for any model of the forward transition rate $\gamma_{\alpha}\left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right)\geq0$ that is non-negative. Physically, the latter one must represent a parametrization of a microscopically derived transition rate (using Fermi’s Golden Rule [@Alicki1977]) in terms of the averaged macroscopic potentials. In particular, this enables the direct inclusion of microscopically calculated capture rates e.g. from Refs. [@Magnusdottir2002; @Nielsen2004; @Malic2007; @Dachner2010; @Wilms2013b].
Entropy production and the second law of thermodynamics\[subsec:Entropy-production-and\]
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From Eq. (\[eq: entropy production (general)\]) we obtain the entropy production rate as (see Appendix \[sec:Entropy-production-rate\] for the derivation) $$\begin{aligned}
\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t} & =\frac{1}{T}\int_{\Omega}\mathrm{d}^{3}r\,\left(\mu_{c}-\mu_{v}\right)R\nonumber \\
& \phantom{=}+\frac{1}{qT}\int_{\Omega}\mathrm{d}^{3}r\,\left(\mathbf{j}_{n}\cdot\nabla\mu_{c}+\mathbf{j}_{p}\cdot\nabla\mu_{v}\right)\nonumber \\
& \phantom{=}-k_{B}\mathrm{tr}\left(\left(\beta H+\log{\rho}\right)\mathcal{D}_{0}\left(\rho;\chi_w \right)\right)\label{eq: entropy production rate}\\
& \phantom{=}-k_{B}\mathrm{tr}\left(\big(\beta\left(H-\mu_{c}^{\text{eff}}N\right)+\log{\rho}\big)\mathcal{D}_{e}\left(\rho;\chi_w\right)\right)\nonumber \\
& \phantom{=}-k_{B}\mathrm{tr}\left(\big(\beta\left(H-\mu_{v}^{\text{eff}}N\right)+\log{\rho}\big)\mathcal{D}_{h}\left(\rho;\chi_w\right)\right)\nonumber \end{aligned}$$ with $\mu_{c/v}^{\text{eff}}=\left\langle \mu_{c/v}\right\rangle _{w}+q\left\langle \psi\right\rangle _{w}$. The dependency of the dissipators on the state of the classical environment is indicated by the abbreviation $\chi_w = \left( \langle \mu_c\rangle_w,\langle \mu_v\rangle_w,\langle \psi \rangle_w\right)$. The first two lines describe the entropy production rate of the van Roosbroeck system [@Gajewski1996] and the third line is the entropy production rate of an open quantum system coupled to a heat bath [@Spohn1978]. The fourth and fifth line represent the contributions arising from the coupling of the QD with its macroscopic environment via capture and escape. All terms are products of abstract thermodynamic forces and their corresponding fluxes, which is in agreement with the general theory of linear irreversible thermodynamics [@DeGroot1984]. Using *Spohn’s inequality* [@Spohn1978], it can be shown that all individual lines of Eq. (\[eq: entropy production rate\]) are non-negative and therefore it holds $$\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t}\geq0,$$ where the equality holds only in the case of thermodynamic equilibrium. A proof is given in the Appendix \[subsec:Positivity-of-the\]. This results relies on the specific coupling imposed in the previous sections, which involves the spatially averaged macroscopic potentials. We emphasize, that if e.g. averaged carrier densities were used instead, a non-negative entropy production rate could not be guaranteed in general. Finally, we conclude that our hybrid quantum-classical modeling approach is consistent with the second law of thermodynamics.
Our approach can also be interpreted as a damped Hamiltonian system in the framework of GENERIC *(general equation for the non-equilibrium reversible-irreversible coupling)* [@Grmela1997], which automatically ensures a non-negative entropy production rate and the existence of an unique thermodynamic equilibrium. It can be applied to a wide range of physical problems [@Oettinger2011; @Mielke2015; @Mittnenzweig2017].
Application to electrically driven single-photon sources\[sec:Application\]
===========================================================================
In this section we demonstrate the usefulness of our approach for applications in semiconductor device simulation. As an example we consider an electrically driven single-photon source based on a p-i-n diode including a single QD. Such devices have been shown to act as single-photon emitters and are promising candidates for applications in quantum communication networks [@Yuan2002; @Bennett2008; @Unrau2012; @Schlehahn2016a].
Model specification
-------------------
The model equations are described in Sec. \[sec:Model-equations\] and \[sec:Thermodynamics\]. For the hybrid system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]), we have to specify the Hamiltonian $H$ as well as the quantum jump operators $A_{\alpha}$ and the transition rates $\gamma_{\alpha}$, which constitute the dissipative interactions with the macroscopic environment. In particular, they need to satisfy the conditions (\[eq: commutator N and H\]) and (\[eq: charge conservation of D0\]) that guarantee charge conservation and the eigenoperator relations (\[eq: jump operator requirements\]).
![Cross section of the example device considered in the numerical simulations: A single QD is placed on the symmetry axis within the center of the intrinsic zone of a cylindrical p-i-n diode with etched mesa structure on top. The bottom mirror leads to a directed emission in vertical direction. Due to the lack of a top mirror, the device represents a leaky photonic cavity with low $Q$ factor. The device has electric contacts at the top and the bottom facets.[]{data-label="fig: device sketch"}](fig3){width="1\columnwidth"}
### Hamiltonian
We consider a single QD embedded in a very leaky dielectric cavity with low $Q$ factor, which is sketched in Fig. \[fig: device sketch\]. In such devices, the light-matter interaction is governed by spontaneous emission and thus can be described by a Lindblad dissipator. Hence, we can model the quantum system by a purely electronic Hamiltonian. We aim for a description of the electronic QD states in terms of many-body states covering single particle states, excitons, trions and the biexciton as shown in Fig. \[fig: electronic states\](a). We assume a single one-particle level (ground state) for the electrons and holes each, labeled by $\varepsilon_{c}$ and $\varepsilon_{v}$, respectively. The Hamiltonian $$\begin{aligned}
H=H_{0}+H_{I}\end{aligned}$$ contains the single-particle contributions $$\begin{aligned}
H_{0}=\sum_{\sigma}\varepsilon_{c}e_{\sigma}^{\dagger}e_{\sigma}-\sum_{\sigma}\varepsilon_{v}h_{\sigma}^{\dagger}h_{\sigma}\end{aligned}$$ and their Coulomb interaction $$\begin{aligned}
H_{I} & =\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\big(V_{c,c}e_{\sigma}^{\dagger}e_{\sigma^{\prime}}^{\dagger}e_{\sigma^{\prime}}e_{\sigma}+V_{v,v}h_{\sigma}^{\dagger}h_{\sigma^{\prime}}^{\dagger}h_{\sigma^{\prime}}h_{\sigma}-\\
& \phantom{=\frac{1}{2}\sum\big(}-2V_{c,v}e_{\sigma}^{\dagger}h_{\sigma^{\prime}}^{\dagger}h_{\sigma^{\prime}}e_{\sigma}\big).\end{aligned}$$ The operators $e_{\sigma}^{\dagger}$ $(e_{\sigma})$ and $h_{\sigma}^{\dagger}$ $(h_{\sigma})$ create (annihilate) an electron or hole with total angular momentum quantum number in $z$-direction $\sigma$. We consider a single valence band describing heavy holes with a pseudo spin $\pm3/2$ indicated by $\left\{ \Uparrow,\Downarrow\right\} $. Here, only Hartree-like Coulomb matrix elements $V_{i,j}=V_{i,j,j,i}$ occur, which are of the order of several tens of meV (see Appendix \[sec:Parameters-and-auxiliary\]). The creation and annihilation operators obey the fermionic anti-commutator relations $\lbrace e_{\sigma},e_{\sigma^{\prime}}^{\dagger}\rbrace=\lbrace h_{\sigma},h_{\sigma^{\prime}}^{\dagger}\rbrace=\delta_{\sigma,\sigma^{\prime}}$ and $\lbrace e_{\sigma},e_{\sigma^{\prime}}\rbrace=\lbrace h_{\sigma},h_{\sigma^{\prime}}\rbrace=0$. The single-particle energy levels and the Coulomb matrix elements are obtained from Schrödinger’s equation with an effective confinement potential for InGaAs-QDs [@Wojs1996]. With the number operators $n_{e,\sigma}=e_{\sigma}^{\dagger}e_{\sigma}$, $n_{h,\sigma}=h_{\sigma}^{\dagger}h_{\sigma}$ and the abbreviations $$\begin{aligned}
n_{e}=\sum_{\sigma=\left\{ \uparrow,\downarrow\right\} }n_{e,\sigma},\qquad n_{h}=\sum_{\sigma=\left\{ \Uparrow,\Downarrow\right\} }n_{h,\sigma} & ,\end{aligned}$$ we can express the Hamiltonian in the occupation number representation as $$\begin{aligned}H & =\left(\varepsilon_{c}-\frac{1}{2}V_{c,c}\right)n_{e}-\left(\varepsilon_{v}+\frac{1}{2}V_{v,v}\right)n_{h}\\
& \phantom{=}+\frac{1}{2}V_{c,c}n_{e}^{2}+\frac{1}{2}V_{v,v}n_{h}^{2}-V_{c,v}n_{e}n_{h}.
\end{aligned}
\label{eq: application Hamiltonian}$$ By diagonalization, we obtain the spectral representation of $H$ in terms of multi-particle states $$H=\sum_{k}\varepsilon_{k}\big\vert k\big\rangle\big\langle k\big\vert,$$ where $k=\left(n_{e,\uparrow},n_{e,\downarrow},n_{h,\Uparrow},n_{h,\Downarrow}\right)$ is a multi-index labeling the 16 different electronic configurations which are illustrated in Fig. \[fig: electronic states\](a, b). If excited states are included and full configuration interaction is taken into account, the diagonalization of $H$ is in general a non-trivial task. In this case, an approximative representation of the Coulomb interaction in terms of number operators as in Eq. (\[eq: application Hamiltonian\]) can be obtained by the Hartree-Fock approximation [@Baer2004].
### Dissipators
We describe the spontaneous emission and the capture and escape of carriers by dissipators of the type (\[eq: Dissipator (quantum detailed balance)\]). Even though the Hamiltonian of the quantum system Eq. (\[eq: application Hamiltonian\]) has a degenerate energy spectrum (due to spin degeneracy), the coherences are decoupled from the populations because of the selection rules. Hence, the resulting dynamical system reduces to a master equation for the populations [@Cuetara2016]. In this case, a jump operator $A_{\alpha}$ describes a transition between two multi-particle states $\left|i\right\rangle $ and $\left|f\right\rangle $ is given by the projector $\left|f\right\rangle \left\langle i\right|$. The allowed transitions are indicated by arrows in Fig. \[fig: electronic states\](a), e.g. the dissipator connected with $A_{\alpha}=\left|X_{1}\right\rangle \left\langle e^{\uparrow}\right|$ describes the capture of a hole into a QD occupied by a single electron leading to the formation of the bright exciton $\left|X_{1}\right\rangle $. By using adjacency matrices to encode the allowed transitions shown in Fig. \[fig: electronic states\](a), the dissipation superoperators for all processes can be written in a compact form as
\[eq: application dissipators\] $$\begin{aligned}
\mathcal{D}_{e}\left(\rho;\chi_w\right) & =\sum_{i,f}\mathcal{A}_{i,f}^{e}\gamma_{i\to f}^{e}(\chi_w)\times \label{eq: application dissipator electron capture}\\
&\phantom{ =\sum_{i,f}}\times\left(L_{\left|f\right\rangle \left\langle i\right|}\left(\rho\right)+e^{-\beta\Delta\varepsilon_{i,f}^{e}(\chi_w)}L_{\left|i\right\rangle \left\langle f\right|}\left(\rho\right)\right), \nonumber\\
\mathcal{D}_{h}\left(\rho;\chi_w\right) & =\sum_{i,f}\mathcal{A}_{i,f}^{h}\gamma_{i\to f}^{h}(\chi_w)\times \label{eq: application dissipator hole capture}\\
&\phantom{ =\sum_{i,f}}\times\left(L_{\left|f\right\rangle \left\langle i\right|}\left(\rho\right)+e^{-\beta\Delta\varepsilon_{i,f}^{h}(\chi_w)}L_{\left|i\right\rangle \left\langle f\right|}\left(\rho\right)\right),\nonumber \\
\mathcal{D}_{0}\left(\rho\right) & =\sum_{i,f}\mathcal{A}_{i,f}^{0}\gamma_{i\to f}^{0}\times \label{eq: application dissipator spontaneous emission}\\
&\phantom{ =\sum_{i,f}}\times\left(L_{\left|f\right\rangle \left\langle i\right|}\left(\rho\right)+e^{-\beta\Delta\varepsilon_{i,f}^{0}}L_{\left|i\right\rangle \left\langle f\right|}\left(\rho\right)\right), \nonumber\end{aligned}$$
where the indices $i$ and $f$ run over all multi-particle eigenstates. Again, the dependency of the dissipation superoperators on the state of the classical environment is indicated by the abbreviation $\chi_w = \left( \langle \mu_c\rangle_w,\langle \mu_v\rangle_w,\langle \psi \rangle_w\right)$. In accordance with Eq. (\[eq: quantum detailed balance rates\]), the effective transition energies are given as $$\begin{aligned}
\Delta\varepsilon_{i,f}^{e}(\chi_w) & = \varepsilon_{i}-\varepsilon_{f}-q\left\langle \psi\right\rangle _{w}-\left\langle \mu_{c}\right\rangle _{w},\\
\Delta\varepsilon_{i,f}^{h}(\chi_w) & =\varepsilon_{i}-\varepsilon_{f}+q\left\langle \psi\right\rangle _{w}+\left\langle \mu_{v}\right\rangle _{w},\\
\Delta\varepsilon_{i,f}^{0} & =\varepsilon_{i}-\varepsilon_{f}\end{aligned}$$ and the adjacency matrix elements encoding Pauli blocking and the optical selection rules (conservation of total angular momentum) read $$\begin{aligned}
\mathcal{A}_{i,f}^{e} & =\delta_{\left\langle i\right|n_{e}\left|i\right\rangle +1,\left\langle f\right|n_{e}\left|f\right\rangle }\prod_{\sigma=\left\{ \Uparrow,\Downarrow\right\} }\delta_{\left\langle i\right|n_{h,\sigma}\left|i\right\rangle ,\left\langle f\right|n_{h,\sigma}\left|f\right\rangle },\\
\mathcal{A}_{i,f}^{h} & =\delta_{\left\langle i\right|n_{h}\left|i\right\rangle +1,\left\langle f\right|n_{h}\left|f\right\rangle }\prod_{\sigma=\left\{ \uparrow,\downarrow\right\} }\delta_{\left\langle i\right|n_{e,\sigma}\left|i\right\rangle ,\left\langle f\right|n_{e,\sigma}\left|f\right\rangle },\\
\mathcal{A}_{i,f}^{0} & =\delta_{\left\langle i\right|n_{e,\uparrow}\left|i\right\rangle ,\left\langle f\right|n_{e,\uparrow}\left|f\right\rangle }\delta_{\left\langle i\right|n_{e,\downarrow}\left|i\right\rangle -1,\left\langle f\right|n_{e,\downarrow}\left|f\right\rangle }\times\\
& \phantom{=+}\times\delta_{\left\langle i\right|n_{h,\Uparrow}\left|i\right\rangle -1,\left\langle f\right|n_{h,\Uparrow}\left|f\right\rangle }\delta_{\left\langle i\right|n_{h,\Downarrow}\left|i\right\rangle ,\left\langle f\right|n_{h,\Downarrow}\left|f\right\rangle }+\\
& \phantom{=}+\delta_{\left\langle i\right|n_{e,\uparrow}\left|i\right\rangle -1,\left\langle f\right|n_{e,\uparrow}\left|f\right\rangle }\delta_{\left\langle i\right|n_{e,\downarrow}\left|i\right\rangle ,\left\langle f\right|n_{e,\downarrow}\left|f\right\rangle }\times\\
& \phantom{=+}\times\delta_{\left\langle i\right|n_{h,\Uparrow}\left|i\right\rangle ,\left\langle f\right|n_{h,\Uparrow}\left|f\right\rangle }\delta_{\left\langle i\right|n_{h,\Downarrow}\left|i\right\rangle -1,\left\langle f\right|n_{h,\Downarrow}\left|f\right\rangle }.\end{aligned}$$ Please note that the above adjacency matrices are non-symmetric $\mathcal{A}_{i,f}\neq \mathcal{A}_{f,i}$. Thereby, they contain a *directionality* which refers to the primary processes indicated by the arrow directions shown in Fig. \[fig: electronic states\](a). This is employed in the notation of the dissipators in Eq. (\[eq: application dissipators\]), which explicitly accounts for the actual net-transition rates by hard-wiring the relation (\[eq: quantum detailed balance rates\]) between forward and backward rates. Consequently, the quantum detailed balance relation is guaranteed in the thermodynamic equilibrium independent of the model for the forward rate $\gamma_{i\to f}(\chi_w)$. The respective backward transition rate is obtained according to Eq. (\[eq: quantum detailed balance rates\]). An alternative representation of the master equation for the populations can be found in Appendix \[sec:Projection-on-eigenstates\].
![(a) Diagram of electronic states of the QD-Hamiltonian (\[eq: application Hamiltonian\]) and possible (irreversible) transitions. The arrows indicate capture and recombination, for the corresponding reverse processes (escape, generation) the arrows need to be reversed. We use short notations for the multi-particle states $\big\vert n_{e,\uparrow},n_{e,\downarrow},n_{h,\Uparrow},n_{h,\Downarrow}\big\rangle$: empty QD $\big\vert0\big\rangle=\big\vert0,0,0,0\big\rangle$, single-electron states $\big\vert e^{\uparrow}\big\rangle=\big\vert1,0,0,0\big\rangle$, $\big\vert e^{\downarrow}\big\rangle=\big\vert0,1,0,0\big\rangle$, single-hole states $\big\vert h^{\Uparrow}\big\rangle=\big\vert0,0,1,0\big\rangle$, $\big\vert h^{\Downarrow}\big\rangle=\big\vert0,0,0,1\big\rangle$, two-electron state $\big\vert ee\big\rangle=\big\vert1,1,0,0\big\rangle$, two-hole state $\big\vert hh\big\rangle=\big\vert0,0,1,1\big\rangle$, bright excitons $\big\vert X_{1}\big\rangle=\big\vert1,0,0,1\big\rangle$, $\big\vert X_{2}\big\rangle=\big\vert0,1,1,0\big\rangle$, dark excitons $\big\vert D_{1}\big\rangle=\big\vert1,0,1,0\big\rangle$, $\big\vert D_{2}\big\rangle=\big\vert0,1,0,1\big\rangle$, negative trions $\big\vert X_{-}^{\Uparrow}\big\rangle=\big\vert1,1,1,0\big\rangle$, $\big\vert X_{-}^{\Downarrow}\big\rangle=\big\vert1,1,0,1\big\rangle$, positive trions $\big\vert X_{+}^{\uparrow}\big\rangle=\big\vert1,0,1,1\big\rangle$, $\big\vert X_{+}^{\downarrow}\big\rangle=\big\vert0,1,1,1\big\rangle$ and the biexciton state $\big\vert B\big\rangle=\big\vert1,1,1,1\big\rangle$. (b) Schematic representation of the QD occupation for some example states. (c) Illustration of the effective scattering cascade in the reduced model involving only the single-particle ground states.[]{data-label="fig: electronic states"}](fig4){width="1\columnwidth"}
### Transition rate models
The spontaneous decay rates of the various (bright) electronic states of the quantum system can be modeled by the Weisskopf-Wigner rate [@Weisskopf1930] $$\begin{aligned}
\gamma_{i\to f}^{0} & =\frac{P_{i,f}d_{c,v}^{2}n_{r}}{6\pi\hbar\varepsilon_{0}c_{0}^{3}}\left(\frac{\varepsilon_{i}-\varepsilon_{f}}{\hbar}\right)^{3}\left(1+n_{\text{pt}}\left(\frac{\varepsilon_{i}-\varepsilon_{f}}{\hbar}\right)\right)\label{eq: radiative decay rate}\end{aligned}$$ for allowed index pairs ${i\to f}$ giving ${\mathcal{A}_{i,f}^0 = 1}$. Here, $n_{\text{pt}}\left(\omega\right)=\left(e^{\beta\hbar\omega}-1\right)^{-1}$ is the thermally induced photon number, $n_{r}$ is the refractive index of the material, $d_{c,v}$ denotes the interband dipole moment and $c_{0}$ is the vacuum speed of light. Due to cavity effects, the decay rate is slightly modified with respect to the free space decay rate, which is accounted for by the Purcell factors $P_{i,f}$. The Weisskopf-Wigner rate is applicable in low $Q$ optical resonators, where the photonic density of states varies insignificantly over the linewidth of the emitter [@Lodahl2015; @Florian2012]. Using the parameters given in Appendix \[sec:Parameters-and-auxiliary\], all decay rates are found to be approximately $10^{9}\,\text{s}^{-1}$.
For semiconductor QDs, the Fröhlich coupling and Auger scattering typically constitute the dominant capture processes. As a rule of thumb, at low carrier densities, the LO-phonon assisted Fröhlich coupling provides the dominant scattering channel, whereas at elevated carrier densities the Auger scattering becomes increasingly efficient [@Dachner2010; @Chow2013; @Ferreira2015]. Due to the relatively large Coulomb matrix elements in semiconductor QDs, the scattering rates into charged states differ significantly from those into neutral states. This effect is known as *Coulomb suppression* or *Coulomb enhancement*, respectively [@Ferreira2015].
The scattering rates can be calculated microscopically by Fermi’s Golden rule and then always satisfy the detailed balance relation between the forward and backward process [@Magnusdottir2002; @Nielsen2004; @Malic2007; @Dachner2010; @Wilms2013b]. However, here we restrict ourselves to phenomenological laws for the effective capture rate of continuum carriers into the QD. The respective escape rates follow via the detailed balance relation. The effective capture rate approximates the entire scattering cascade, see Fig. \[fig: electronic states\](c). We model the effective electron capture rates entering Eq. (\[eq: application dissipator electron capture\]) as
\[eq: capture rate model\] $$\begin{aligned}
\gamma_{i\to f}^{e} & \left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right)=\label{eq: capture rate model electrons}\\
& =\frac{1+n_{\text{LO}}}{\tau_{\text{LO}}^{e}}\frac{1}{e^{\beta\left(E_{c}-q\left\langle \psi\right\rangle _{w}-\left\langle \mu_{c}\right\rangle _{w}+a_{\text{LO}}^{e}+C_{i,f}^{e}\right)}+1}+\nonumber \\
& \phantom{=}+\frac{1}{\tau_{\text{Au}}^{e,e}}\frac{\bar{n}_{w}^{2}}{1+\bar{n}_{w}^{2-\gamma_{\text{Au}}^{e,e}}}+\frac{1}{\tau_{\text{Au}}^{e,h}}\frac{\bar{n}_{w}\bar{p}_{w}}{1+\left(\bar{n}_{w}\bar{p}_{w}\right)^{1-\gamma_{\text{Au}}^{e,h}/2}},\nonumber \end{aligned}$$ for the admitted index pairs ${i\to f}$ giving ${\mathcal{A}_{i,f}^e = 1}$ and the effective hole capture rates in Eq. (\[eq: application dissipator hole capture\]) as $$\begin{aligned}
\gamma_{i\to f}^{h} & \left(\left\langle \mu_{c}\right\rangle _{w},\left\langle \mu_{v}\right\rangle _{w},\left\langle \psi\right\rangle _{w}\right)=\label{eq: capture rate model holes}\\
& =\frac{1+n_{\text{LO}}}{\tau_{\text{LO}}^{h}}\frac{1}{e^{-\beta\left(E_{v}-q\left\langle \psi\right\rangle _{w}-\left\langle \mu_{v}\right\rangle -a_{\text{LO}}^{h}-C_{i,f}^{h}\right)}+1}+\nonumber \\
& \phantom{=}+\frac{1}{\tau_{\text{Au}}^{h,h}}\frac{\bar{p}_{w}^{2}}{1+\bar{p}_{w}^{2-\gamma_{\text{Au}}^{h,h}}}+\frac{1}{\tau_{\text{Au}}^{h,e}}\frac{\bar{n}_{w}\bar{p}_{w}}{1+\left(\bar{n}_{w}\bar{p}_{w}\right)^{1-\gamma_{\text{Au}}^{h,e}/2}}.\nonumber \end{aligned}$$
for the admitted index pairs ${i\to f}$ yielding ${\mathcal{A}_{i,f}^h = 1}$. Here we have used the abbreviations $\bar{n}_{w}=n_{w}/n_{\text{Au}}^{\text{crit}}$ and $\bar{p}_{w}=p_{w}/p_{\text{Au}}^{\text{crit}}$. Please note that the ambient continuum carrier densities ${n_{w}=N_{c}F_{1/2}\left(\beta\left(\left\langle \mu_{c}\right\rangle _{w}-E_{c}+q\left\langle \psi\right\rangle _{w}\right)\right)}$ and ${p_{w}=N_{v}F_{1/2}\left(\beta\left(E_{v}-q\left\langle \psi\right\rangle _{w}-\left\langle \mu_{v}\right\rangle _{w}\right)\right)}$ are functions of the averaged macroscopic potentials. The first terms in Eq. (\[eq: capture rate model\]) describe the LO-phonon assisted relaxation of continuum carriers and the last lines are each attributed to Auger scattering. The number of thermally excited LO-phonons is given by $n_{\text{LO}}=\left(e^{\beta\hbar\omega_{\text{LO}}}-1\right)^{-1}$. The time constants $\tau_{\text{LO}}^{\lambda}$ and the parameters $a_{\text{LO}}^{\lambda},\gamma_{\text{LO}}^{\lambda}$, $\lambda\in\left\{ e,h\right\} $ are considered as fitting factors that can be extracted from microscopic calculations or experimental data. The phonon assisted capture rates involve the Coulomb enhancement/suppression factors $$\begin{aligned}
C_{i,f}^{e} & =\varepsilon_{f}-\varepsilon_{i}-\varepsilon_{c},\\
C_{i,f}^{h} & =\varepsilon_{f}-\varepsilon_{i}+\varepsilon_{v},\end{aligned}$$ which describe the additional attractive or repulsive Coulomb shifts and thereby either enhance (if $C_{i,f}^{\lambda}<0$, $\lambda\in\left\{ e,h\right\} $) or decrease (if $C_{i,f}^{\lambda}>0$, $\lambda\in\left\{ e,h\right\} $) the capture rate. At low temperatures the effect of Coulomb enhancement or suppression becomes increasingly important. For the Auger-like capture processes the modifications of the capture rates due to Coulomb shifts are assumed to be negligible due to strong screening effects at high carrier densities. The expressions in Eq. (\[eq: capture rate model\]) take saturation effects at high carrier densities into account. The functional form is motivated from microscopically computed results presented in Refs. [@Wilms2013b; @Ferreira2015]. In the low density limit (Maxwell–Boltzmann approximation) the capture rate models asymptotically take the form $$\begin{aligned}
\gamma_{i\to f}^{e} & \approx\frac{\left(n_{\text{LO}}+1\right)e^{-\beta a_{\text{LO}}^{e}}}{\tau_{\text{LO}}^{e}N_{c}}e^{-\beta C_{i,f}^{e}}n_{w}^{\text{MB}}\\
& \phantom{=}+\frac{\left(\bar{n}_{w}^{\text{MB}}\right)^{2}}{\tau_{\text{Au}}^{e,e}}+\frac{\bar{n}_{w}^{\text{MB}}\bar{p}_{w}^{\text{MB}}}{\tau_{\text{Au}}^{e,h}},\\
\gamma_{i\to f}^{h} & \approx\frac{\left(n_{\text{LO}}+1\right)e^{-\beta a_{\text{LO}}^{h}}}{\tau_{\text{LO}}^{h}N_{v}}e^{-\beta C_{i,f}^{h}}p_{w}^{\text{MB}}\\
& \phantom{=}+\frac{\left(\bar{p}_{w}^{\text{MB}}\right)^{2}}{\tau_{\text{Au}}^{h,h}}+\frac{\bar{n}_{w}^{\text{MB}}\bar{p}_{w}^{\text{MB}}}{\tau_{\text{Au}}^{h,e}},\end{aligned}$$ showing a linear dependency on the continuum carrier density in the case of LO-phonon assisted capture and a quadratic dependency for the Auger capture processes. Moreover, the Coulomb enhancement and suppression effect becomes apparent in this form. The expression for $n_{w}^{\text{MB}}$ is obtained by replacing $F_{1/2}\left(\cdot\right)\to\exp{\left(\cdot\right)}$ in the above definition of $n_{w}$ (analogous for $p_{w}^{\text{MB}}$). The parameters $n_{\text{Au}}^{\text{crit}}$, $p_{\text{Au}}^{\text{crit}}$ and $\gamma_{\text{Au}}^{\lambda,\lambda^{\prime}}$ $\lambda,\lambda^{\prime}\in\left\{ e,h\right\} $ are fitting factors.
Numerical simulation method
---------------------------
The van Roosbroeck system (\[eq: Poisson equation\])–(\[eq: hole transport\]) is discretized using a Voronoï box based finite volumes method [@Selberherr1984; @Farrell2017] along with a modified Scharfetter–Gummel scheme [@Scharfetter1969; @Bessemoulin-Chatard2012; @Koprucki2014] for the discretization of the current densities. The latter one properly reflects the strong degeneration effects of the electron-hole plasma at cryogenic temperatures and takes the Fermi–Dirac statistics and nonlinear diffusion via a generalized Einstein relation fully into account [@Kantner2016]. For time-dependent simulations, we use an implicit Euler discretization and an adaptive time stepping method.
The discretized van Roosbroeck system is solved along with the QME (\[eq: quantum master equation\]) by a full Newton iteration using the electrostatic potential $\psi$, the quasi-Fermi energies $\mu_{c}$, $\mu_{v}$ and the density matrix elements $\langle k\vert\rho\vert l\rangle$ as independent variables. In order to obtain a system of ordinary differential equations, the QME is projected on the Hilbert space basis spanned by the multi-particle eigenstates of $H$ (see Appendix \[sec:Projection-on-eigenstates\]).
The coupling terms $Q$ and $S_{n/p}$ given by Eq. (\[eq: charge density of the quantum system\]) and (\[eq: loss terms\]) introduce a non-local coupling of the van Roosbroeck system with the QME via the spatial profile function $w$. This has an impact on the sparsity pattern of the Jacobian of the discretized system, since the quantum system interacts in general with a large number of control volumes in its environment. Since the discretized spatial profile function $w_{K}=\left|\Omega_{K}\right|^{-1}\int_{\Omega_{K}}\mathrm{d}^{3}r\,w\left(\mathbf{r}\right)$ (where $\left|\Omega_{K}\right|$ is the volume of the $K$-th Voronoï cell), quickly decays, we discard small matrix elements below a chosen threshold. This preserves the quadratic convergence of Newton’s iteration while the numerical effort is reduced.
Single-photon sources are typically operated at cryogenic temperatures, which causes serious convergence issues during the numerical solution of the van Roosbroeck system because of the strong depletion of minority carrier densities [@Selberherr1987; @Richey1994; @Kantner2016]. By using the temperature embedding method described in Ref. [@Kantner2016], the problem becomes tractable in the vicinity of flat band conditions.
Device specification
--------------------
In the numerical simulations presented in the following, we consider the cylindrical GaAs-based p-i-n structure depicted in Fig. \[fig: device sketch\], where a single QD is placed on the symmetry axis within the center of the intrinsic zone. The total height of the device is $800\,\text{nm}$, the intrinsic layer has a thickness of $200\,\text{nm}$ and the doped layers both are $300\,\text{nm}$ in height. The doping concentrations are $C=N_{D}=2\times10^{18}\,\text{cm}^{-3}$ and $C=-N_{A}=-10^{19}\,\text{cm}^{-3}$ in the n- and p-domain, respectively. The top radius of the mesa is 0.5m and the total radius (at the bottom) is 2.5m. The bottom facet is assumed to consist of a highly reflective metal such that it simultaneously acts as an electric contact and a mirror leading to a directed emission in vertical direction. The ohmic contact on the top facet is assumed to consist of an optically transparent material, such that the structure forms a leaky cavity with a low $Q$ factor. The remaining facets are modeled by homogeneous Neumann boundary conditions. The wetting layer (WL) indicated in Fig. \[fig: device sketch\] is neglected in the simulation. The device is assumed to operate under cryogenic conditions at $T=50\,\text{K}$.
The numerical simulation exploits the rotational symmetry of the device, such that the computational domain reduces to a 2D cross section with adapted cell volumes.
Stationary operation
--------------------
The device operates as a p-i-n diode, which can be seen from the current-voltage curve shown in Fig. \[fig: stationary injection\](c). At cryogenic temperatures the Fermi energy levels in the doped domains are very close to the band edges and therefore the diode’s threshold voltage approximately equals the energy band gap of the material (around 1.52V). The population of the QD states $\langle k\rangle =\langle k\vert\rho\vert k\rangle$ can be controlled by the externally applied bias as shown in Fig. \[fig: stationary injection\](a). Since the QD is located within the intrinsic zone of the device, it is most probably unoccupied in the low bias regime. When the applied bias approaches the diode’s threshold voltage, the QD population turns into a non-equilibrium distribution: At first, due to the increased continuum carrier densities in the vicinity of the QD, the single-particle and excitonic states are populated. In particular, due to the lack of an radiative decay channel, the dark excitons $\langle D_{1/2}\rangle$ have a high occupation probability. Finally, beyond the threshold, the QD is quickly driven into saturation and the population is dominated by the biexciton state $\langle B\rangle$. Due to Coulomb enhancement and suppression, the population of neutral states is favored in the whole bias range. In particular, Fig. \[fig: stationary injection\](a) shows that the population of the doubly charged states $\langle ee\rangle$ and $\langle hh\rangle$ is strongly suppressed.
![Numerical results at stationary injection. (a) Occupation of the QD states vs. applied bias. (b) Single-photon generation rates of the different emission lines vs. injection current. (c) Current-voltage curve of the diode. (d) Second order correlation function of the photons generated on the exciton line. (e) Comparison of recombination and capture rates of free carriers along the symmetry axis of the device. For the labeling of QD states we refer to the caption of Fig. \[fig: electronic states\].[]{data-label="fig: stationary injection"}](fig5){width="1\columnwidth"}
The single-photon generation rates of the different emission lines are given by $$\Gamma_{k}=\sum_{l}\mathcal{A}_{k,l}^{0}\gamma_{k \to l }^{0}\langle k\rangle .\label{eq: single photon generation rate}$$ Since the decay rates for all radiative processes are approximately equal, the occupation probabilities are directly proportional to the single-photon generation rates, which are depicted in Fig. \[fig: stationary injection\](b). At low injection currents, the emission spectrum is dominated by photons generated via the decay of bright excitons. Close to the threshold voltage the bright exciton line reaches a maximum and then decreases while the intensity of the biexciton line grows until it finally saturates. In this regime, the capture rates exceed the radiative decay rates by several orders of magnitude. This simulation result agrees with experimental observations presented in Ref. [@Yuan2002].
![Carrier transport at pulsed excitation. (a) Illustration of the voltage ramp used in the simulations. (b, c) Snapshots of the carrier density distribution on a 2D cross-section at several instances of time. The carrier density is color-coded, the arrows indicate the current density vector field (arrows point into the direction of particle motion). The peak voltage in the simulation was set to $1.6\,\text{V}$.[]{data-label="fig: pulsed injection transport"}](fig6){width="1\columnwidth"}
Another important figure of merit for single-photon emitters is the second order intensity correlation function of the generated photons $$g^{\left(2\right)}\left(\tau\right)=\frac{\left\langle a^{\dagger}\left(0\right)a^{\dagger}\left(\tau\right)a\left(\tau\right)a\left(0\right)\right\rangle }{\left\langle a^{\dagger}\left(0\right)a\left(0\right)\right\rangle ^{2}},\label{eq: second order correlation function}$$ where the operator $a^{\dagger}\left(a\right)$ creates (annihilates) a photon and $\tau$ is a time delay. A value of $g^{\left(2\right)}\left(0\right)<0.5$ indicates the presence of a single-photon Fock state in the radiation field. In our model the decay of an optically active QD state is equivalent to the generation of a corresponding photon. Therefore, the electronic operators can be used to evaluate Eq. (\[eq: second order correlation function\]), cf. Ref. [@Florian2012]. For the bright exciton line, we identify the photon creation operator with the projector $a^{\dagger}=\vert0\rangle\langle X_{i}\vert$ (with $i=1$ or 2) and use the quantum regression theorem [@Breuer2002] to evaluate Eq. (\[eq: second order correlation function\]). The result is presented in Fig. \[fig: stationary injection\](d) and recovers the characteristic dip around $\tau = 0$ for high-quality single-photon sources [@Yuan2002; @Santori2010]. Since the present model assumes an ideal quantum emitter and an instantaneous extraction of the generated photons from the cavity, the value of $g^{\left(2\right)}\left(0\right)$ is exactly zero. For a refined description at this stage, a coherent light-matter interaction must be included in the Hamiltonian and $\mathcal{D}_{0}$ has to be extended by a photon outcoupling mechanism.
Finally, in Fig. \[fig: stationary injection\](e) we show the recombination rate $R$ of the continuum carriers and the capture rates $S_{n/p}$ along the vertical (symmetry) axis of the device. Close to the threshold voltage, the transition of carriers into the QD imposes the dominant loss mechanism of continuum carriers in the vicinity of the QD.
![QD occupation and single-photon generation under pulsed excitation. (a) Comparison of the transient exciton and biexciton occupation probabilities in the low and high injection case. In the low injection case the occupation probabilities have been multiplied by a factor 2 for better visibility. (b) Number of generated photons per pulse on the different emission lines for different values of the peak bias. (c) Single-photon generation rate on the bright exciton line vs. repetition frequency of the time-periodic excitation pulse (for different values of the peak bias).[]{data-label="fig: pulsed injection quantum dot"}](fig7){width="1\columnwidth"}
Pulsed operation
----------------
For many applications, the generation of single photons at certain instances of time is required. Electrically driven QD-based single-photon sources offer an easy off-resonant excitation scheme [@Michler2009; @Florian2012], where the QD is excited by short voltage pulses. This process shall be simulated in the following, where we apply rectangular voltage pulses with a fixed duration of 100ps superimposed on a DC bias of 1.35V as illustrated in Fig. \[fig: pulsed injection transport\](a). We investigate the impact of the pulse repetition time and the peak bias, which are the key external control parameters. The results of a numerical carrier transport simulation for a single pulse with a peak voltage of $1.6\,\text{V}$ are shown in Fig. \[fig: pulsed injection transport\](b, c). Due to the high carrier mobilities at low temperatures (cf. Appendix \[sec:Parameters-and-auxiliary\]), the carriers quickly spread out within the device such that the intrinsic zone is highly populated at the end of the excitation pulse (100ps). Subsequently, when the applied voltage is switched back to the resting DC bias, the carriers are quickly withdrawn from the intrinsic zone. In the snapshots taken at $116\,\text{ps}$ and $200\,\text{ps}$ we observe that in particular the vicinity of the QD (which is located on the center of the symmetry axis at 0.4m, cf. Fig. \[fig: device sketch\]) is depleted first. Moreover, a conducting channel underneath the insulating region is formed. The plot at 10ns shows the stationary state reached after a long time.
The impact of the voltage pulse on the occupation of the QD is shown in Fig. \[fig: pulsed injection quantum dot\](a). In the case of an excitation with a peak voltage of $1.6\,\text{V}$ (high injection), one first observes a fast occupation of the biexciton state which subsequently decays radiatively. Via the so-called *biexciton-cascade*, the bright exciton states are populated in the following. Comparing the time scales of the carrier transport with the life times of the bright QD states, see Fig. \[fig: pulsed injection transport\](b, c) and Fig. \[fig: pulsed injection quantum dot\](a), it is apparent that the decay of the bright exciton happens a long time after the continuum carriers have left the vicinity of the QD. This separation of time scales is of particular importance for the generation of indistinguishable photons [@Bennett2008], since fluctuations of the carrier density in the vicinity of the emitter might shift the generated photon’s energy.
Next, we study the impact of the peak bias value. Figure \[fig: pulsed injection quantum dot\](b) shows the number of generated photons for different peak voltages after 10ns. The number of generated photons on line $k$ until time $t$ is obtained from $$N_{k}\left(t\right)=\int_{0}^{t}\mathrm{d}t^{\prime}\,\Gamma_{k}\left(t^{\prime}\right),$$ using the single-photon generation rate defined in Eq. (\[eq: single photon generation rate\]). The plot clearly reveals the existence of two regimes: A subthreshold (low injection) regime, where the peak voltage is insufficient for the excitation of the QD (cf. Fig. \[fig: pulsed injection quantum dot\](a)), and a high injection regime where the biexciton-cascade can be observed practically after each pulse. For the exciton-photons, this implies a generation efficiency of around 50% for both polarizations. The generation efficiency of the two differently polarized photons on the biexciton-line is a little higher than 50%, due to additional recombination during the excitation period, see Fig. \[fig: pulsed injection quantum dot\](a, b).
![Entropy production rate during pulsed excitation (peak bias 1.6V). The plot shows the five contributions arising from the individual lines in Eq. (\[eq: entropy production rate\]) and the total entropy production rate. The inset is a zoom on the first 200ps.[]{data-label="fig: pulsed injection entropy production"}](fig8){width="1\columnwidth"}
Finally, we investigate the optimal repetition frequency of the excitation cycle for the generation of single exciton-photons. The optimal repetition frequency $f^{\ast}=1/t^{\ast}$ maximizes the number of generated photons per time: $$\bar{\Gamma}_{X}\left(t^{\ast}\right)=\frac{N_{X}\left(t^{\ast}\right)}{t^{\ast}}=\frac{1}{t^{\ast}}\int_{0}^{t^{\ast}}\mathrm{d}t^{\prime}\,\Gamma_{X}\left(t^{\prime}\right)\to\max.$$ Figure \[fig: pulsed injection quantum dot\](c) shows a clear maximum at a pulse repetition rate of $f^{\ast}\approx650\,\text{MHz}$ ($t^{\ast}\approx1.5\,\text{ns}$), which corresponds to a maximum single-photon generation rate of $\bar{\Gamma}_{X}\left(t^{\ast}\right)\approx185\,\text{MHz}$. Even though in this optimal case the photon generation efficiency per pulse shrinks to 28%, the high repetition frequency leads to an enhanced overall performance. Moreover, Fig. \[fig: pulsed injection quantum dot\](c) indicates that this result is practically independent of the peak voltage. In order to obtain the actual emission rate, the generation rate must be multiplied with the extraction efficiency [@Zwiller2004].
We conclude this section with a consideration of the entropy production during an excitation cycle, which is depicted in Fig. \[fig: pulsed injection entropy production\]. The plot shows, that during the first 2ns the entropy production rate is clearly governed by the contributions arising from the macroscopic system, whereas at later times the slow decay of the QD-exciton becomes dominant. The numerical result is in agreement with the theory presented in Sec. \[subsec:Entropy-production-and\], which predicts a positive entropy production rate at all times.
Discussion and outlook\[sec:Outlook\]
=====================================
The electrically driven single-photon source considered in the previous section is a realistic application that fits into the framework of the new model system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) introduced in this paper. This example is a *proof of concept* that shows the computational tractability of our approach regarding its potential for applications in numerical semiconductor device simulation. As demonstrated in Sec. \[sec:Application\], the hybrid approach enables a comprehensive analysis of QD-based devices even in the case of complex, multi-dimensional device geometries as well as the investigation of transient processes.
If the feedback of the quantum system on its classical environment is weak, i.e. if the capture rates $S_{n/p}$ are small compared to the recombination rate $R$, simplified approaches can be considered. A first option is to merely consider the semi-classical transport while neglecting the quantum system as carried out e.g. in Ref. [@Kantner2016a]. In this case, however, the model gives no access on the quantum optical figures of merit, of course. A second option is to treat the quantum master equation alone by choosing an appropriate parametrization of the transition rates in the dissipation superoperators as done e.g. in Ref. [@Florian2012]. However, in electrically driven devices, the carrier densities, the electric field and the current densities, which usually drive the transition rates, strongly depend on the applied voltage and can vary over many orders of magnitude. In general, their detailed behavior is not apriori known and requires full device simulation since the evolution of these quantities is determined by specific design parameters such as the device geometry, doping profiles, heterostructures etc. In conclusion, the hybrid modeling approach described in this paper goes beyond existing ones.
In the case of weak feedback, the coupling of both subsystems in the hybrid model becomes effectively uni-directional. This means that the dynamics of the quantum system is slaved by the evolution of its classical environment, which can be exploited to reduce the computational effort in a two-step method: First, the transport simulation is carried out whilst omitting coupling terms to the quantum system. In a second step, the solution of the classical system is used to determine the time-dependent dissipators that drive the evolution of the open quantum system. Hence, the quantum master equation is solved in a “post-processing” step, which finally gives access to the quantum optical figures of merit. Via the explicit dependency of the microscopic transition rates on the state of the classical environment (spatially averaged macroscopic potentials), the hybrid model provides a consistent link between the two steps of the unidirectionally coupled simulation approach. Nevertheless, even in the case of weak feedback, where one-way coupled approaches are admissible, the fully coupled hybrid model allows to assess the approximation errors. Thereby it helps to justify simplified simulation approaches.
The application considered in Sec. \[sec:Application\] is an example for a quantum system with a weak feedback on the classical environment, which in principle would allow for the one-way method outlined above. The essential reason for this is the slow radiative decay in comparison to the fast electronic processes, which keeps the capture rates $S_{n/p}$ small once the QD is occupied. However, this is not always the case. For example, in electrically driven QD nanolasers, where the QD is placed inside a resonant cavity, the Purcell-enhanced light-matter interaction strongly decreases the radiative carrier lifetimes [@Strauf2011]. As a consequence, the capture rates $S_{n/p}$ are expected to increase by some orders of magnitude such that the quantum system significantly couples back to its classical environment and contributes to current guiding. We suspect that in this case the predictions of the hybrid model differ clearly from a decoupled treatment.
An interesting extension of the system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]) concerns reservoirs with different temperatures, as frequently studied in quantum thermodynamics [@Kosloff2013; @Strasberg2017]. We are confident that it is possible to achieve a thermodynamically consistent coupling of the quantum master equation (\[eq: quantum master equation\]) with energy transport models [@Albinus2002] or other transport models taking higher moments of the semi-classical Boltzmann equation [@Juengel2009] into account. The latter extend the isothermal van Roosbroeck system by one or multiple heat flow equations that determine the spatial temperature distribution of the crystal lattice and the continuum carriers. The construction of the corresponding hybrid system should be analogous to the case considered in this paper. The essential difference is that the coupling of both subsystems involves spatially averaged thermodynamic forces instead of chemical potentials, e.g. $\langle \mu_c\rangle_w \to \langle \mu_c/T\rangle_w$ etc. What might be interesting in the non-isothermal case is the impact of the quantum-classical interactions on the heat generation.
Summary
=======
Nowadays, quantum optical technologies are on their way from the lab to real world applications. To advance this development, device engineers will need simulation tools, which combine classical device physics with models from cavity quantum electrodynamics. As a step on this route, we have presented a new modeling approach for the simulation of single and few quantum dot devices.
By connecting semi-classical carrier transport theory with a quantum master equation in Lindlad form, our approach has lead to a hybrid quantum-classical system, that allows for a comprehensive description of electrically driven quantum dot devices on multiple scales: It enables the computation of the spatially resolved carrier transport together with the calculation of quantum optical figures of merit (e.g. photon generation rates, higher order correlation functions) in realistic semiconductor structures in a unified way. This has been demonstrated by numerical simulations of an electrical single-photon source based on a single quantum dot. We have presented a thorough theoretical analysis of the approach and showed that it guarantees the conservation of charge and the consistency with the thermodynamic equilibrium. Finally, we have proven that our hybrid quantum-classical system obeys the second law of thermodynamics.
We believe that our approach serves as a blueprint for the simulation of further quantum dot based photonic devices, in particular nanolasers.
The work of M.K. has been support by the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center 787 *Semiconductor Nanophotonics* under grant B4. M.M. was supported by the ERC via AdG 267802 *AnaMultiScale*. The authors acknowledge valuable discussions with H.-J. Wünsche, U. Bandelow, D. Peschka and A. Mielke. The authors are grateful to the reviewer for the detailed and helpful comments.
Boundary conditions\[sec:Boundary-conditions\]
==============================================
We assume a decomposition of the domain boundary $$\partial\Omega=\Big(\bigcup_{i}\Gamma_{i}\Big)\cup\partial\Omega_{N}$$ into several ohmic contacts and artificial boundaries of the device [@Selberherr1984]. On the artificial boundaries $\partial\Omega_{N}$, we assume homogeneous Neumann conditions $$\mathbf{n}\cdot\nabla\psi=0,\quad\mathbf{n}\cdot\nabla\mu_{c}=0,\quad\mathbf{n}\cdot\nabla\mu_{v}=0,$$ where $\mathbf{n}$ denotes the outer normal vector. The ohmic contacts are modeled by Dirichlet boundary conditions $$\psi=\psi_{\text{eq}}+U_{\text{appl},i},\quad\mu_{c}=\mu_{i},\quad\mu_{v}=\mu_{i},$$ on $\Gamma_{i}$, where $U_{\text{appl},i}$ represents the applied voltage at the $i$-th ohmic contact and $\mu_{i}=\mu_{\text{eq}}-qU_{\text{appl},i}$. The value of the built-in potential $\psi_{\text{eq}}$ is obtained from the local charge neutrality condition at the ohmic boundaries and zeros bias conditions ($\mu_{i}\equiv\mu_{\text{eq}}$ $\forall i$) [@Farrell2017].
Electrostatic field energy\[sec:Electrostatic-field-energy\]
============================================================
Following [@Albinus1996], we split the electrostatic potential $$\psi=\psi_{\text{int}}+\psi_{\text{ext}}$$ into an internal field $\psi_{\text{int}}$ generated by the internal charge density and an external field $\psi_{\text{ext}}$, which arises from the built-in doping profile and the applied voltages. Consequently, the Poisson problem (\[eq: Poisson equation\]) is decomposed into $$\begin{aligned}
-\nabla\cdot\varepsilon\nabla\psi_{\text{int}} & =q\rho_{\text{int}},\\
-\nabla\cdot\varepsilon\nabla\psi_{\text{ext}} & =qC,\end{aligned}$$ such that the internal field $\psi_{\text{int}}=\psi_{\text{int}}\left(\rho_{\text{int}}\right)$ can be written as a functional of the total internal carrier density $$\rho_{\text{int}}=p-n+Q\left(\rho\right).$$ On the domain boundaries it holds $$\begin{aligned}
\mathbf{n}\cdot\varepsilon\nabla\psi_{\text{int}} & =0 & & \text{on }\partial\Omega_{N},\\
\psi_{\text{int}} & =0 & & \text{on }\Gamma_{i},\end{aligned}$$ and $$\begin{aligned}
\mathbf{n}\cdot\varepsilon\nabla\psi_{\text{ext}} & =0 & & \text{on }\partial\Omega_{N},\\
\psi_{\text{ext}} & =\psi_{\text{eq}}+U_{\text{appl},i} & & \text{on }\Gamma_{i}.\end{aligned}$$ A variation of the internal carrier density $\rho_{\text{int}}\to\rho_{\text{int}}+a\delta\rho$ ($0<a\ll1$ is a small parameter) in the interior of the domain yields a variation of the electrostatic field $\delta\psi$ according to $$-\nabla\cdot\varepsilon\nabla\delta\psi=q\delta\rho\quad\text{on }\Omega$$ with the same boundary conditions for $\delta\psi$ as for $\psi_{\text{int}}$ stated above. The variation of the internal energy given by Eq. (\[eq: internal energy electric field\]) leads to $$\begin{aligned}
U_{\psi}\left(\rho_{\text{int}}+a\delta\rho\right) & =U_{\psi}\left(\rho_{\text{int}}\right)\\
& \phantom{=}+a\int_{\Omega}\mathrm{d}^{3}r\,\varepsilon\nabla\psi_{\text{int}}\left(\rho_{\text{int}}\right)\cdot\nabla\delta\psi\\
& \phantom{=}+aq\int_{\Omega}\mathrm{d}^{3}r\,\delta\rho\psi_{\text{ext}}+\mathcal{O}\left(a^{2}\right).\end{aligned}$$ Finally, using the identity $$\begin{aligned}
\int_{\Omega}\mathrm{d}^{3}r\,\varepsilon\nabla\psi_{\text{int}}\left(\rho_{\text{int}}\right)\cdot\nabla\delta\psi & =q\int_{\Omega}\mathrm{d}^{3}r\,\psi_{\text{int}}\left(\rho_{\text{int}}\right)\delta\rho,\end{aligned}$$ one obtains the Gâteaux-derivative $$\begin{aligned}
\lim_{a\to0} & \frac{U_{\psi}\left(\rho_{\text{int}}+a\delta\rho\right)-U_{\psi}\left(\rho_{\text{int}}\right)}{a}=\\
& \quad\quad\quad=q\int_{\Omega}\mathrm{d}^{3}r\,\left(\psi_{\text{int}}\left(\rho_{\text{int}}\right)+\psi_{\text{ext}}\right)\delta\rho.\end{aligned}$$ The central feature of the field’s internal energy expression Eq. (\[eq: internal energy electric field\]) is [@Albinus1996; @Albinus2002] $$\frac{\delta U_{\psi}}{\delta\rho}=q\text{\ensuremath{\psi}}.\label{eq: functional derivative of electric field energy}$$
Entropy production rate \[sec:Entropy-production-rate\]
=======================================================
This section gives some details on the derivation of the expression (\[eq: entropy production rate\]) for the entropy production rate. Starting from Eq. (\[eq: entropy production (general)\]), one obtains by using Eq. (\[eq: total thermodynamic potentials\]) and (\[eq: classical entropy and energy density\]) the entropy production rate as $$\begin{aligned}
\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t} & =-\frac{1}{T}\int_{\Omega}\mathrm{d}^{3}r\,\left(\frac{\partial u_{\text{cl}}\left(n,p\right)}{\partial t}-T\frac{\partial s_{\text{cl}}\left(n,p\right)}{\partial t}\right)\\
& \phantom{=}-\frac{1}{T}\mathrm{tr}\left(H\mathcal{L}\left(\rho;\chi_w\right)\right)-k_{B}\mathrm{tr}\left(\log\left(\rho\right)\mathcal{L}\left(\rho;\chi_w\right)\right)\\
& \phantom{=}-\frac{1}{T}\frac{\mathrm{d}U_{\psi}}{\mathrm{d}t}+\sum_{i\geq1}\frac{\mu_{i}}{qT}\int_{\Gamma_{i}}\mathrm{d}\mathbf{A}\cdot\left(\mathbf{j}_{n}+\mathbf{j}_{p}\right).\end{aligned}$$ Taking the partial time derivatives, using the state equations (\[eq: carrier densities\]), Eq. (\[eq: functional derivative of electric field energy\]) and $$\frac{\mathrm{d}U_{\psi}}{\mathrm{d}t}=\int_{\Omega}\mathrm{d}^{3}r\,q\psi\frac{\partial\left(p-n+Q\left(\rho\right)\right)}{\partial t},$$ we arrive at $$\begin{aligned}
\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t} & =-\frac{1}{T}\int_{\Omega}\mathrm{d}^{3}r\,\left(\mu_{c}\frac{\partial n}{\partial t}-\mu_{v}\frac{\partial p}{\partial t}\right)\\
& \phantom{=}-\frac{1}{T}\mathrm{tr}\left(H\mathcal{L}\left(\rho;\chi_w\right)\right)-k_{B}\mathrm{tr}\left(\log\left(\rho\right)\mathcal{L}\left(\rho;\chi_w\right)\right)\\
& \phantom{=}+\frac{q}{T}\left\langle \psi\right\rangle _{w}\mathrm{tr}\left(N\mathcal{L}\left(\rho;\chi_w\right)\right)\\
& \phantom{=}+\sum_{i\geq1}\frac{\mu_{i}}{qT}\int_{\Gamma_{i}}\mathrm{d}\mathbf{A}\cdot\left(\mathbf{j}_{n}+\mathbf{j}_{p}\right),\end{aligned}$$ where we have explicitly used Eq. (\[eq: charge density of the quantum system\]) for the charge density of the quantum system. For different $Q\left(\rho\right)$ and multiple QDs, the calculation follows the same lines. With the help of the carrier transport equations (\[eq: electron transport\])–(\[eq: hole transport\]), the macroscopic capture rates (\[eq: loss terms\]), partial integration and the boundary conditions given in Appendix \[sec:Boundary-conditions\], this is $$\begin{aligned}
\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t} & =\frac{1}{T}\int_{\Omega}\mathrm{d}^{3}r\,\left(\mu_{c}-\mu_{v}\right)R\\
& \phantom{=}+\frac{1}{qT}\int_{\Omega}\mathrm{d}^{3}r\,\left(\mathbf{j}_{n}\cdot\nabla\mu_{c}+\mathbf{j}_{p}\cdot\nabla\mu_{v}\right)\\
& \phantom{=}+\frac{1}{T}\left\langle \mu_{c}\right\rangle_{w}\mathrm{tr}\left(N \mathcal{D}_{e}(\rho;\chi_w)\right)\\
& \phantom{=}+\frac{1}{T}\left\langle \mu_{v}\right\rangle_{w}\mathrm{tr}\left(N \mathcal{D}_{h}(\rho;\chi_w)\right)\\
& \phantom{=}-\frac{1}{T}\mathrm{tr}\left(H\mathcal{D}\left(\rho;\chi_w\right)\right) -k_B\mathrm{tr}\left(\log\left(\rho\right)\mathcal{D}\left(\rho;\chi_w\right)\right)\\
& \phantom{=}+\frac{q}{T}\left\langle \psi\right\rangle _{w}\mathrm{tr}\left(N\mathcal{D}\left(\rho;\chi_w\right)\right).\end{aligned}$$ In the above expression, the surface integrals have canceled out. Using Eqns. (\[eq: dissipator decomposition\])–(\[eq: charge conservation of D0\]), one arrives at Eq. (\[eq: entropy production rate\]).
Second law of thermodynamics \[subsec:Positivity-of-the\]
==========================================================
In this section we proof the non-negativity of the entropy production rate (\[eq: entropy production rate\]) of the hybrid system (\[eq: Poisson equation\])–(\[eq: quantum master equation\]). First, we introduce the (auxiliary) density matrices
\[eq: auxiliary stationary states\] $$\begin{aligned}
\rho_{0}^{\ast} & =\frac{1}{Z_{0}^{\ast}}e^{-\beta H},\label{eq: auxiliary stationary states 0}\\
\rho_{e}^{\ast}(\chi_w) & =\frac{1}{Z_{e}^{\ast}}e^{-\beta\left(H-\mu_{c}^{\text{eff}}(\chi_w)N\right)},\label{eq: auxiliary stationary states e}\\
\rho_{h}^{\ast}(\chi_w) & =\frac{1}{Z_{h}^{\ast}}e^{-\beta\left(H-\mu_{v}^{\text{eff}}(\chi_w)N\right)}\label{eq: auxiliary stationary states h}\end{aligned}$$
with $\mu_{c/v}^{\text{eff}}(\chi_w)=\left\langle \mu_{c/v}\right\rangle _{w}+q\left\langle \psi\right\rangle _{w}$. Using Eq. (\[eq: quantum detailed balance rates\]) it can be shown by direct calculation that $\mathcal{D}_{\nu}\left(\rho_{\nu}^{\ast}(\chi_w);\chi_w\right)=0$, $\nu\in\left\{ 0,e,h\right\} $, for the dissipators given in Eq. (\[eq: dissipator decomposition\]). Then, *Spohn’s inequaility* [@Spohn1978] states that $$\mathrm{tr}\left(\left(\log\rho_{\nu}^{\ast}(\chi_w)-\log\rho\right)\mathcal{D}_{\nu}\left(\rho;\chi_w\right)\right)\geq0\label{eq: entropy production rate inequality}$$ for $\nu\in\left\{ 0,e,h\right\} $.
The entropy production rate Eq. (\[eq: entropy production rate\]) can be written in the form $$\begin{aligned}
\frac{\mathrm{d}S_{\text{tot}}}{\mathrm{d}t} & =k_{B}\int_{\Omega}\mathrm{d}^{3}r\,\beta\left(\mu_{c}-\mu_{v}\right)\left(1-e^{-\beta\left(\mu_{c}-\mu_{v}\right)}\right)\sum_{j}r_{j}\nonumber \\
& \phantom{=}+\frac{1}{q^{2}T}\int_{\Omega}\mathrm{d}^{3}r\,\left(\sigma_{n}\left|\nabla\mu_{c}\right|^{2}+\sigma_{p}\left|\nabla\mu_{v}\right|^{2}\right)\nonumber \\
& \phantom{=}+k_{B}\mathrm{tr}\left(\left(\log{\rho_{0}^{\ast}}-\log{\rho}\right)\mathcal{D}_{0}\left(\rho;\chi_w\right)\right)\label{eq: entropy production rate (rearranged)}\\
& \phantom{=}+k_{B}\mathrm{tr}\left(\left(\log{\rho_{e}^{\ast}(\chi_w)}-\log{\rho}\right)\mathcal{D}_{e}\left(\rho;\chi_w\right)\right)\nonumber \\
& \phantom{=}+k_{B}\mathrm{tr}\left(\left(\log{\rho_{h}^{\ast}(\chi_w)}-\log{\rho}\right)\mathcal{D}_{h}\left(\rho;\chi_w\right)\right),\nonumber \end{aligned}$$ where we have used Eqns. (\[eq: current densities\]), (\[eq: auxiliary stationary states\]), the trace conservation property of the dissipator and a recombination rate of the form (\[eq: recombination rate\]). Here, $j$ labels the recombination channels and the functions $r_{j}=r_{j}\left(n,p,\psi\right)$ are non-negative by construction (cf. Appendix \[sec: Parameters and auxiliary - van Roosbroeck\]). Using the inequalities (\[eq: entropy production rate inequality\]) and $x\left(1-e^{-x}\right)\geq0$ $\forall x\in\mathbb{R}$, it is easy to see that each line of Eq. (\[eq: entropy production rate (rearranged)\]) is non-negative.
Projection on eigenstates\[sec:Projection-on-eigenstates\]
==========================================================
In order to obtain a system of ODEs from Eq. (\[eq: quantum master equation\]), it must be projected on a basis of the quantum system’s Hilbert space. We use the eigenbasis of the Hamiltonian $H$, for which we assume the spectral representation $$H=\sum_{k}\varepsilon_{k}\vert\varphi_{k}\rangle\langle\varphi_{k}\vert.$$ For the sake of simplicity, we consider the energy spectrum $\left\{ \varepsilon_{k}\right\} $ to be non-degenerate here. Moreover, the Lamb-Shift contribution to $H$ is neglected. Then, the jump operators are projectors between energy eigenstates $A_{\alpha}\to A_{i,j}=\vert\varphi_{i}\rangle\langle\varphi_{j}\vert$. The equations of motion for the diagonal elements of the density matrix are obtained as $$\begin{aligned}
\partial_{t}\langle\varphi_{k}\vert\rho\vert\varphi_{k}\rangle & =\sum_{j}\left(\mathcal{M}_{k,j}\langle\varphi_{j}\vert\rho\vert\varphi_{j}\rangle-\mathcal{M}_{j,k}\langle\varphi_{k}\vert\rho\vert\varphi_{k}\rangle\right),\end{aligned}$$ whereas the off-diagonal elements $k\neq l$ obey $$\begin{aligned}
\partial_{t}\langle\varphi_{k}\vert\rho\vert\varphi_{l}\rangle= & -\frac{i}{\hbar}\left(\varepsilon_{k}-\varepsilon_{l}\right)\langle\varphi_{k}\vert\rho\vert\varphi_{l}\rangle\\
& -\frac{1}{2}\sum_{j}\left(\mathcal{M}_{j,k}+\mathcal{M}_{j,l}\right)\langle\varphi_{k}\vert\rho\vert\varphi_{l}\rangle\end{aligned}$$ with the (non-negative) transition rate matrix elements $$\begin{aligned}
\mathcal{M}_{i,j} & =\gamma_{i,j}+\hat{\gamma}_{j,i}\\
& =\gamma_{i,j}\left(1+e^{-\beta\left(\varepsilon_{j}-\varepsilon_{i}-\left(\left\langle \mu_{i,j}\right\rangle _{w}+q\left\langle \psi\right\rangle _{w}\right)\ell_{i,j}\right)}\right)\geq0.\end{aligned}$$ Obviously, in the case of non-degenerate energy spectra the diagonal elements decouple from the off-diagonal elements. The off-diagonal elements are fully decoupled each and show damped oscillations (dephasing). This has important implications on the complexity of the numerical simulations: Starting from the thermodynamic equilibrium state, where only diagonal elements of the density matrix are occupied, the dynamics never excite any off-diagonal elements (in the energy eigenbasis representation). Hence, the off-diagonal elements can be omitted from the simulation. Thereby the number of degrees of freedom of the quantum system grows only with $N$ instead of $N^{2}$, where $N$ is the dimension of the (possibly truncated) Hilbert space. However, if the spectrum of $H$ is degenerate, this feature is lost in general and one has to account for degenerate eigenstate coherences (i.e. off-diagonal elements contribute to the dynamics) [@Cuetara2016].
Parameters and auxiliary models\[sec:Parameters-and-auxiliary\]
===============================================================
This section lists auxiliary models and parameters used in the numerical simulations presented in Sec. \[sec:Application\].
Van Roosbroeck system {#sec: Parameters and auxiliary - van Roosbroeck}
---------------------
We use GaAs parameters at $T=50\,\text{K}$. The effective masses are $m_{e}^{\ast}=0.068\,m_{0}$, $m_{h}^{\ast}=0.503\,m_{0}$, where $m_{0}$ denotes the (free) electron mass and the band edge energies are taken as $E_{v}=0\,\text{eV}$ and $E_{c}=1.516\,\text{eV}$. The (static) relative permittivity is set to $\varepsilon_{r}=12.9$, the LO-phonon energy is $\hbar\omega_{\text{LO}}=36.5\,\text{meV}$ and the refractive index is $n_{r}=3.55$ (around $\text{950\,nm}$). The (net-)recombination rate in Eqns. (\[eq: electron transport\]), (\[eq: hole transport\]) is modeled as [@Selberherr1984; @Farrell2017] $$\begin{aligned}
\label{eq: recombination rate}
\begin{aligned}
R&=R_{\text{SRH}}+R_{\text{sp}}+R_{\text{Au}} \\
&= \left(1-e^{-\beta\left(\mu_{c}-\mu_{v}\right)}\right) \sum_j r_j(n,p,\psi)
\end{aligned}\end{aligned}$$ where $j\in\{ \text{SRH},\text{sp},\text{Au} \}$ labels the different channels and $$\begin{aligned}
r_{\text{SRH}} & =\frac{np}{\tau_{p}\left(n+n_{d}\right)+\tau_{n}\left(p+p_{d}\right)},\\
r_{\text{sp}} & =Bnp,\\
r_{\text{Au}} & =\left(C_{\text{Au}}^{n}n+C_{\text{Au}}^{p}p\right)np\end{aligned}$$ with $n_{d}=ne^{\beta\left(E_{T}-q\psi-\mu_{c}\right)}$, $p_{d}=pe^{-\beta\left(E_{T}-q\psi-\mu_{v}\right)}$. The recombination rates $r_j(n,p,\psi)$ of the individual channels are non-negative by construction. The non-radiative life times are sensitive to the impurity concentration and modeled via $\tau_{n/p}=\tau_{n/p,0}/\big(1+\big(\vert C\vert/C_{\text{ref}}\big)^{\gamma_{\text{SRH}}}\big)$ with $\tau_{n,0}=\tau_{p,0}=10\,\text{ns}$, $\gamma_{\text{SRH}}=1.72$ and $C_{\text{ref}}=9\times10^{17}\,\text{cm}^{-3}$ [@Palankovski2004]. The trap energy level $E_{T}$ is assumed to be in the center of the energy gap. The radiative recombination coefficient is taken as $B=1.06\times10^{-8}\,\text{cm}^{-3}\,\text{s}^{-1}$ and the Auger recombination coefficients are set to $C_{\text{Au}}^{n}=6\times10^{-30}\,\text{cm}^{-6}\,\text{s}^{-1}$, $C_{\text{Au}}^{p}=1.6\times10^{-29}\,\text{cm}^{-6}\,\text{s}^{-1}$ [@Palankovski2004]. The carrier mobilities $M_{n/p}$ are taken from the model given in Ref. [@Mnatsakanov2004], which is reported to hold down to $T=50\,\text{K}$. Despite the low temperatures, we assume complete ionization due to the metal-insulator transition at heavy doping [@Mott1968].
Open quantum system
-------------------
The eigenenergies of the Hamiltonian (\[eq: application Hamiltonian\]) are obtained from the parabolic/step-like confinement potential (relative to the respective continuum band edge) $U_{\lambda}\left(r,z\right)=-U_{0}^{\lambda}\Theta\left(h/2-\left|z\right|\right)+\frac{1}{2}m_{\lambda}^{\ast}\omega_{\lambda,0}^{2}r^{2}$, $\lambda\in\left\{ e,h\right\} $, by solving the stationary Schrödinger equation at flat band conditions [@Wojs1996; @Nielsen2004]. The parameters for the InGaAs-QD are taken as $U_{e}=\text{350\,\text{meV}}$, $U_{h}=\text{170\,\text{meV}}$, $m_{e}^{\ast}=0.067\,m_{0}$, $m_{h}^{\ast}=0.15\,m_{0}$ [@Nielsen2004] and $\hbar\omega_{e,0}=45.5\,\text{meV}$, $\hbar\omega_{h,0}=20.3\,\text{meV}$. The QD height is assumed as $h=3\,\text{nm}$. For the computation of the Coulomb matrix elements we set the background dielectric permittivity to $\varepsilon_{r}=12.5$ [@Nielsen2004].
With the parameters above, the QD conduction band ground state $\varepsilon_{c}$ is found at $137.7\,\text{meV}$ below the continuum band edge and the QD valence band ground state $\varepsilon_{v}$ is $44.7\,\text{meV}$ above the valence band edge. The Coulomb matrix elements are obtained as $V_{c,c}=23.2\,\text{meV}$, $V_{v,v}=24.5\,\text{meV}$ and $V_{c,v}=23.7\,\text{meV}$. The interband dipole moment is assumed as $d_{c,v}=q\times0.6\,\text{nm}$ and the Purcell factor is set to $P_{i,f}=1.8$ for all allowed optical transitions. The emission energies are obtained around $1.31\,\text{eV}$ with radiative life times of approximately $1\,\text{ns}$ according to Eq. (\[eq: radiative decay rate\]). The fitting parameters in the carrier scattering rates are set to $\tau_{\text{LO}}^{e}=\tau_{\text{LO}}^{h}=10\,\text{ps}$, $a_{\text{LO}}^{e}=25\,\text{meV}$, $a_{\text{LO}}^{h}=7\,\text{meV}$, $\tau_{\text{Au}}^{\lambda,\lambda^{\prime}}=1\,\text{ps}$, $\gamma_{\text{Au}}^{\lambda,\lambda^{\prime}}=0.7$ (for all $\lambda,\lambda^{\prime}\in\left\{ e,h\right\} $), $n_{\text{Au}}^{\text{crit}}=1\times10^{19}\,\text{cm}^{-3}$ and $p_{\text{Au}}^{\text{crit}}=5\times10^{18}\,\text{cm}^{-3}$.
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---
abstract: 'It is theoretically demonstrated that parallel weakly tunnel coupled quantum dots exhibit non-equilibrium blockade regimes caused by a full occupation in the spin triplet state, in analogy to the Pauli spin blockade in serially weakly coupled quantum dots. Charge tends to accumulate in the two-electron triplet for bias voltages that support transitions between the singlet and three-electron states.'
author:
- 'J. Fransson'
title: 'Non-equilibrium triplet blockade in parallel coupled quantum dots'
---
From fundamental aspects of spin and charge correlations the two-level system in a double quantum dot (DQD) has recently become highly attractive. It has been demonstrated that spin correlations lead to Pauli spin blockade in serially coupled quantum dots (QDs), where the current is suppressed because of spin triplet correlations,[@ono2002; @rogge2004; @johnson2005; @franssoncm2005] something which may be applied in spin-qubit readout technologies.[@bandyopadhyay2003] Pauli spin blockade has also been reported for general DQDs with more than two electrons.[@liu2005] Recently, the Pauli spin blockade with nearly absent singlet-triplet splitting has been employed in studies of hyperfine couplings between electron and nuclear spins.[@ono2004; @johnson_nature2005; @erlingsson2005; @koppens2005; @petta2005] Besides being present in serially coupled QDs, it is relevant to ask whether an analog of the Pauli spin blockade is obtainable in parallel QDs.
The purpose of this paper is to demonstrate that parallel coupled QDs, see Fig \[fig-system\], exhibit regimes of non-equilibrium triplet blockade. Here only one of the QDs is tunnel coupled to the external leads while the second QD functions as a perturbation to the first QD. Important quantities in order to find the non-equilibrium triplet blockade regime is that the QDs are coupled through charge interactions, e.g. interdot Coulomb repulsion and exchange interaction, and weakly through tunnelling. In absence of interdot exchange interaction there may be regimes of usual Coulomb blockade in a finite bias voltage range around equilibrium.
In presence of a sufficiently large ferromagnetic interdot exchange interaction the triplet states $\ket{\sigma}_A\ket{\sigma}_B,\ \sigma=\up,\down$ (one electron in each QD with equal spins) and $[\ket{\up}_A\ket{\down}_B+\ket{\down}_A\ket{\up}_B]/\sqrt{2}$ acquire a lower energy than the lowest two-electron singlet (the singlet states being superpositions of the Fock states $\{[\ket{\up}_A\ket{\down}_B-\ket{\down}_A\ket{\up}_B]/\sqrt{2},\ket{\up\down}_A\ket{0}_B,\ket{0}_A\ket{\up\down}_B\}$). Then, the triplet naturally becomes the equilibrium ground state with a unit occupation probability, provided that the two-electron triplet state has a lower energy than all other states. The triplet persists in being fully occupied for bias voltages smaller than the energy separation between the triplet and singlet states, although transitions between the one-electron states and the singlets may open for conduction. However, for larger bias voltages this low bias triplet blockade is lifted as the transitions between the triplet and the one-electron states become resonant with the lower of the chemical potentials of the leads. At this lifting, the current through the system is mediated via transitions between the two-electron singlets and the one-electron states.
![(Colour online) Left panel: The coupled QDs of which only one is tunnel coupled to the leads. Right panel: Processes leading to the non-equilibrium triplet blockade. Faint and bold lines signify low and high transition probabilities, respectively. See text for notation.[]{data-label="fig-system"}](system_trans_cm.eps){width="8.5cm"}
The non-equilibrium triplet blockade regime is entered at bias voltages such that transitions between three-electron states and, at least, one of the singlet states become resonant, see Fig. \[fig-system\], while transitions between the three-electron states and the triplet lie out of resonance. At those conditions, an electron can enter the DQD from the lead with the higher chemical potential, through transitions between the singlet and the three-electron states. Transitions from the triplet to the three-electron state are suppressed since the bias is lower than the energy barriers between those states. However, electron tunnelling from the three-electron states in the DQD to the lead with the lower chemical potential are supported through transitions from those states to the triplet, since the tunnelling barrier to this lead is sufficiently low. Thereto, the probability for such transitions are about unity whereas the probability for transitions between the three-electron states and the singlet is at most a half. Finally, charge that end up in the triplet through this process is trapped in this state because of the negligible probability for transitions between the triplet and the one-electron states.
It is noticed that, a finite (ferromagnetic) interdot exchange interaction is not a necessary condition for the existence of a non-equilibrium triplet blockade regime. Nevertheless, a ferromagnetic exchange yields a larger degree of freedom in the variation of the interdot tunnelling and, also, allows a higher temperature.
For quantitative purposes, consider two single level QDs $(\dote{A},\dote{B}$, spin-degenerate) with intradot charging energies $(U_A,U_B)$, which are coupled by interdot charging $(U')$, exchange $(J\geq0)$, and tunnelling $(t)$ interactions. Specifically, the DQD is modelled by[@sandalov1995; @inoshita2003; @cota2005] $\Hamil_{DQD}=\sum_{i=A,B}(\sum_\sigma\dote{i\sigma}\ddagger{i\sigma}\dc{i\sigma}+Un_{i\up}n_{i\down})+(U'-J/2)(n_{A\up}+n_{A\down})(n_{B\up}+n_{B\down})-2J\bfs_A\cdot\bfs_B+\sum_\sigma(t\ddagger{A\sigma}\dc{B\sigma}+H.c.)$, where $\bfs_i=(1/2)\sum_{\sigma\sigma'}\ddagger{i\sigma}\hat{\sigma}_{\sigma\sigma'}\dc{i\sigma'}$, $\sigma,\sigma'=\up,\down$, $i=A,B$, are the spins of the two levels. In analogy with the Pauli spin blockade in serially coupled QDs,[@ono2002; @franssoncm2005] it is required that the lowest one-electron states, the triplet, and the two lowest singlets are nearly aligned, and that the lowest three-electron states lie below the equilibrium chemical potential $\mu$. Hence, $E_T\approx E_{S1}\approx E_{S2}\approx\min_{n=1}^4\{E_{1n}\}<\min_{n=1}^4\{E_{3n}\}<\mu<E_4$, where $E_T$ and $E_{Sn},\ 1,2$, are eigenenergies for the triplet and the two lowest singlet states, respectively, whereas $E_{1n}$, $E_{3n}$, and $E_4$ are the energies for the one-, three-, and four-electron states, respectively. This requires that $\mu-\dote{B}\approx\Delta\dote{}$, $U'\approx\Delta\dote{}$, and $U_A\approx2\Delta\dote{}\leq U_B$, where $\Delta\dote{}=\dote{B}-\dote{A}$. The inequality $U_A\leq U_B$ points out that the QDs do not have to be identical, merely that the charging energy of the second QD should be lower bounded by the charging energy of the first. It should be emphasized, however, that the presence of the second QD is essential in order to obtain the effect discussed in this paper. Finally, weakly coupled QDs, e.g. $\xi=2t/\Delta\dote{}\ll1$ implies that the energies for the lowest one- and three-electron states acquire their main weight on QD$_A$. This condition yields a low (large) probability for transitions between the triplet and the lowest one-electron (three-electron) states.
In general there are 16 eigenstates of $\Hamil_{DQD}$, labelled $\{\ket{N,n},E_{Nn}\}$ denoting the $n$th state of the $N$-electron ($N=1,\ldots,4$) configuration at the energy $E_{Nn}$.[@franssoncm2005] In diagonal form, the DQD is thus described by $\Hamil_{DQD}=\sum_{Nn}E_{Nn}\ket{N,n}\bra{N,n}$. Taking the leads to be free-electron like metals and the (single-electron) tunnelling between the DQD with rate $v_{k\sigma}$, the full system can be written as[@franssoncm2005] $$\begin{aligned}
\lefteqn{
\Hamil=\sum_{k\sigma\in L,R}\leade{k}\cdagger{k}\cc{k}+\Hamil_{DQD}
}
\label{eq-Ham}\\&&
+\sum_{k\sigma,Nnn'}[v_{k\sigma}(\dc{A\sigma})_{NN+1}^{nn'}
\cdagger{k}\ket{N,n}\bra{N+1,n'}+H.c.],
\nonumber\end{aligned}$$ where $(\dc{A\sigma})_{NN+1}^{nn'}=\bra{N,n}\dc{A\sigma}\ket{N+1,n'}$ is the matrix element for the transitions $\ket{N,n}\bra{N+1,n'}$. The operator $\dc{A\sigma}$ signify that electrons tunnel from molecular like orbitals in the DQD through QD$_A$ to the leads, which appropriately describes the physical tunnelling processes.
Following the procedure in Ref. , the occupation of the eigenstates are described by a density matrix $\rho=\{\ket{N,n}\bra{N,n}\}$. In the Markovian approximation (sufficient for stationary processes) one thus derives that the equations for $P_{Nn}\equiv\av{\ket{N,n}\bra{N,n}}$ to the first order in the couplings $\Gamma^{L/R}=2\pi\sum_{k\in L/R}|v_{k\sigma}|^2\delta(\omega-\leade{k})=\Gamma_0/2$ between the DQD and the leads, can be written as
$$\begin{aligned}
%\lefteqn{
\ddt P_{Nn}&=&
\frac{1}{\hbar}\sum_{\alpha=L,R}\biggl(
\sum_{n'}\Gamma_{N-1n',Nn}^\alpha
%}
%\nonumber\\&&\vphantom{\sum_\sigma}\times
[f^+_\alpha(\Delta_{Nn,N-1n'})P_{N-1n'}
-f^-_\alpha(\Delta_{Nn,N-1n'})P_{Nn}]
\nonumber\\&&
-\sum_{n'}\Gamma^\alpha_{Nn,N+1n'}
[f^+_\alpha(\Delta_{N+1n',Nn})P_{Nn}
%\nonumber\\&&\vphantom{\sum_\sigma}
-f^-_\alpha(\Delta_{N+1n',Nn})P_{N+1n'}]\biggr)=0,
%\ N=1,\ldots,4,
\label{eq-dtN}\\
N&=&1,\ldots,4,
\nonumber\end{aligned}$$
where $P_{-1n}=P_{5n}\equiv0$. Here, $\Delta_{N+1n',Nn}=E_{N+1n'}-E_{Nn}$ denote the energies for the transitions $\ket{N,n}\bra{N+1,n'}$, while $\Gamma_{Nn,N+1n'}^{L/R}=\sum_\sigma\Gamma^{L/R}(\dc{A\sigma})_{NN+1}^{nn'}$. Also, $f^+_{L/R}(\omega)=f(\omega-\mu_{L/R})$ is the Fermi function at the chemical potential $\mu_{L/R}$ of the left/right $(L/R)$ lead, and $f^-_{L/R}(\omega)=1-f^+_{L/R}(\omega)$. Effects from off-diagonal occupation numbers $\av{\ket{N,n}\bra{N,n'}}$, which only appear in the second order (and higher) in the couplings, are neglected since these include off-diagonal transition matrix elements to the second order (or higher) which generally are small for $\xi\ll1$.
Since the low bias triplet blockade can be found for weakly coupled QDs whenever $J>0$ is sufficiently large, the following derivation focus on the non-equilibrium blockade. The non-equilibrium blockade discussed here, is driven by opening transitions between the two- and three-electron states. For simplicity, assume that the bias voltage $V=(\mu_L-\mu_R)/e$ is applied such that $\mu_{L/R}=\mu\pm eV/2$. Then for $|eV|<7\Delta\dote{}/4$, $k_BT<0.01U_A$, and $\xi<0.2$, which is sufficient for the present purposes, only the population numbers $P_{1n},\ n=1,2$, $N_T=P_{2n}/3,\ n=1,2,3$, $P_{24},\ P_{25}$, and $P_{3n},\ n=1,2$, are non-negligible. The other populations are negligible since the corresponding transition energies lie out of resonance. Because of spin-degeneracy it is noted that $P_{1n}=N_1/2,\ n=1,2$, and $P_{3n}=N_3/2,\ n=1,2$, which reduces the system to five equations for the population numbers. As discussed in the introduction, the non-equilibrium blockade arises when transitions between a singlet and the three-electrons state are resonant. Therefore, the bias voltage is tuned into the regime where $\mu_L$ lies around these transition energies, e.g.[@Delta32] $\min_{nn'}\{\Delta_{3n',2n}\}<\mu_L<\max_{nn'}\{\Delta_{3n',2n}\}$, $n=1,\ldots,5$, $n'=1,2$ (here $eV>0$, the case $eV<0$ follows by symmetry of the system). For such biases it is clear that $f_L^+(\Delta_{2n,1n'})=f_R^-(\Delta_{2n,1n'})=1$, $n=1,\ldots,5$, $n'=1,2$, and that $f_R^+(\Delta_{3n',2n})=0$, $n=1,\ldots,5$, $n'=1,2$. It is also clear that the charge accumulation in the triplet is lifted for biases that supports transitions from the triplet to the three-electron states, hence, the bias voltage has to be such that $f_{L/R}^+(\Delta_{3n',2n})\approx0$, that is $\Delta_{3n',2n}=E_{3n'}-E_T>\mu_L+k_BT$, $n=1,2,3$, $n'=1,2$. Thus, the equations for the population numbers can be written as
\[eq-Pneq\] $$\begin{aligned}
N_1&=&\frac{1}{p}N_3=\frac{2/3}{1+2p(\kappa/\beta)^2}N_T
\label{eq-P1neq}\\
P_{2n}&=&\frac{1}{2}\frac{L_n^2
+\Lambda_n^2p\sum_\alpha f_\alpha^-(\Delta_{31,2n})}
{L_n^2+\Lambda_n^2f_L^+(\Delta_{31,2n})}N_1,\ n=4,5,
\label{eq-P2nneq}\\
p&=&\sum_{n=4}^5L_n^2
\frac{\Lambda_n^2f_L^+(\Delta_{31,2n})}
{L_n^2+\Lambda_n^2f_L^+(\Delta_{31,2n})}
\biggl\{3\kappa^2+\sum_{\alpha,n=4}^5\Lambda_n^2
\nonumber\\&&\times
f_\alpha^-(\Delta_{31,2n})\left[1-
\frac{\Lambda_n^2f_L^+(\Delta_{31,2n})}
{L_n^2+\Lambda_n^2f_L^+(\Delta_{31,2n})}
\right]\biggr\}^{-1}.
\label{eq-p}\end{aligned}$$
Here, ($n'=1,2$, $n=4,5$) $\beta^2\equiv\sum_\sigma|(\dc{A\sigma})^{n'1}_{12}|^2=\xi^2/[(1+\sqrt{1+\xi^2})^2+\xi^2]$, $L_n^2\equiv\sum_\sigma|(\dc{A\sigma})_{12}^{n'n}|^2$, $\kappa^2\equiv\sum_\sigma|(\dc{A\sigma})_{23}^{1n'}|^2=(1+\xi^2)/[(1+\sqrt{1+\xi^2})^2+\xi^2]$, and $\Lambda_n^2\equiv\sum_\sigma|(\dc{A\sigma})_{23}^{nn'}|^2$ are the matrix elements for the relevant transitions. The above relations are due to spin-degeneracy, e.g. $\Delta_{2n,11}=\Delta_{2n,12}$ and $\Delta_{31,2n}=\Delta_{32,2n}$, $n=1,\ldots,5$. Using Eq. (\[eq-Pneq\]), charge conservation ($1=\sum_{Nn}P_{Nn}=N_1+N_T+\sum_nP_{2n}+N_3$) thus implies that $$\begin{aligned}
N_\text{T}&=&\biggl\{1+\frac{2/3}{1+2p(\kappa/\beta)^2}
\Bigl(1+p
\nonumber\\&&
+\frac{1}{2}\sum_{n=4}^5\frac{L_n^2
+p\Lambda_n^2\sum_\alpha f_\alpha^-(\Delta_{31,2n})}
{L_n^2+\Lambda_n^2f_L^+(\Delta_{31,2n})}
\Bigr)\biggr\}^{-1}.
\label{eq-NTneq}\end{aligned}$$
Now, the matrix elements $L_n^2,\Lambda_n^2,\ n=4,5$, are finite and bounded, however, $L_4^2,2\Lambda_5^2\rightarrow1$ and $L_5^2,\Lambda_4^2\rightarrow0$ as $\xi\rightarrow0$, hence, the last term in Eq. (\[eq-NTneq\]) is at most 1/2 for weakly coupled QDs since $p\rightarrow0$, $\xi\rightarrow0$, in the considered bias regime (see discussion below). However, the ratio $2p(\kappa/\beta)^2$ is finite for all $\xi$ and $J>0$, while it diverges as $\xi\rightarrow0$ for $J=0$, see main panel in Fig. \[fig-Jvar\]. For weakly coupled QDs one thus finds that $N_T\approx1/(1+[1+2p(\kappa/\beta)^2]^{-1})\approx1$ whenever $2p(\kappa/\beta)^2\gg1$. The inset of Fig. \[fig-Jvar\] illustrates a subset in $(t,J)$-space where this ratio is larger than $10^2$. At this condition, the boundary is approximately given by $J(t)=J_0-15t^2[1+(10t)^2]$.
![(Colour online) Variation of the ratio $2p(\kappa/\beta)^2$ as function of $J$ for different $t$ at constant $\Delta\dote{},\ U',\ U_{A/B}$. The inset shows the region in $(t,J)$-space where $2p(\kappa/\beta)^2>10^2$.[]{data-label="fig-Jvar"}](Jvar_cm.eps){width="8.5cm"}
Using the transport equation derived in Ref. , identifying $G^<_{Nn,N+1n'}(\omega)=i2\pi P_{N+1n'}\delta(\omega-\Delta_{N+1n',Nn})$ and $G^>_{Nn,N+1n'}(\omega)=-i2\pi P_{Nn}\delta(\omega-\Delta_{N+1n',Nn})$, the current in the considered regime is given by $$\begin{aligned}
I&=&\frac{e\Gamma_0}{6\hbar}\biggl[3(\beta^2-\kappa^2)
+\sum_{n=4}^5[L_n^2-\Lambda_n^2f_L^-(\Delta_{31,2n})]
\nonumber\\&&
+2\sum_{n=4}^5\Lambda_n^2f_L^+(\Delta_{31,2n})
\frac{L_n^2+p\Lambda_n^2\sum_\alpha f_\alpha^-(\Delta_{31,2n})}
{L_n^2+\Lambda_n^2f_L^+(\Delta_{31,2n})}\biggr]
\nonumber\\&&\times
\frac{N_T}{1+2p(\kappa/\beta)^2}.
\label{eq-Jneq}\end{aligned}$$ This expression clearly shows that a large value of $2p(\kappa/\beta)^2$ yields a suppression of the current, that is, at the formation of a unit occupation in the triplet state. For biases such that $\mu_L<\min_{nn'}\{\Delta_{3n,2n}\}$ is follows that $f_L^+(\Delta_{3n',2n})\approx0\ \Rightarrow\ p\approx0$, which accounts for a lifting of the triplet blockade where the current is $\sim2p(\kappa/\beta)^2$ larger than in the blockaded regime.
The non-equilibrium triplet blockade depends on the interplay between $J$ and $t$. A reduced $t$ leads to a strong localisation of the odd number states in either of the QDs, which for $\Delta\dote{}>0$ leads to that the lowest odd number states are strongly localised on QD$_A$. Then, the probability for transitions between the triplet, and the one-/three-electron states is small/large $(\beta\rightarrow0/\kappa\rightarrow1$).
The singlets, on the other hand, are expanded in terms of the Fock states $\{[\ket{\up}_A\ket{\down}_B-\ket{\down}_A\ket{\up}_B]/\sqrt{2},\ket{\up\down}_A\ket{0}_B,\ket{0}_A\ket{\up\down}_B\}$ with weights that are slowly varying functions of $t$, however, strongly dependent on $J$. A negligible $J$ yields that the two lowest singlets are almost equally weighted on the states $[\ket{\up}_A\ket{\down}_B-\ket{\down}_A\ket{\up}_B]/\sqrt{2}$ and $\ket{\up\down}_A\ket{0}_B$. Increasing $J>0$ redistributes the weights such that the lowest singlet ($\ket{2,4}$) acquires an increasing weight on $\ket{\up\down}_A\ket{0}_B$, whereas the second singlet ($\ket{2,5}$) becomes stronger weighted on $[\ket{\up}_A\ket{\down}_B-\ket{\down}_A\ket{\up}_B]/\sqrt{2}$. Hence, for a finite $J>0$ and $t\rightarrow0$, this redistribution leads to that transitions between the lowest one-electron states and $\ket{2,4}\ (\ket{2,5})$ occur with an enhanced (reduced) probability, e.g. $L_4^2\rightarrow1,\ (L_5^2\rightarrow0$), and oppositely for transitions between the singlets and the three-electron states, e.g. $\Lambda_4^2\rightarrow0,\ (\Lambda_5^2\rightarrow1/2$). This implies that $p\rightarrow0$ as $t\rightarrow0$ while $p(\kappa/\beta)^2$ remains almost constant. This constant, however, becomes larger (smaller) for smaller (larger) $J$.
![(Colour online) Variation of the triplet occupation number $N_\text{T}$ a) and the modulus of the current (units of $e\Gamma_0/h$) b) as function of the bias voltage and the equilibrium chemical potential $\mu$. Here, $\xi=0.01$, $k_BT=0.01U_A=4t$, and $J=0.2(U_A-U')/2$.[]{data-label="fig-NT"}](JV_NT_copper.eps){width="8.5cm"}
The typical variation of the triplet state occupation number $N_\text{T}$, calculated from Eq. (\[eq-dtN\]), as function of the bias voltage and the equilibrium chemical potential for $0<J<J_0-15t^2[1+(10t)^2]$ and $t/(k_BT)<2$ is plotted in Fig. \[fig-NT\] a). Here, varying the equilibrium chemical potential mimics the effect of applying an external gate voltage $V_g$ by means of which the levels of the DQD are shifted relatively the equilibrium chemical potential. The extended diamond marks the region where the occupation of the triplet is nearly unity and where the transport through the DQD is blockaded. The calculated current is displayed in Fig. \[fig-NT\] b), from which it is legible that the triplet blockade regime is subset of a larger domain of a nearly vanishing current through the DQD. The two diamonds within the low current regime are caused by a lifting of the triplet blockade (see the introduction), where the current is mediated by transitions between the one-electron states and the singlets.
As is seen in Fig. \[fig-NT\], shifting $\mu$ in the range $\dote{B}+(\Delta\dote{}-J,2\Delta\dote{})$ causes an extension of the low bias triplet regime since the transitions between the triplet and the one-electron states become resonant at higher biases. On the other hand, the non-equilibrium triplet blockade is shifted to lower biases since $\mu$ lies closer to the transition between the three-electron states and the singlets. The two blockade regimes merge into single as $|\mu-\Delta_{3n',2n}|<|\mu-\Delta_{21,1n'}|$, $n=4,5$, $n'=1,2$, e.g. for $\mu-\dote{B}\in(3\Delta\dote{}/2,2\Delta\dote{})$, see Fig. \[fig-NT\]. Shifting $\mu$ in the interval $\dote{B}+(\Delta\dote{}/2,\Delta\dote{}-J)$ removes the low bias blockade since the one-particle states become the equilibrium ground state. The non-equilibrium blockade is shifted to lower biases, here caused by transitions between the one- and two-electron states which tend to accumulate the occupation in the triplet.
While the case $\Delta\dote{}>0$ is considered here, the non-equilibrium blockade is also found in the opposite case, e.g. $\Delta\dote{}<0$ and $\mu-\dote{A}\approx\Delta\dote{}$. In this case, however, the system has to be gated such that only the four-electron state lies above $\mu$, whereas the charge accumulation of the triplet state is governed by the same processes as described here.
It should be noted that higher order effects, as well as singlet-triplet relaxation, have been neglected in the equation for the population probabilities $P_{Nn}$. However, in many aspects the situation discussed here corresponds to the experiment reported in Ref. , hence the effect considered should be measurable under much the same conditions. Therefore, as in the case of the serially coupled DQD, the higher order effects give contributions that are at least two orders of magnitude smaller than the second order contributions. Therefore, these can be neglected in the present study. On the same basis as in the description of the serially coupled DQD,[@franssoncm2005] the singlet-triplet relaxation may be neglected here.
The conditions required for the existence of non-equilibrium triplet blockade, concerning the intra- and interdot charge interactions for weakly coupled QDs, have been experimentally obtained for serially coupled QDs.[@ono2002; @rogge2004; @johnson2005] The additional requirement, i.e. a ferromagnetic interdot exchange interaction which is larger than the interdot tunnelling and the thermal excitation energy, is accessible within the present state-of-the-art technology.[@kouwenhoven2001; @vanbeveren2005; @johnson2005; @petta2005]
Support from Carl Trygger’s Foundation is acknowledged. The Institute of Physics and Deutsche Physikalische Gessellschaft is gratefully acknowledged for covering the publications costs.
[20]{} K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science, [**297**]{}, 1313 (2002). M. C. Rogge, C. Fühner, U. F. Keyser, and R. J. Haug, Appl. Phys. Lett. [**85**]{}, 606 (2004). A. C. Johnson, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. B, [**72**]{}, 165308 (2005). J. Fransson and M. Råsander, Phys. Rev. B, [**73**]{}, 205333 (2006). S. Bandyopadhyay, Phys. Rev. B, [**67**]{}, 193304 (2003). H. W. Liu, T. Fujisawa, T. Hayashi, and Y. Hirayama, Phys. Rev. B, [**72**]{}, 161305(R) (2005). M. Eto, T. Ashiwa, and M. Murata, J. Phys. Soc. Jap. [**73**]{}, 307 (2004). K. Ono and S. Tarucha, Phys. Rev. Lett. [**92**]{}, 256803 (2004). A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Nature, [**435**]{}, 925 (2005). S. I. Erlingsson, O. N. Jouravlev, and Y. V. Nazarov, Phys. Rev. B, [**72**]{}, 033301 (2005). F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, Science, [**309**]{}, 1346 (2005). J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. B, [**72**]{}, 161301(R) (2005). I. S. Sandalov, O. Hjortstam, B. Johansson, and O. Eriksson, Phys. Rev. B, [**51**]{}, 13987 (1995). T. Inoshita, K. Ono, and S. Tarucha, J. Phys. Soc. Jpn. Suppl. A, [**72**]{}, 183 (2003). E. Cota, R. Aguado, and G. Platero, Phys. Rev. Lett. [**94**]{}, 107202 (2005). There is one (spin-degenerate) transition energy $(\Delta_{3n,26},\ n=3,4)$ in the vicinity of $\mu$ which is only relevant for $eV\geq U$ since $\min_{n=1}^4\{\Delta_{26,1n}-\mu\}\gtrsim U/2$. A. -P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B, [**50**]{}, 5528 (1994). L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog. Phys. [**64**]{} 701 (2001). L. H. Willems van Beveren, R. Hanson, I. T. Vink, F. H. L. Koppens, L. P. Kouwenhoven, and L. M. K. Vandersypen, New Journal of Physics, [**7**]{} 182 (2005).
|
---
author:
- Liwei Song
- Reza Shokri
- Prateek Mittal
bibliography:
- 'refers.bib'
title: Privacy Risks of Securing Machine Learning Models against Adversarial Examples
---
<ccs2012> <concept> <concept\_id>10002978.10003022</concept\_id> <concept\_desc>Security and privacy Software and application security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
|
---
abstract: 'Deep neural networks have been successfully applied in many different fields like computational imaging, medical healthcare, signal processing, or autonomous driving. In a proof-of-principle study, we demonstrate that computational optical form measurement can also benefit from deep learning. A data-driven machine learning approach is explored to solve an inverse problem in the accurate measurement of optical surfaces. The approach is developed and tested using virtual measurements with known ground truth.'
author:
- 'Lara Hoffmann [^1]$\ $ and Clemens Elster'
bibliography:
- 'ref.bib'
date: 'Dated: '
title: Deep Neural Networks for Computational Optical Form Measurements
---
Introduction
============
Deep neural networks and machine learning in general are experiencing an ever greater impact on science and industry. Their application has proven beneficial in many different domains, including autonomous driving [@grigorescu], anomaly detection in quality management [@staar], computational imaging [@barbastathis], signal processing [@mousavi], analysis of raw sensor data [@moraru], or medical health care [@esteva], [@kretz]. Machine learning methods have also been successfully employed in optics. Examples comprise the compensation of lens distortions [@chung], or correcting abberated wave fronts in adaptive optics [@vdovin].
Machine learning has been used for misalignment corrections [@baermiss], [@zhang], aberration detection [@yan], or phase predicitons [@rivenson2]. But to the best of our knowledge, deep learning has not yet been applied for the accurate computational measurement of optical aspheres and freeform surfaces predicting the surface under test from its optical path length differences. The precise reconstruction of aspheres and freeform surfaces is currently limited by the accuracy of optical form measurements with an uncertainty range of approximately $50$ nm [@schachtschneider]. The aim of this paper is to demonstrate through a proof-of-principle study that this field in optics can also significantly benefit from machine learning techniques.
Our investigations were conducted using the SimOptDevice [@simopt] simulation toolbox. The toolbox provides realistic, virtual experiments with known ground truth. We concentrated on the tilted-wave interferometer (TWI) [@baer] for the experimental realization. It is a promising technique for the accurate computational measurement of optical aspheres and freeform surfaces using contact-free interferometric measurements. The TWI combines a special measurement setup with model-based evaluation procedures. Four CCD images with several interferograms are generated by using multiple light sources to illuminate the surface under test. A simplified scheme is shown in Figure \[twischeme\]. The test topography is then reconstructed by solving a numerically expensive nonlinear inverse problem by comparing the measured optical path length differences to simulated ones using a computer model and the known design of the surface under test. In this study, PTB’s realization of the TWI evaluation procedure is considered [@ptbtwi].
While the great success of deep networks is based on their ability to learn complex relations from data without knowing the underlying physical laws, including existing physical knowledge into the models can further improve results (cf. [@bezenac], [@karpatne] or [@raissi]). In our study, we also follow such an approach by developing a hybrid method which combines physical knowledge with data-driven deep neural networks. The employed scientific knowledge is twofold; training data is generated by physical simulations and a conventional calibration method is used to generalize the trained network to non-perfect systems.
This paper is organized as follows. Section $2$ briefly introduces neural networks and presents the proposed deep learning framework. The means of generating the training data and the details of training the network are explained, combining this approach with a conventional calibration method for better generalization. The results obtained for independent test data are then presented and discussed in Section $3$. Finally, some conclusions are drawn from our findings and possible future research is suggested.
Hybrid method
=============
This section provides an overview of deep neural networks and introduces the hybrid method that was developed which combines the TWI procedure with a data-driven deep learning approach. Without loss of generality, each specimen can be assumed to have a known design topography. The overall goal of form measurement is to determine the deviation $\Delta T$ of a specimen $T_s$ to the given design topography $T_d$, i.e. : $ T_s = T_d + \Delta T$. To this end, the TWI provides measurements of the optical path length differences $L_s$ of the specimen under test. Simultaneously, a computer model [@simopt] computes the optical path length differences $L$ of a given topography $T$. The inverse problem is to reconstruct the specimen topography $T_s = T_d + \Delta T$ from its measured optical path length differences $L_s$.
The conventional evaluation procedure of the TWI is numerically expensive and relies on linearization. A general advantage of neural networks is the ability to produce instant results once they are trained. Furthermore, it is interesting to explore whether deep learning could also improve the quality of the inverse reconstruction as a nonlinear approach. We aim to address the inverse problem described above using deep networks, i.e. by reconstructing a difference topography $\Delta T$ from given differences of optical path length differences $\Delta L = L_s - L_d$.
Data generation
---------------
When solving an inverse problem with neural networks, it is a common practice to generate data through physical simulations [@lucas], [@mccann]. Here, various difference topographies $\Delta T$ are generated through randomly chosen weighted Zernike polynomials. They are then added to a specific design topography at a fixed measurement position to create different virtual specimens. The sequence of Zernike polynomials yields an orthogonal basis of the unit disc and is a popular tool in optics to model wave fronts [@wang]. Following the forward pass, the computer model is used to compute optical path length differences of the design topography and the modeled specimens. Note that the data is generated assuming perfect system conditions, i.e. the computer model is undisturbed. An example can be seen in Figures \[difftopo\] and \[opdd\].
Data were generated for two different design topographies with about $22.000$ data points for each design. It should be noted, that $10$% of the data was used exclusively for testing and was not included in the network training.
Deep neural network architecture
--------------------------------
A simple, fully connected neural network with a single hidden layer is represented by a nonlinear function $f_\Phi : \mathbb{R}\rightarrow\mathbb{R}$ with parameters $\Phi = \lbrace \omega_i,b_i\in\mathbb{R}|i=1,\ldots,n\rbrace$, where $n$ is the number of neurons in the hidden layer. The univariate output of the network is modeled as $f_\Phi (x) = \sum_{i=1}^n\sigma (\omega_ix+b_i),\ x\in\mathbb{R},$ where $\sigma$ is a nonlinear activation function. In general, input and output are higher dimensional, and the architecture can become arbitrarily deep by adding more layers. Neural networks with high complexity are called “deep neural networks”. An example of this type of architecture is shown in Figure \[dnn\]. There, two outputs are predicted based on three given inputs after processing the information through several hidden layers. Also, different types of layers - convolutional layers [@cun], for example- can be used instead of fully connected ones. The network parameters can be optimized via backpropagation on the given training data by minimizing a chosen loss function between the predicted and known output.
At this point, it is important to consider the network as being an image-to-image regression function $f_\Phi$ which maps the differences of optical path length differences $\Delta L$ (see Fig. \[opdd\]) onto a difference topography $\Delta T$ (see Fig. \[difftopo\]), i.e.: $f_\Phi:\ \mathbb{R}^{M\times M\times K}\rightarrow\mathbb{R}^{M\times M},\ \Delta L\mapsto\Delta T,$ where $\Phi$ are the network parameters to be trained, $M\times M$ is the dimension of the images, and $K$ is the number of channels in the input. Note that the image dimension of the input equals the image dimension of the output here. This is not mandatory, but it suits the network architecture described below very well. While the CCD gives a resolution of $2048\times 2048$ pixels, we chose $M=64$. For the asphere and the multi-spherical freeform artefact [@multisphere] as seen in Figure \[opdd\], $K=4$ and $K=1$ were used, respectively. This is because the multi-spherical freeform artefact has a big patch in the first channel which almost covers the entire CCD for the selected measurement position. Furthermore, even though some pixels are missing, the first channel sufficed for the purpose of this deep learning proof-of-principle study.
We chose a U-Net as the network architecture. U-Nets have been successfully applied in various image-to-image deep learning applications [@isil]. An example of a structure is shown in Figure \[unet\]. The input passes through several convolution and rectified linear unit layers on the left side before being reduced in dimension in every vertical connection. After reaching its bottleneck at the bottom, the original data dimension is restored step by step through transposed convolution layers on the right side. During each dimensional increase step, a depth concatenation layer is added which links the data of the current layer to the data of the former layer with same dimension. These skip connections are depicted as horizontal lines in Figure \[unet\].
Here, the chosen U-Net architecture consists of a total of $69$ layers. The training set was used to normalize all input and output data prior to feeding them into the network. The U-Net was trained using an Adam optimizer [@adam] and the mean squared error as the loss function. About two hours of training were carried out for the multi-spherical freeform artefact with an initial learning rate of $0.0005$, a drop factor of $0.75$ every five periods, and a mini batch size of $64$. In addition, a dual norm regularization of the network parameters with a regularization parameter of $0.004$ was employed to stabilize the training. For the asphere, training was carried out for $15$ epochs with a mini batch size of eight samples, an initial learning rate of $0.0005$ which decreased every three epochs by a learning rate drop factor of $0.5$, and a regularization parameter of $0.0005$.
Generalization to non-perfect systems
-------------------------------------
In real world applications there no perfect systems exist. Thus, the computer model needs to be adapted phenomenologically. In the conventional calibration procedure, the beam path is calibrated through the computer model by using known, well fabricated spherical topographies at different measurement positions to compare the optical path length differences measured by the TWI and its computer model [@fortmeier].
It is not feasible to generate an entire new data base and train a new network after each system calibration when using deep networks. Nonetheless, the ultimate goal is to apply the trained deep network to real-world data. To this aim, we propose a hybrid method that trains the selected U-Net on data generated under perfect system conditions but also generalizes well to non-perfect systems by evaluating data derived through the conventional calibration method. A workflow chart of the hybrid method is shown in Figure \[hybrid\].
Results
=======
The following results are all based on simulated data. As mentioned above, two different design topographies are considered as mentioned above: an asphere and a multi-spherical freeform artefact. First, the results of data acquired from a perfect system environment are presented. The networks which were trained for the design topography of an asphere and a multi-spherical freeform artefact are addressed, respectively. Next, additional strategies which could improve the models are discussed as well. Finally, the application of the hybrid method which was developed is presented in a non-perfect system environment.
The topographies have a circle as the base area. Since the required input and output of the network are images, the area outside of the circle shape is defined with zeros which the network learns to predict. Nonetheless, only the difference topography pixels inside the circle shape are considered in the presented results.
Perfect system
--------------
About $2200$ samples were used for testing. They were not included in the training. First, the multi-spherical freeform artefact was considered as the design topography. Three randomly chosen prediction examples are shown in Figure \[exval\]. The root mean squared error of the U-Net predictions on the test set is $33$ nm. For comparison, the difference topographies in the test set have a total root mean squared deviation of $559$ nm. The median of the absolute errors of the U-Net is about $18$ nm, while the median of total absolute deviations in the test set is $428$ nm.
For the asphere as the design topography the root mean squared error is $102$ nm, while the test set has a root mean squared deviation of $589$ nm. The median of the absolute errors of the U-Net is $52$ nm and the median absolute deviation of the test set is $451$ nm for comparison. One possible explanation for the discrepancy in the accuracy of the predictions between the network for the asphere and multi-spherical freeform artefact as the design topographies is the following. The input of the respective U-Nets and their resulting architecture vary widely. As mentioned above, the network concerning the asphere has four input channels. These can be seen in Figure \[opdd\]. In each channel, various different areas are illuminated at the CCD, resulting in a distribution of information into different and smaller patches. The multi-spherical freeform artefact on the other side, illuminates one big circle shaped patch in the first channel for the selected measurement position. This channel, which contains most of the important information in a single patch, forms the only input to the corresponding network.
However, the results for the asphere can be improved further. One way to do so is to increase the amount of training data (cf. Fig. \[amounttestdata\]). As the input has four channels for the asphere, it seems natural that more data is needed for training than for the multi-spherical freeform artefact. A second approach is to use a network ensemble [@zhou] rather than a single trained network. To this end, $15$ U-Nets were trained from scratch and the ensemble output was taken as the mean of the ensemble predictions. The results are shown in Figure \[amounttestdata\]. In this way, the accuracy was already improved to a root mean squared error of $80$ nm using an ensemble of $15$ U-Nets, each trained from scratch on almost $28.000$ data points. It should be noted that a further improvement seems possible as the amount of data is crucial for training and the network’s architecture of the asphere is more complex due to more input channels.
Non-perfect system
------------------
In any real world application, no experiment is carried out under perfect system conditions. This motivated the idea of disturbing the perfect simulated forward pass and of generalizing the model to non-perfect systems. The network now needed to cope with data coming from a non-perfect TWI after having trained on a perfect simulation environment in the first stage. This was achieved by using a conventional calibration to determine the correct model of the interferometer.
Here, we focused on the multi-spherical freeform artefact as the design topography. Thirty difference topographies were randomly chosen from the former test set, i.e. not included in U-Net training. They had a total root mean squared deviation of $545$ nm and are ranged from $296$ nm to $6.1\ \mu$m in their absolute maximal deviation from peak to valley. The results are shown in Table \[calib\], where the same trained network was used for differently produced inputs. The root mean squared error of the network predictions was $30$ nm on the perfect TWI system. This increased to $538$ nm after having disturbed the TWI system. The trained network is incapable of predicting properly. However, the error can be reduced to $67$ nm by using a calibrated forward pass to produce the input data. Hence, our proposed hybrid method can also generalize to non-perfect systems.
------------ ------- -------- -------
**RMSE** 30 nm 538 nm 67 nm
**Median** 16 nm 298 nm 33 nm
------------ ------- -------- -------
: Root mean squared and median of absolute errors for the predictions of the same U-Net using different inputs. The perfect TWI system which was also used to generate the training data is in the first column, the disturbed TWI system without calibration is in the second column, and the hybrid method is in the third column.[]{data-label="calib"}
Conclusion
==========
The obtained results are promising and suggest that deep learning can be successfully applied in the context of computational optical form measurements. The presented results are based on simulated data only and they constitute a proof-of-principle study rather than a final method that is ready for application. An extensive comparison with conventional methods is the next step. Testing the approach on real measurements and accounting for fine-tuning (such as the calibration of the numerical model of the experiment) is reserved for future work as well. Nevertheless, these initial results are encouraging and once trained, a neural network solves the inverse problem orders of magnitudes much faster than the currently applied conventional methods. We conclude from our findings that computational optical form metrology can also greatly benefit from deep learning.
Acknowledgements
================
The authors would like to thank Manuel Stavridis for providing the SimOptDevice software tool and Ines Fortmeier and Michael Schulz for their helpful discussions about optical form measurements.
[^1]: Electronic address: `lara.hoffmann@ptb.de`; Corresponding author
|
---
abstract: 'The complex exponential function $e^z$ is a local homeomorphism and gives therefore rise to an étale groupoid and a $C^*$-algebra. We show that this $C^*$-algebra is simple, purely infinite, stable and classifiable by K-theory, and has both K-theory groups isomorphic to $\mathbb Z$. The same methods show that the $C^*$-algebra of the anti-holomorphic function $\overline{e^z}$ is the stabilisation of the Cuntz-algebra $\mathcal O_3$.'
address: 'Institut for matematiske fag, Ny Munkegade, 8000 Aarhus C, Denmark'
author:
- Klaus Thomsen
date:
-
-
title: 'The $C^*$-algebra of the exponential function'
---
Introduction
============
The crossed product of a locally compact Hausdorff space by a homeomorphism has been generalised to local homeomorphisms in the work of Renault, Deaconu and Anantharaman-Delaroche, [@Re],[@De], [@An]. In many cases the algebra is both simple and purely infinite, and can be determined by the use of the Kirchberg-Phillips classification result. The purpose with the present note is to demonstrate how methods and results about the iteration of complex holomorphic functions can be used for this purpose. This will be done by determining the $C^*$-algebra of an entire holomorphic function $f$ when $f'(z) \neq
0$ and $\# f^{-1}(f(z)) \geq
2$ for all $z \in \mathbb C$, and the Julia set $J(f)$ of $f$ is the whole complex plane $\mathbb C$. A prominent class of functions with these properties is the family $\lambda e^z$ where $\lambda >
\frac{1}{e}$. These functions commute with complex conjugation and we can therefore use the same methods to determine the $C^*$-algebra of the anti-holomorphic function $\overline{f}$. The results are as stated in the abstract for $f(z) = e^z$.
The $C^*$-algebra of a local homeomorphism
==========================================
The definition
--------------
We describe in this section the construction of a $C^*$-algebra from a local homeomorphism. It was introduced in increasing generality by J. Renault [@Re], V. Deaconu [@De] and Anantharaman-Delaroche [@An].
Let $X$ be a second countable locally compact Hausdorff space and $\varphi : X \to X$ a local homeomorphism. Set $$\Gamma_{\varphi} = \left\{ (x,k,y) \in X \times \mathbb Z \times X :
\ \exists n,m \in \mathbb N, \ k = n -m , \ \varphi^n(x) =
\varphi^m(y)\right\} .$$ This is a groupoid with the set of composable pairs being $$\Gamma_{\varphi}^{(2)} \ = \ \left\{\left((x,k,y), (x',k',y')\right) \in \Gamma_{\varphi} \times
\Gamma_{\varphi} : \ y = x'\right\}.$$ The multiplication and inversion are given by $$(x,k,y)(y,k',y') = (x,k+k',y') \ \text{and} \ (x,k,y)^{-1} = (y,-k,x)
.$$ Note that the unit space of $\Gamma_{\varphi}$ can be identified with $X$ via the map $x \mapsto (x,0,x)$. Under this identification the range map $r: \Gamma_{\varphi} \to X$ is the projection $r(x,k,y) = x$ and the source map the projection $s(x,k,y) = y$.
To turn $\Gamma_{\varphi}$ into a locally compact topological groupoid, fix $k \in \mathbb Z$. For each $n \in \mathbb N$ such that $n+k \geq 0$, set $${\Gamma_{\varphi}}(k,n) = \left\{ \left(x,l, y\right) \in X \times \mathbb
Z \times X: \ l =k, \ \varphi^{k+n}(x) = \varphi^n(y) \right\} .$$ This is a closed subset of the topological product $X \times \mathbb Z
\times X$ and hence a locally compact Hausdorff space in the relative topology. Since $\varphi$ is locally injective $\Gamma_{\varphi}(k,n)$ is an open subset of $\Gamma_{\varphi}(k,n+1)$ and hence the union $${\Gamma_{\varphi}}(k) = \bigcup_{n \geq -k} {\Gamma_{\varphi}}(k,n)$$ is a locally compact Hausdorff space in the inductive limit topology. The disjoint union $$\Gamma_{\varphi} = \bigcup_{k \in \mathbb Z} {\Gamma_{\varphi}}(k)$$ is then a locally compact Hausdorff space in the topology where each ${\Gamma_{\varphi}}(k)$ is an open and closed set. In fact, as is easily verified, $\Gamma_{\varphi}$ is a locally compact groupoid in the sense of [@Re], i.e. the groupoid operations are all continuous, and an étale groupoid in the sense that the range and source maps are local homeomorphisms.
To obtain a $C^*$-algebra, consider the space $C_c\left(\Gamma_{\varphi}\right)$ of continuous compactly supported functions on $\Gamma_{\varphi}$. They form a $*$-algebra with respect to the convolution-like product $$f \star g (x,k,y) = \sum_{z,n+ m = k} f(x,n,z)g(z,m,y)$$ and the involution $$f^*(x,k,y) = \overline{f(y,-k,x)} .$$ To obtain a $C^*$-algebra, let $x \in X$ and consider the Hilbert space $l^2\left(s^{-1}(x)\right)$ of square summable functions on $s^{-1}(x) = \left\{ (x',k,y')
\in \Gamma_{\varphi} : \ y' = x \right\}$ which carries a representation $\pi_x$ of the $*$-algebra $C_c\left(\Gamma_{\varphi}\right)$ defined such that $$\label{pirep}
\left(\pi_x(f)\psi\right)(x',k, x) = \sum_{z, n+m = k}
f(x',n,z)\psi(z,m,x)$$ when $\psi \in l^2\left(s^{-1}(x)\right)$. One can then define a $C^*$-algebra $C^*_r\left(\Gamma_{\varphi}\right)$ as the completion of $C_c\left(\Gamma_{\varphi}\right)$ with respect to the norm $$\left\|f\right\| = \sup_{x \in X} \left\|\pi_x(f)\right\| .$$ Since we assume that $X$ is second countable it follows that $C^*_r\left(\Gamma_{\varphi}\right)$ is separable. Note that this $C^*$-algebra can be constructed from any locally compact étale groupoid $\Gamma$ in the place of $\Gamma_{\varphi}$, see e.g. [@Re], [@An]. Note also that $C^*_r\left(\Gamma_{\varphi}\right)$ is the classical crossed product $C_0(X) \times_{\varphi}
\mathbb Z$ when $\varphi$ is a homeomorphism.
The generalised Pimsner-Voiculescu exact sequence
=================================================
There is a six-terms exact sequence which can be used to calculate the $K$-theory of $C^*_r\left(\Gamma_{\varphi}\right)$. It was obtained from the work of Pimsner, [@Pi], by Deaconu and Muhly in a slightly different setting in [@DM]. In particular, Deaconu and Muhly require $\varphi$ to be surjective and essentially free, but thanks to the work of Katsura in [@Ka] we can now establish it for arbitrary local homeomorphisms. This generalisation will be important here because $e^z$ is not surjective.
Consider the set $$\Gamma_{\varphi}(1,0) = \left\{ \left(x,1,y\right) \in
\Gamma_{\varphi}(1) : \ y =
\varphi(x) \right\} ,$$ which is an open subset of $\Gamma_{\varphi}(1)$ and hence of $\Gamma_{\varphi}$. Set $E_0 = C_c\left(\Gamma_{\varphi}(1,0)\right)$. Note that $f^*g \in C_c(\varphi(X))
\subseteq C_c\left(\Gamma_{\varphi}\right)$ when $f,g \in E_0$. In fact $$f^*g(x,k,y) = \begin{cases} 0, & \ k \neq 0 \ \vee \ x \neq y \ \vee \
x \notin \varphi(X) \\ \sum_{z \in
\varphi^{-1}(x)} \overline{f(z,1,x)}g(z,1,x), & \ k = 0 \ \wedge \ x =
y \in \varphi(X). \end{cases}$$ It follows that the closure $E$ of $E_0$ in $C^*_r\left(\Gamma_{\varphi}\right)$ is a Hilbert $C_0(X)$-module with an $C_0(X)$-valued inner product $\left<
\cdot, \ \cdot \right>$ defined such that $\left< f,g\right> = f^*g, \ f,g \in E$. Since $C_0(X) E \subseteq E$ we can consider $E$ as a $C^*$-correspondence over $C_0(X)$ in the obvious way, cf. Definition 1.3 of [@Ka]. Let $\iota :
C_0(X) \to C^*_r\left(\Gamma_{\varphi}\right)$ and $t : E \to
C^*_r\left(\Gamma_{\varphi}\right)$ be the inclusion maps. Then $(\iota,t)$ is an injective representation of the $C^*$-correspondence $E$ in the sense of Katsura, cf. Definitions 2.1 and 2.2 of [@Ka].
Let $\mathbb K(E)$ be the $C^*$-algebra of adjointable operators on $E$ generated by the elementary operators $\Theta_{f,g}, f,g \in E$, where $\Theta_{f,g}(k) = f \left<g,k\right>$.
\[rep\] $C_0(X) \subseteq \mathbb K(E)$.
Since $\Theta_{f,g}(k) = fg^*k$ when $f,g,k \in E$ it suffices to show that the elements of $C_0(X)$ of the form $fg^*$ for some $f,g \in E$ span a dense subspace of $C_0(X)$. Let $U$ be an open subset of $X$ where $\varphi$ is injective and consider a non-negative function $h \in C_c(X)$ supported in $U$. Then $$W = \left\{ (x,1,\varphi(x)) : \ x \in U \right\}$$ is an open subset of $\Gamma_{\varphi}(1,0)$ and we define $f \in E_0$ such that ${\operatorname{supp}}f \subseteq W$ and $f(x,1,\varphi(x)) = \sqrt{h(x)}, \ x
\in U$. Then $h = \Theta_{f,f}$ and we are done.
It follows that $(\iota, t)$ is covariant in the sense of [@Ka], cf. Proposition 3.3 and Definition 3.4 of [@Ka], and there is therefore an associated $*$-homomorphism $\rho : \mathcal O_E \to
C^*_r\left(\Gamma_{\varphi}\right)$.
\[cuntzpimsner\] $\rho : \mathcal O_E \to
C^*_r\left(\Gamma_{\varphi}\right)$ is an isomorphism.
By construction $C^*_r\left(\Gamma_{\varphi}\right)$ carries an action $\beta$ by the circle $\mathbb T$ defined such that $$\beta_{\lambda}(f)(x,k,y) = \lambda^k f(x,k,y)$$ when $f \in C_c\left(\Gamma_{\varphi}\right)$. This is the *gauge action*. This action ensures that $(\iota,t)$ admits a gauge action in the sense of [@Ka]. By Theorem 6.4 of [@Ka] it suffices therefore to show that $C^*_r\left(\Gamma_{\varphi}\right)$ is generated, as a $C^*$-algebra, by $C_0(X)$ and $E$.
Let $\mathcal A$ be the $*$-subalgebra of $C_c\left(\Gamma_{\varphi}\right)$ generated by $C_c(X)$ and $E_0$. Let $k \in \mathbb N$. We claim that $C_c\left(\Gamma_{\varphi}(k,0)\right) \in \mathcal A$. Since $C_c\left(\Gamma_{\varphi}(0,0)\right) = C_c(X)$ and $E_0 =
C_c\left(\Gamma_{\varphi}(1,0)\right)$ it suffices to prove this when $k \geq 2$. To this end it suffices, since $C_c(X) \subseteq \mathcal A$, to consider an open subset $U$ of $X$ on which $\varphi^k$ is injective and show that any non-negative continuous function $h$ compactly supported in $\left\{ (x,k,\varphi^k(x)) : \ x \in U\right\}$ is in $\mathcal A$. To this end, let $f_j \in E_0$ be supported in $$\left\{ (y,1,\varphi(y)) : \ y \in \varphi^{j-1}(U)\right\}$$ and satisfy that $f_j\left(\varphi^{j-1}(x),1,\varphi^{j}(x)\right) =
h(x,k,\varphi^k(x))^{\frac{1}{k}}$ for all $x \in U$. Then $h =
f_1f_2\cdots f_{k}$ and hence $h \in \mathcal A$.
We will next prove by induction that $C_c\left(\Gamma_{\varphi}(k,n)\right) \subseteq
\mathcal A$ for all $n \in \mathbb N$. The assertion holds when $n
=0$ as we have just shown, so assume that it holds for $n$. To show that $C_c\left(\Gamma_{\varphi}(k,n+1)\right) \subseteq \mathcal A$, let $U$ and $V$ be open subsets in $X$ such that $\varphi^n$ is injective on both $U$ and $V$. It suffices to consider a continuous function $h$ compactly supported in $\Gamma_{\varphi}(k,n+1) \cap (U \times \{k\}
\times V)$ and show that $h \in \mathcal A$. Note that $W =\Gamma_{\varphi}(k,n) \cap \left(\varphi(U) \times \{k\} \times
\varphi(V)\right)$ is open in $\Gamma_{\varphi}(k,n)$ and that we can define $\tilde{h} : W \to \mathbb C$ such that $\tilde{h}(x,k,y) =
h\left(x',k,y'\right)$, where $x'\in U, y' \in V$ and $\varphi(x') =x, \
\varphi(y') = y$. Then $\tilde{h}$ is continuous and has compact support in $W$; in fact, the support is the image of the support, $K$, of $h$ under the continuous map $\Gamma_{\varphi}(k,n+1) \ni (x,k,y) \mapsto
(\varphi(x),k,\varphi(y)) \in \Gamma_{\varphi}(k,n)$. Hence $\tilde{h}
\in \mathcal A$ by assumption. Note that $r(K)$ is a compact subset of $U$ and $s(K)$ a compact subset of $V$. Let $a \in C_c(X)$ be supported in $U$ such that $a(x) = 1, \ x \in r(K)$, and $b \in C_c(X)$ be supported in $V$ such that $b(x) = 1, \ x \in s(K)$. Define $\tilde{a},\tilde{b} \in C_c\left(\Gamma_{\varphi}(1,0)\right) =E_0$ with supports in $\left\{ (x,1,\varphi(x)) : \ x \in U\right\}$ and $\left\{ (x,1,\varphi(x)) : \ x \in V\right\}$, respectively, such that $\tilde{a}(x,1,\varphi(x)) =
a(x)$ when $x \in U$ and $\tilde{b}(x,1,\varphi(x)) = b(x)$ when $x
\in V$. Since $h = \tilde{a}\tilde{h}\tilde{b}^*$ we conclude that $h
\in \mathcal A$. Thus $C_c\left(\Gamma_{\varphi}(k,n)\right) \subseteq \mathcal A$ for all $k
\geq 0, n \geq 0$. Since $C_c\left(\Gamma_{\varphi}(-k,n)\right)^* =
C_c\left(\Gamma_{\varphi}(n,n-k)\right)$ when $n \geq k \geq 0$, we conclude that $\mathcal A =
C_c\left(\Gamma_{\varphi}\right)$.
By combining Proposition \[cuntzpimsner\] and Lemma \[rep\] with Theorem 8.6 of [@Ka] we obtain the following.
\[6terms\] (Deaconu and Muhly, [@DM]) Let $[E] \in KK\left(C_0(X),C_0(X)\right)$ be the element represented by the embedding $C_0(X) \subseteq \mathbb
K(E)$. There is an exact sequence $$\begin{xymatrix}{
K_0\left(C_0(X)\right) \ar[r]^-{{\operatorname{id}}_* - [E]_*} & K_0\left(C_0(X)\right)
\ar[r]^-{\iota_*} & K_0\left(C^*_r\left(\Gamma_{\varphi}\right)\right) \ar[d]
\\
K_1\left(C^*_r\left(\Gamma_{\varphi}\right)\right) \ar[u] &
K_1\left(C_0(X)\right) \ar[l]^{\iota_*} & K_1\left(C_0(X)\right) \ar[l]^-{{\operatorname{id}}_* - [E]_*}}
\end{xymatrix}$$
Simple purely infinite $C^*$-algebras from entire functions without critical points in the Julia set
====================================================================================================
Throughout this section $f : \mathbb C \to \mathbb C$ is an entire function of degree at least 2; i.e. either a polynomial of degree at least 2 or a transcendental function. An $n$-periodic point $z \in \mathbb C$ is *repelling* when $\left|(f^n)'(z)\right| > 1$. The *Julia set* $J(f)$ of $f$ can then be defined as the closure of the repelling periodic points. Although this is not the standard definition it emphasises one of the properties that will be important here. Others are
1. $J(f)$ is non-empty and perfect, and
2. $J(f)$ is totally $f$-invariant, i.e. $f^{-1}(J(f)) = J(f)$.
We refer to the survey by Bergweiler, [@Be], for the proof of these properties.
Let $\mathcal E(f)$ denote the set of points $x \in \mathbb C$ such that $f^{-1}(x) = \{x\}$. For example, when $f(z) = 2ze^z$ the point $0$ will be in $\mathcal E(f) \cap J(f)$.
\[1\] $\# \mathcal E(f) \leq 1$.
Let $x,y \in \mathcal E(f)$ and assume for a contradiction that $x \neq y$. Since $J(f)$ is infinite, $J(f)
\backslash \{x,y\}$ is not empty. Let $U$ be an open subset of $\mathbb C$ such that $$\label{nonint}
U \cap \left(J(f) \backslash \{x,y\}\right) \neq \emptyset .$$ Then $\bigcup_{i=0}^{\infty} f^i\left( U \backslash
\{x,y\}\right)$ is open, non-empty, $f$-invariant and does not contain $\{x,y\}$. It follows therefore from Montel’s theorem, cf. Theorem 3.7 in [@Mi], that $f^n, n \in \mathbb N$, is a normal family when restricted to $\bigcup_{i=0}^{\infty} f^i\left( U \backslash
\{x,y\}\right)$. This contradicts (\[nonint\]) since $U \backslash
\{x,y\}$ contains a repelling periodic point.
The set $J(f) \backslash \mathcal E(f)$ is locally compact in the relative topology inherited from $\mathbb C$ and $f^{-1}\left(J(f)
\backslash \mathcal E(f)\right) = J(f) \backslash \mathcal E(f)$. If we now assume that $f'(z) \neq 0$ when $z \in J(f)$, it follows that the restriction $$F : J(f) \backslash \mathcal E(f) \to J(f) \backslash \mathcal E(f)$$ of $f$ to $J(f) \backslash \mathcal E(f)$ is a local homeomorphism on the second countable locally compact Hausdorff space $J(f) \backslash
\mathcal E(f)$. Note that $F$ is not always surjective - it is not when $f(z) = e^z$.
Following [@An] we say that an étale groupoid $\Gamma$ with range map $r$ and source map $s$ is *essentially free* when the points $x$ of the unit space $\Gamma^0$ for which the isotropy group $s^{-1}(x) \cap r^{-1}(x)$ is trivial (i.e. only consists of $\{x\}$) is dense in $\Gamma^0$. For the groupoid $\Gamma_f$ of a local homeomorphism $f : X \to X$ this occurs if and only if $\left\{ x \in X : \ f^i(x) = x\right\}$ has empty interior for all $i \in \mathbb N$.
We say that $\Gamma$ is *minimal* when there is no open non-empty subset $U$ of $\Gamma^0$, other than $\Gamma^0$, which is $\Gamma$-invariant in the sense that $r(\gamma) \in U \Leftrightarrow s(\gamma) \in U$ for all $\gamma \in
\Gamma$. This holds for the groupoid $\Gamma_f$ if and only if the full orbit $\bigcup_{i,j\in \mathbb N} f^{-i}(f^j(x))$ is dense in $X$ for all $x \in X$.
Finally, we say that $\Gamma$ is *locally contracting* when every open non-empty subset of $\Gamma^0$ contains an open non-empty subset $V$ with the property that there is an open bisection $S$ in $\Gamma$ such that $\overline{V} \subseteq s(S)$ and $\alpha_{S}^{-1}\left(\overline{V}\right) \subsetneq V$ when $\alpha_S
: r(S) \to s(S)$ is the homeomorphism defined by $S$, cf. Definition 2.1 of [@An] (but note that the source map is denoted by $d$ in [@An]).
\[julia\] Assume that $f'(z) \neq 0$ for all $z \in
J(f)$. Then $\Gamma_{F}$ is minimal, essentially free and locally contracting in the sense of [@An].
To show that $\Gamma_F$ is essentially free we must show that $$\label{eq12}
\left\{ z \in J(f) \backslash \mathcal E(f) : \ F^i(z) = z \right\}$$ has empty interior in $J(f) \backslash \mathcal E(f)$ for all $i
\in \mathbb N$. Assume that $U$ is open in $\mathbb C$ and that $U \cap J(f)
\backslash \mathcal E(f)$ is a non-empty subset of (\[eq12\]). Since $J(f)$ is perfect it follows that every point $z_0$ of $U \cap J(f)
\backslash \mathcal E(f)$ is the limit of a sequence from $$\label{eq13}
\left\{ z \in \mathbb C: \ f^i(z) = z \right\} \backslash \{z_0\}.$$ Since $f$ is entire it follows that $f^i(z) = z$ for all $z \in
\mathbb C$, contradicting that $J(f) \neq \emptyset$. Hence $\Gamma_F$ is essentially free.
To show that $\Gamma_F$ is minimal, consider an open subset $U
\subseteq \mathbb C$ such that $$\label{eq14}
U \cap
J(f) \backslash \mathcal E(f) \neq \emptyset.$$ Let $W = \bigcup_{i,j \in \mathbb N}
f^{-j}\left( f^i(U \backslash \mathcal E(f))\right)$. Since $W$ is open (in $\mathbb C$), non-empty, $f$-invariant and has non-trivial intersection with $J(f)$ it follows from Montel’s theorem, cf. Theorem 3.7 in [@Mi], that $\mathbb C \backslash W$ contains at most one element. Note that this element must be in $\mathcal E(f)$ because $W$ and hence also $\mathbb C \backslash W$ is totally $f$-invariant. It follows therefore that $W \cap J(f) \backslash
\mathcal E(f) = J(f) \backslash
\mathcal E(f)$. Hence $$\bigcup_{i,j \in \mathbb N}
f^{-j}\left( f^i(U \cap J(f) \backslash \mathcal E(f))\right) = W
\cap J(f) = J(f) \backslash
\mathcal E(f) ,$$ proving that $\Gamma_F$ is minimal.
To show that $\Gamma_F$ is locally contracting consider an open subset $U$ of $\mathbb C$ such that $U \cap J(f) \backslash \mathcal E(f)
\neq \emptyset$. There is then a repelling periodic point $z_0 \in U
\cap J(f) \backslash \mathcal E(f)$. There is therefore an $n \in
\mathbb N$, a positive number $\kappa > 1$ and an open neighbourhood $W
\subseteq U$ of $z_0$ such that $f^n(z_0) =
z_0$, $f^n$ is injective on $W $ and$$\label{eu1}
\left|f^n(y) -z_0\right| \geq \kappa |y -z_0|$$ for all $y \in W$. Let $\delta_0 > 0$ be so small that $$\label{eu2}
\left\{y \in \mathbb C : |y - z_0| \leq \delta_0 \right\} \subseteq
f^n(W) \cap W .$$ $z_0$ is not isolated in $J(f) \backslash \mathcal E(f)$ since $J(f)$ is perfect. There is therefore an element $z_1\in J(f) \backslash
\mathcal E(f)$ such that $0 < \left|z_1-z_0\right| < \delta_0$. Choose $\delta$ strictly between $\left| z_1 -z_0\right|$ and $\delta_0$ such that $$\label{cru}
\kappa
\left|z_1-z_0\right| > \delta.$$ Set $V_0 = \left\{y \in \mathbb C : |y -z_0| < \delta \right\}$. Then $$\label{eu102}
\overline{V_0} \cap J(f) \backslash
\mathcal E(f) \subsetneq f^n\left({V_0} \cap J(f) \backslash
\mathcal E(f)\right).$$ Indeed, if $\left|y-z_0\right| \leq \delta$ (\[eu2\]) implies that there is a $y' \in W$ such that $f^n(y') = y$ and then (\[eu1\]) implies that $\left|y' - z_0\right|
< \delta$. Since $y' \in J(f)\backslash \mathcal E(f)$ when $y \in
J(f)\backslash \mathcal E(f)$ it follows that $\overline{V_0} \cap J(f) \backslash
\mathcal E(f) \subseteq f^n\left({V_0} \cap J(f) \backslash
\mathcal E(f)\right)$. On the other hand, it follows from (\[cru\]) and (\[eu1\]) that $f^n(z_1) \notin \overline{V_0}$. This shows that (\[eu102\]) holds. Then $$S = \left\{ (z,n,f^n(z)) \in \Gamma_F(n,0) : \ z \in V_0 \right\}$$ is an open bisection in $\Gamma_F$ such that $\overline{V_0} \cap J(f) \backslash
\mathcal E(f) \subseteq s(S)$ and $$\alpha_{S^{-1}} \left(\overline{V_0} \cap J(f) \backslash
\mathcal E(f)\right) \subsetneq {V_0} \cap J(f) \backslash
\mathcal E(f).$$ Then $V = V_0 \cap J(f) \backslash \mathcal
E(f)$ is an open subset of $U \cap J(f) \backslash \mathcal
E(f)$ such that $\alpha_{S^{-1}}(\overline{V}) \subsetneq V$. This shows that $\Gamma_F$ is locally contracting.
\[simpleinf\] The $C^*$-algebra $C^*_r\left(\Gamma_F\right)$ is simple and purely infinite.
By Theorem 4.16 of [@Th] simplicity is a consequence of the minimality and essential freeness of $\Gamma_F$. Pure infiniteness follows from Proposition 2.4 in [@An] because $\Gamma_F$ is essentially free and locally contracting.
The $C^*$-algebra of the exponential function
=============================================
For the statement of the next theorem, which is the main result of the note, recall that the separable, stable, simple purely infinite $C^*$-algebras which satisfy the universal coefficient theorem (UCT) of Rosenberg and Schochet, [@RS], is exactly the class of $C^*$-algebras known from the Kirchberg-Phillips results, [@Ph], to be classified by their $K$-theory groups alone.
\[main\] Let $f : \mathbb C \to \mathbb C$ be an entire transcendental function such that
1. $f'(z) \neq 0 \ \forall z \in \mathbb C$,
2. the Julia set $J(f)$ of $f$ is $\mathbb C$, and
3. $\# f^{-1}(f(x)) \geq 2$ for all $x \in \mathbb C$.
Then $C^*_r\left(\Gamma_f\right)$ is the separable stable simple purely infinite $C^*$-algebra which satisfies the UCT, and $K_0(C^*_r\left(\Gamma_f\right))
\simeq K_1(C^*_r\left(\Gamma_f\right)) \simeq \mathbb Z$.
First observe that $f$ is a local homeomorphism because it is holomorphic with no critical points by assumption i). Hence $C^*_r\left(\Gamma_f\right)$ is defined. As we pointed out above the separability of $C^*_r\left(\Gamma_f\right)$ follows because $\mathbb C$ has a countable base for its topology. It follows from Proposition \[cuntzpimsner\], Lemma \[rep\] and Proposition 8.8 of [@Ka] that $C^*_r\left(\Gamma_f\right)$ satisfies the UCT. Since iii) implies that $\mathcal E(f) = \emptyset$ it follows from Corollary \[simpleinf\] that $C^*_r\left(\Gamma_f\right)$ is simple and purely infinite. Since $C^*_r\left(\Gamma_f\right)$ is not unital (because $\mathbb C$ is not compact), it follows from Theorem 1.2 of [@Z] that $C^*_r\left(\Gamma_f\right)$ is stable.
It remains now only to calculate the $K$-theory of $C^*_r\left(\Gamma_f\right)$. We use Theorem \[6terms\] for this and we need therefore to determine the action on $K$-theory of the $KK$-element $[E]$. Let $\Delta$ be a small open disc centered at $0 \in \mathbb C$ such that $f$ is injective on $\overline{\Delta}$. Set $V = f(\Delta)$ and let $i : C_0(\Delta) \to
C_0(\mathbb C)$ and $j : C_0(V) \to C_0(\mathbb C)$ denote the natural embeddings. Define $\psi_f : C_0(\Delta) \to C_0(V)$ such that $\psi_f(g) =
g \circ f^{-1}$. It is easy to see that $$i^*[E] = j_*[\psi_f] = [j \circ \psi_f]$$ in $KK\left(C_0(\Delta),C_0(\mathbb C)\right)$. To proceed we apply Schoenfliess’ theorem to get a homeomorphism $F : \mathbb C\to \mathbb C$ extending $f : U \to
V$. Note that $F$ must be orientation preserving since $f$ is. It follows therefore that $F$ is isotopic to the identity, cf. Theorem 2.4.2 on page 92 in [@L]. This shows that $j \circ \psi_f$ is homotopic to $i$ and we conclude therefore that $[j \circ \psi_f] =
[i] = i^*\left[{\operatorname{id}}_{C_0(\mathbb C)}\right]$. Since $i^* :
KK\left(C_0(\mathbb C),C_0(\mathbb C)\right) \to
KK\left(C_0(\Delta),C_0(\mathbb C)\right)$ is an isomorphism it follows that $[E] = \left[{\operatorname{id}}_{C_0(\mathbb C)}\right]$. The conclusion that $K_0(C^*_r\left(\Gamma_f\right))
\simeq K_1(C^*_r\left(\Gamma_f\right)) \simeq \mathbb Z$ follows now straightforwardly from the generalised Pimsner-Voiculescu exact sequence of Theorem \[6terms\].
The function $f(z) = \lambda e^z$ clearly satisfies assumptions i) and iii) of Theorem \[main\] when $\lambda \neq 0$. Furthermore, when $\lambda > \frac{1}{e}$ it is shown in [@De] that also assumption ii) holds, extending the result of Misiurewics, [@M], dealing with the case $\lambda
= 1$.
The $C^*$-algebra of $e^{\overline{z}}$
=======================================
Let $\mathbb K$ be the $C^*$-algebra of compact operators on an infinite-dimensional separable Hilbert space.
\[main2\] Let $f : \mathbb C \to \mathbb C$ be an entire transcendental function such that
1. $f'(z) \neq 0 \ \forall z \in \mathbb C$,
2. the Julia set $J(f)$ of $f$ is $\mathbb C$,
3. $\# f^{-1}(f(x)) \geq 2$ for all $x \in \mathbb C$, and
4. $\overline{f(z)} = f\left(\overline{z}\right), \ z \in
\mathbb C$.
Define $\overline{f} : \mathbb C \to \mathbb C$ such that $\overline{f}(z) = \overline{f(z)}$. Then $C^*_r\left(\Gamma_{\overline{f}}\right) \simeq \mathcal O_3
\otimes \mathbb K$ where $\mathcal O_3$ is the Cuntz-algebra with $K_0(\mathcal O_3) \simeq \mathbb Z_2$ and $K_1\left(\mathcal O_3\right) =
0$, cf. [@C].
$C^*_r\left(\Gamma_{\overline{f}}\right)$ is separable and satisfies the UCT for the same reason that $C^*_r\left(\Gamma_{{f}}\right)$ has these properties. Since $J(f^2)
= J(f) = \mathbb C$ and $\mathcal E(f^2) = \emptyset$ by iii) we conclude from Lemma \[julia\] that $\Gamma_{f^2}$ is minimal, essentially free and locally contracting. Since $\Gamma_{f^2}
\subseteq \Gamma_{\overline{f}}$ it follows that $\Gamma_{\overline{f}}$ is minimal and locally contracting. Furthermore, by using that $$\left\{ z \in \mathbb C : \ \overline{f}^i(z) = z \right\} \subseteq
\left\{ z \in \mathbb C : \ f^{2i}(z) = z\right\},$$ it follows also that $\Gamma_{\overline{f}}$ is essentially free because $\Gamma_{f^2}$ is. As in the proof of Corollary \[simpleinf\] we conclude now that $C^*_r\left(\Gamma_{\overline{f}}\right)$ is simple and purely infinite. Finally, since $\overline{f}$ is orientation reversing the calculation of the $K$-theory in the proof of Theorem \[main\] now yields the conclusion that $[E] = - \left[{\operatorname{id}}_{C_0(\mathbb C)}\right]$, leading to the result that $K_0\left(C^*_r\left(\Gamma_{\overline{f}}\right)\right) \simeq
\mathbb Z_2$ while $K_1\left(C^*_r\left(\Gamma_{\overline{f}}\right)\right) = 0$. Hence the theorem of Zhang, [@Z], and the Kirchberg-Phillips classification theorem, Theorem 4.2.4 of [@Ph], imply that $C^*_r\left(\Gamma_{\overline{f}}\right) \simeq \mathcal O_3 \otimes
\mathbb K$.
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|
---
abstract: 'Tumblr is one of the largest and most popular microblogging website on the Internet. Studies shows that due to high reachability among viewers, low publication barriers and social networking connectivity, microblogging websites are being misused as a platform to post hateful speech and recruiting new members by existing extremist groups. Manual identification of such posts and communities is overwhelmingly impractical due to large amount of posts and blogs being published every day. We propose a topic based web crawler primarily consisting of multiple phases: training a text classifier model consisting examples of only hate promoting users, extracting posts of an unknown tumblr micro-blogger, classifying hate promoting bloggers based on their activity feeds, crawling through the external links to other bloggers and performing a social network analysis on connected extremist bloggers. To investigate the effectiveness of our approach, we conduct experiments on large real world dataset. Experimental results reveals that the proposed approach is an effective method and has an F-score of $0.80$. We apply social network analysis based techniques and identify influential and core bloggers in a community.'
author:
- |
Swati Agarwal\
\
Ashish Sureka\
\
\
title: 'Spider and the Flies : Focused Crawling on Tumblr to Detect Hate Promoting Communities'
---
Introduction
============
Tumblr is the second largest microblogging platform, has gained phenomenal momentum recently. It is widely used by fandoms: communities of users having similar interests in various TV shows and movies [@renwick2014audience]. Therefore, it is especially popular among young generation users and provides them a platform to discuss daily events. They communicate by blogging and publishing GIF images as their reactions and emotions on several topics [@Bourlai:2014:MCT:2615569.2615697]. According to Tumblr statistics $2015$ [^1], over $219$ million blogs are registered on Tumblr and $420$ million are the active users. $80$ million posts are being published everyday, while the number of new blogs and subscriptions are $0.1$ million and $45$ thousands respectively.\
Tumblr is also posed as a social networking website that facilitates users to easily connect to each other by following other users and blogs without having a mutual confirmation. Bloggers can also communicate via direct messages that can be sent privately or can be posted publicly using ’ask box’. It facilitates bloggers to send these messages anonymously if they don’t want to reveal their Tumblr identity [@marquart2010microblog]. Similar to other social networking websites, Tumblr has very low publication barriers. A blogger can publish a new post and can re-blog an existing public post which is automatically broadcasted to it’s followers unless it is enabled as a private post [@DBLP:journals/corr/ChangTIL14]. The type of posts can be chosen among seven different categories including multi-media and other content: text, quote, link, photo, audio, video and URL. Unlike Twitter, Tumblr has no limit on the length of textual posts. Similar to hashtags in Twitter, there are separated tags associated with the blog content that make a post easier to be searchable on Tumblr [^2]. Tumblr also allows users to update their other connected social networking profiles when something is posted.
The simplicity of navigation, high reachability across wide range of viewers, low publication barriers, social networking and anonymity has led users to misuse Tumblr in several ways. Previous studies shows that these features of Tumblr are exclusive factors to gain the attention of modern extremist groups [@bates2014psychological][@sureka2014learning]. This is because Tumblr provides every kind of multimedia posts which is a great medium to share your views with your target audiences. These groups form their own communities that share a common propaganda. They post rude comments against a religion to express their hatred and spread extremist content. Social networking facilitates these groups to recruit more people to promote their beliefs and ideology among global audiences [@decary2011gang][@mahmood2012online]. Figure \[tumblr\_post\_example\] illustrates a concrete example of various types of hate promoting posts and their associated tags on Tumblr. The number of notes shows the number of times that post has been liked and re-blogged by other blogs.\
Online-radicalization and posting hateful speech is a crime against the humanity and mainstream morality; it has a major impact on society [@decary2011gang]. Presence of such extremist content on social media is a concern for law enforcement and intelligence agencies to stop such promotion in country as it poses the threat to the society [@Agarwal2015][@devore2012exploring]. It also degrades the reputation of the website and therefore is a concern for website moderators to identify and remove such communities. Due to the dynamic nature of website, automatic identification of extremist posts and bloggers is a technically challenging problem [@wilner2011transformative]. Tumblr is a large repository of text, pictures and other multimedia content which makes it impractical to search for every hate promoting post using keyword based flagging. The textual posts are user generated data that contain noisy content such as spelling, grammatical mistakes, presence of internet slangs and abbreviations. Presence of low quality content in contextual metadata poses technical challenges to text mining and linguistic analysis [@Agarwal2015][@marquart2010microblog]. The work presented in this paper is motivated by the need of investigating solutions to counter and combat the online extremism on Tumblr.\
The research aim of the work presented in this paper is the following:
1. To investigate the application of topical crawling based algorithm for retrieving hate promoting bloggers on Tumblr. Our aim is to examine the effectiveness of random walk in social network graph graph traversal and measuring its performance.
2. To investigate the effectiveness of contextual metadata such as content of the body, tags and caption or title of a post for computing the similarity between nodes in graph traversal. To examine the effectiveness of *re-blogging* and *like* on a post as the links between two bloggers.
3. To conduct experiments on large real world dataset and demonstrate the effectiveness of proposed approach in order to locate virtual and hidden communities of hate and extremism promoting bloggers and apply Social Network Analysis based techniques to locate central and influential users.
Literature Survey
=================
In this section, we discuss closely related work to the study presented in this paper. Based on our review of existing work, we observe that most of the researches for detecting online radicalization are performed on Twitter, YouTube and various discussion forums [@agarwal2015applying]. We conduct a literature survey in the area of identifying hate promoting communities on social networking websites and short text classification of Tumblr microblog . O’Callaghan et. al. [@o2013uncovering] describe an approach to identify extreme right communities on multiple social networking websites. They use Twitter as a possible gateway to locate these communities in a wider network and track dynamic communities. They perform a case study using two different datasets to investigate English and German language communities. They implement a heterogeneous network within a homogeneous network and use four different social networking platforms (Twitter accounts, Facebook profiles, YouTube channels and all other websites) as extreme right entities or peers and edges are the possible interactions among these accounts.\
Mahmood S. [@mahmood2012online] describes several mechanisms that can be useful in order to detect presence of terrorists on social networking websites by analyzing their activity feeds. They use Google search and monitor terror attack using keyword-based flagging mechanism. They monitor sentiments and opinions of users following several terrorism groups on on-line social networks and propose a counter-terrorism mechanism to identify those users who are more likely to commit a violent act of terror. They also discuss honeypots and counter-propaganda techniques that can be used to rehabilitate radicalized users back to normal users. The disadvantage of keyword based flagging approach is the large number of false alarms. David and Morcelli [@decary2011gang] present a keyword based search to detect several criminal organizations and gangs on Twitter & Facebook. They discuss a study of analyzing the presence of organized crime and how these gangs use social media platforms to recruit new members, broadcast their messages and coordinate their illegal activities on web $2.0$. They perform a qualitative analysis on $28$ groups and compare their organized crime between $2010$ and $2011$ on Facebook.\
Agarwal et. al. propose a one-class classification model to identify hate and extremism promoting tweets [@sureka2014learning]. They conducted a case study on Jihad and identified several linguistic and stylistic features from free form text such as presence of war, religious, negative emotions and offensive terms. They conduct experiments on large real world dataset and demonstrate a correlation between hate promoting tweets and discriminatory features. They also perform a leave-p-out strategy to examine the influence of each feature on classification model.\
In context to existing work, the study presented in this paper makes the following unique contributions extending our previous work [@agarwal2015topical]:
1. We present an application of topical crawler based approach for locating extremism promoting bloggers on Tumblr. While there has been work done in the area of topical based crawling of social media platforms, to the best of our knowledge this paper is the first study on topical crawling for navigating connections between Tumblr bloggers.
2. We conduct experiments on large real world dataset to demonstrate the effectiveness of one class classifier and filtering hate promoting blog posts (text). We retrieve Tumblr blogger profiles and their links with other hate promoting bloggers and apply Social Network Analysis to locate strongly connected communities and core bloggers.
Experimental Setup {#experimental_setup}
==================
We conduct our experiments on an open source and real time data extracted from Tumblr micro-blogging website. In a social networking website, a topical crawler extracts the external link to a profile and returns the nodes that are relevant to a defined topic. We define the relevance of a node based on extent of similarity of it’s activity feeds and training document. Topical crawler learns the features and characteristics from these training documents and classify a profile to be relevant. Figure \[exemplary\_doc\] illustrates the general framework to obtain these documents. As shown in Figure \[exemplary\_doc\], we implement a bootstrapping methodology to collect the training samples. We perform a manual search on Tumblr and create a lexicon of popular and commonly used tags associated with hate promoting posts. Figure \[cloud\_terms\] shows a word cloud of such terms. To collect our training samples, we perform a keyword (search tag) based flagging and extract their associated textual posts. We also acquire the related tags and the linked profiles (users who made these posts). We expand our list of keywords by extracting associated tags from these posts and their related tags. We run this framework iteratively until we get a reasonable number of exemplary documents ($400$ training samples). As mentioned above, we train our classifier for only hate promoting users. Therefore, the training documents contain the content and caption of only extremist posts.\
We use these linked bloggers and posts to compute the threshold value for language modeling. We take a sample of $30$ bloggers and compare their posts with the exemplary documents. For each blogger we get a relevance score. To compute the threshold value for similarity computation we take an average of these scores. Figure \[relscore\_threshold\] illustrates the relevance score statistics of each blogger (Sorted in increasing order). We notice that $80\%$ of the bloggers have relevance score between $-2.7$ and $-1.5$. We take average (turns out to be $-2.58$) of these scores to avoid the under-fitting and over-fitting of bloggers during classification.
Research Methodology {#methodology}
====================
In this section, we present the general research framework and methodology of proposed approach for classifying extremist bloggers on Tumblr (refer to Figure \[framework\]). The proposed approach is an iterative multi-step that uses a hate promoting blogger as a seed channel and results a connected graph where nodes represents the extremist bloggers and links represents the relation between two bloggers (like and re-blog). As shown in Figure \[framework\], proposed framework is a multi-step process primarily consists of four phases: i) extraction of activity feeds of a blogger, ii) training a text classification model and filtering hate promoting and unknown bloggers, iii) navigating through external links to bloggers and extracting linked frontiers and iv) traversing through spider network for selecting next blogger. In Phase $1$, we use a positive blogger *$U_{i}$* (annotated as hate promoting during manual inspection) called as ’seed’. We extract $n$ number of textual posts (either re-blogged or newly posted by user)of *$U_{i}$* by using Tumblr API [^3]. We further use Jsoup Java library [^4] to extract the content and caption of these posts. Tumblr allows users to post content in multiple languages. However, our focus of this paper is to mine only English language posts. Therefore, we perform data-preprocessing on all extracted posts and by using Java language detection library [^5], we filter all non-English language posts. In Phase $2$, we train our classification model over training samples (refer to Section \[experimental\_setup\]). We perform character level n-gram language modeling[^6] on English language posts and compute their extent of similarity against training samples. We classify a blogger as hate promoting based on the relevance score and computed threshold value (refer to Section \[experimental\_setup\]).\
If a channel is classified as hate promoting or relevant, we further proceed to Phase $3$ and extract the notes information for each posts (collected in phase $1$). Notes in a Tumblr post contains the information about bloggers who liked or re-blogged a post. These user hits on a post indicates the similar interest among bloggers who may or may not be direct followers of each other. We extract the Tumblr ids of profiles from notes for the following reasons: i) due to privacy policy, Tumblr does not allow users/developers to extract the followers list unless the list is public and ii) Tumblr facilitates users to track any number of search tags or keywords. Whenever a new post is published on Tumblr containing any of these tags, it appears on the dashboard of user and a blogger no longer need to follow the original poster. We manage a queue of all extracted bloggers and traverse through the network using Random Walk algorithm. We use uniform distribution to select next blogger and extract it’s frontiers. We extract these bloggers until the graph converges without re-visiting a blogger. The proposed framework results into a connected graph that represents a Tumblr network. We perform social network analysis on the output graph to locate hidden virtual groups and extremist bloggers playing major roles in community.
Generate URL of post to fetch post content and caption SetParameters() { Authenticate the client via API Keys $C_{k}$ and $C_{s}$ params.put(“type”, “text”) params.put(“filter”, “text”) params.put(“reblog info”, true) params.put(“notes info”, true) } TaggedPost() { Posts = client.tagged($tag\_name$, params) } BloggerPost(){ Posts = client.tagged($tag\_name$, params) }
Solution Implementation
=======================
A topical crawler starts from a seed node, traverses in a graph navigating through some links and returns all relevant nodes to a given topic. In proposed solution approach we divide our problem into three sub-problems. First we classify the given seed node $S$ as hate promoting or unknown according to the published post (originally posted by blogger or re-blogged from other Tumblr users). Second, if the node is relevant then we extend this node into it’s frontiers and it further leads us to more hate promoting bloggers. In third sub-problem, we perform topical crawling on Tumblr network and use random walk algorithm to traverse along the graph.
Retrieval of Published Posts
----------------------------
Algorithm \[get\_post\] describes the method to search Tumblr posts using keyword based flagging and extraction of posts published by a given blogger. The work presented in this paper focus on mining textual metadata on Tumblr therefore we set a few parameters and extract only text based posts for further analysis. For each blogger we set the limit of $100$ posts published recently. Function (steps $12$ and $13$) filters the search results and displays only the textual posts (quote, chat, text, url). Function with given parameters search for text posts that exclusively contain given tag name. fetches the textual posts published by given blogger ID. Both the functions make a Tumblr API request to fetch these data. Function filters the response and extract body content & caption of each post. Tumblr API allows us to only extract the summary of large posts. Therefore we use HTML parsing for extracting the whole message in blog post. In steps $4$ to $7$, we generate the URL from post summary and id to fetch the remaining post details. ID is a unique identifier of Tumblr posts and slug is a short text summary of that post which is appended in the end of every URL. We invoke this URL using Jsoup library and parse the HTML document to get the post content.
ExtractFrontiers[$U$]{}{ Posts= }
$U_{i}=S$, F.add($S$) TopicalCrawler[$S$]{}{ }
Retrieval of External Links to Bloggers
---------------------------------------
Algorithm \[get\_frontiers\] describes the steps to extract frontiers of a given node $U$. Due to the privacy policies, Tumblr API does not allow developers to extract subscriptions and followers of a Tumblr user. The link between two bloggers indicates the similar interest between them so that number of frontiers vary for every post published by a blogger. For each user, we extract $25$ bloggers for each relation i.e. users who have liked and re-blogged that post recently. If a blogger $B_{1}$ re-blog and as well as likes a post published by another blogger $B_{2}$ then in the graph $G$, we create an edge with both labels i.e. ($B_{1}$, $B_{2}$, <like, re-blog>). To avoid the redundancy we extract one more frontier who have either liked or re-blogged the post recently. To extract the linked bloggers of a Tumblr user we first need to extract the posts made by $U$. We can extract notes information only when notes and re-blog information parameters are set to be true (refer to steps $14$ and $15$ of Algorithm \[get\_post\]). As described in Algorithm \[get\_frontiers\], in step $4$, we extract notes for each textual post (hate promoting) made by User $U$. In steps $5$ to $8$, we extract the name of unique bloggers who liked and re-blogged the post $P$. In step $8$, $F$ represents the list of frontiers and relation of $U$ with each frontier. We maintain a list of all processed bloggers and the number of hit counts on their recent $100$ posts. These number of notes varied from $0$ to $25$K therefore we perform smoothing on data points and plot median of these values. Figure \[notes\] shows the statistics of number of notes collected on $100$ posts of each blogger extracted during topical crawler. Figure \[notes\] reveals that overall number of hit counts (number of reblog and like) for extremism promoting users is very high. These hit counts reveals the popularity of extremist content and the number of viewers connected to such bloggers.
Topical Crawler Using Random Walk
---------------------------------
Algorithm \[algo\_random\_walk\] describes the proposed crawler for locating a group of hidden extremist bloggers on Tumblr. The goal of this algorithm is to compare each blogger against training examples and then connecting all positive class (hate promoting or relevant) bloggers. Algorithm \[algo\_random\_walk\] takes several inputs: seed blogger (positive class user) $S$, size of the graph $S_{g}$ i.e. maximum number of nodes in a graph, width of the graph $W_{g}$ i.e. the maximum number of frontiers or adjacency nodes for each blogger, a set of exemplary documents $D_{e}$, threshold $th$ and n-gram value $N_{g}$ for relevance computation. We create a list of $30$ positive class bloggers extracted during experimental setup (refer to section \[experimental\_setup\]) and compute their relevance score against the exemplary documents. We take an average of these scores and compute the threshold value for language modeling. We use n-gram language modeling ($N_{g}$=$3$) to build our statistical model. Algorithm \[algo\_random\_walk\] is a recursive process that results into a cyclic directed graph. We run this algorithm until we get a graph of size $S$ ($1000$ bloggers) or there is no node left in the queue for further extension. We perform a self-avoiding random walk that means we make sure a node is never being re-visited. If a node re-appears in the frontiers list then there are two possibilities: 1) the frontier has already been processed (extended or discarded based upon the relevance score- Steps $4$ to $7$). If it exists in the processed nodes list then we create a directed edge between the node and it’s parent and avoid further extension. 2) If the re-appearing node is in frontiers list and is not yet processed, we created a directed edge in the graph and continue the traversal.\
The topical crawler is a recursive process that adds and removes nodes after each iteration. The resultant graph is dynamic and not irreducible that means given a graph $G(V, E)$, if there is a directed edge between two nodes $u$ and $v$, it is not necessary that there exists a directed path from $v$ to $u$. Consider that object (topical crawler) processed node $i$ at time $t-1$. In the next iteration object moves to an adjacency node of $i$. The probability that object moves to node $j$ at time $t$ is $\frac{1}{d^{+}_{i}}$ when there exists a direct edge from $i$ to $j$. $M_{ij}$=$\frac{1}{d^{+}_{i}}$ denotes the probability to reach from $i$ to $j$ in one step where $d^{+}_{i}$ is the out-degree of node $i$. Therefore we can define:\
$$\label{probability}
M_{i,j}=
\left\{
\begin{aligned}
\frac{1}{d^{+}_{i}},\ \ \ if \ (i,j) \ is \ an \ edge \ in \ digraph \ G
\\
0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise
\end{aligned}
\right\}$$
Therefore for each vertex $i$, the sum of the probability to traverse an adjacency node of $i$ is $1$.\
$$\label{Mij}
\forall i \sum_{j \in A(i)} M_{ij}=1\\$$ Where, A(i) denotes the list of adjacency nodes $i$. In random walk on graph $G$ topical crawler traverse along the nodes according to the probability of $M_{ij}$. Graph $G$ is a dynamic social networking graph, therefore we compute a Markov chain M after each iteration and compute the probability matrix over graph $G$. Markov chain is a random process where the probability distribution of node $j$ depends on the current state of matrix. The probability matrix $M^{k}$ gives us a picture of graph $G$ after $k$ iterations of topical crawler. Using this matrix, we compute the probability distribution $P$ that object moves to a particular vertex. $P^{k}$ is the probability distribution of a node $j$ after $k$ iteration then probability of $i$ to be traversed in $k+1^{th}$ iteration is the following: $$\label{prob_dis}
P^{k+1}=P^{k}.M \\
where, P^{k}= P^{0}*M^{k}\\$$
Where $P^{0}$ is the initial distribution fixed for the seed node.
Experimental Results
====================
Topical Crawler Results
-----------------------
We hired $30$ graduate students from different departments as volunteers to annotate all $600$ bloggers processed or traversed during the topical crawling. We provided them guidelines for annotation and to remove the bias, we performed horizontal and vertical partitioning on the bloggers’ dataset. We divided annotators into $10$ different groups, $3$ members each. We asked each group to to annotate $60$ bloggers. Therefore, we got $3$ reviews for each blogger. We used majority voting approach (a blogger labeled as X by at least two annotators) for annotating each blogger as hate promoting or unknown. We compute the effectiveness of our classification by using precision, recall and f-measure as accuracy matrices. Table \[results\] shows the confusion matrix and accuracy results for unary classification performed during graph traversal. Table \[conf\_matrix\] reveals that among $600$ bloggers, our model classifies $382$($290$+$92$) bloggers as hate promoting and $218$($173$+$45$) bloggers as unknown with a misclassification of $13$% and $34$% in predicting extremist and unknown bloggers respectively. Table \[acc\_results\] shows the standard information retrieval matrices for accuracy results. our results shows that the both the precision and recall are high, as it is important to reduce the number of false alarms and also not to miss an extremist blogger in order to locate their communities. We use F-score as the accuracy metrics for our classifier and the results reveal that we are able to predict extremist blogger with an f-score of $80$%.
**Graph** **\#Nodes** **\#Edges** **Dia** **\#SCC** **\#ACC** **\#Mod** **IBC** **ICC**
----------- ------------- ------------- --------- ----------- ----------- ----------- --------- ---------
**TC** 382 275 4 137 0.026 12.00 11.36 0.20
**LB** 27 60 1 21 0.0231 1.307 0 0.38
**RB** 355 215 6 185 0.021 7.01 6.284 0.40
Social Network Analysis
-----------------------
We perform social network analysis on topical crawler’s network resulted into a directed graph $G(V, E)$, where $V$ represent a set of Tumblr bloggers accounts and $E$ represent a directed edge between two bloggers. We define this edge as a relation having two labels ’posts liked by’ and ’post re-blogged by’. To examine the effectiveness of these relations we generate two independent networks exclusively for ’like’ and ’re-blog’ links between bloggers. Figure \[community\] illustrates the representations of these networks. In each graph, size of the node is directionally proportional to its out-degree. A node with maximum number of adjacency vertices is biggest in size. Colours in the graph represents the clusters of nodes having similar properties. Here, we define the similarity measure as the ratio of out-degree and in-degree.\
We also perform several network level measurements on these graphs. Table \[social\_network\_measurements\] reveals that re-blogging is a strong indicator of links between two Tumblr profiles. Here, we observe that the graphs generated for topical crawler and re-blogging link have same pattern in network measurements. Both graphs are dense (also evident from the Figure \[community\]) and have higher modularity in comparison to the graph created for ’liked’ link. Table \[social\_network\_measurements\] and Figure \[community\] also reveal that by navigating through re-blogging links we can locate large number of connected components in a extreme right community. While following ’like’ as a link we are able to detect small number of connected blogs. Though as illustrated in Figure \[graph\_like\], we can not completely avoid this feature as a set of blogs extracted using this link are irreducible. Table \[social\_network\_measurements\] also shows that the graph created for ’like’ relation has slightly larger value for average clustering co-efficient. This is because the number of nodes in the graph is very less and a major set of these nodes is strongly connected. Higher value of In-between centrality shows the presence of bloggers who are being watched by a large number of users. As the Figure \[graph\_re-blog\] shows there are many users which are not directly connected to each other (shown in red colour) but has a huge network of common bloggers. These disjoint bloggers are two or three hop away and are connected via other bloggers (having second largest number of adjacency nodes). These bloggers are connected with maximum number of other bloggers present in the graph and has a wide spread network in extreme right communities. These nodes have the maximum closeness centrality and play central role in the community. Nodes represented as black dots have minimum number of out-degree. They don’t have a directed path to the central users or original source of extremist posts. Based upon our study we find these bloggers to be the target audiences who share these posts in their own network. These users are very crucial for such communities though they don’t play a major role in the network.
Conclusions
===========
In this paper, we perform a case study on Jihadist groups and locate their existing extreme right communities on Tumblr. We conduct experiments on real world dataset and use topical crawler based approach to collect textual data (published posts) from Tumblr users. We perform one class classification and identify hate promoting bloggers according to the content present in their posts. We use random walk algorithm for graph traversal and extract exclusive links to these bloggers. We conclude that by performing social networking analysis on a graph (vertices are the Tumblr bloggers and edges are the links among these bloggers: re-blog and like) we are able to uncover hidden virtual communities of extremist bloggers with an accuracy of $77$%. We compute various centrality measures to locate the influential bloggers playing major roles in extremist groups. We also investigate the effectiveness of link features (likes and re-blogs) in order to find the communities. Our results reveal re-blogging is a strong indicator and a discriminatory feature to mine strongly connected communities on Tumblr.\
We perform a manual inspection on Tumblr and perform a characterization on several hate promoting posts. Our study reveals that these posts are very much popular among extremist bloggers and get large number of hits. These posts are published targeting some specific audiences. Keywords present in the blog content, tags associated with post and comments by other bloggers are clear evidence of hate promotion among their viewers.
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[^1]: https://www.tumblr.com/about
[^2]: https://www.tumblr.com/docs/en/using\_tags
[^3]: https://www.tumblr.com/docs/en/api/v2
[^4]: http://jsoup.org/apidocs/
[^5]: https://code.google.com/p/language-detection/
[^6]: http://alias-i.com/lingpipe/index.html
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---
abstract: 'We derive a quantum master equation which describes the dynamics of the ensemble-averaged state of homogeneous disorder models at short times, and mediates a transition from coherent superpositions into classical mixtures. While each single realization follows unitary dynamics, this decoherence-like behavior arises as a consequence of the ensemble average. The master equation manifestly reflects the translational invariance of the disorder correlations and allows us to relate the disorder-induced dynamics to a collisional decoherence process, where the disorder correlations determine the spatial decay of coherences. We apply our theory to the (one-dimensional) Anderson model.'
author:
- Clemens Gneiting
- 'Felix R. Anger'
- Andreas Buchleitner
title: Incoherent ensemble dynamics in disordered systems
---
Introduction
============
It was the insight of Anderson that disorder can substantially modify the dynamical behavior of quantum particles: The destructive quantum interference due to multiple scattering off impurities in the wire potentially brings the electrons to a halt, giving then rise to Anderson localization [@Anderson1958absence]. Even when the consequences of disorder are less drastic, its interplay with quantum interference can still alter the mobility pattern, causing, e.g., a transition from ballistic propagation to weak localization [@Wellens2008nonlinear]. While these interference effects already occur on the level of single disorder realizations, they even prevail under an average over many disorder realizations, this way stripping off individual peculiarities and defining a statistically robust effect. Anderson localization, e.g., unveils its characteristic trait, exponential wave function tails, on the level of the ensemble average.
The possibility to implement disorder models with highly-controllable cold atomic gases has brought it into reach to access disorder phenomena and their underlying quantum origin even on the level of the spatially resolved atomic density $n(\vec{r},t)$ [@Billy2008direct; @Roati2008anderson]. It was, for instance, observed that the ensemble-averaged correlation function of density fluctuations exhibits, at long times, characteristic long-range correlations, which can be traced back to the macroscopic coherence in the gas [@Henseler2008density; @Cherroret2008long]. Here, we investigate the evolution of quantum coherence under the disorder average at [*short times*]{}. We find that the spatial pattern of the coherence loss of the ensemble-averaged state is directly related to the correlations in the disorder potential. This loss already happens at ballistic times much shorter than the mean free time $\tau$, where the disorder does not yet have a significant effect on the level of single realizations.
We emphasize that this effective decoherence of the ensemble average state does not correspond to a loss of information as it generally occurs in the presence of an environment. In our case, single disorder realizations follow the unitary dynamics of isolated quantum systems, i.e. the occurrence of quantum interference phenomena which survive the ensemble average, such as Anderson or weak localization, remains untouched. The coherent nature of the dynamics of single realizations can for instance be recovered by considering higher-order correlators, such as, e.g., intensity correlations. The loss of coherence of the ensemble-averaged state, on the other hand, is a consequence of the fact that different disorder realizations propagate an initially pure state into different evolved states, and that their averaging generally results in a mixed state.
To establish our results, we derive a general Lindblad master equation for the evolution of the disorder-averaged state on short time scales, allowing us to investigate the transient dynamics for arbitrary initial states $\rho_0$. In this approach, the dynamical impact of the disorder is reflected by the structure of the resulting master equation. In particular, coherent and incoherent contributions to the dynamics of the ensemble-averaged state are consistently separated. As we show, the evolution generated by the master equation for the one-dimensional (1D) Anderson model perfectly agrees with the short-time dynamics of numerically exact simulations thereof.
Let us stress that our approach lies at the interface between quantum transport theory of disordered systems and the theory of open systems. It complements other perturbative methods to treat disorder dynamics, e.g. based on averaged propagators and/or diagrammatic methods [@Lloyd1969exactly; @Akkermans2007mesoscopic]. Alternative evolution equations for the ensemble average state have been proposed in [@Rammer1991quantum; @Mueller2009diffusive].
A comprehensive understanding of the disorder-induced dynamics at short times is also of practical relevance, as it permits one to access the detrimental impact of perturbations on the functioning of quantum devices. To see this, let us consider a simple example, the double-slit experiment. There, the observed fringe pattern, which represents the purpose of the device, strongly relies on the delicate interplay between the delocalized state prepared by the slits and the phases accumulated on the way to the screen. What happens if the particles are disturbed along their way, e.g. if they propagate across a disordered scattering potential towards the screen? As we show in Fig. 1, averaging over many realizations of the disorder potential gives rise to a continuous-in-time decay of coherences, i.e. the visibility of the interference pattern in momentum is monotonously reduced as time elapses. In other words, while single realizations exhibit distorted interference fringes, the ensemble average recovers the structure of the undisturbed pattern, however with an increasingly reduced visibility.
\(a) (b)\
![\[Fig:Double-slit\_decoherence\] (Color online) Decoherence dynamics induced by the disorder average: Evolution of an initial spatial superposition state in the (one-dimensional) Anderson model, mimicking the double-slit experiment in the presence of disorder. (a) Density matrix of the spatial superposition of two Gaussian wave packets at $t=0 \hbar/J$ (initial state). (b) Ensemble average state $\rho_{\rm ens}(t)$ at $t=0.8 \hbar/J$ ($J$ denotes the hopping constant), with disorder strength $W=5 J$, and averaged over $K=100$ realizations. One observes a decay of the coherences between the two peaks, as well as of each individual peak. The loss of coherence is also reflected in the reduced visibility of the interference pattern in momentum (c – blue solid at $t=0.8 \hbar/J$, orange dashed at $t=0 \hbar/J$), and in the decay of the purity $p_{\rm ens}(t)={\rm tr}[\rho_{\rm ens}(t)^2]$ of the state (d).](./Fig1a.pdf "fig:"){width="0.48\columnwidth"} ![\[Fig:Double-slit\_decoherence\] (Color online) Decoherence dynamics induced by the disorder average: Evolution of an initial spatial superposition state in the (one-dimensional) Anderson model, mimicking the double-slit experiment in the presence of disorder. (a) Density matrix of the spatial superposition of two Gaussian wave packets at $t=0 \hbar/J$ (initial state). (b) Ensemble average state $\rho_{\rm ens}(t)$ at $t=0.8 \hbar/J$ ($J$ denotes the hopping constant), with disorder strength $W=5 J$, and averaged over $K=100$ realizations. One observes a decay of the coherences between the two peaks, as well as of each individual peak. The loss of coherence is also reflected in the reduced visibility of the interference pattern in momentum (c – blue solid at $t=0.8 \hbar/J$, orange dashed at $t=0 \hbar/J$), and in the decay of the purity $p_{\rm ens}(t)={\rm tr}[\rho_{\rm ens}(t)^2]$ of the state (d).](./Fig1b.pdf "fig:"){width="0.48\columnwidth"} (c) (d)\
![\[Fig:Double-slit\_decoherence\] (Color online) Decoherence dynamics induced by the disorder average: Evolution of an initial spatial superposition state in the (one-dimensional) Anderson model, mimicking the double-slit experiment in the presence of disorder. (a) Density matrix of the spatial superposition of two Gaussian wave packets at $t=0 \hbar/J$ (initial state). (b) Ensemble average state $\rho_{\rm ens}(t)$ at $t=0.8 \hbar/J$ ($J$ denotes the hopping constant), with disorder strength $W=5 J$, and averaged over $K=100$ realizations. One observes a decay of the coherences between the two peaks, as well as of each individual peak. The loss of coherence is also reflected in the reduced visibility of the interference pattern in momentum (c – blue solid at $t=0.8 \hbar/J$, orange dashed at $t=0 \hbar/J$), and in the decay of the purity $p_{\rm ens}(t)={\rm tr}[\rho_{\rm ens}(t)^2]$ of the state (d).](./Fig1c.pdf "fig:"){width="0.48\columnwidth"} ![\[Fig:Double-slit\_decoherence\] (Color online) Decoherence dynamics induced by the disorder average: Evolution of an initial spatial superposition state in the (one-dimensional) Anderson model, mimicking the double-slit experiment in the presence of disorder. (a) Density matrix of the spatial superposition of two Gaussian wave packets at $t=0 \hbar/J$ (initial state). (b) Ensemble average state $\rho_{\rm ens}(t)$ at $t=0.8 \hbar/J$ ($J$ denotes the hopping constant), with disorder strength $W=5 J$, and averaged over $K=100$ realizations. One observes a decay of the coherences between the two peaks, as well as of each individual peak. The loss of coherence is also reflected in the reduced visibility of the interference pattern in momentum (c – blue solid at $t=0.8 \hbar/J$, orange dashed at $t=0 \hbar/J$), and in the decay of the purity $p_{\rm ens}(t)={\rm tr}[\rho_{\rm ens}(t)^2]$ of the state (d).](./Fig1d.pdf "fig:"){width="0.48\columnwidth"}
Homogeneous disorder
====================
We consider a single quantum particle subject to a homogeneous disorder potential, i.e. correlations among different locations are translationally invariant. For simplicity, we focus here on a one-dimensional, discrete (infinitely extended) configuration space, comprised of sites $|j\rangle$ with lattice spacing $a$; however, as will become clear in the course of the article, our theory works as well for continuous configuration spaces, higher dimensions, finite-size topologies, and many particles. In the case considered here, the Hamiltonian for a single disorder realization is given by $$\label{Eq:Anderson_model_Hamiltonian}
\hat{H}_{\vec{\varepsilon}} = - J \sum_{j \in \mathbb{Z}} (|j \rangle \langle j+1| + |j+1 \rangle \langle j|) + \sum_{j \in \mathbb{Z}} \varepsilon_j |j \rangle \langle j| ,$$ where the (infinite-dimensional) vector $\vec{\varepsilon}$ comprised of the random on-site energies $\varepsilon_j$ distinguishes different disorder realizations. The tunneling or hopping term is characterized by the tunneling/hopping constant $J$, which controls the maximal propagation speed in the system.
The distribution $p(\vec{\varepsilon})$ of the on-site energies is assumed to be homogeneous. Besides the normalization $\left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} p(\vec{\varepsilon}) = 1$, we thus require that the expectation values and two-point correlation functions satisfy
\[Eq:Translational-invariant\_distribution\] $$\begin{aligned}
&\left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} p(\vec{\varepsilon}) \, \varepsilon_j = \overline{\varepsilon}_j = \overline{\varepsilon} = 0 , \label{Eq:Vanishing_expectation_values} \\
&\left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} p(\vec{\varepsilon}) \, \varepsilon_j \varepsilon_{j'} = J^2 \, C(j-j') .\end{aligned}$$
Note that we assume without loss of generality that the expectation value $\overline{\varepsilon}$ of the onsite energies vanishes. For convenience (as will become clear below), the two-point correlation function $C(\Delta j)$ is measured in units of the hopping constant $J$ (as well as all other quantities with the dimension of an energy). As we will see, it is sufficient to characterize the expectation values and the two-point correlations of the disorder distribution, as only these appear in the master-equation description at short times. Of course, homogeneity requires all higher-order correlation functions to be translationally invariant as well. We restrict ourselves to disorder which is diagonal in the site basis; however, other forms of homogeneous disorder are also conceivable.
In the case of the Anderson model [@Anderson1958absence], the on-site energies of different sites are completely uncorrelated, i.e. the disorder distribution $p(\vec{\varepsilon})$ decomposes into a product of identical single-site distributions, $p(\vec{\varepsilon}) = \prod_{i \in \mathbb{Z}} p_s(\varepsilon_i)$. The box-shaped single-site distributions $p_s(\varepsilon) = \Theta(W/2 + \varepsilon) \Theta(W/2 - \varepsilon)/W$ are characterized by the disorder strength $W$ ($\Theta$ denotes the Heaviside function). The translation-invariant correlation function is accordingly given by $$\label{Eq:Anderson_correlation_function}
C(j-j') = \frac{1}{12} \left( \frac{W}{J} \right)^2 \delta_{j-j',0} .$$
Short-time evolution.
=====================
We now derive a quantum master equation which accurately describes the ensemble average dynamics of disorder models such as (\[Eq:Anderson\_model\_Hamiltonian\]) at short times. In particular, it renders the, in general, incoherent nature of the ensemble average dynamics manifest in terms of the emerging Lindblad terms. To this end, we first consider the [*unitary*]{} time evolution for a single realization of the disorder potential, $\rho_{\vec{\varepsilon}}(t) = \hat{U}_{\vec{\varepsilon}}(t) \rho_0 \hat{U}_{\vec{\varepsilon}}^{\dagger}(t)$, with the initial state $\rho_0$ at $t_0 = 0$ and the time-evolution operator $\hat{U}_{\vec{\varepsilon}}(t) = \exp(-{\rm i} \hat{H}_{\vec{\varepsilon}} \, t/\hbar)$. Since we are interested in the evolution on short time scales, we expand to second order in ${\rm d}t$: $$\begin{aligned}
\label{Eq:Second_order_expansion_single_realization}
\rho_{\vec{\varepsilon}}({\rm d}t) =& \rho_0 + \frac{\rm i}{\hbar} {\rm d}t \, [\rho_0, \hat{H}_{\vec{\varepsilon}}] \\
&+ \frac{{\rm d}t^2}{\hbar^2} \left( \hat{H}_{\vec{\varepsilon}} \rho_0 \hat{H}_{\vec{\varepsilon}} - \frac{1}{2} \hat{H}_{\vec{\varepsilon}}^2 \rho_0 - \frac{1}{2} \rho_0 \hat{H}_{\vec{\varepsilon}}^2 \right) + \mathcal{O}({\rm d}t^3) .\nonumber\end{aligned}$$ The second-order term is structurewise reminiscent of a Lindblad term, and, indeed, upon averaging over different realizations, the leading incoherent contributions to the disorder dynamics arise at second order in time. To see this, we take the average $\overline{\rho} = \left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} p(\vec{\varepsilon}) \rho_{\vec{\varepsilon}}$ of (\[Eq:Second\_order\_expansion\_single\_realization\]), which yields $$\begin{aligned}
\label{Eq:Second_order_expansion_ensemble_average}
\overline{\rho}({\rm d}t) = \rho_0 &+ \frac{\rm i}{\hbar} {\rm d}t \, [\rho_0, \hat{\overline{H}}] + {\rm d}t \left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} \frac{p(\vec{\varepsilon}) {\rm d}t}{\hbar^2} \\
& \times \left( \hat{H}_{\vec{\varepsilon}} \rho_0 \hat{H}_{\vec{\varepsilon}} - \frac{1}{2} \hat{H}_{\vec{\varepsilon}}^2 \rho_0 - \frac{1}{2} \rho_0 \hat{H}_{\vec{\varepsilon}}^2 \right) + \mathcal{O}({\rm d}t^3) . \nonumber\end{aligned}$$ In the first-order von Neumann term we exploited that the initial state is independent of the disorder realization. It therefore commutes with the ensemble average, resulting in the average Hamiltonian $\hat{\overline{H}} = \left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} p(\vec{\varepsilon}) \hat{H}_{\vec{\varepsilon}}$. Such reduction is, in general, impossible for the second order term, which ultimately gives rise to incoherent dynamics.
To convert (\[Eq:Second\_order\_expansion\_ensemble\_average\]) into a differential equation of Lindblad form, we must not restrict our treatment to the leading contributions in ${\rm d}t$, since we would thus lose the incoherent part of the dynamics and end with the coherent evolution induced by the average Hamiltonian $\hat{\overline{H}}$ alone. To consistently identify next-to-leading order contributions, we replace $\hat{H}_{\vec{\varepsilon}} \rightarrow (\hat{H}_{\vec{\varepsilon}}-\hat{\overline{H}}) + \hat{\overline{H}}$. Equation (\[Eq:Second\_order\_expansion\_ensemble\_average\]) can then be rewritten as $$\begin{aligned}
\label{Eq:Second_order_expansion_ensemble_average_consistent}
\overline{\rho}({\rm d}t) =& \rho_0 + \frac{\rm i}{\hbar} {\rm d}t \, [\rho_0, \hat{\overline{H}}] \\
&+ \frac{{\rm d}t^2}{\hbar^2} \left( \hat{\overline{H}} \rho_0 \hat{\overline{H}} - \frac{1}{2} \hat{\overline{H}}^2 \rho_0 - \frac{1}{2} \rho_0 \hat{\overline{H}}^2 \right) \nonumber \\
&+ {\rm d}t \left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} \frac{p(\vec{\varepsilon}) {\rm d}t}{\hbar^2} \Big( (\hat{H}_{\vec{\varepsilon}}-\hat{\overline{H}}) \rho_0 (\hat{H}_{\vec{\varepsilon}}-\hat{\overline{H}}) \nonumber \\
&- \frac{1}{2} (\hat{H}_{\vec{\varepsilon}}-\hat{\overline{H}})^2 \rho_0 - \frac{1}{2} \rho_0 (\hat{H}_{\vec{\varepsilon}}-\hat{\overline{H}})^2 \Big) + \mathcal{O}({\rm d}t^3) , \nonumber\end{aligned}$$ where the first two lines represent the von Neumann commutator, and the last two lines the Lindblad terms, respectively, each to second order in time. The decoherence rates associated with the Lindblad terms increase linearly in time.
It follows that Eq. (\[Eq:Second\_order\_expansion\_ensemble\_average\_consistent\]) solves, to second order in time, a Lindblad master equation for the ensemble average state, $$\begin{aligned}
\label{Eq:Short-time_master_equation}
\dot{\overline{\rho}} =& -\frac{\rm i}{\hbar} [\hat{\overline{H}}, \overline{\rho}] + \left\{ \prod_{i \in \mathbb{Z}} \int {\rm d}\varepsilon_i \right\} \gamma_{\vec{\varepsilon}}(t) \\
& \times \left( \hat{L}_{\vec{\varepsilon}} \overline{\rho} \hat{L}_{\vec{\varepsilon}}^{\dagger} - \frac{1}{2} \hat{L}_{\vec{\varepsilon}}^{\dagger} \hat{L}_{\vec{\varepsilon}} \overline{\rho} - \frac{1}{2} \overline{\rho} \hat{L}_{\vec{\varepsilon}}^{\dagger} \hat{L}_{\vec{\varepsilon}} \right) , \nonumber\end{aligned}$$ which captures the disorder dynamics at short times. The (time-independent) Lindblad operators $\hat{L}_{\vec{\varepsilon}}$ and the corresponding (time-dependent) decoherence rates $\gamma_{\vec{\varepsilon}}(t)$ read $$\hat{L}_{\vec{\varepsilon}} = \frac{\hat{H}_{\vec{\varepsilon}} - \hat{\overline{H}}}{E_0} \qquad ; \qquad \gamma_{\vec{\varepsilon}}(t) = \frac{2 p(\vec{\varepsilon}) E_0^2}{\hbar^2} t \, ,$$ where the characteristic energy scale $E_0$ is introduced in order to obtain the appropriate dimensions; as stated before, in the case of the model (\[Eq:Anderson\_model\_Hamiltonian\]) it is conveniently chosen to be the hopping constant $J$. We thus find that the ensemble average accounts for each disorder realization by an independent Lindblad term, where the Hermitian Lindblad operators are given by the offset of the disorder Hamiltonian from the average Hamiltonian. The associated decoherence rates are proportional to the probability $p(\vec{\varepsilon})$ for the realization to occur and scale linearly in time, i.e. the rates vanish at $t=0$. The latter expresses that there is no incoherent contribution to the dynamics at first order in time. The validity range of the short-time approximation (\[Eq:Short-time\_master\_equation\]) depends on the composition of the underlying disorder ensemble and must be determined case by case. While the master equation (\[Eq:Short-time\_master\_equation\]) does not require, e.g., weak disorder, the time scale on which it yields reliable predictions in general depends on the disorder strength. Below we will give a numerical estimate for the Anderson model.
We emphasize that the disorder master equation (\[Eq:Short-time\_master\_equation\]) still holds for arbitrary systems and general disorder distributions, since we have not yet made use of the Hamiltonian (\[Eq:Anderson\_model\_Hamiltonian\]) and/or of the homogeneous distribution (\[Eq:Translational-invariant\_distribution\]). In the Appendix, we thus evaluate the short-time disorder dynamics (\[Eq:Short-time\_master\_equation\]) for two unrelated, yet instructive examples: a particle of mass $m$ in one-dimensional, continuous space, subject to a random i) linear or ii) harmonic potential. In these cases one finds that the short-time dynamics of the ensemble-averaged state are governed by the well-known Caldeira-Leggett master equation [@Caldeira1983path; @Diosi1993high].
In the case of the disorder model (\[Eq:Anderson\_model\_Hamiltonian\]), the average Hamiltonian is given by the discrete hopping term, $\hat{\overline{H}} = - J \sum_{j \in \mathbb{Z}} (|j \rangle \langle j+1| + |j+1 \rangle \langle j|)$ (we used (\[Eq:Vanishing\_expectation\_values\])), and the Lindblad operators are given by the disorder potentials, $\hat{L}_{\vec{\varepsilon}} = \sum_{j \in \mathbb{Z}} (\varepsilon_j/J) |j \rangle \langle j|$ (with $E_0 = J$). This is already conceptually appealing, since it demonstrates that the Lindblad operators are diagonal in the site basis; moreover, Eq. (\[Eq:Short-time\_master\_equation\]) predicts (confirmed by observation) an initially quasi-free, dispersive evolution of the ensemble average state in addition to the loss of coherence. However, this representation is not yet viable from a practical point of view, in the sense that it is not amenable to transparent approximations or efficient numerical simulation. In the following, we derive an alternative representation for homogeneous disorder models (\[Eq:Translational-invariant\_distribution\]) which resolves these issues and, in addition, reveals a connection to collisional decoherence.
Collisional decoherence master equation.
========================================
To obtain an alternative representation for the short-time dynamics (\[Eq:Short-time\_master\_equation\]) of the homogeneous disorder models (\[Eq:Translational-invariant\_distribution\]), we exploit that Lindblad master equations are invariant w.r.t unitary transformations of the Lindblad operators. In our case this corresponds to a transformation from the position to the momentum basis. To this end, we perform the disorder integrals in (\[Eq:Short-time\_master\_equation\]) and are left with the double sum over the sites appearing in the Lindblad operators, $$\begin{aligned}
\dot{\overline{\rho}} = -\frac{\rm i}{\hbar} [\hat{\overline{H}}, \overline{\rho}] &+ \sum_{j,j' \in \mathbb{Z}} \frac{2 J^2 t}{\hbar^2} C(j-j') \Big(|j \rangle \langle j| \overline{\rho} |j' \rangle \langle j'| \nonumber \\
&- \frac{1}{2} |j \rangle \langle j|j' \rangle \langle j'| \overline{\rho} - \frac{1}{2} \overline{\rho} |j \rangle \langle j|j' \rangle \langle j'| \Big) ,\end{aligned}$$ where we already made use of the translational invariance (\[Eq:Translational-invariant\_distribution\]). If we then rewrite the correlation function $C(j)$ in terms of its Fourier transform $G(q)$, $C(j) = \int_{-h/2 a}^{h/2 a} {\rm d}q \, {\rm e}^{{\rm i} q j a/\hbar} G(q)$ ($a$ denotes the lattice spacing), we obtain with $\hat{x} = \sum_{j \in \mathbb{Z}} j a |j \rangle \langle j|$ $$\label{Eq:Collisional_decoherence_equation}
\dot{\overline{\rho}} = -\frac{\rm i}{\hbar} [\hat{\overline{H}}, \overline{\rho}] + \frac{2 J^2 t}{\hbar^2} \int\limits_{-h/2 a}^{h/2 a} {\rm d}q \, G(q) \left( {\rm e}^{{\rm i} q \hat{x}/\hbar} \overline{\rho} \, {\rm e}^{-{\rm i} q \hat{x}/\hbar} - \overline{\rho} \right) .$$ This is our main result. We find that the ensemble average dynamics of homogeneous disorder models (\[Eq:Translational-invariant\_distribution\]) are at short times described by the discrete version of the collisional decoherence master equation. The (non-Hermitian, but unitary) Lindblad operators $\hat{L}_q = \exp(\mathrm{i} q \hat{x}/\hbar)$ describe momentum kicks, whose occurrence is weighted by the momentum transfer distribution $G(q)$. The latter follows by Fourier transform from the two-point disorder correlation function $C(j)$, $G(q) = (a/h) \sum_{j \in \mathbb{Z}} \exp(-\mathrm{i} q j a/\hbar) C(j)$ [^1].
The collisional decoherence master equation (\[Eq:Collisional\_decoherence\_equation\]) is usually known from an open-system context [@Gallis1990environmental; @Hornberger2003collisional], where it describes the decoherence that a heavy test particle undergoes due to scattering in a background gas of light particles, i.e. no appreciable energy exchange occurs. It represents the simplest manifestation of a translational-covariant Lindblad master equation [@Kossakowski1972quantum; @Manita1991properties; @Botvich1991translation; @Holevo1995translation].
The master equation (\[Eq:Collisional\_decoherence\_equation\]) allows us to deduce the decoherence dynamics of the homogeneous disorder models (\[Eq:Translational-invariant\_distribution\]). One can best understand the spatial decoherence behavior of (\[Eq:Collisional\_decoherence\_equation\]) by neglecting the coherent dynamics according to the von Neumann commutator and solving the remaining equation in the position representation. One then obtains $\langle j| \overline{\rho}(t)|j' \rangle = \exp \left( -\frac{J^2 t^2}{\hbar^2} F(j-j') \right) \langle j| \rho_0|j' \rangle$, where the localization function $F(j-j')$ (not to be confused with the exponential localization of the particle density in the Anderson model) follows from a Fourier (back-)transformation of $G(q)$ and evaluates as $$\label{Eq:Localization_function}
F(j-j') = \int_{-h/2 a}^{h/2 a} {\rm d}q \, G(q) - C(j-j') .$$ We thus find that the disorder two-point correlations $C(j-j')$ directly translate into the spatial decay of coherences, in the sense that the stronger the correlation between two sites, the longer their coherence survives. This again reflects the fact that the disorder, i.e. the deviations among different ensemble members, gives rise to the decoherence-like behavior.
In the case of the 1D Anderson model with the correlation function (\[Eq:Anderson\_correlation\_function\]), one obtains a constant momentum transfer distribution, $G(q) = \frac{a W^2}{12 h J^2}$, and the localization function reads $F(j-j') = \frac{W^2}{12 J^2} (1-\delta_{j-j', 0})$, i.e., while the populations remain unaffected, all spatial coherences undergo the same decay, independent of the separation of the two respective sites, since the sites are uncorrelated.
As a second example we consider a Gaussian random potential with the correlation function $C(j-j') = \frac{\xi}{J^2} \exp \left(- \frac{(j-j')^2 a^2}{L^2} \right)$, where $\xi$ denotes the correlation strength and $L$ the correlation length. Such correlations may, for example, emerge from a collection of Gaussian scattering potentials $v(j-j_n)$ with randomly distributed scattering centers $j_n$. This then yields the momentum transfer distribution $G(q) = \frac{\sqrt{\pi} L \xi}{h J^2} \exp \left( -\frac{L^2 q^2}{4 \hbar^2} \right)$ and the localization function $F(j-j') = \frac{\xi}{J^2} \Big( 1-\exp \Big[ - \left( \frac{(j-j') a}{L} \right)^2 \Big] \Big)$, i.e. there is a smooth, Gaussian transition into the regime of constant decoherence ($|j-j'| a \gg L$). The coherence loss at short times caused by such Gaussian disorder correlations was also investigated in [@Boonpan2012loss] in terms of path-integral techniques (for Gaussian initial states in the continuum and a harmonic average potential $\hat{\overline{H}}$). In our language, the authors derive the localization function $F(x-x') = (\xi/J^2) (1-(1+2 (\frac{x-x'}{L})^2)^{-1/2})$, which coincides in the short-range region ($|x-x'| < L/2$) with our result and shows qualitatively the same behavior in the long-range region. As the short-time master equation (\[Eq:Short-time\_master\_equation\]) is derived without reference to a Hilbert space basis and therefore holds over the range of all sites, we interpret the quantitative deviation in the long-range region in terms of a breakdown of the path-integral approach.
Numerical comparison.
=====================
In order to estimate the range of validity $t_{\rm max}$ of the short-time disorder master equation (\[Eq:Collisional\_decoherence\_equation\]), we compare it in case of the 1D Anderson model to the numerical ensemble average over a finite sample of disorder realizations. In Fig. \[Fig:Master\_equation\_comparison\] we show, in terms of an initial Gaussian state and for strong disorder with $W=10 J$, that the master equation correctly predicts (relative error $\pm 5\%$) the spatially homogeneous decay of the coherences up to about $t_{\rm max}=0.2 \hbar/J$, where the state has lost about $45\%$ of its initial purity $p={\rm tr}[\rho^2]$. Similarly, one obtains for disorder strengths $W=1 J$ and $W=0.1 J$ validity ranges of about $t_{\rm max}=0.9 \hbar/J$ at a purity loss of $10\%$ and $t_{\rm max}=6 \hbar/J$ at a purity loss of $1\%$, respectively. A more detailed analysis confirms that $t_{\rm max}$ roughly scales inversely with $W$, $t_{\rm max} \propto 1/W$, or, in terms of the mean free path $\ell$, $t_{\rm max} \propto \sqrt{\ell}$ (similarly, the momentum-independent decoherence rate $\gamma(t)$ scales inversely with the mean free time $\tau$, $\gamma(t) \propto t/\tau$). This suggests to interpret our theory in terms of an expansion in $W t$. Notwithstanding, we can probe the regime of strong decoherence as induced by large $W$. Let us also emphasize that we could have chosen any initial state for this analysis.
\(a) (b)\
![\[Fig:Master\_equation\_comparison\] (Color online) Dynamics of a Gaussian initial state in the 1D Anderson model: Comparison of the time evolution $\rho_{\rm me}(t)$ predicted by the disorder master equation (\[Eq:Collisional\_decoherence\_equation\]) with the numerically exact dynamics $\rho_{\rm ens}(t)$, obtained by averaging over a finite number of realizations. (a) Density matrix of the evolved ensemble average state $\rho_{\rm ens}(t)$ at $t=0.2 \hbar/J$ with strong disorder, $W=10 J$, and $K=200$ realizations. The state displays a spatially homogeneous decay of the coherences, as predicted by the master equation. (b) This is also confirmed by the ratio of the two density matrices, which remains everywhere close to one, with local fluctuations on the order of a few percent, due to the finite sampling. (c) The short-time approximation becomes poor beyond $t=0.2 \hbar/J$, which can easily be seen by inspecting the ratio of the two purities $p_{\rm ens}(t)$ and $p_{\rm me}(t)$, which starts to increasingly deviate from one. The purity provides a global measure of the decoherence and is robust, in the sense that it averages out local fluctuations due to the finite sampling. (d) At $t=0.2 \hbar/J$ the initially pure state has lost about 45% of its purity. The latter continues to decrease monotonically and eventually converges to $p_{\rm ens} = 0.074$ beyond $t=0.5 \hbar/J$, reflecting the remaining coherence in the asymptotic state.](./Fig2a.pdf "fig:"){width="0.48\columnwidth"} ![\[Fig:Master\_equation\_comparison\] (Color online) Dynamics of a Gaussian initial state in the 1D Anderson model: Comparison of the time evolution $\rho_{\rm me}(t)$ predicted by the disorder master equation (\[Eq:Collisional\_decoherence\_equation\]) with the numerically exact dynamics $\rho_{\rm ens}(t)$, obtained by averaging over a finite number of realizations. (a) Density matrix of the evolved ensemble average state $\rho_{\rm ens}(t)$ at $t=0.2 \hbar/J$ with strong disorder, $W=10 J$, and $K=200$ realizations. The state displays a spatially homogeneous decay of the coherences, as predicted by the master equation. (b) This is also confirmed by the ratio of the two density matrices, which remains everywhere close to one, with local fluctuations on the order of a few percent, due to the finite sampling. (c) The short-time approximation becomes poor beyond $t=0.2 \hbar/J$, which can easily be seen by inspecting the ratio of the two purities $p_{\rm ens}(t)$ and $p_{\rm me}(t)$, which starts to increasingly deviate from one. The purity provides a global measure of the decoherence and is robust, in the sense that it averages out local fluctuations due to the finite sampling. (d) At $t=0.2 \hbar/J$ the initially pure state has lost about 45% of its purity. The latter continues to decrease monotonically and eventually converges to $p_{\rm ens} = 0.074$ beyond $t=0.5 \hbar/J$, reflecting the remaining coherence in the asymptotic state.](./Fig2b.pdf "fig:"){width="0.48\columnwidth"} (c) (d)\
![\[Fig:Master\_equation\_comparison\] (Color online) Dynamics of a Gaussian initial state in the 1D Anderson model: Comparison of the time evolution $\rho_{\rm me}(t)$ predicted by the disorder master equation (\[Eq:Collisional\_decoherence\_equation\]) with the numerically exact dynamics $\rho_{\rm ens}(t)$, obtained by averaging over a finite number of realizations. (a) Density matrix of the evolved ensemble average state $\rho_{\rm ens}(t)$ at $t=0.2 \hbar/J$ with strong disorder, $W=10 J$, and $K=200$ realizations. The state displays a spatially homogeneous decay of the coherences, as predicted by the master equation. (b) This is also confirmed by the ratio of the two density matrices, which remains everywhere close to one, with local fluctuations on the order of a few percent, due to the finite sampling. (c) The short-time approximation becomes poor beyond $t=0.2 \hbar/J$, which can easily be seen by inspecting the ratio of the two purities $p_{\rm ens}(t)$ and $p_{\rm me}(t)$, which starts to increasingly deviate from one. The purity provides a global measure of the decoherence and is robust, in the sense that it averages out local fluctuations due to the finite sampling. (d) At $t=0.2 \hbar/J$ the initially pure state has lost about 45% of its purity. The latter continues to decrease monotonically and eventually converges to $p_{\rm ens} = 0.074$ beyond $t=0.5 \hbar/J$, reflecting the remaining coherence in the asymptotic state.](./Fig2c.pdf "fig:"){width="0.49\columnwidth"} ![\[Fig:Master\_equation\_comparison\] (Color online) Dynamics of a Gaussian initial state in the 1D Anderson model: Comparison of the time evolution $\rho_{\rm me}(t)$ predicted by the disorder master equation (\[Eq:Collisional\_decoherence\_equation\]) with the numerically exact dynamics $\rho_{\rm ens}(t)$, obtained by averaging over a finite number of realizations. (a) Density matrix of the evolved ensemble average state $\rho_{\rm ens}(t)$ at $t=0.2 \hbar/J$ with strong disorder, $W=10 J$, and $K=200$ realizations. The state displays a spatially homogeneous decay of the coherences, as predicted by the master equation. (b) This is also confirmed by the ratio of the two density matrices, which remains everywhere close to one, with local fluctuations on the order of a few percent, due to the finite sampling. (c) The short-time approximation becomes poor beyond $t=0.2 \hbar/J$, which can easily be seen by inspecting the ratio of the two purities $p_{\rm ens}(t)$ and $p_{\rm me}(t)$, which starts to increasingly deviate from one. The purity provides a global measure of the decoherence and is robust, in the sense that it averages out local fluctuations due to the finite sampling. (d) At $t=0.2 \hbar/J$ the initially pure state has lost about 45% of its purity. The latter continues to decrease monotonically and eventually converges to $p_{\rm ens} = 0.074$ beyond $t=0.5 \hbar/J$, reflecting the remaining coherence in the asymptotic state.](./Fig2d.pdf "fig:"){width="0.475\columnwidth"}
Experimental verification.
==========================
Besides the conceptual insight provided by our theory into the incoherent ensemble average dynamics of disordered quantum systems, direct experimental verifications thereof are conceivable, for example based on experiments with ultracold atoms subject to optical speckle potentials. These systems have already been successfully employed to probe the Anderson localization in the asymptotic time regime [@Billy2008direct; @Roati2008anderson]. Moreover, it is possible to imprint various homogeneous disorder distributions on the speckle potential [@Deissler2010delocalization]. The restriction to short times would be implemented by simply switching the speckle potential off after the desired exposure time. Time-of-flight measurements then reveal the momentum distribution of the state. Producing an initial spatial superposition state, one may in this way observe the disorder-induced transition from a superposition into a mixture in terms of the loss of visibility of the interference pattern in momentum.
Conclusions.
============
We developed a theory which describes the ensemble average dynamics of disordered quantum systems at short times in terms of Lindblad master equations, with the statistical properties of the disorder potential encoded in the Lindblad terms. While this effective evolution equation accurately captures the onset of the disorder-induced coherence loss of the ensemble-averaged state in the 1D Anderson model, our theory is not yet capable to explain other disorder effects such as diffusive propagation or localization. However, a (translation-covariant) master equation which also captures the ensemble average dynamics of such disorder-induced phenomena must, in principle, exist. These must then emerge as a feature of the, in general, incoherent evolution of the ensemble-averaged state. Indeed, Fig. \[Fig:Master\_equation\_comparison\](d) illustrates the monotonic decay of the averaged state’s purity towards that of the (localized) asymptotic state. The asymptotic value of the purity decreases with increasing disorder strength $W$, which reflects that the remaining coherence in the asymptotic state is related to the localization length $\xi \propto 1/W^2$ [@Roemer2004weak].
Our results represent a first step towards a treatment of disordered quantum systems in terms of quantum master equations. The impact of spectral and of unitarily invariant disorder on the dynamics of the ensemble-averaged state of finite-dimensional quantum systems at arbitrary times $t$ is the subject of [@Kropf2015effective].
Acknowledgments.
================
C.G. thanks Valentin Volchkov for insightful discussions on the state of the art of disorder experiments with ultracold atoms. A.B. acknowledges financial support from the EU Collaborative project QuProCS (Grant Agreement 641277). Moreover, we thank Thomas Wellens, Alberto Rodriguez, Rodolfo Jalabert, and Cord Müller for helpful comments on the manuscript.
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While Eq. (\[Eq:Collisional\_decoherence\_equation\]) is completely determined by the two-point correlations, we expect that a master equation that goes beyond short times systematically includes higher-order correlation functions, as well.
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Appendix
========
In the following we evaluate the short-time disorder dynamics (\[Eq:Short-time\_master\_equation\]) for two simple, yet relevant examples: a particle of mass $m$ in one-dimensional, continuous space, subject to a random i) linear or ii) harmonic potential.
#### i) In the linear-potential case, {#i-in-the-linear-potential-case .unnumbered}
we consider a Hamiltonian of the form $\hat{H}_{\varepsilon} = \hat{p}^2/2 m + \varepsilon \hat{x}$, i.e. the randomness lies in the strength of the constant force exerted on the particle. This describes for example experiments where a charged particle is exposed to a homogeneous, but not fully controlled electric field, i.e. the field strength varies from run to run. If we assume for simplicity that $\overline{\varepsilon} = 0$, the average Hamiltonian corresponds to the free Hamiltonian, $\hat{\overline{H}} = \hat{p}^2/2 m$, and for the Lindblad operators we obtain $\hat{L}_{\varepsilon} = \varepsilon \hat{x}/E_0$, with $E_0$ an arbitrary energy scale (which is again introduced for dimensional reasons and irrelevant for the final result (\[Eq:Caldeira-Leggett\_equation\])). Since all Lindblad operators are proportional to $\hat{x}$, we can perform the disorder integral in (\[Eq:Short-time\_master\_equation\]) and obtain the simplified master equation $$\label{Eq:Caldeira-Leggett_equation}
\dot{\overline{\rho}} = -\frac{\rm i}{\hbar} \left[ \frac{\hat{p}^2}{2 m}, \overline{\rho} \right] - \frac{\overline{\varepsilon^2}}{\hbar^2} t \, [\hat{x}, [\hat{x}, \overline{\rho}]] .$$ This is the well-known Caldeira-Leggett master equation [@Caldeira1983path; @Diosi1993high], which usually emerges in an open-system context from a linear coupling model. The incoherent part of (\[Eq:Caldeira-Leggett\_equation\]) predicts an exponential decay of spatial coherences according to (as for the derivation of the collisional decoherence localization function (\[Eq:Localization\_function\]), we neglect for the moment the von Neumann commutator) $\overline{\rho}_t(x, x') = \exp \left[ - \frac{\overline{\varepsilon^2}}{2 \hbar^2} t^2 (x-x')^2 \right] \rho_0(x, x')$. While such a localization rate which grows above all bounds for $|x-x'| \rightarrow \infty$ is usually considered as unphysical in the open-system context, it arises here as a natural and unavoidable consequence of the disorder average.
#### ii) In the harmonic-potential example, {#ii-in-the-harmonic-potential-example .unnumbered}
we could in principle allow for both a random frequency and a random center point. We focus here on the latter and keep the frequency fixed, $\hat{H}_{\varepsilon} = \hat{p}^2/2 m + (m \omega^2/2) (\hat{x} - \varepsilon)^2$. This may describe experiments where a particle is harmonically trapped, but where the trap center is subject to fluctuations. In this case (again assuming that $\overline{\varepsilon} = 0$), the short-time dynamics (\[Eq:Short-time\_master\_equation\]) result, once again, in Caldeira-Leggett decoherence, $\dot{\overline{\rho}} = - ({\rm i}/\hbar) [\hat{\overline{H}}, \overline{\rho}] - m \omega^2 (\overline{\varepsilon^2}/\hbar^2) t [\hat{x}, [\hat{x}, \overline{\rho}]]$, but this time with a harmonic average potential $\hat{\overline{H}} = \hat{p}^2/2 m + (m \omega^2/2) \hat{x}^2$. In this example, we can even anticipate the evolution of the ensemble average beyond the short-time approximation: Since all random potentials share the same frequency $\omega$, any initial state will at multiples of the period $T = 2 \pi/\omega$ recur, and in particular it will regain the purity lost in the early stage. On the level of the disorder master equation, this indicates periodic, partly negative decoherence rates $\gamma_{\varepsilon}(t)$. While such time dependence comprising (at least partial) purity revivals is likely the generic pattern of the ensemble average dynamics, Fig. \[Fig:Master\_equation\_comparison\](d) indicates that the Anderson model exhibits a strictly monotonic decay of coherences, also beyond the short-time approximation.
[^1]: While Eq. (\[Eq:Collisional\_decoherence\_equation\]) is completely determined by the two-point correlations, we expect that a master equation that goes beyond short times systematically includes higher-order correlation functions, as well.
|
---
abstract: 'We present first-principles studies of the adsorption of Sb and Ag on clean and Sb-covered Ag (111). For Sb, the [*substitutional*]{} adsorption site is found to be greatly favored with respect to on-surface fcc sites and to subsurface sites, so that a segregating surface alloy layer is formed. Adsorbed silver adatoms are more strongly bound on clean Ag(111) than on Sb-covered Ag. We propose that the experimentally reported surfactant effect of Sb is due to Sb adsorbates reducing the Ag adatom mobility. This gives rise to a high density of Ag islands which coalesce into regular layers.'
address: 'Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany'
author:
- 'Sabrina Oppo,[@p_add] Vincenzo Fiorentini,[@p_add] and Matthias Scheffler'
date: Submitted to Physical Review Letters on 13 July 1993
title: 'Theory of adsorption and surfactant effect of Sb on Ag (111)'
---
=10000
The goal of epitaxial crystal growth is to achieve atomically-flat and defect-free surfaces of specified crystallographic orientation, under the widest possible range of growth conditions. Significant efforts are devoted since many years to the growth of semiconductors. The epitaxial growth of metals on metal substrates has also attracted considerable interest (see for example Ref.[@mgrowth] and references therein).
Layer-by-layer, or two-dimensional (2D), growth is such that the epitaxial layer being currently deposited is completed before further layers begin to grow on top of it; this mode is also named Frank-van der Merwe. In the three-dimensional (3D) or cluster growth, many overlayers grow at the same time, none of them being completed, so that the surface exhibits 3D islands. For heteroepitaxy, depending on whether the 3D mode manifests itself immediately or only after the formation of a few 2D overlayers, the 3D growth is named either after Volmer and Weber, or Stransky and Krastanov.
With a view at extending the external conditions for the growth of high-quality 2D surfaces towards lower temperatures and higher deposition rates, use has been made recently of surface contaminants which purportedly act as surfactants. Although by definition[@oxf] a surfactant should reduce the surface formation energy[@ssurfact], there is at present no consensus as to the actual mechanisms of surfactant action, [*e.g.*]{} as to whether the contaminant affects the surface energy or the kinetics of growth[@mass]; the term surfactant is thus often used in the broader sense that it promotes 2D growth as opposed to 3D growth.
The surfactant technique, although still in its infancy, has been by now rather widely applied[@ssurfact; @mass] in the field of semiconductors to help regular growth of heteroepitaxial strained layers. On the other hand, we are only aware of one report of surfactant-assisted growth of metals[@vdv]. This is concerned with the homoepitaxy of the (111) surface of Ag. The growth mode was found to be drastically altered, from 3D to layer-by-layer 2D, by the one-time deposition of Sb at the beginning of the growth process, at coverages $\Theta$ between 0.05 and 0.2. Clean Ag (111) was observed to grow in a 3D fashion between 250 and 400 K, as signaled by the exponential decrease of the reflected x-ray beam intensity which monitors the degree of coherence of the upper layers of the sample [@vdv]. A crossover to step-flow growth (corresponding to constant reflected intensity) was observed above 450-500 K. In the presence of Sb, an oscillatory behaviour of the reflected intensity was observed instead, which is a fingerprint of 2D growth. The layer-by-layer growth of Sb-precovered Ag (111) continues for a rather long time (typically equivalent to the growth of 25 monolayers or more), at a nominal Ag deposition rate of 0.02 monolayer per second, even at 280-300 K.
The actual growth mode of an ideal clean Ag surface is still unclear. Indeed, Ag would be expected to grow layer-by-layer even at rather low temperatures; 3D cluster growth might be initiated [*e.g.*]{} by nucleation at surface defects. This notwithstanding, Sb unambiguously promotes the 2D growth of Ag, and it is therefore important to investigate the pertaining mechanism. With this aim, we have performed [*ab initio*]{} studies of the energetics of Sb and Ag adsorption on Ag (111). The calculations presented in this Letter are performed at Sb coverages down to $\Theta=1/4$. Among the considered adsorption geometries, the most stable one is the substitutional site, Sb being bound into a Ag surface vacancy. This site is considerably more favourable than the conventional on-surface fcc sites, and than subsurface sites. Thus, an Sb-Ag alloy layer forms in the surface layer. Dissolution of adsorbed Sb into bulk Ag is energetically disfavoured. When covered with Ag, the substitutional surface alloy reforms as the topmost surface layer by segregation of Sb. Due to the need of forming surface vacancies, the formation of the substitutional surface alloy needs thermal activation; we predict that at the relevant temperature, at low enough coverages, disordering of the surface alloy should take place.
Calculations for Ag adsorbed on clean and substitutional Sb-covered Ag (111) give informations on the growth mode of Ag. The main result is that Ag is more bound on clean portions of the surface, while the vicinity of substitutional Sb centers is less favorable. This implies that the average barrier for Ag diffusion is increased, thus Sb reduces the surface mobility of Ag. This causes a high Ag island density, which reduces the probability of 3D growth. The segregation of Sb to the surface layer allows the process to continue.
The calculations were performed within density-functional theory[@gd] in the local density approximation (LDA), using the all-electron full-potential linear muffin-tin orbitals (LMTO) method[@fp]. We used a non-relativistic code, which gives a very good description of Ag (bulk equilibrium properties: $a_0^{\rm th}=7.73$ bohr, $B^{\rm th}=1.10$ Mbar, zero-point energy not included, to be compared with the low-temperature experimental values $a_0^{\rm exp}=7.74$ bohr, $B^{\rm exp}=1.01$ Mbar) and Sb, which is only slightly heavier. The (111) surface of fcc Ag, clean or with Sb coverages of $\Theta$=1, $\Theta$=1/3, and $\Theta$=1/4 (whereby the 1$\times$1, ($\sqrt{3}$$\times$$\sqrt{3}$) $R$30$^{\circ}$, and 2$\times$2 cells were used respectively), was simulated by slabs of thickness ranging from 5 to 13 atomic layers, separated by 10 layers of vacuum. The supercells contained a number of atoms ranging from 7 to 30. The [**k**]{}-summation was done on a uniform mesh in the irreducible part of the surface Brillouin zone, encompassing 19 points for the 1$\times$1 cell, 7 and 13 points for the clean and adsorbate-covered $\sqrt{3}\times\sqrt{3}$ cell, 5 and 9 points for the clean and adsorbate-covered 2$\times$2 cell. The vertical position of the adsorbates was optimized. Substrate relaxation is neglected, but is expected to change the adsorption energies only marginally. Full details of this study will be presented elsewhere[@noi].
The binding energy for Sb on-surface fcc adsorption is $$E_{\rm ad}^{ \rm fcc} = -(\frac{1}{2}E^{\rm Sb/Ag(111)} -
\frac{1}{2}E^{\rm Ag(111)} - E^{\rm Sb}_{\rm atom})$$ with $E^{\rm Sb/Ag(111)}$, $E^{\rm Ag(111)}$, and $E^{\rm Sb}_{\rm atom}$ being the total energies of the adsorbate-covered slab, of the clean Ag slab, and of the spin-polarized Sb free atom; the factor 1/2 accounts for the facts that we adsorb on both slab sides. In the case of substitutional Sb adsorption, a slightly different definition applies: $$E_{\rm ad}^{ \rm sub} = -[(\frac{1}{2}E^{\rm Sb/Ag(111) sub}
+ E^{\rm Ag}_{\rm bulk}) \\
- (\frac{1}{2}E^{\rm Ag(111)} - E^{\rm Sb}_{\rm atom})]$$ where $E^{\rm Ag}_{\rm bulk}$ is the bulk total energy per atom of fcc Ag. The substitutional process implies in fact that a surface vacancy be created, and the kicked-out Ag atom migrates to a kink site at a surface step, thus gaining the cohesive energy[@neu]. While the formation of a surface vacancy costs energy, the subsequent binding of the adsorbate into the vacancy leads to a net energy gain. While at $\Theta$=1 we only have on-surface adsorption, substitutional adsorption with Sb adatoms being not nearest neighbours is possible for all $\Theta\leq$1/3 (we did not consider the coverage 0.3$<\Theta<$1).
The calculated adsorption energies for the substitutional, fcc on-surface, and sublayer adsorption are given in Table \[tab1\] for all coverages studied. As seen from the Table, the substitutional site is greatly favoured with respect to “normal” on-surface fcc adsorption, and also against sublayer adsorption. Sb is thus expected to be adsorbed in substitutional sites[@surfe].
An obstacle to the establishment of a substitutional adsorbate superstructure is the energy barrier which may exist for vacancy formation. To estimate the barrier, we calculated the formation energy of a distant Frenkel pair[@neu], consisting of an isolated Ag adatom on Ag (111) plus a vacancy. The results are summarized in Table \[tab2\]. The resulting maximum barrier of about 1.5 eV corresponds to an activation temperature of about 500 K. At that temperature the surface mobility of Ag atoms on Ag (111) is very high, so that migration of the atom released from the vacancy to a kink site is easily achieved. We note that, if dissipated locally, the adsorption energy of Sb in the fcc site would be more than enough to create a surface vacancy.
As a check as to whether Sb might be incorporated into the bulk of Ag, we calculated the adsorption energy for Sb in a sublayer site, [*i.e.*]{} below one overlayer of Ag. As seen from Table \[tab1\], this site is strongly disfavored with respect to Sb sitting in a substitutional site [*in*]{} the surface layer. For Sb below two Ag overlayers, the adsorption energy decreases further. We conclude that Sb segregates to the surface and is not incorporated into Ag. Indeed, the segregation of Sb (and the ensuing reduction of surface energy) in transition and noble metals and alloys has been known for some time in metallurgy[@seg]. In the present case, segregation is essentially due to the size difference of Ag and Sb, as has been checked by additional calculations at a 5% increased lattice constant. For $\Theta$=1/3, this gives that the difference of substitutional and sublayer adsorption energies drops from 1.1 eV/atom to about 0.65 eV/atom. Sb is just about the right size to fit into a surface vacancy, but it is somewhat too large for a bulk vacancy. Since Sb is confined into the surface layer, an Sb-Ag alloy layer will form at the (111) surface of Ag upon submonolayer Sb deposition. It is worth noticing that the substitutional configuration on the fcc (111) surface has a first-neighbour geometry very close to that of SbAg$_3$, the only stable ordered Sb-Ag compound known, having a tetragonally-distorted fcc structure[@vill].
The substitutional Sb adsorbate sits in the surface vacancy in a position very close to the ideal fcc location of the substituted Ag atom, with an outward relaxation of only 5 to 8 % of the interlayer spacing, [*ie.*]{} about 0.25-0.35 Å. Due to the effective in-plane screening thus provided by the surrounding substrate atoms, the substitutional Sb adatoms interact only weakly with each other; the adsorption energy for the substitutional site does not change much at low coverage if the local environment for substitutional Sb is conserved (see Table \[tab1\]). If we assume the $\Theta$=1/4 adsorption energy to be the low coverage limit value, and the coverage to be low enough, the entropic contribution to the free energy can overcome the internal energy difference at relatively low temperatures. At a coverage of $\Theta\simeq$0.1, the annealing temperature (say, 600 K) is sufficient to cause disordering with respect to the $\sqrt{3}$$\times$$\sqrt{3}$ arrangement. If the substitutional adsorption is activated by annealing, we therefore expect that the substitutional surface alloy thus obtained will be disordered.
To clarify the effects of Sb adsorption on the growth mode of Ag, we studied Ag adsorption on clean and Sb-covered Ag (111). We used the 2$\times$2 cell for these studies, both because neighboring adsorbates are reasonably decoupled from each other, and because Ag can be adsorbed on the substitutional Sb-covered surface either as a nearest neighbour to Sb, or not. We call these two sites “near” and “far”. The adsorption energies are summarized in Table \[tab3\]. The main result is that Ag has a higher adsorption energy on clean Ag than at both of the sites on Sb-covered Ag. Among the latter sites, the “near” site is marginally disfavored, and it would be probably more so if Sb had been allowed to relax outwards (see Table \[tab1\]).
It is thus energetically preferrable for Ag to sit on clean portions of the surface, while the vicinity of substitutional Sb centers is unfavourable. This could be called long-range “site” blocking, as the interaction giving rise to it is apparently long-ranged. We put “site” in quotes because the adsorbate potential energy is expected to change gradually as the adsorbate approaches the Sb centers, so that the average diffusion barrier for Ag increases already at some distance from, and not only [*at*]{}, the Sb centers. Diffusion barriers for Ag on Ag (111) are smaller than 0.1 eV: near an Sb center, they increase significantly, namely to about 0.4-0.5 eV. As a consequence, adsorbed Sb in the substitutional configuration reduces the surface mobility of Ag. The presence of substitutional Sb should therefore favor the growth of small-sized Ag islands. If the island density is high, one expects that they coalesce into a single layer before overgrowth on the islands can occur, as it is generally believed that small islands have lower energy barriers at descending steps. As deposited Ag covers the Ag:Sb surface alloy layer, Sb atoms find themselves in the disfavoured sublayer configuration, and will thus tend to segregate to the new surface layer. The alloy surface layer is thus reestablished, and the process can start again.
Recent STM experiments[@vdv2] on this system have indeed shown that on annealed Sb-covered surfaces, Sb induces a high density of small Ag islands on the surface, and that it efficiently segregates upon deposition of Ag. Another observation is that upon annealing at about 550 K, Sb is adsorbed substitutionally and, at very low coverages, it forms a disordered 2D array, in agreement with our prediction. For the unannealed surface, the on-surface fcc site is occupied at room temperature; this agrees with our estimate of the activation of substitutional adsorption. At very low coverage, Sb is observed to form islands. Our largest calculated adsorption energy for on-surface adsorption is that of the 2$\times$2 superstructure; we cannot exclude however that the adsorption energy may increase further in the extreme low-coverage limit, which is computationally very demanding and has not been addressed here.
In summary, we presented [*ab initio*]{} calculations of Sb and Ag adsorption on clean and Sb-covered Ag (111). For Sb, the substitutional adsorption site is energetically highly favoured with respect to “normal” on-surface sites; in addition, subsurface positions are also strongly disfavoured. Sb is thus effectively confined into the surface and forms a segregating surface alloy. This alloy should disorder, at low coverages, for typical annealing temperatures. As to Ag, we find it to be sizably more bound on clean Ag(111) than on substitutional Sb-covered Ag: this indicates that Sb produces a site blocking, or more precisely, a significant increase of the diffusion barrier for Ag adatoms approaching the Sb centers. Based on these results, we offered an explanation of the recently observed Sb-induced layer-by-layer homoepitaxial growth of Ag (111): substitutionally-adsorbed Sb induces, by mobility reduction, a high density of small-sized Ag islands which coalesce into a regular 2D layer; as Ag covers the surface, Sb segregates to the newly formed layer, thus reestablishing the alloy layer at the surface, and the process starts again. Most of our results seem to be confirmed by recent STM experiments[@vdv2].
We thank R. Stumpf for helpful discussions, and J. Vrijmoet for communicating his results prior to publication. This work was partly supported by the Deutsche Forschungsgemeinschaft within Sonderforschungsbereich 1421.
Present address: Dipartimento di Scienze Fisiche, Università di Cagliari, via Ospedale 72, I-09124 Cagliari, Italy. F. J. A. den Broeder, D. Kuiper, A. P. van den Mosselaer, and W. Hoving, Phys. Rev. Lett. [**60**]{}, 2769 (1988); W. F. Egelhoff and I. Jacob,[*ibid.*]{} [**62**]{}, 921 (1989); M. Bott, T. Michely, and G. Comsa, Surf. Sci, [**272**]{}, 161 (1992). (Oxford UP, Oxford 1988). D. J. Eaglesham, F. C. Unterwald, and D. C. Jacobson, Phys. Rev. Lett. [**70**]{}, 966 (1993), and references therein. C. W. Snyder and B. G. Orr, Phys. Rev. Lett. [**70**]{}, 1030 (1993); N. Grandjean and J. Massies, [*ibid.*]{}, 1031, and references therein. H. A. van der Vegt, H. M. van Pinxteren, M. Lohmeier, E. Vlieg, and J. M. C. Thornton, Phys. Rev. Lett. [**68**]{}, 3335 (1992). See [*e.g.*]{} R. Dreizler and E. Gross, [*Density functional theory*]{}, (Springer, Berlin, 1990). The exchange-correlation energy is by D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. [**45**]{}, 566 (1980), as parametrized by S. Wosko, L. Wilk, and M. Nusair, Can. J. Phys. [**58**]{}, 1200 (1980). M. Methfessel, Phys. Rev. B [**38**]{}, 1537 (1988); M. Methfessel, C. O. Rodriguez, and O. K. Andersen, [*ibid.*]{} [**40**]{}, 2009 (1989). For surface studies, see M. Methfessel, D. Hennig, and M. Scheffler, Phys. Rev. [**B 46**]{}, 4816 (1992). S. Oppo, V. Fiorentini, and M. Scheffler, to be published. J. Neugebauer and M. Scheffler, Phys. Rev [**46**]{}, 16067 (1992); A. Schmalz, S. Aminpirooz, L. Becker, J. Haase, J. Neugebauer, M. Scheffler, D. R. Batchelor, D. L. Adams, and E. B${\rm \o}$gh, Phys. Rev. Lett. [**67**]{}, 2163 (1991). The results may be reformulated in terms of surface energy reduction caused by adsorption. This requires the definition of a chemical potential for the atoms to be adsorbed. Here we adopt the adsorption energy terminology. See [*e.g.*]{} M. Jenko, F. Vodopivec, and B. Pra$\check{\rm c}$ek, Appl. Surf. Sci. [**70/71**]{}, 118 (1993) and references therein; M. Hondros and A. Mc Lean, in [*Surface Phenomena of Metals*]{}, Soc. Chem. Ind. Monograph Nr. 28 (Soc. Chem. Ind., London 1969), p. 39. P. Villars, K. Matthis, and F. Hulliger, in [*The structure of binary compounds*]{}, F. R. De Boer and D. G. Pettifor eds., p.1 (Elsevier, Amsterdam, 1989). J. Vrijmoeth, private communication.
----------------------------- ------ ------ ------
E$_{\rm ad}^{\rm sub}$ 4.49 4.37 —
[relaxation]{} +5% +8% —
E$_{\rm ad}^{\rm fcc}$ 3.26 3.34 3.22
[relaxation]{} –11% –5% +6%
E$_{\rm ad}^{\rm sublayer}$ 3.41 3.45 2.71
----------------------------- ------ ------ ------
: Adsorption energies (in eV/atom) of Sb on Ag (111) for the adsorption sites and coverages studied here. Vertical adsorbate relaxations compared to ideal silver fcc position (in percentage of interlayer spacing) are also given.[]{data-label="tab1"}
----------------------- ------ ------
E$_{\rm f}^{\rm vac}$ 0.69 0.66
E$_{\rm f}^{\rm Fp}$ 1.46 1.43
----------------------- ------ ------
: Vacancy and Frenkel pair formation energies (eV)[]{data-label="tab2"}
------------------------ ------ ------ ------
$E_{\rm ads}^{\rm Ag}$ 2.41 2.02 1.99
[relaxation]{} –9% –9% –5%
------------------------ ------ ------ ------
: Adsorption energy (eV/atom) and relaxation compared to the ideal Ag position (in percentage of ideal interlayer spacing) for Ag on clean and Sb-covered Ag (111). Clean: Ag on Ag (111); far: Ag on Sb:Ag(111), “far” site; near: Ag on Sb:Ag(111), “near” site.[]{data-label="tab3"}
|
---
abstract: 'We present an [*ab initio*]{} GW calculation to study dynamical effects on an organic compound (TMTSF)$_2$PF$_6$. Calculated polarized reflectivities reproduce experimental plasma edges at around 0.2 eV for $E\|b''$ and 1.0 eV for $E\|a$. The low-energy plasmons come out from the low-energy narrow bands energetically isolated from other higher-energy bands, and affect the low-energy electronic structure via the GW-type self-energy. Because of the quasi-one-dimensional band structure, a large plasmon-induced electron scattering is found in the low-energy occupied states along the X-M line.'
author:
- 'Kazuma Nakamura$^{1}$'
- 'Shiro Sakai$^{2}$'
- 'Ryotaro Arita$^{2}$'
- 'Kazuhiko Kuroki$^{3}$'
date: '10 April, 2013'
title: '[*Ab initio*]{} GW calculation for organic compounds (TMTSF)$_2$PF$_6$'
---
Introduction {#sec:intro}
============
Physics and chemistry of organic conductors have recently attracted much attention owing to its low dimensionality, strong electron correlation, and material variety and flexibility. Despite the complicated structure of the molecules themselves, rather simple and energetically isolated band structure commonly seen in these materials provides ideal basis for studying fundamental physics of electron correlation. Among a number of organic conductors, (TMTSF)$_2$PF$_6$ has been of particular interest ever since its discovery as the first organic superconductor. [@Bechgaard] In this material, the molecules are stacked along the $a$-axis, and this is the most conductive axis because the molecular orbitals are elongated in the stack direction. A small overlap between neighboring molecular orbitals in the interstack direction gives rise to a weak two-dimensionality. The importance of electron correlation can readily be seen from the presence of the spin density wave (SDW) phase, which takes place below 12K at ambient pressure. [@Ishiguro-Yamaji-Saito] The SDW transition temperature decreases upon applying hydrostatic pressure, and superconductivity sits next to the SDW phase in the temperature-pressure phase diagram. The superconducting state also shows some interesting features suggesting unconventional pairing, which may also be a manifestation of electron correlation. [@Kuroki]
Theoretically, the microscopic origin of the density waves and superconductivity has been intensively investigated on simplified Hubbard type models where on-site and/or short-ranged off-site repulsions are taken into account. [@Kuroki] On the other hand, recent [*ab initio*]{} studies on organic materials show that the long-range part of the Coulomb interactions is appreciably present, [@Nakamura-organic-1; @Nakamura-organic-2] suggesting that dielectric properties can also be of interest. In fact, reflectance measurements of (TMTSF)$_2$PF$_6$ show presence of plasma edge, [@Dressel; @Jacobsen] indeed indicating the importance of the long range Coulomb interaction.
In general in solids, the energy scale of the plasmon excitation is of the order of 10 eV and therefore it is believed that such excitations are irrelevant to the low-energy physics of the order of 0.1-1 eV. However, in the organic materials, low-energy bands around the Fermi level tend to be isolated from other high-energy bands, resulting in a plasmon characterized by the bandwidth and occupancy of the isolated low-energy bands. Since their bandwidth is typically of the order of $\sim$1 eV, the plasma frequency can also be in this energy scale. Then, the plasmon excitation may produce new aspects in the low-energy electronic states through self-energy effects. Since recent progress in angle-resolved photo-emission spectroscopy [@ARPES-1; @ARPES-2; @ARPES-3] has made it possible to measure quasiparticle band structure and to perform detailed self-energy analyses, corresponding first-principle calculations are highly desired.
In the present study, we present an [*ab initio*]{} GW calculation to study dynamical effects on the electronic structure of the organic compound (TMTSF)$_2$PF$_6$. The GW calculation takes into account the effect of plasmon excitation. [@Hedin; @Hybertsen; @Aryasetiawan; @Onida; @GW-1; @GW-2; @GW-3; @GW-4; @GW-5; @GW-6; @GW-7; @GW-8; @GW-9; @GW-10; @GW-11; @GW-12; @GW-13; @GW-14; @GW-15; @GW-16; @GW-17; @GW-18; @GW-19; @GW-20; @Nohara] The calculated reflectances well reproduce experimental results, identifying the experimentally-observed plasma edges to the plasmons within the low-energy bands. By calculating GW self energy and spectral function, we will show that this low-energy plasmon excitation affects the low-energy electronic structure. Since the isolated band character inducing the low-energy plasmon is ubiquitous in strongly-correlated electron systems such as organic compounds and transition-metal compounds, the present result will provide a general basis for analyzing various correlated materials.
Method {#sec:method}
======
Here we describe our scheme. The non-interacting Green’s function is given by $$\begin{aligned}
G_0({\bf r,r'},\omega) = \sum_{\alpha{\bf k}}
\frac{\psi_{\alpha{\bf k}}({\bf r}) \psi^{*}_{\alpha{\bf k}}({\bf r'})}
{\omega-\epsilon_{\alpha{\bf k}}+i \delta {\rm sgn}(\epsilon_{\alpha{\bf k}}-\epsilon_f)}, \end{aligned}$$ where $\psi_{\alpha {\bf k}} ({\bf r})$ and $\epsilon_{\alpha {\bf k}}$ are the Kohn-Sham (KS) wavefunction and its eigenvalue of band $\alpha$ and wavevector ${\bf k}$, and $\epsilon_f$ is the Fermi level. $\delta$ is chosen to be a small but finite positive value to stabilize numerical calculations. A polarization function of a type $-i G_0 G_0$ is written in a matrix form in the plane wave basis as $$\begin{aligned}
\chi_{{\bf GG'}}&({\bf q},\omega)&
\!=\!2\sum_{{\bf k}}\sum^{vir}_{\alpha}\sum^{occ}_{\beta}
M_{\alpha\beta}^{{\bf G}}({\bf k,q})
M_{\alpha\beta}^{{\bf G'}}({\bf k,q})^{*} \nonumber \\
\!\!\!&\times&\!\!\!\Biggl\{\!\!
\frac{1}{\omega\!-\!\epsilon_{\alpha{\bf k\!+\!q}}\!+\!\epsilon_{\beta{\bf k}}\!+\!i\delta}\!-\!
\frac{1}{\omega\!+\!\epsilon_{\alpha{\bf k\!+\!q}}\!-\!\epsilon_{\beta{\bf k}}\!-\!i\delta}\!\!\Biggr\}
\label{eq:chi} \end{aligned}$$ with $M_{\alpha\beta}^{{\bf G}}({\bf k,q})=\langle \psi_{\alpha {\bf k+q}}|e^{i({\bf q+G}){\bf r}}|\psi_{\beta {\bf k}} \rangle$. Optical properties in a metal are related to the symmetric dielectric function [@Hybertsen] in the ${\bf q}\to0$ limit $$\begin{aligned}
\epsilon_{{\bf G\!G'}}(\omega)\!=\!\delta_{{\bf G\!G'}}\!-\!\frac{(\omega_{pl,\mu\mu})^2}{\omega(\omega\!+\!i\delta)}\delta_{{\bf G0}}\delta_{{\bf G'0}}\!-\!\!\lim_{{\bf q}\to0}\!\frac{4\pi}{N\Omega}\!\frac{\chi_{{\bf GG'}}^{inter}({\bf q},\omega)}{|{\bf q\!+\!G}||{\bf q\!+\!G'}|} \nonumber \\
\label{eq:eps}\end{aligned}$$ with ${\bf q}$ approaching zero along the Cartesian $\mu$ direction. $N$ is the total number of sampling $k$ points and $\Omega$ is the unitcell volume. The second term is the Drude term due to the intraband transition around the Fermi level. The third term represents the interband contribution, where $\chi_{{\bf GG'}}^{inter}({\bf q},\omega)$ is a polarization matrix due to the interband transitions. Plasma frequency in the second term is given in a tensor form by [@Draxl-1; @Draxl-2] $$\begin{aligned}
\omega_{pl,\mu\nu}=\sqrt{\frac{8\pi}{\Omega N} \sum_{\alpha{\bf k}} p_{\alpha {\bf k},\mu} p_{\alpha {\bf k},\nu} \delta(\epsilon_{\alpha {\bf k}}-\epsilon_f}) \label{wpl} \end{aligned}$$ with $p_{\alpha{\bf k},\mu}$ being a matrix element of a momentum as $$\begin{aligned}
p_{\alpha {\bf k},\mu} = - i \langle \psi_{\alpha {\bf k}}| \frac{\partial}{\partial x_{\mu}}+[V_{NL}, x_{\mu}] |\psi_{\alpha {\bf k}} \rangle,
\label{p_ij} \end{aligned}$$ where $V_{NL}$ is the non-local part of the pseudopotential.
The GW self-energy is given by $$\begin{aligned}
\Sigma({\bf r,r'},\omega)\!=\!i\!\int \frac{d\omega'}{2\pi} G_0({\bf r,r'},\omega+\omega') W({\bf r,r'},\omega').
\label{GW_SIGMA}\end{aligned}$$ The screened Coulomb interaction $W(\omega)=v^{\frac{1}{2}}\epsilon^{-1}(\omega)v^{\frac{1}{2}}$ is decomposed into the bare Coulomb interaction $v$ and the frequency-dependent part $W_C(\omega)=W(\omega)-v$. The frequency integral of $iG_0 v$ gives the bare exchange term $\Sigma_X$ and that of $iG_0 W_C$ gives the correlation term $\Sigma_C(\omega)$ including the retardation effect. The calculation of $\Sigma_X$ is straightforward while that of $\Sigma_C(\omega)$ is somewhat technical. In the present calculation, we fit the following function to [*ab initio*]{} $W_C(\omega)$, [@Nohara] $$\begin{aligned}
\tilde{W}_C({\bf r,r'},\omega)
= \sum_{j} \Biggl( \frac{1}{\omega-z_j} + \frac{1}{\omega+z_j} \Biggr) a_j( {\bf r,r'}), \end{aligned}$$ where $z_j$ and $a_j({\bf r,r'})$ are the pole and amplitude of the model interactions, respectively. Since the frequency-dependent part is decoupled from the amplitude in $\tilde{W}_C$, the frequency integral in $iG_0\tilde{W}_C$ can be analytically performed. The resulting matrix elements of $\Sigma_C(\omega)$ is $$\begin{aligned}
\langle \psi_{\alpha {\bf k}}| \Sigma_C(\omega) |\psi_{\alpha {\bf k}} \rangle\!\!=\!\!\sum_{jn{\bf q}}
\frac{\langle \psi_{\alpha {\bf k}} \psi_{n {\bf k-q}}| a_j |\psi_{n {\bf k-q}} \psi_{\alpha {\bf k}} \rangle}
{\omega\!-\!\epsilon_{n{\bf k-q}}\!-\!(z_j\!-\!i\delta)\!{\rm sgn}\!(\epsilon_{n{\bf k-q}}\!-\!\epsilon_f)}. \label{eq:SGM} \nonumber \\ \end{aligned}$$
The spectral function is calculated by $$\begin{aligned}
A({\bf k},\omega)=\frac{1}{\pi} \sum_{\alpha} \Biggl| {\rm Im} \frac{1}{\omega-(\epsilon_{\alpha {\bf k}} + \Delta \Sigma_{\alpha {\bf k}} (\omega) + \Delta)} \Biggr|, \label{Akw}\end{aligned}$$ where $\Delta \Sigma_{\alpha {\bf k}} (\omega)\!=\!\langle \psi_{\alpha {\bf k}}| \Sigma({\bf r,r'},\omega)\!-\!v_{{\rm XC}}({\bf r})\delta({\bf r}\!-\!{\bf r'}) |\psi_{\alpha {\bf k}} \rangle$, and $v_{{\rm XC}}$ is the exchange correlation potential. A shift $\Delta$ is introduced to keep the electron density to be the same as that of KS, $N_{elec}$, i.e., $\int_{-\infty}^{\epsilon_f} d\omega \int d{\bf k} A({\bf k},\omega)=N_{elec}$.
results and discussions {#sec:result}
=======================
Our density-functional calculations are based on Tokyo Ab-initio Program Package (TAPP) (Ref. ) with plane-wave basis sets, where we employ norm-conserving pseudopotentials [@PP-1; @PP-2] and generalized gradient approximation (GGA) for the exchange-correlation potential. [@PBE96] The experimental structure of (TMTSF)$_2$PF$_6$ obtained by a neutron measurement [@structure-TMTSF] at 20 K is adopted. The cutoff energies in wavefunction and in charge densities are 36 Ry and 144 Ry, respectively, and an 11$\times$11$\times$3 $k$-point sampling is employed. The maximally localized Wannier function (MLWF) (Refs. and ) is used for interpolation of the self-energy and spectral function to a finer $k$ grid. The cutoff of polarization function is set to be 3 Ry and 198 bands are considered, which cover an energy range from the bottom of the occupied states near $-$30 eV to the top of the unoccupied states near 15 eV . The integral over the Brillouin-zone (BZ) is evaluated by the generalized tetrahedron method. [@Fujiwara; @Nohara] The polarization up to $\omega$=86 eV is calculated in a logarithmic mesh with 110 energy points. The frequency dependence of the self-energy for the states near the Fermi level is calculated for \[$-$30 eV: 30 eV\] with the interval of 0.01 meV. The broadening $\delta$ in Eqs. (\[eq:chi\]), (\[eq:eps\]) and (\[eq:SGM\]) is set to 0.02 eV. The self energy at ${\bf q}$=${\bf G}$=${\bf 0}$ is treated in the manner in Ref. .
Figure \[Fig1\] (a) shows the calculated GGA band structure of (TMTSF)$_2$PF$_6$. We find two narrow bands around the Fermi level, which are well separated in energy from other higher-energy bands. The appearance of such isolated low-energy bands is common to various organic conductors. [@Nakamura-organic-1; @Nakamura-organic-2; @organic-1; @organic-2; @organic-3; @organic-4; @organic-5] We assign these bands to “target bands" for which the self-energy effects are considered below. We construct MLWFs for the target bands and evaluate the transfer integrals as shown in the panel (b). The transfer integrals well reproduce the original bands as shown by the blue-dotted curves in the panel (a). Note that the transfers along the $a$ axis are about four times larger than those along the $b$ axis, reflecting a quasi-one-dimensional structure of the compound.
![(Color online) (a) GGA band structure of (TMTSF)$_2$PF$_6$, along the high-symmetry lines in the $ab$ plane, where $\Gamma$=(0, 0, 0), X=($a^{*}$/2, 0, 0), M=($a^{*}$/2, $b^{*}$/2, 0), Y=(0, $b^{*}$/2, 0). The Fermi level is at zero energy. (b) Schematic crystal structure in the $ab$ plane, where the unit cell contains two TMTSF molecules denoted by the ellipsoids. The number in the ellipsoids represent the transfer integral (in the unit of meV) between the highest-occupied molecular orbitals at the starred site and each site. The tight-binding bands \[blue dotted curves in the panel (a)\] are calculated from these transfers.[]{data-label="Fig1"}](Fig1.eps){width="50.00000%"}
Figure \[Fig2\] (a) shows the calculated energy loss function $-{\rm Im}\epsilon^{-1}(\omega)$. The red-solid and green-dotted curves are results for the light polarization $E$ parallel to $a$ (along $x$ axis) and $b'$ (along $y$ axis), respectively. In the low-energy region, we find two plasmon peaks, at $\sim$0.2 eV for $E\|b'$ and $\sim$1.0 eV for $E\|a$. These peaks result from the plasmon excitations within the isolated target bands which have a low carrier density and small band widths. These plasmons are distinct from the plasmon seen at around 20 eV, which is relevant to the total charge density and the bare electron mass. We note that the difference between the plasmon peaks for $E\|a$ and $E\|b'$ reflects the difference in the transfer integrals ($\sim$260 meV along the $a$ axis and $\sim$60 meV along the $b$ axis).
Figure \[Fig2\] (b) compares the reflectance, $$\begin{aligned}
R_{\mu}(\omega)=\Biggl| \frac{1-\sqrt{\epsilon_{\mu}^{-1}(\omega)}}{1+\sqrt{\epsilon_{\mu}^{-1}(\omega)}} \Biggr|, \end{aligned}$$ between our theory (circles) and the experiment (crosses) of Ref. . The calculated result reproduces the experimental one fairly well, in particular for $E\|a$ (dark red). The smaller energy scale of $E\|b'$ (light green) is also qualitatively reproduced.
To quantify the comparison, we fit the following function to theoretical and experimental reflectance data $$\begin{aligned}
\epsilon_{\mu}(\omega)=\epsilon_{{\rm core,\mu}}-\frac{\omega_{pl,\mu\mu}^2}{\omega(\omega+i\delta)}.
\label{Drude-model}\end{aligned}$$ For the theoretical data, $\omega_{pl,\mu\mu}$ is calculated with Eq. (\[wpl\]) and $\delta$ is fixed at 0.02 eV, so that the effective dielectric constant $\epsilon_{\rm core,\mu}$ is the only free parameter in the fitting. Table \[PARAM-Drude\] summarizes the values of $\epsilon_{{\rm core},\mu}$ and $\omega_{pl,\mu\mu}$. The calculated results well reproduce the experimental ones, [@Note_delta] although the theoretical plasma edge of $E\|b'$ is by $\sim$1.5 times higher than that of experiment.
![(Color online) (a) Calculated energy loss function of (TMTSF)$_2$PF$_6$. The red-solid and green-dotted curves represent the spectra for $E\|a$ and $E\|b'$, respectively. (b) [*Ab initio*]{} reflectivity (circles) and experimental one (crosses) at 25 K (Ref. ). The results for $E\|a$ and $E\|b'$ are displayed by dark red and light green, respectively. The solid and dotted curves are the fits by Eq. (\[Drude-model\]) with parameters shown in Table \[PARAM-Drude\].[]{data-label="Fig2"}](Fig2-new.eps){width="40.00000%"}
[lc@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c]{}\
\[-8pt\] & & &\
\
\[-8pt\] & $\epsilon_{{\rm core}}$ & $\omega_{pl}$ & & $\epsilon_{{\rm core}}$ & $\omega_{pl}$\
\
\[-8pt\] Theory & 2.5 & 10074 & & 3.7 & 3331\
Expt. & 2.9 & 11400 & & 4.5 & 2360\
\[PARAM-Drude\]
The low-energy plasmons found in Fig. \[Fig2\] can affect the low-energy electronic structure. To see this effect, we show in Fig. \[Fig3\] (a) the GW spectral function $A({\bf k},\omega)$ \[Eq. (\[Akw\])\]. While the quasiparticle band structure around $-$1.0-0.1 eV is similar to that of KS (red solid curves), we see an appreciable weight transfer to higher energy due to the self-energy effects. Along the Y-$\Gamma$ line, the new states emerge around by 1 eV above the unoccupied states, and around by 1 eV below the occupied states. Along the X-M line, the spectra are more broadened and spread in the range from $-$1.5 to 0 eV. In the panel (b), the density of states calculated by KS (red-solid curve) and by GW (green-dotted one) is displayed, from which we see that a considerable amount of weight is transferred to higher energy due to the self-energy effect.
![(Color online) (a) Calculated spectral function $A({\bf k},\omega)$ of (TMTSF)$_2$PF$_6$, superposed by the Kohn-Sham band structure (red curves). The Fermi level is at zero energy. (b) Density of states obtained by KS (red-solid curve) and GW (green-dotted one). []{data-label="Fig3"}](Fig3-new-low.eps){width="50.00000%"}
To get insight into the relation between the dielectric function (Fig. \[Fig2\]) and the spectral function (Fig. \[Fig3\]), we plot in Fig. \[Fig4\] ${\rm Im}\Sigma({\bf k},\omega)$=$\sum_{\alpha}|{\rm Im}\Sigma_{\alpha{\bf k}}(\omega)|$ \[(a)\] and ${\rm Im}\Sigma(\omega)$=$\int d{\bf k}{\rm Im}\Sigma({\bf k},\omega)$ \[(b)\]. Im$\Sigma({\bf k},\omega)$ is directly related to the dielectric function through Eq. (\[GW\_SIGMA\]), and to the spectral function through Eq. (\[Akw\]). We see that Im$\Sigma({\bf k},\omega)$ has strong intensities at 0.5$\sim$1.0 eV and $-$2.0$\sim$$-$0.5 eV; about 0.5 eV above the unoccupied states and about 0.5 eV below the occupied states. The energy scale of 0.5 eV roughly corresponds to the average of $\omega_{pl,b'}$$\sim$$0.2$ eV and $\omega_{pl,a}$$\sim$$1.0$ eV. Since the plasmons are known to make a peak in Im$\Sigma$ at energy $\epsilon_{unocc}$+$\omega_{pl}$ or $\epsilon_{occ}$$-$$\omega_{pl}$, [@Aryasetiawan] the bright region in Fig. \[Fig4\](a) can be interpreted as an effect of the low-energy plasmons in Fig. \[Fig2\], although the electron-electron scattering other than the plasmon excitation can also contribute to Im$\Sigma({\bf k},\omega)$. Since the strong peak of Im$\Sigma({\bf k},\omega)$ causes a large variation of Re$\Sigma({\bf k},\omega)$ through the Kramers-Kronig relation, new poles of the Green’s function can be created just outside of the peak of Im$\Sigma({\bf k},\omega)$. This is consistent with $A({\bf k},\omega)$ in Fig. \[Fig3\].
![(Color online) Calculated spectrum of (a) ${\rm Im}\Sigma({\bf k},\omega)$ and (b) ${\rm Im}\Sigma(\omega)$ of (TMTSF)$_2$PF$_6$. The Fermi level is at zero energy. The Kohn-Sham band structure (red solid curves) is superposed in the panel (a).[]{data-label="Fig4"}](Fig4-new-low.eps){width="50.00000%"}
Conclusion {#sec:conclusion}
==========
In summary, we study low-energy dynamical properties of an organic compound (TMTSF)$_2$PF$_6$ from first principles. Theoretical reflectance reproduces experimentally-observed plasma edges, and their anisotropy due to the quasi-one dimensional nature. The low-energy plasmons come out from the energetically-isolated bands around the Fermi level. The self-energy effect due to these plasmon excitations on the low-energy electronic structure is studied within the GW approximation. We have found that the self-energy effect is appreciable at energy by $\sim$0.5 eV above unoccupied states and below occupied states, suggesting that the plasmons can influence low-energy physics. Since organic conductors, or more generally, strongly-correlated electron materials, often have such isolated bands around the Fermi level, we expect that similar low-energy plasmon excitation can be relevant to physical properties of these materials. A detection of these effects in experiments such as photoem ission spectroscopy is an interesting future issue.
We would like to thank Takahiro Ito, Kyoko Ishizaka, Yoshiro Nohara, Yoshihide Yoshimoto, and Yusuke Nomura for useful discussions. Calculations were done at Supercomputer center at Institute for Solid State Physics, University of Tokyo. This work was supported by Grants-in-Aid for Scientific Research (No. 22740215, 22104010, 23110708, 23340095, 23510120, 25800200) from MEXT, Japan.
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|
---
abstract: 'It is understood that the Supernovae (SNe) associated to Gamma Ray Bursts (GRBs) are of type Ib/c. The temporal coincidence of the GRB and the SN represents still a major enigma of Relativistic Astrophysics. We elaborate here, from the earlier paradigm, that the concept of induced gravitational collapse is essential to explain the GRB-SN connection. The specific case of a close (orbital period $<1$ h) binary system composed of an evolved star with a Neutron Star (NS) companion is considered. We evaluate the accretion rate onto the NS of the material expelled from the explosion of the core progenitor as a type Ib/c SN, and give the explicit expression of the accreted mass as a function of the nature of the components and binary parameters. We show that the NS can reach, in a few seconds, the critical mass and consequently gravitationally collapses to a Black Hole. This gravitational collapse process leads to the emission of the GRB.'
author:
- 'Jorge A. Rueda and Remo Ruffini'
title: 'On the Induced Gravitational Collapse of a Neutron Star to a Black Hole by a Type Ib/c Supernova'
---
The systematic and spectroscopic analysis of the Gamma Ray Burst (GRB)-Supernova (SN) events, following the pioneering discovery of the temporal coincidence of GRB 980425 [@2000ApJ...536..778P] and SN 1998bw [@1998Natur.395..670G], has evidenced the association of other nearby GRBs with Type Ib/c SNe [see e.g. @2011IJMPD..20.1745D; @2011arXiv1104.2274H for a recent review on GRB-SN systems]. It has been also clearly understood that the most likely explanation of SN Ib/c, which lack Hydrogen (H)/Helium (He) in their spectra, is that the SN core progenitor star, likely a He, CO, or a Wolf-Rayet star, is in a binary system with a compact companion, a Neutron Star (NS) (see e.g. [@1988PhR...163...13N; @1994ApJ...437L.115I] and [@2007PASP..119.1211F; @2010ApJ...725..940Y] for more recent calculations).
In the current literature there has been the attempt to explain both the SN and the GRB as two aspects of the same astrophysical phenomenon. Hence, GRBs have been assumed to originate from a specially violent SN process, a hypernova or a collapsar . Both of these possibilities imply a very dense and strong wind-like CircumBurst Medium (CBM) structure. Such a dense medium appears to be in contrast with the CBM density found in most GRBs (see e.g. Fig. 10 in ). In fact, the average CBM density, inferred from the analysis of the afterglow, has been shown to be in most of the cases of the order of 1 particle cm$^{-3}$ [see e.g. @2011IJMPD..20.1797R]. The only significant contribution to the baryonic matter component in the GRB process is the one represented by the baryon load . In a GRB, the electron-positron plasma, loaded with a certain amount of baryonic matter, is expected to expand at ultra-relativistic velocities with Lorentz factors $\Gamma\gtrsim 100$ [@1990ApJ...365L..55S; @1993MNRAS.263..861P; @1993ApJ...415..181M]. Such an ultra-relativistic expansion can actually occur if the amount of baryonic matter, quantifiable through the baryon load parameter, $B=M_B c^2/E_{e^+e-}$, where $M_B$ is the engulfed baryon mass from the progenitor remnant and $E_{e^+e-}$ is the total energy of the $e^+e-$ plasma, does not exceed the critical value $B \sim 10^{-2}$ .
In our approach we have assumed that the GRB consistently has to originate from the gravitational collapse to a Black Hole (BH). The SN follows instead the complex pattern of the final evolution of a massive star, possibly leading to a NS or to a complete explosion but never to a BH. There is a further general argument in favor of this explanation, namely the extremely different energetics of SNe and GRBs. While the SN energy range is $10^{49}$–$10^{51}$ erg, the GRBs are in a larger and wider range of energies $10^{49}$–$10^{54}$ erg. It is clear that in no way a GRB, being energetically dominant, can originate from the SN.
There are however scenarios for GRB-SN systems that invoke a single progenitor, e.g. the collapse of massive stars [see e.g. @reviewzhang2011 for a recent review]. In these models the core of the star must rotate at very high rates in order to produce, during the gravitational collapse, a collimated (e.g. jet) emission with a beaming angle $\theta_j$. In this way, an event with observed isotropic energy $E_{iso}$, corresponds to an actual energy released at the source reduced by the beaming factor $f_b =(1-\cos\theta_j)\sim \theta^2_j/2<1$, namely $E_{s}=f_b E_{iso}<E_{iso}$ . Outstandingly small beaming factors of order $f_p\sim 1/500$, corresponding to jet angles $\theta_j \sim 1^\circ$, are needed to bring the most energetics GRBS with $E_{iso}\sim 10^{54}$ erg to standard energies $\sim 10^{51}$ erg [@2001ApJ...562L..55F]. However, observational evidence of the existence of so narrow beaming angles in GRBs, as suggested by these models, is inconclusive [see e.g. @2006NCimB.121.1171C; @2007ApJ...657..359S; @2009cfdd.confE..23B]. We explain the temporal coincidence of the two phenomena, the SN explosion and the GRB, within the concept of *induced gravitational collapse* [@2001ApJ...555L.117R; @2008mgm..conf..368R]. In the recent years we have outlined two different possible scenarios for the GRB-SN connection. In the first version [@2001ApJ...555L.117R], we have considered the possibility that the GRBs may have caused the trigger of the SN event. For the occurrence of this scenario, the companion star had to be in a very special phase of its thermonuclear evolution [see @2001ApJ...555L.117R for details].
More recently, we have proposed [@2008mgm..conf..368R] a different possibility occurring at the final stages of the evolution of a close binary system: the explosion in such a system of a Ib/c SN leads to an accretion process onto the NS companion. The NS will reach the critical mass value, undergoing gravitational collapse to a BH. The process of gravitational collapse to a BH leads to the emission of the GRB (see Fig. \[fig:scenario\]). In this Letter we evaluate the accretion rate onto the NS and give the explicit expression of the accreted mass as a function of the nature of the components and the binary parameters.
![Sketch of the accretion induced collapse scenario. An evolved star in close binary with a NS explodes as a SN Ib/c. The NS rapidly accretes a part of the SN ejecta and reaches in a few seconds the critical mass undergoing gravitational collapse to a BH, emitting the GRB.[]{data-label="fig:scenario"}](binary_scenario)
We turn now to the details of the accretion process of the SN material onto the NS. In a spherically symmetric accretion process, the magnetospheric radius is given by [see e.g. @2012MNRAS.420..810T] $R_m = B^2 R^6/(\dot{M} \sqrt{2 G M_{\rm NS}})^{2/7}$, where $B$, $M_{\rm NS}$, $R$ are the NS magnetic field, mass, radius, and $\dot{M}\equiv dM/dt$ is the mass-accretion rate onto the NS. We now estimate the relative importance of the NS magnetic field on the accretion process. At the beginning of a SN explosion, the ejecta moves at high velocities $v\sim 10^9$ cm s$^{-1}$, and the NS will capture matter at a radius approximately given by $R^{\rm sph}_{\rm cap} \sim 2 G M/v^2$. For $R_m << R^{\rm sph}_{\rm cap}$, we can neglect the effects of the magnetic field. In Fig. \[fig:RmRcap\] we have plotted the ratio between the magnetospheric radius and the gravitational capture radius as a function of the mass accretion rate onto a NS of $B=10^{12}$ Gauss, $M_{\rm NS}=1.4 M_\odot$, $R=10^6$ cm, and for a flow with velocity $v=10^9$ cm s$^{-1}$. It can be seen how for high accretion rates the influence of the magnetosphere is negligible.
![Magnetospheric to gravitational capture radius ratio of a NS of $B=10^{12}$ Gauss, $M_{\rm NS}=1.4 M_\odot$, $R=10^6$ cm, in the spherically symmetric case. The flow velocity has been assumed as $v=10^9$ cm s$^{-1}$.[]{data-label="fig:RmRcap"}](RmRc)
**
We therefore assume hereafter, for simplicity, that the NS is non-rotating and neglect the effects of the magnetosphere. The NS captures the material ejected from the core collapse of the companion star in a region delimited by the radius $R_{\rm cap}$ from the NS center $$\label{eq:Rcap}
R_{\rm cap} = \frac{2 G M_{\rm NS}}{v^2_{\rm rel,ej}}\, ,$$ where $v_{\rm rel,ej}$ is the ejecta velocity relative to the orbital motion of the NS $$\label{eq:vrel}
v_{\rm rel,ej}=\sqrt{v_{\rm orb}^2+v^2_{\rm ej}}\, ,\quad v_{\rm orb}= \sqrt{\frac{G (M_{\rm SN-prog}+M_{\rm NS})}{a}}\, ,$$ with $v_{\rm ej}$ the velocity of the ejecta and $v_{\rm orb}$ the orbital velocity of the NS, where $a$ is the binary separation. Here we have assumed that the velocity of the SN ejecta $v_{\rm ej}$ is much larger than the sound speed $c_s$ of the material, namely that the Mach number of the SN ejecta satisfies ${\cal M}=v_{\rm ej}/c_s>>$ 1, which is a reasonable approximation in the present case. The orbital period of the binary system is $$\label{eq:period}
P=\sqrt{\frac{4\pi^2 a^3}{G (M_{\rm SN-prog}+M_{\rm NS})}}\, ,$$ where $M_{\rm SN-prog}$ is the mass of the SN core progenitor.
The NS accretes the material that enters into its capture region defined by Eq. (\[eq:Rcap\]). The mass-accretion rate is given by [see @1944MNRAS.104..273B for details] $$\label{eq:Mdot}
\dot{M}= \xi \pi \rho_{\rm ej} R^2_{\rm cap} v_{ej} = \xi \pi \rho_{ej} \frac{(2 G M_{\rm NS})^2}{(v_{\rm orb}^2+v^2_{ej})^{3/2}}\, ,$$ where the parameter $\xi$ is comprised in the range $1/2\leq\xi\leq 1$, $\rho_{\rm ej}$ is the density of the accreted material, and in the last equality we have used Eqs. (\[eq:Rcap\]) and (\[eq:vrel\]). The upper value $\xi=1$ corresponds to the Hoyle-Lyttleton accretion rate [@1939PCPS...35..405H]. The actual value of $\xi$ depends on the properties of the medium on which the accretion process occurs, e.g. vacuum, wind. In Fig. \[fig:scenario\] we have sketched the accreting process of the SN ejected material onto the NS.
The density of the ejected material can be assumed to decrease in time following the simple power-law [see e.g. @1989ApJ...346..847C] $$\label{eq:rhoej}
\rho_{\rm ej}(t)=\frac{3 M_{\rm ej}(t)}{4\pi r^3_{\rm ej}(t)}=\frac{3 M_{\rm ej}}{4\pi \sigma^3 t^{3 n}}\, ,$$ where, without loss of generality, we have assumed that the radius of the SN ejecta expands as $r_{\rm ej}=\sigma t^n$, with $\sigma$ and $n$ constants. The velocity of the ejecta is thus $v_{\rm ej}=n r_{\rm ej}/t$.
If the accreted mass onto the NS is much smaller than the initial mass of the ejecta, i.e. $M_{acc}(t)/M_{\rm ej}(0)<<1$, one can assume $M_{\rm ej}(t)\approx M_{\rm ej}(0)$ and thus the integration of Eq. (\[eq:Mdot\]) gives
$$\label{eq:deltaM}
\Delta M (t)=\left.\int_{t^{\rm acc}_0}^t \dot{M} dt = \pi \xi (2 G M_{\rm NS})^2\frac{3 M_{\rm ej}(0)}{4\pi n^3 \sigma^6} {\cal F}\right|_{t^{\rm acc}_0}^t\, ,$$
where
$${\cal F} = \frac{t^{-3 (n+1)} \left[-4 n (2 n-1) t^{4 n} \sqrt{k t^{2-2 n}+1} \, _2F_1\left(\frac{1}{2},\frac{1}{n-1};\frac{n}{n-1};-k
t^{2-2 n}\right)-k^2 \left(n^2-1\right] t^4+2 k (n-1) (2 n-1) t^{2 n+2}+4 n (2 n-1) t^{4 n}\right)}{k^3 (n-1) (n+1) (3 n-1)\sqrt{k+t^{2 n-2}}}\, ,$$
with $k=v^2_{\rm orb}/(n\,\sigma)^2$, $_{2}F_{1}(a,b;c;z)$ is the Hypergeometric function, and $t^{\rm acc}_0$ is the time at which the accretion process starts, namely the time at which the SN ejecta reaches the NS capture region (see Fig. \[fig:scenario\]), i.e. so $\Delta M (t)=0$ for $t \leq t^{\rm acc}_0$. The above expression increases its accuracy for massive NSs close to the critical value, since the amount of mass needed to reach the critical mass by accretion is much smaller than $M_{\rm ej}$. In general, the total accreted mass must be computed from the full numerical integration of Eq. (\[eq:Mdot\]). We turn now to the maximum stable mass of a NS. Non-rotating NS equilibrium configurations have been recently constructed by [@belvedere2012] taking into account the strong, weak, electromagnetic, and gravitational interactions within general relativity. The equilibrium equations are given by the general relativistic Thomas-Fermi equations, coupled with the Einstein-Maxwell system of equations, the Einstein-Maxwell-Thomas-Fermi system of equations, which must be solved under the condition of global charge neutrality. The strong interactions between nucleons have been modeled through the exchange of virtual mesons ($\sigma$, $\omega$, $\rho$) within the Relativistic Mean Field (RMF) model, in the version of @1977NuPhA.292..413B. These self-consistent equations supersede the traditional Tolman-Oppenheimer-Volkoff ones that impose the condition of local charge neutrality throughout the configuration.
The uncertainties in the behavior of the nuclear equation of state (EOS) at densities about and larger than the nuclear saturation density $n_{\rm nuc}\approx 0.16$ fm$^{-3}$, lead to a variety of EOS with different nuclear parameters. A crucial parameter in this respect is the so-called nuclear symmetry energy, $\left.E_{\rm sym}=d^2 ({\cal E}/n_b)/d\delta^2\right|_{\delta=0,n_{\rm nuc}}$, where ${\cal E}$ is the nuclear matter energy-density, and $\delta=(n_n-n_p)/n_b$ is the asymmetry parameter with $n_n$, $n_p$, $n_b=n_p+n_p$ the neutron, proton, baryon densities; we refer to [@2012PhRvC..86a5803T] for a recent review. The symmetry energy is relevant for the determination of the value and density dependence of the particle abundances (e.g. $n_n/n_p$ ratio) in the NS interior [see e.g. @1987PhLB..199..469M; @2007PhRvC..76b5801K; @2009PhLB..682...23S; @2010PhRvL.105p1102H; @2011PhRvC..83f5809L]. The differences in the behavior of $E_{\rm sym}(n)$ for different nuclear EOS models and parameterizations lead to a variety of NS mass-radius relations and consequently to different values the NS critical mass $M_{\rm crit}$ and the corresponding radii [see e.g. @2012PhRvC..85c2801G]. We have plotted in Fig. \[fig:esymrho\] the behavior nuclear symmetry energy for the RMF parameterizations used in [@belvedere2012] in a wide range of baryon densities expanding from $n_b\sim 0.7 n_{\rm nuc}$ all the way up to high densities $n_b\sim 10 n_{\rm nuc}$ found in the cores of NSs; we have included in the legend the values of the mass and radius of the critical neutron star configuration, $M_{\rm crit}$ and $R$. Concerning our induced gravitational collapse scenario, the precise value of the time needed for the NS to reach $M_{\rm crit}$ by accretion of the SN material depends, for fixed binary parameters $(M_{\rm NS}, a, M_{\rm prog}, v_{\rm ej})$, on the adopted EOS which lead to a specific value of $M_{\rm crit}$.
![Nuclear symmetry energy as a function of the baryon density for selected parameterizations of the RMF nuclear model [see @belvedere2012 for details].[]{data-label="fig:esymrho"}](esymrho)
The high and rapid accretion rate of the SN material can lead the NS mass to reach the critical value $M_{\rm crit}$. This system will undergo gravitational collapse to a BH, producing a GRB. The initial NS mass is likely to be rather high due to the highly non-conservative mass transfer during the previous history of the evolution of the binary system [see e.g. @1988PhR...163...13N; @1994ApJ...437L.115I for details]. Thus, the NS could reach the critical mass in just a few seconds. Indeed, Eq. (\[eq:Mdot\]) shows that for an ejecta density $10^6$ g cm$^{-3}$ and ejecta velocity $10^9$ cm s$^{-1}$, the accretion rate might be as large as $\dot{M} \sim 0.1 M_\odot s^{-1}$.
The occurrence of a GRB-SN event in the scenario presented in this Letter is subjected to some specific conditions of the binary progenitor system, such as a short binary separation and orbital period $P<1$ h. This is indeed the case of GRB 090618 and GRB 970828 [@2012arXiv1205.6651I], which we are going to analyze within the framework presented here in forthcoming publications. In addition of offering an explanation to the GRB-SN temporal coincidence, the considerations presented in this Letter leads to an astrophysical implementation of the concept of proto-BH, generically introduced in our previous works on of GRBs 090618, 970828, and 101023 . The proto-BH represents the first stages, $20 \lesssim t \lesssim 200$ s, of the SN evolution.
It is also worth noticing that the condition $B \lesssim 10^{-2}$ on the baryon load parameter of a GRB might constrain on the binary separation $a$ for the occurrence a GRB-SN event. When the NS reaches the critical mass, the distance between the location of the front of the undisturbed SN ejecta and the NS center should be $<< a$, otherwise the emitted $e^+e^-$ plasma in the GRB might engulf a large amount of baryonic matter from the SN ejecta, reaching or even overcoming the critical value $B \sim 10^{-2}$.
It is appropriate now to discuss the possible progenitors of such binary systems. A viable progenitor is represented by X-Ray Binaries such as Cen X-3 and Her X-1 [@1972ApJ...172L..79S; @1972ApJ...174L..27W; @1972ApJ...174L.143T; @1973ApJ...180L..15L; @1975ASSL...48.....G; @2011ApJ...730...25R]. The binary system is expected to follow an evolutionary track [see @1988PhR...163...13N; @1994ApJ...437L.115I for details]: the initial binary system is composed of main-sequence stars 1 and 2 with a mass ratio $M_2/M_1\gtrsim 0.4$. The initial mass of the star 1 is likely $M_1 \gtrsim 11 M_\odot$, leaving a NS through a core-collapse event. The star 2, now with $M_2\gtrsim 11 M_\odot$ after some almost conservative mass transfer, evolves filling its Roche lobe. It then starts a spiral in of the NS into the envelope of the star 2. If the binary system does not merge, it will be composed of a Helium star and a NS in close orbit. The Helium star expands filling its Roche Lobe and a non-conservative mass transfer to the NS, takes place. This scenario naturally leads to a binary system composed of a CO star and a massive NS, as the one considered in this Letter.
We point out that the systems presenting a temporal coincidence of GRB-SN form a special class of GRBs:
\(1) There exist Ib/c SNe not associated to a GRB, e.g. the observations of SN 1994I [@2002ApJ...573L..27I] and SN 2002ap . Also this class of apparently isolated SNe may be in a binary system with a NS companion at a large binary separation $a$ and long orbital period $P$ (\[eq:period\]), and therefore the accretion rate (\[eq:Mdot\]) is not highly enough to trigger the process of gravitational collapse of the NS. A new NS binary system may be then formed and lead the emission of a short GRB in a NS merger after the shrinking of the binary orbit by gravitational waves emission. (2) There are GRBs that do not show the presence of an associated SN. This is certainly the case of GRBs at large cosmological distances $z\gtrsim 0.6$ when the SN is not detectable even by the current high power optical telescopes. This is likely the case of GRB 101023 .
\(3) There is the most interesting case of GRBs that do not show a SN, although it would be detectable. This is the case of GRB 060614 in which a possible progenitor has been indicated in a binary system formed of a white dwarf and a NS, which clearly departs from the binary class considered in this Letter. Finally, there are systems originating genuinely short GRBs which have been proved to have their progenitors in binary NSs, and clearly do not have an associated SN, e.g. GRB 090227B [@2012arXiv1205.6600M; @2012arXiv1205.6915R].
Before closing, we like to look to the problem of the remnants of the class of GRBs considered in this Letter. It is clear that after the occurrence of the SN and the GRB emission, the outcome is represented, respectively, by a NS and a BH. A possible strong evidence of the NS formation is represented by the observation of a characteristic late ($t=10^8$–$10^9$ s) X-ray emission (called URCA sources, see [@2005tmgm.meet..369R]) that has been interpreted as originated by the young ($t \sim$ 1 minute–$(10$–$100)$ years), hot ($T \sim 10^7$–$10^8$ K) NS, which we have called neo-NS . This has been indeed observed in GRB 090618 and also in GRB 101023 . If the NS and the BH are gravitationally bound they give origin to a new kind of binary system, which can lead itself to the merging of the NS and the BH and consequently to a new process of gravitational collapse of the NS into the BH. In this case the system could originate a yet additional process of GRB emission and possibly a predominant emission in gravitational waves.
We thank the anonymous referee for many suggestions which have improved the presentation of our results.
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abstract: |
Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato, Janssen, and Roshanbin \[Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22\] define the burning number $b(G)$ of a graph $G$ as the smallest integer $k$ for which there are vertices $x_1,\ldots,x_k$ such that for every vertex $u$ of $G$, there is some $i\in \{ 1,\ldots,k\}$ with ${\rm dist}_G(u,x_i)\leq k-i$, and ${\rm dist}_G(x_i,x_j)\geq j-i$ for every $i,j\in \{ 1,\ldots,k\}$.
For a connected graph $G$ of order $n$, they prove that $b(G)\leq 2\left\lceil\sqrt{n}\right\rceil-1$, and conjecture $b(G)\leq \left\lceil\sqrt{n}\right\rceil$. We show that $b(G)\leq \sqrt{\frac{32}{19}\cdot \frac{n}{1-\epsilon}}+\sqrt{\frac{27}{19\epsilon}}$ and $b(G)\leq \sqrt{\frac{12n}{7}}+3\approx 1.309 \sqrt{n}+3$ for every connected graph $G$ of order $n$ and every $0<\epsilon<1$. For a tree $T$ of order $n$ with $n_2$ vertices of degree $2$, and $n_{\geq 3}$ vertices of degree at least $3$, we show $b(T)\leq \left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil$ and $b(T)\leq \left\lceil\sqrt{n}\right\rceil+n_{\geq 3}$. Furthermore, we characterize the binary trees of depth $r$ that have burning number $r+1$.
author:
- |
Stéphane Bessy$^1$\
Anthony Bonato$^2$\
Jeannette Janssen$^3$\
Dieter Rautenbach$^4$\
Elham Roshanbin$^3$
title: Bounds on the Burning Number
---
[$^1$ Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM),\
Montpellier, France, `stephane.bessy@lirmm.fr`\
$^2$ Department of Mathematics, Ryerson University, Toronto, ON,\
Canada, M5B 2K3, `abonato@ryerson.ca`\
$^3$ Department of Mathematics and Statistics, Dalhousie University,\
Halifax, NS, Canada, B3H 3J5, `jeannette.janssen,e.roshanbin@dal.ca`\
$^4$ Institute of Optimization and Operations Research, Ulm University,\
Ulm, Germany, `dieter.rautenbach@uni-ulm.de` ]{}
--------------- ------------------------------
**Keywords:** burning; distance domination
**MSC2010:** 05C57; 05C69
--------------- ------------------------------
Introduction
============
Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato, Janssen, and Roshanbin [@bjr1; @bjr2] define a [*burning sequence*]{} of a graph $G$ as a sequence $(x_1,\ldots,x_k)$ of vertices of $G$ such that $$\begin{aligned}
\forall u\in V(G): \exists i\in [k]: & {\rm dist}_G(u,x_i)\leq & k-i\mbox{ and}\label{e1}\\
\forall i,j\in [k]: & {\rm dist}_G(x_i,x_j)\geq & j-i,\label{e2}\end{aligned}$$ where $[k]$ denotes the set of the positive integers at most $k$. Furthermore, they define the [*burning number*]{} $b(G)$ of $G$ as the length of a shortest burning sequence of $G$.
A burning sequence is supposed to model the expansion of a fire within a graph: At each discrete time step, first a new fire starts at a vertex that is not already burning, and then the fire spreads from burning vertices to all their neighbors that are not already burning. Condition (\[e1\]) ensures that putting fire to the vertices of a burning sequence $(x_1,\ldots, x_k)$ in the order $x_1,\ldots,x_k$, all vertices of $G$ are burning after $k$ steps. Condition (\[e2\]) ensures that one never puts fire to a vertex that is already burning.
We consider only finite, simple, and undirected graphs, and use standard terminology and notation [@d]. For a graph $G$, a vertex $u$ of $G$, and an integer $k$, let $N_G^k[u]=\{ v\in V(G):{\rm
dist}_G(u,v)\leq k\}$. Note that $N_G^0[u]=\{ u\}$ and $N_G^1[u]=N_G[u]=\{ u\}\cup N_G(u)$.
With this notation (\[e1\]) is equivalent to $$\begin{aligned}
V(G) &=& N_G^{k-1}[x_1]\cup N_G^{k-2}[x_2]\cup \cdots \cup N_G^0[x_k].\label{e1b}\end{aligned}$$ As previously said, condition (\[e2\]) is motivated by the considered graph process, which in each step puts fire to a vertex that is not already burning. Our first result is that condition (\[e2\]) is redundant.
\[lemma1\] The burning number of a graph $G$ is the minimum length of a sequence $(x_1,\ldots,x_k)$ of vertices of $G$ satisfying (\[e1b\]).
[*Proof:*]{} Let $k$ be the minimum length of a sequence satisfying (\[e1b\]). By definition, $b(G)\geq k$. It remains to show equality. For a contradiction, suppose $b(G)>k$. Let the sequence $s=(x_1,\ldots,x_k)$ be chosen such that (\[e1b\]) holds, and $j(s)=\min\{ j\in [k]:{\rm dist}_G(x_i,x_j)<j-i\mbox{ for some }i\in [j-1]\}$ is as large as possible. Since $b(G)>k$, the index $j(s)$ is well defined. Let $i(s)\in [j(s)-1]$ be such that ${\rm dist}_G(x_{i(s)},x_{j(s)})<j(s)-i(s)$. Since $k>j(s)-1$, there is a vertex $y$ in $$V(G)\setminus \left(N_G^{(j(s)-1)-1}[x_1]\cup N_G^{(j(s)-1)-2}[x_2]\cup \cdots \cup N_G^{0}[x_{j(s)-1}]\right).$$ Since $N_G^{k-j(s)}[x_{j(s)}]\subseteq N_G^{k-i(s)}[x_{i(s)}]$, the sequence $s'=(x_1,\ldots,x_{j(s)-1},y,x_{j(s)+1},\ldots,x_k)$ satisfies (\[e1b\]) and $j(s')>j(s)$, which is a contradiction. $\Box$
In view of Lemma \[lemma1\], the burning number can be considered a variation (but distinct from) of well known distance domination parameters [@h]. For a graph $G$ and an integer $k$, a set $D$ of vertices of $G$ is a [*distance-$k$-dominating set*]{} of $G$ if $\bigcup_{x\in D}N_G^k[x]=V(G)$. The [*distance-$k$-domination number $\gamma_k(G)$*]{} of $G$ is the minimum cardinality of a distance-$k$-dominating set of $G$.
The following bound on the distance-$k$-domination number will be of interest.
\[theoremmm\] If $G$ is a connected graph of order $n$ at least $k+1$, then $\gamma_k(G)\leq\frac{n}{k+1}$.
As observed in [@bjr1; @bjr2] the burning number can be bounded above in terms of the distance-$k$-domination number. In fact, if $\{ x_1,\ldots,x_{\gamma}\}$ is a distance-$k$-dominating set of $G$, then $$\begin{aligned}
V(G) &=& N_G^{k}[x_1]\cup N_G^{k}[x_2]\cup \cdots \cup N_G^{k}[x_{\gamma}]\\
& = & N_G^{k+\gamma-1}[x_1]\cup N_G^{k+\gamma-2}[x_2]\cup \cdots \cup N_G^{k}[x_{\gamma}].\end{aligned}$$ Appending any $k$ vertices to the sequence $(x_1,\ldots,x_{\gamma})$ yields a sequence of length $k+\gamma$ satisfying (\[e1b\]), which, by Lemma \[lemma1\], implies $b(G)\leq \gamma_k(G)+k$. Using Theorem \[theoremmm\] and choosing $k=\left\lceil\sqrt{n}\right\rceil-1$, this implies the following.
\[theoremjbr\] If $G$ is a connected graph of order $n$, then $b(G)\leq 2\left\lceil\sqrt{n}\right\rceil-1$.
One of the most interesting open problems concerning the burning number is the following.
\[conjecturebjr\] If $G$ is a connected graph of order $n$, then $b(G)\leq \left\lceil\sqrt{n}\right\rceil$.
Since the path $P_n$ of order $n$ has burning number $\left\lceil\sqrt{n}\right\rceil$ [@bjr1; @bjr2], the bound in Conjecture \[conjecturebjr\] would be tight.
Let ${\rm rad}(G)$ denote the radius of a graph $G$. Since $V(G)=N_G^{{\rm rad}(G)}[x]$ for every connected graph $G$ and every vertex $x$ of $G$ of minimum eccentricity, Lemma \[lemma1\] implies the following.
\[theoremradius\] If $G$ is a connected graph, then $b(G)\leq {\rm rad}(G)+1$.
In the present note, we improve the bound of Theorem \[theoremjbr\] by showing several upper bounds on the burning number, thereby contributing to Conjecture \[conjecturebjr\]. Furthermore, we characterize the extremal binary trees for Theorem \[theoremradius\].
Results
=======
We begin with two straightforward results that lead to a first improvement of Theorem \[theoremjbr\], and rely on arguments that are typically used to prove Theorem \[theoremmm\]. For a vertex $u$ of a rooted tree $T$, let $T_u$ denote the subtree of $T$ rooted in $u$ that contains $u$ as well as all descendants of $u$. Recall that the height of $T_u$ is the eccentricity of $u$ in $T_u$.
\[lemma2\] Let $T$ be a tree. If the non-negative integer $d$ is such that $N_T^d[u]\not= V(T)$ for every vertex $u$ of $T$, then there is a vertex $x$ of $T$ and a subtree $T'$ of $T$ with $n(T')\leq n(T)-(d+1)$ and $V(T)\setminus V(T')\subseteq N_T^d[x]$.
[*Proof:*]{} Root $T$ at a vertex $r$. Since $N_T^d[r]\not= V(T)$, the height of $T$ is at least $d+1$. The desired properties follow for a vertex $x$ such that $T_x$ has height exactly $d$ and the tree $T'=T-V(T_x)$. $\Box$
\[theorem1\] Let $T$ be a tree. If the non-negative integers $d_1,\ldots,d_k$ are such that $\sum\limits_{i=1}^k(d_i+1)\geq n(T)$, then there are vertices $x_1,\ldots,x_k$ of $T$ such that $\bigcup\limits_{i=1}^kN_T^{d_i}[x_i]=V(T)$.
[*Proof:*]{} For a contradiction, suppose that such vertices do no exist. Repeatedly applying Lemma \[lemma2\], yields a sequence $x_1,\ldots,x_k$ of vertices of $T$ as well as a sequence $T_1,\ldots,T_k$ of subtrees of $T$ such that $n(T_i)\leq n(T_{i-1})-(d_i+1)$ and $V(T_{i-1})\setminus V(T_i)\subseteq N_{T_{i-1}}^{d_i}[x_i]\subseteq N_T^{d_i}[x_i]$ for every $i\in [k]$, where $T_0=T$. Note that after $j-1<k$ applications of Lemma \[lemma2\], our assumption implies that $N_{T_{j-1}}^{d_j}[u]\not= V(T_{j-1})$ for every vertex $u$ of $T_{j-1}$, because otherwise $$\begin{aligned}
V(T) & \subseteq &
(V(T_0)\setminus V(T_1))\cup(V(T_1)\setminus V(T_2))\cup \cdots \cup (V(T_{j-2})\setminus V(T_{j-1}))\cup V(T_{j-1})\\
& \subseteq &
\bigcup\limits_{i=1}^{j-1}N_T^{d_i}[x_i]\cup N_{T_{j-1}}^{d_j}[u]\\
& \subseteq &
\bigcup\limits_{i=1}^{j-1}N_T^{d_i}[x_i]\cup N_{T}^{d_j}[u]\end{aligned}$$ for some vertex $u$ of $T$, contradicting our assumption. Therefore, the hypothesis of Lemma \[lemma2\] remains satisfied throughout its repeated applications. Now, $V(T)\setminus V(T_k)\subseteq \bigcup_{i=1}^kN_T^{d_i}[x_i]$. Since $n(T_k)\leq n(T)-\sum_{i=1}^k(d_i+1)\leq 0$, it follows that $V(T_k)$ is empty, again contradicting our assumption. $\Box$
The previous result already allows to improve Theorem \[theoremjbr\].
\[corollary1\] If $G$ is a connected graph of order $n$, then $b(G)\leq \left\lceil \sqrt{2n+\frac{1}{4}}-\frac{1}{2}\right\rceil$.
[*Proof:*]{} If $H$ is a spanning subgraph of $G$, then $b(G)\geq b(H)$. Hence, we may assume that $G$ is a tree. If $k=\left\lceil \sqrt{2n+\frac{1}{4}}-\frac{1}{2}\right\rceil$, then $((k-1)+1)+((k-2)+1)+\cdots+(0+1)={k+1\choose 2}\geq n(G)$. By Theorem \[theorem1\], there are vertices $x_1,\ldots,x_k$ in $G$ with $\bigcup\limits_{i=1}^kN_G^{k-i}[x_i]=V(G)$. By Lemma \[lemma1\], $b(G)\leq k$. $\Box$
Note that Theorem \[theoremmm\] is tight for any graph that arises by attaching a path of order $k$ to each vertex of a connected graph. In fact, also Theorem \[theorem1\] is tight for the same kind of graph. Therefore, in order to further improve Theorem \[theoremjbr\], one really has to leverage the full spectrum of different distances associated with the different vertices in a burning sequence. The following lemma offers some way of doing this.
\[lemma3\] Let $T$ be a tree. If the positive integers $d_1$ and $d_2$ are such that $d_2\geq \left\lceil\frac{3d_1}{2}\right\rceil$ and $N_T^{d_1}[u]\cup N_T^{d_2}[v]\not= V(T)$ for every two vertices $u$ and $v$ of $T$, then there are two vertices $x$ and $z$ of $T$ and a subtree $T'$ of $T$ with $n(T')\leq n(T)-\left(\left\lceil\frac{3d_1}{2}\right\rceil+d_2+2\right)$ and $V(T)\setminus V(T')\subseteq N_T^{d_1}[x]\cup N_T^{d_2}[z]$.
[*Proof:*]{} Root $T$ at a vertex $r$. Since $N_G^{d_2}[r]\not= V(T)$, the height of $T$ is at least $d_2+1$. Let the vertex $z$ be such that $T_z$ has height exactly $d_2$. Note that $V(T_z)\subseteq N_T^{d_2}[z]$ and $|V(T_z)|\geq d_2+1$. Let $x$ be a descendant of $z$ such that ${\rm dist}_T(x,z)=d_2-d_1$ and $T_x$ has height exactly $d_1$. Note that $d_2-d_1\geq \left\lceil\frac{d_1}{2}\right\rceil$. Let the vertex $y$ on the path in $T$ between $x$ and $z$ be such that ${\rm dist}_T(x,y)=\left\lceil\frac{d_1}{2}\right\rceil$.
If $V(T_y)\subseteq N_T^{d_1}[x]$, then Lemma \[lemma2\] applied to the tree $\tilde{T}=T-V(T_y)$ and the value $d_2$ implies the existence of a vertex $z'$ and a subtree $T'$ of $\tilde{T}$ with $n(T')\leq n(\tilde{T})-(d_2+1)$ and $V(\tilde{T})\setminus V(T')\subseteq N_{\tilde{T}}^{d_2}[z']$. Now, we have that $$\begin{aligned}
n(T') & \leq & n(\tilde{T})-(d_2+1)\\
& = & n(T)-|V(T_y)|-(d_2+1)\\
& \leq & n(T)-\left(\left\lceil\frac{3d_1}{2}\right\rceil+d_2+2\right)\end{aligned}$$ and $$\begin{aligned}
V(T)\setminus V(T') & = & (V(T)\setminus V(\tilde{T}))\cup (V(\tilde{T})\setminus V(T'))\\
& \subseteq & V(T_y)\cup N_{\tilde{T}}^{d_2}[z']\\
& \subseteq & N_T^{d_1}[x]\cup N_T^{d_2}[z'].\end{aligned}$$ Hence, we may assume that $V(T_y)\not\subseteq N_T^{d_1}[x]$. This implies the existence of a descendant $y'$ of $y$ that is not a descendant of $x$ and satisfies ${\rm dist}_T(x,y')>d_1$. By the choice of $x$, $y$, and $z$, this implies $|V(T_z)|\geq d_2+1+\left\lceil\frac{d_1}{2}\right\rceil$. Lemma \[lemma2\] applied to the tree $\tilde{T}=T-V(T_z)$ and the value $d_1$ implies the existence of a vertex $x'$ and a subtree $T'$ of $\tilde{T}$ with $n(T')\leq n(\tilde{T})-(d_1+1)$ and $V(\tilde{T})\setminus V(T')\subseteq N_{\tilde{T}}^{d_1}[x']$. Now, we have that $$\begin{aligned}
n(T') & \leq & n(\tilde{T})-(d_1+1)\\
& = & n(T)-|V(T_z)|-(d_1+1)\\
& \leq & n(T)-\left(\left\lceil\frac{3d_1}{2}\right\rceil+d_2+2\right)\end{aligned}$$ and $$\begin{aligned}
V(T)\setminus V(T') & = & (V(T)\setminus V(\tilde{T}))\cup (V(\tilde{T})\setminus V(T'))\\
& \subseteq & V(T_z)\cup N_{\tilde{T}}^{d_1}[x']\\
& \subseteq & N_T^{d_1}[x'] \cup N_T^{d_2}[z],\end{aligned}$$ which completes the proof. $\Box$
\[theorem2\] If $G$ is a connected graph and $0<\epsilon<1$, then $b(G)\leq \sqrt{\frac{32}{19}\cdot \frac{n(G)}{1-\epsilon}}+\sqrt{\frac{27}{19\epsilon}}$.
[*Proof:*]{} As in the proof of Corollary \[corollary1\], we may assume that $G$ is a tree $T$.
Let $\ell=\left\lceil\log_9\left(\frac{3}{19\epsilon}\right)\right\rceil$. Note that $\left(1-\frac{3}{19}\cdot \left(\frac{1}{9}\right)^{\ell}\right)\geq 1-\epsilon$ and $3^{\ell}<\sqrt{\frac{27}{19\epsilon}}.$ Let $k$ be the smallest integer such that $(1-\epsilon)\cdot \frac{19k^2}{32}+(1-\epsilon)\cdot \frac{3k}{8}\geq n(T)$ and $k\equiv 0$ ( mod $3^{\ell}$). Note that $$k\leq
\left\lceil\sqrt{\frac{32}{19}\cdot \frac{n(T)}{1-\epsilon}+\left(\frac{6}{19}\right)^2}-\frac{6}{19}+3^{\ell}-1\right\rceil
\leq \sqrt{\frac{32}{19}\cdot \frac{n(T)}{1-\epsilon}}+\sqrt{\frac{27}{19\epsilon}}.$$ For a contradiction, suppose that $b(G)>k$.
For $j\in [\ell]$, let $I_j=\left[\frac{2k}{3^j}-1\right]\setminus \left[\frac{k}{3^j}-1\right]=\left\{\frac{k}{3^j},\frac{k}{3^j}+1,\ldots,\frac{2k}{3^j}-1\right\}$. Since $\frac{k}{3^j}$ is an integer, it follows that $\left\lceil\frac{3d}{2}\right\rceil\leq \frac{k}{3^j}+d$ for every $d\in I_j$. Repeatedly applying Lemma \[lemma3\] to the $\left(1-\frac{1}{3^{\ell}}\right)k$ disjoint pairs $\left\{d,\frac{k}{3^j}+d\right\}$ for $j\in [\ell]$ and $d\in I_j$, yields pairs of vertices $\left\{ x_d,x_{\frac{k}{3^j}+d}\right\}$ as well as a subtree $T'$ of $T$ such that $$\begin{aligned}
n(T')
& \leq & n(T)
-\sum_{j=1}^{\ell}\sum_{d=\frac{k}{3^j}}^{\frac{2k}{3^j}-1}\left(\left\lceil\frac{3d}{2}\right\rceil+\left(\frac{k}{3^j}+d\right)+2\right)\\
& \leq & n(T)
-\sum_{j=1}^{\ell}\sum_{d=\frac{k}{3^j}}^{\frac{2k}{3^j}-1}\left(\frac{5d}{2}+\frac{k}{3^j}+2\right)\\
& = & n(T)
-\sum_{j=1}^{\ell}\left(\frac{1}{9^{j-1}}\cdot\frac{19k^2}{36}+\frac{1}{3^{j-1}}\cdot \frac{k}{4}\right)\\
& = & n(T)
-\left(1-\left(\frac{1}{9}\right)^{\ell}\right)\cdot \frac{19k^2}{32}
-\left(1-\left(\frac{1}{3}\right)^{\ell}\right)\cdot \frac{3k}{8}\end{aligned}$$ and $$\begin{aligned}
V(T)\setminus V(T')
& \subseteq & \bigcup_{j=1}^{\ell}\bigcup_{d=\frac{k}{3^j}}^{\frac{2k}{3^j}-1}
\left(N_T^d[x_d]\cup N_T^{\left(\frac{k}{3^j}+d\right)}\left[x_{\frac{k}{3^j}+d}\right]\right)\\
& = & \bigcup_{i=\frac{k}{3^{\ell}}}^{k-1}N_T^i[x_i].\end{aligned}$$ Note that, similarly as in the proof of Theorem \[theorem1\], the assumption $b(G)>k$ implies that the hypothesis of Lemma \[lemma3\] remains satisfied throughout its repeated applications.
Now, repeatedly applying Lemma \[lemma2\] for all $\frac{k}{3^{\ell}}$ values $d$ in $\{ 0\}\cup [\frac{k}{3^{\ell}}-1]$, yields vertices $x_0,\ldots, x_{\frac{k}{3^{\ell}}-1}$ and a subtree $T''$ of $T'$ such that $$\begin{aligned}
n(T'') & \leq & n(T')-\sum_{d=0}^{\frac{k}{3^{\ell}}-1}(d+1)\\
& = & n(T')-\left(\frac{1}{9}\right)^{\ell}\cdot\frac{k^2}{2}-\left(\frac{1}{3}\right)^{\ell}\cdot\frac{k}{2}\end{aligned}$$ and $$V(T')\setminus V(T'')\subseteq\bigcup_{i=0}^{\frac{k}{3^{\ell}}-1}N_T^i[x_i].$$ Altogether, the vertices $x_0,\ldots,x_{k-1}$ satisfy $$V(T)\setminus V(T'')\subseteq\bigcup_{i=0}^{k-1}N_T^i[x_i].$$ Since $$\begin{aligned}
n(T'') & \leq & n(T)
-\left(1-\left(\frac{1}{9}\right)^{\ell}\right)\cdot \frac{19k^2}{32}
-\left(1-\left(\frac{1}{3}\right)^{\ell}\right)\cdot \frac{3k}{8}
-\left(\frac{1}{9}\right)^{\ell}\cdot\frac{k^2}{2}
-\left(\frac{1}{3}\right)^{\ell}\cdot\frac{k}{2}\\
& = & n(T)
-\left(1-\frac{3}{19}\cdot \left(\frac{1}{9}\right)^{\ell}\right)\cdot \frac{19k^2}{32}
-\left(1+\left(\frac{1}{3}\right)^{\ell+1}\right)\cdot \frac{3k}{8}\\
& \leq & n(T)
-(1-\epsilon)\cdot \frac{19k^2}{32}
-(1-\epsilon)\cdot \frac{3k}{8}\\
& \leq & 0,\end{aligned}$$ it follows that $V(T'')$ is empty, which implies the contradiction $b(T)\leq k$. $\Box$
Choosing in the above proof $\ell=1$, and $k$ as the smallest multiple of $3$ that satisfies $\frac{7}{12}k^2+\frac{5}{12}k\geq n(T)$, allows to deduce a similar contradiction, which implies $b(G)\leq \sqrt{\frac{12n(G)}{7}}+3\approx 1.309 \sqrt{n(G)}+3$ for every connected graph $G$.
The following results generalize the equality $b(P_n)=\left\lceil\sqrt{n}\right\rceil$, and establish approximate versions of Conjecture \[conjecturebjr\] under additional restrictions.
\[lemma4\] If $n_1,\ldots,n_p$ and $k$ are positive integers such that $n_1+\cdots+n_p+k(p-1)\leq k^2$, then $b(P_{n_1}\cup\cdots\cup
P_{n_p})\leq k$.
[*Proof:*]{} The proof is by induction on $n=n_1+\cdots+n_p$. Let $G=P_{n_1}\cup\cdots\cup P_{n_p}$ and $n_1\leq \ldots \leq n_p$. Note that $p\leq k$. If $n_p\leq k-p+1$, let the set $\{ x_1,\ldots,x_p\}$ contain a vertex from each component of $G$. We have $V(G)=N_G^{k-1}[x_1]\cup N_G^{k-2}[x_2]\cup \cdots \cup
N_G^{k-p}[x_p]$, and Lemma \[lemma1\] implies $b(G)\leq k$. Hence, we may assume that $n_p\geq k-p+2$, which implies $n\geq
(p-1)+(k-p+2)=k+1$.
If $n_p\geq 2k$, let $x_1$ be a vertex at distance $k-1$ from an endvertex of a component of $G$ of order $n_p$. The graph $G'=G-N^{k-1}_G[x_1]$ has $p$ components and $|N^{k-1}_G[x_1]|=2k-1$. Since $$\begin{aligned}
n_1+\cdots+n_{p-1}+(n_p-(2k-1))+(k-1)(p-1)
& \leq & n_1+\cdots+n_{p-1}+(n_p-(2k-1))+k(p-1)\\
&\leq & k^2-(2k-1)\\
&=& (k-1)^2,\end{aligned}$$ there are, by induction, vertices $x_2,\ldots,x_k$ such that $$\begin{aligned}
\label{e3}
V(G')=N_{G'}^{(k-1)-1}[x_2]\cup N_{G'}^{(k-1)-2}[x_3]\cup \cdots \cup N_{G'}^0[x_k].\end{aligned}$$ This implies (\[e1b\]), and Lemma \[lemma1\] implies $b(G)\leq k$.
Hence, we may assume that $n_p\leq 2k-1$. In this case we choose as $x_1$ a vertex of minimal eccentricity in a component of $G$ of order $n_p$. This implies that $G'=G-N^{k-1}_G[x_1]$ has $p-1$ components. Since $$\begin{aligned}
n_1+\cdots+n_{p-1}+(k-1)(p-2) &\leq& k^2-n_p-\left(k(p-1)-(k-1)(p-2)\right)\\
& \leq & k^2-(k-p+2)-(k+p-2)\\
&=& k^2-2k\\
&<& (k-1)^2,\end{aligned}$$ there are, by induction, vertices $x_2,\ldots,x_k$ that satisfy (\[e3\]), which again implies $b(G)\leq k$. $\Box$
Since $n+\left(\left\lceil\sqrt{n}\right\rceil+(p-1)\right)(p-1)\leq \left(\left\lceil\sqrt{n}\right\rceil+(p-1)\right)^2$ for positive integers $n$ and $p$, Lemma \[lemma4\] implies the following.
\[corroshanbin\] If the forest $T$ of order $n$ is the union of $p$ paths, then $b(T)\leq \left\lceil\sqrt{n}\right\rceil+(p-1)$.
We derive further consequences of Lemma \[lemma4\].
\[theorem3\] If $T$ is a tree of order $n$ that has $n_{\geq 3}$ vertices of degree at least $3$, then $b(T)\leq \left\lceil\sqrt{n}\right\rceil+n_{\geq 3}$.
[*Proof:*]{} Clearly, we may assume that $n_{\geq 3}\geq 1$. Let $k=\left\lceil\sqrt{n}\right\rceil+n_{\geq 3}$. Let $x_1,\ldots,x_{n_{\geq 3}}$ be the vertices of degree at least $3$. Let $T'=T-\{ x_1,\ldots,x_{n_{\geq 3}}\}$, and let $T''=T-N_T^{k-1}[x_1]\cup \cdots \cup N_T^{k-n_{\geq 3}}[x_{n_{\geq 3}}]$. Every component of $T'$ is a path $P$ such that at least one endvertex of $P$ has a neighbor in $\{ x_1,\ldots,x_{n_{\geq 3}}\}$. Therefore, the distinct components of $T''$ arise by removing at least $k-n_{\geq 3}=\left\lceil\sqrt{n}\right\rceil$ vertices from distinct components of $T'$. This implies that if $T''=P_{n_1}\cup\cdots\cup P_{n_p}$, then $$n_1+\cdots+n_p+\left\lceil\sqrt{n}\right\rceil (p-1)<
\left(n_1+\left\lceil\sqrt{n}\right\rceil\right)+\cdots+\left(n_p+\left\lceil\sqrt{n}\right\rceil\right)
\leq n-n_{\geq 3}<\left\lceil\sqrt{n}\right\rceil^2.$$ Now, Lemma \[lemma4\] implies the existence of vertices $y_1,\ldots,y_{\left\lceil\sqrt{n}\right\rceil}$ such that $$V(T'')=N_{T''}^{\left\lceil\sqrt{n}\right\rceil-1}[y_1]\cup \cdots \cup N_{T''}^{0}[y_{\left\lceil\sqrt{n}\right\rceil}].$$ We obtain $$V(T)=N_T^{k-1}[x_1]\cup \cdots \cup N_T^{\left\lceil\sqrt{n}\right\rceil}[x_{n_{\geq 3}}]
\cup N_T^{\left\lceil\sqrt{n}\right\rceil-1}[y_1]\cup \cdots \cup N_T^{0}[y_{\left\lceil\sqrt{n}\right\rceil}],$$ and Lemma \[lemma1\] implies $b(T)\leq k$. $\Box$
\[theorem4\] If $T$ is a tree of order $n$ that has $n_2$ vertices of degree $2$, then $$b(T)\leq \left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil.$$
[*Proof:*]{} Let $k=\left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil$. Note that $k(k-1)\geq n+n_2$. For a contradiction, suppose that $b(T)>k$. Root $T$ at a vertex $r$. As before, we obtain that the height of $T$ is at least $k$. Let $x_d$ be a vertex of $T$ such that $T_{x_d}$ has height exactly $d$ for some $d\in \{ 0\}\cup [k-1]$. Let $V(T_{x_d})$ contain exactly $p_d$ vertices that have degree $2$ in $T$. If $P$ is a path of length $d$ between $x_d$ and a leaf of $T_{x_d}$, then at least $d-p_d$ vertices of $P$ have a child that does not lie on $P$. Therefore, $|V(T_{x_d})\setminus \{ x_d\}|\geq 2d-p_d$, and $T'=T-(V(T_{x_d})\setminus \{ x_d\})$ is a tree with $n_2-p_d$ vertices of degree $2$ such that $V(T)\setminus V(T')\subseteq N_T^d[x_d]$. Note that $x_d$ has degree $1$ in $T'$. Iteratively repeating this argument similarly as in the previous proofs, we obtain vertices $x_0,\ldots,x_{k-1}$ and integers $p_0,\ldots,p_{k-1}$ such that $p_0+\cdots+p_{k-1}\leq n_2$ and $\sum_{d=0}^{k-1}(2d-p_d)\leq n$. Since $\sum_{d=0}^{k-1}(2d-p_d)\geq k(k-1)-n_2\geq n$, we obtain $V(T)=N_T^0[x_0]\cup\cdots\cup N_T^{k-1}[x_{k-1}]$, which implies the contradiction $b(G)\leq k$. $\Box$
In view of the simple argument that shows Theorem \[theoremradius\], the extremal graphs for this bound might have a rather special structure. Our final result supports this intuition for binary trees.
Recall that a rooted tree is [*binary*]{} if every vertex has at most two children, and that a binary tree is [*perfect*]{} if every non-leaf vertex has exactly two children, and all leaves have the same depth, that is, the same distance from the root. Let $T_1$ be the rooted tree of order $2$, and, for an integer $r$ at least $2$, let $T_r$ be the rooted tree that arises from the perfect binary tree of depth $r-1$ by subdividing all edges that are incident with a leaf. Alternatively, $T_r$ arises by attaching a new leaf to each of the $2^{r-1}$ leaves of the perfect binary tree of depth $r-1$.
\[theoremradiusextr\] If $r$ is a positive integer and $T$ is a binary tree of depth $r$, then $b(T)=r+1$ if and only if $T$ contains $T_r$ as a subtree.
[*Proof:*]{} Since the statement is trivial for $r=1$, we may assume that $r\geq 2$.
First, we show that $T=T_r$ has burning number $r+1$. For a contradiction, suppose that $b(T)\leq r$. Let $u$ be the root of $T$, and let $v^1$ and $v^2$ be the two children of $u$. For $i\in [2]$, let $T^i$ be the subtree of $T$ rooted in $v^i$ that contains $v^i$ as well as all descendants of $v^i$ in $T$. By Lemma \[lemma1\], there are vertices $x_1,x_2,\ldots,x_r$ with $V(T)=N_T^{r-1}[x_1]\cup N_T^{r-2}[x_2]\cup \cdots \cup N_T^0[x_r]$. By symmetry, we may assume that $x_1\not\in V(T^1)$. Let $L$ be the set of leaves of $T$ that belong to $T^1$. Since $T^1$ is isomorphic to $T_{r-1}$, we have $|L|=2^{r-2}$. Note that $N_G^{r-1}[x_1]$ does not intersect $L$. Furthermore, for every $i\in [r-1]\setminus \{ 1\}$, the set $N_T^{r-i}[x_i]$ contains at most $2^{r-i-1}$ vertices from $L$. In fact, the set $N_T^{r-i}[x_i]$ contains exactly $2^{r-i-1}$ vertices from $L$ if and only if $x_i\in V(T^1)$ and $x_i$ has depth $i$ in $T$. Since $N_T^0[x_r]=\{ x_r\}$, the set $N_T^0[x_r]$ contains at most one vertex from $L$. Since $|L|=2^{r-2}=\sum_{i=2}^{r-1}2^{r-i-1}+1$, every vertex in $L$ belongs to exactly one of the sets $N_T^{r-i}[x_i]$ for $i=[r]\setminus \{ 1\}$. This implies that $x_2,\ldots,x_r\in V(T^1)$, $x_i$ has depth $i$ in $T$ for $i\in [r-1]\setminus \{ 1\}$, and $x_r$ is a leaf of $T$. Let $u_0\ldots u_r$ be the path in $T$ from the root $u=u_0$ to the leaf $x_r=u_r$. Note that $u_1=v^1$. Since $x_2$ belongs to $T^1$, $x_2$ has depth $2$ in $T$, and $x_r\not\in N_T^{r-2}[x_2]$, the vertex $x_2$ is the child of $u_1$ distinct from $u_2$. Moreover, as every vertex of $L$ belongs to exactly one of the sets $N_T^{r-i}[x_i]$ for $i\in [r]\setminus \{ 1\}$, no vertex $x_i$ with $i\in [r]\setminus \{ 1,2\}$ is a descendant of $x_2$. Iterating these arguments, it follows that, for every $i\in [r-1]\setminus \{ 1\}$, the vertex $x_i$ is the child of $u_{i-1}$ distinct from $u_i$. However, this implies the contradiction that $u_{r-1}\not\in N_T^{r-1}[x_1]\cup N_T^{r-2}[x_2]\cup \cdots \cup N_T^0[x_r]$. Altogether, we obtain that $T_r$ has burning number $r+1$. Together with Theorem \[theoremradius\], this implies that a binary tree $T$ of depth $r$ has burning number $r+1$ if $T$ contains $T_r$ as a subtree.
For the converse, we assume that $T$ is a binary tree of depth $r$ that does not contain $T_r$ as a subtree. It follows that $T$ has a leaf of depth less than $r$ or that $T$ has a vertex of depth less than $r-1$ that has only one child. In both cases we will show that $b(T)\leq r$. First, we assume that $T$ has a leaf at depth less than $r$. Let $d$ be the minimum depth of a leaf of $T$. Let $u_0\ldots u_d$ be a path in $T$ between the root $u_0$ and a leaf $u_d$. By assumption, we have $d<r$. For $i\in [d]$, let $x_i$ be the child of $u_{i-1}$ that is distinct from $u_i$. Note that the subtree of $T$ rooted in $x_i$ that contains $x_i$ as well as all descendants of $x_i$ in $T$ has depth at most $r-i$. This implies that $V(T)=N_T^{r-1}[x_1]\cup N_T^{r-2}[x_2]\cup \cdots \cup N_T^{r-d}[x_d]\cup N_T^0[u_d]$, and, by Lemma \[lemma1\], we obtain $b(T)\leq r$. Next, we assume that $T$ has a vertex $x$ of depth less than $r-1$ that has only one child. Let $T'$ arise from $T$ by adding a new leaf $y$ as a child of $x$. Clearly, $T'$ is a binary tree of depth $r$ that has a leaf of depth less than $r$, and, hence, $b(T)\leq b(T')\leq r$. $\Box$
[**Acknowledgment**]{} This paper is part of a collaborative work that grew out of [@br] and [@bjr3].
S. Bessy, D. Rautenbach, Bounds, Approximation, and Hardness for the Burning Number, arXiv:1511.06023. A. Bonato, J. Janssen, E. Roshanbin, Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22. A. Bonato, J. Janssen, E. Roshanbin, How to burn a graph, to appear in Internet Mathematics, arXiv:1507.06524. A. Bonato, J. Janssen, E. Roshanbin, Burning a Graph is Hard, arXiv:1511.06774. R. Diestel, Graph Theory, 4th ed., Springer 2010. M.A. Henning, Distance domination in graphs, in: T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998, 321-349. A. Meir, J.W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975) 225-233. E. Roshanbin, Burning a graph as a model of social contagion, PhD Thesis, Dalhousie University, 2016.
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